essays on macroeconomics and credit risk · sudarshan p. gururaj the first chapter of this...

ESSAYS ON MACROECONOMICS AND CREDIT RISK

SUDARSHAN P. GURURAJ

Submitted in partial fulfillment of the

requirements for the degree

of Doctor of Philosophy

under the Executive Committee

of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2009

UMI Number: 3348432

INFORMATION TO USERS

The quality of this reproduction is dependent upon the quality of the copy

submitted. Broken or indistinct print, colored or poor quality illustrations and

photographs, print bleed-through, substandard margins, and improper

alignment can adversely affect reproduction.

In the unlikely event that the author did not send a complete manuscript

and there are missing pages, these will be noted. Also, if unauthorized

copyright material had to be removed, a note will indicate the deletion.

®

UMI UMI Microform 3348432

Copyright 2009 by ProQuest LLC.

All rights reserved. This microform edition is protected against

unauthorized copying under Title 17, United States Code.

ProQuest LLC 789 E. Eisenhower Parkway

PO Box 1346 Ann Arbor, Ml 48106-1346

©2009

SUDARSHAN P. GURURAJ

ALL RIGHTS RESERVED

Abstract

Essays on Macroeconomics and Credit Risk

Sudarshan P. Gururaj

The first chapter of this dissertation empirically examines the impact of macroe-

conomic conditions on credit risk, particularly under shifting regimes. The second

chapter links a new Keynesian macroeconomic model with a model of credit risk to

demonstrate how macroeconomic conditions, namely output growth and inflation af

fect the credit risk of firms, as measured by credit spreads. In the third chapter, we

capture more features of the empirical behavior of credit spreads by including time-

varying preferences in the new Keynesian macroeconomic model with time-varying

preferences, which allow for the regime changes we find empirically.

Contents

1 Macroeconomic Determinants of Credit Spreads 1

1.1 Introduction 1

1.2 How Should Macroeconomic Conditions Affect Credit Spreads? . . . . 3

1.3 Data Description 7

1.3.1 Macroeconomic Data 7

1.3.2 Credit Spread Data 9

1.4 Empirical Properties of Credit Spreads 12

1.4.1 Tests of Persistence 13

1.4.2 Structural Break Tests and Inference from Markov Regime Switch

ing 16

1.4.3 Difference from Previous Empirical Studies 20

1.5 Model Estimation 21

1.5.1 OLS Estimation Results 22

1.5.2 Markov Regime-Switching Results 24

1.5.3 Robustness Check with First Differences 26

1.6 Discussion of Results 28

1.6.1 Regime Switching in Credit Spreads 28

1.6.2 The Impact of Macroeconomic Variables 30

1.7 Conclusions and Future Research 33

2 Credit Spreads in a New Keynesian Macro Model 35

i

2.1 Introduction 35

2.2 An Intuitive Framework 41

2.3 The Model 42

2.3.1 The Macroeconomy 43

2.3.2 Risk-Free and Risky Yields 50

2.4 Model Calibration and Simulation 54

2.4.1 The Macroeconomic Model 54

2.4.2 Calibrating and Simulating Credit Spreads 56

2.5 Results 58

2.5.1 Properties of Model-Generated Credit Spreads and Default Prob

abilities 59

2.5.2 Macroeconomic Factors of Credit Spreads: Contemporaneous

Relationships 60

2.6 Impulse Response Functions and Comparative Statics 61

2.6.1 Impulse Response Functions 62

2.6.2 Correlation with Output Growth, p 63

2.6.3 Idiosyncratic Cash Flow Growth and Volatility, (£, oK) . . . . 64

2.6.4 Relative Risk Aversion, 7 65

2.6.5 Alternative Monetary Policy, (x,wy) 65

2.7 Conclusion 67

3 Credit Spreads in a New Keynesian Macro Model with Habit Per

sistence 71

3.1 Introduction 71

3.2 The Model 73

3.2.1 The Macroeconomy 74

3.2.2 Risk-Free and Risky Yields 81

3.3 Model Calibration and Simulation 85

3.3.1 The Macroeconomic Model 85

11

3.3.2 Calibrating and Simulating Credit Spreads 87

3.4 Results 89

3.4.1 Properties of Model-Generated Credit Spreads and Default Prob

abilities 89

3.4.2 Macroeconomic Factors of Credit Spreads: Contemporaneous

Relationships 91

3.4.3 Macroeconomic Factors of Credit Spreads: Different Regimes . 92

3.5 Conclusion 93

Bibliography 95

A Chap. 1 Tables and Figures 100

A.l Unit Root Tests 100

A.2 Structural Break Tests 101

A.3 OLS Estimation 107

A.4 Markov Regime Switching Estimation I l l

A.5 Robustness Check: Tests with Moody's Data 117

A.6 Figures 118

B Chap. 2 Proofs, Tables, and Figures 122

B.l Credit Spread Expression 122

B.2 The Market Price of Risk and the Risk-Neutral (Q) Measure 123

B.3 Regression Test Coefficients 125

B.4 Impulse Response Functions 137

C Chap. 3 Proofs and Tables 146

C.l Market Price of Risk 146

C.2 Regression Test Coefficients 148

m

List of Figures

A.l 5-year credit spreads and possible break dates, May 1994 to June 2007 119

A.2 VIX, May 1994 to June 2007 120

A.3 Smoothed regime probabilities, 5 year A credit spreads 121

B. 1 Impulse response of macroeconomic conditions to adverse technology /productivity

shock 138

B.2 Impulse response of 1 year credit spreads to adverse technology/productivity

shock 139


shock 140


shock 141

B.5 Impulse response of macroeconomic conditions to positive monetary

policy shock 142

B.6 Impulse response of 1 year credit spreads to positive monetary policy

shock 143


shock 144


shock 145

IV

List of Tables

1.1 Autocorrelation function values for single A credit spreads, May 1994

to June 2007 14

1.2 Autocorrelation function values for 5-year credit spreads, May 1994 to

June 2007 14

1.3 Autocorrelation function values for first differences in single A credit

spreads, May 1994 to June 2007 15

1.4 Autocorrelation function values for first differences in 5-year credit

spreads, May 1994 to June 2007 15

1.5 Bai and Perron (1998) structural break test dates and confidence in

tervals for single A credit spreads 18

1.6 Bai and Perron (1998) structural break test dates and confidence in

tervals for 5-year credit spreads 19

1.7 OLS regression coefficients for single A credit spreads on regressors . 23

1.8 OLS regression coefficients for 5-year credit spreads on regressors . . 23

1.9 Markov regime switching model coefficients for single A credit spreads

on regressors 25

1.10 Markov regime switching model coefficients for 5-year credit spreads

on regressors 26

1.11 Markov regime switching model coefficients for first differences in 5-

year credit spreads on regressors 28

2.1 Macroeconomic model coefficients based on Ravenna and Seppala (2006) 55

v

2.2 Selected variable volatilities and correlations, model vs. historical,

1952-2006 56

2.3 Moments and correlations of output growth quarterly, inflation quar

terly, and interest rates 57

2.4 Firm-specific cash flow parameters, based on Longstaff and Piazzesi

(2004) and recovery rate in Huang and Huang (2003) 57

2.5 Moody's default probabilities and recovery rates (Huang and Huang,

2003 58

2.6 Average credit spread levels, model generated vs. historical and literature 60

2.7 Selected variable volatilities and correlations, model with relative risk

aversion of 10 vs. historical, 1952-2006 65

2.8 Model moments and correlations of output growth quarterly, inflation

quarterly, and interest rates where relative risk aversion is 10 66

2.9 Selected variable volatilities and correlations, model with relative risk

aversion of 25 vs. historical, 1952-2006 66


terly, and interest rates with relative risk aversion is 25 66

2.11 Selected variable volatilities and correlations, model with degree of

monetary policy smoothing of 0.9 vs historical, 1952-2006 67


terly, and interest rates with degree of monetary policy smoothing of

0.9 68

2.13 Selected variable volatilities and correlations, model with Taylor coef

ficient of output of 0.1 vs historical, 1952-2006 68


terly, and interest rates with Taylor coefficient on output of 0.1 . . . . 68


terly, and interest rates 86

vi

3.2 Selected variable volatilities and correlations, model with internal habit

preference vs. historical, 1952-2006 87


terly, and interest rates of internal habit persistence model 87

3.4 Firm-specific cash flow parameters, based on Longstaff and Piazzesi

(2004) and recovery rate in Huang and Huang (2003) 88

3.5 Moody's default probabilities and recovery rates (Huang and Huang,

2003 89

3.6 Average credit spread levels from the model, historical data, and om-

parable models 90

A.l Phillips-Perron test results for single A credit spreads 100

A.2 Phillips-Perron test results for 5-year credit spreads 100

A.3 Bai and Perron (1998) structural break test dates and confidence in

tervals for AAA credit spreads 101


tervals for AA credit spreads 102


tervals for A credit spreads 103


tervals for BBB credit spreads 104


tervals for BB credit spreads 105


tervals for B credit spreads 106

A.9 OLS regression coefficients for AAA credit spreads on regressors . . . 107

A. 10 OLS regression coefficients for AA credit spreads on regressors . . . . 108

A. 11 OLS regression coefficients for A credit spreads on regressors 108

A. 12 OLS regression coefficients for BBB credit spreads on regressors . . . 109

vii

A. 13 OLS regression coefficients for BB credit spreads on regressors . . . . 109

A. 14 OLS regression coefficients for B credit spreads on regressors 110

A. 15 Markov regime switching model coefficients for AAA credit spreads on

regressors I l l

A. 16 Markov regime switching model coefficients for AA credit spreads on

regressors 112

A. 17 Markov regime switching model coefficients for A credit spreads on

regressors 113

A. 18 Markov regime switching model coefficients for BBB credit spreads on

regressors 114

A. 19 Markov regime switching model coefficients for BB credit spreads on

regressors 115

A.20 Markov regime switching model coefficients for B credit spreads on

regressors 116

A.21 Phillips-Perron test results for Moody's Corporate Credit Spread series 117

A.22 OLS model coefficients for Moody's corporate credit spread series . . 117

A.23 Markov regime-switching model coefficients for Moody's corporate credit

spread series 117

B.l Regression of forward-looking default probabilities on contemporane

ous output growth and inflation 125

B.2 Regression of credit spreads on one-quarter lagged credit spreads and

contemporaneous output growth and inflation 126

B.3 Model credit spread regression, output growth = 0.4 127

B.4 Model credit spread regression, output growth = 0.8 128

B.5 Model credit spread regression, mean of idiosyncratic firm cash flow

growth = -2% 129

B.6 Model credit spread regression, mean of idiosyncratic firm cash flow

growth = 2% 130

viii

B.7 Model credit spread regression, vol of idiosyncratic firm cash flow

growth =10% 131

B.8 Model credit spread regression, vol of idiosyncratic firm cash flow

growth = 40% 132

B.9 Model credit spread regression, coefficient of relative risk aversion = 10 133

B.10 Model credit spread regression, coefficient of relative risk aversion = 25 134

B.ll Model credit spread regression, coefficient of monetary policy smooth

ing = 0.9 135

B.12 Model credit spread regression, Taylor coefficient of output = 0.1 . . 136

C.l Regression of forward-looking default probabilities on contemporane

ous output growth and inflation 149

C.2 Regression of credit spreads on one-quarter lagged credit spreads and

contemporaneous output growth and inflation 150

C.3 Regression of A A A-A credit spreads with regime switching on one-

quarter lagged credit spreads and contemporaneous output growth and

inflation 151

C.4 Regression of BBB-B credit spreads with regime switching on one-

quarter lagged credit spreads and contemporaneous output growth and

inflation 152

IX

Acknowledgements

I am deeply indebted to my advisor, Prof. Marc Giannoni, who ushered me through

the long and winding road that is the process of writing a dissertation. I am grateful

for his guidance, insight, and encouragement over the past three years.

I would especially like to thank Prof. John Donaldson, my dissertation committee

chair, for his valuable comments and advice about my research. I also want to thank

Profs. Robert Hodrick, Patrick Bolton, and Martin Lettau and Dr. Bobby Porn-

rojnangkool for their detailed comments before and during the dissertation defense.

I would also like to mention my classmates Binu Balachandran, Sam Cheung, Yael

Eisenthal, Hagit Levy, Jorge Murillo, and Simeon Tsonev, along with the seminar

participants at Columbia Business School and at Platinum Grove Asset Management

who provided me with fruitful discussion and thought that led to these papers. The

empirical part of this paper would not have been possible without the assistance of

Drs. Wai Lee and Bobby Pornrojnangkool in obtaining data from Lehman Brothers.

Finally, the focus of my research is, ultimately, the value of debt. However, I owe

an invaluable debt to my family. I am very thankful to my parents, Drs. Pandu and

Sujatha Gururaj, and my brother Gautam for their unconditional support and love.

I would like to thank my wife Mrinalini for her encouragement during the final stages

of this work.

x

To my parents, Drs. Pandu and Sujatha Gururaj

XI

Chapter 1

Macroeconomic Determinants of

Credit Spreads

1.1 Introduction

Ever since Merton (1974) proposed studying credit risk by considering a firm as a

portfolio of contingent claims, many credit risk studies model the firm's debt as a

put option on the firm's assets. These structural credit models suggest that only

firm-specific variables affect claims on the firm's assets and, therefore, credit risk.

However, empirical studies that use credit spreads, the difference between the yields

of defaultable bonds and risk-free yields of the same maturity, as proxy for credit risk

seem to refute the applicability of such structural models. These papers have reached

the common conclusion that 1) that variables implied by structural models cannot

explain a great deal of the variation in credit spreads and 2) some common, unknown

factor exists that may be able to account for the unexplained variation left over.

Structural credit risk models, therefore, have little practical application due to

their focus on firm specifics alone. Corporate borrowers and lenders are interested in

determining the timing of their financing decisions in the context of wider macroe

conomic risk. Government policymakers want to study the impact of their actions

2

on borrowers and lenders. A more comprehensive empirical study of credit risk is

necessary to explain a greater amount of the variation in credit spreads and relate

that variation to aggregate macroeconomic factors. This paper empirically analyzes

the effects of macroeconomic variables, particularly real activity and inflation which

are considered to affect monetary policy, on credit spread changes

We begin our study by first examining the time-series properties of a new credit

spread dataset obtained from Lehman Brothers, spanning from May 1994 to June

2007. We then estimate a contemporaneous OLS regression of credit spreads on pos

sible factors. Based on the regime shifting behavior of credit spread data established

through econometric tests, we also estimate a simple Markov regime switching model

of the determinants of credit spreads. We find that real activity and inflation have

significant contemporaneous impact on credit spreads, particularly within certain

regimes.

We find, through the course of estimating OLS and Markov regime-switching

models, that output growth and inflation both have significant power in explaining the

variation in credit spreads, particularly in certain regimes and for shorter maturities

up to 10 years. Output generally has positive co-movement on bonds with credit

ratings from AAA and a negative co-movement on lower-rated bonds. Inflation has

a generally positive correlation on all credit spreads. This result provides motivation

for introducing monetary policy into standard structural credit models. The effects

of output growth and inflation on credit spreads can be seen through their effect on

the discounting of cashflows, in particular coupon payments made by the firm to meet

its debt obligations. Based on the "good beta, bad beta" explanation of Campbell

and Vuolteenaho (2004), output has both an effect on the future cashflows of the

firm, as well as the future discount rates the firm will face. Higher output means

both better cashflows to pay off debt, but also possibly higher discount rates in the

future. Smaller and lower credit quality firms have higher sensitivity to cash flow

risk. Inflation has a uniform effect on all credit spreads, as it affects only the discount

3

rate, common to all firms.

Our findings suggest that comprehensive credit risk models should consider ag

gregate, as well as firm-specific, factors and should model the firm's capital structure

in the context of the larger macroeconomy. Indeed, recent credit risk models, termed

"structural equilibrium" models 1 , connect credit risk with macroeconomic condi

tions, although output and inflation are exogenously determined in these models The

results presented in this paper provide motivation for a theoretical model of credit

that endogenously incorporates realistic features of macroeconomic risks. We will

explore this topic further in subsequent papers.

The remainder of the paper is organized as follows. Section 2 examines the empiri

cal implications of structural equilibrium models on the relationship between macroe

conomic conditions and credit risk. Section 3 describes our data, while Section 4

describes the empirical tests we run for the properties of credit spreads. Section 5

presents our results and an analysis of our results follows in Section 6. Sections 7

concludes and suggests further extensions.

1.2 How Should Macroeconomic Condit ions Af

fect Credit Spreads?

Traditional structural models of default, based on Merton (1974) , specify a particular

firm value process and assume that default occurs when firm value falls below some

explicit threshold. The firm's value can be modeled with a risk-neutral process such

as

— = (r- S)dt + adzQ + X(dqQ - p dt)

where V is firm value, r is the spot rate, S is the firm payout rate, a is the firm value

volatility, A is the size of the a firm-value jump, and p is the risk-neutral probability

1 This terminology was first used in Bhamra, Kuehn, and Sterbulaev (2007)

4

of a jump. Default occurs the first time that firm value reaches a threshold K. As

suggested by Longstaff and Schwartz (1995) , a higher spot rate should increase the

risk-neutral drift of the firm value process. A higher drift should reduce the incidence

of default, thereby reducing credit spreads.

This historical approach in the credit literature does not explicitly consider how

macroeconomic conditions should affect firm value and credit spreads. If we impose

the Taylor rule on how a monetary policy agency relates current output and inflation

conditions into a future risk-free rate, we can make an implicit connection between

current changes in output, inflation, and credit spreads. One version of the Taylor

rule is as follows:

r = 4>n7r + (/)yy + 6

where r is the interest rate, TT is the current rate of inflation, and y is current output

growth. The coefficients 0„. and 0y have been empirically estimated to be positive

in the monetary policy literature. Therefore, the risk-free rate is positively related

to output and inflation and, in turn, increases in output and inflation should reduce

credit spreads. Yet, this approach does not also consider how the firm's default

probability or payout ratio may be affected by changes in macroeconomic conditions.

In addition to a lack of connection between credit risk and macroeconomic con

ditions, structural credit risk models also generate lower credit spreads and higher

leverage that is empirically observed. Newer structural equilibrium models of credit

generate these empirically observed features embedding the contingent claim idea of

the firm with a firm value or cash flow process correlated with macroeconomic con

ditions. Examples of such models include Hackbarth, Miao, and Morellec (2006) and

Chen (2007) . They generate higher credit spreads than previous structural credit

models by having a positive correlation of firm cash flow and default recovery rate

with macroeconomic output. In so doing, these models imply that firms can meet

their debt obligations more often and default less in periods of economic boom than

in periods of recession. This, in turn, generates countercyclical behavior of credit

5

spreads with economy-wide output, implying a negative correlation between credit

spreads and GDP growth.

Structural equilibrium models like Hackbarth, Miao, and Morellec (2006) and

Chen (2007) directly incorporate output into credit risk, but they do not explicitly

incorporate inflation risk in their credit models. David (2007) incorporates output and

inflation as factors affecting the risk-free rate, which is a component of the dynamics

of the pricing kernel. He suggests that macroeconomic risks raise asset volatilities,

thereby depressing asset values. However, he does not explicitly incorporate macroe

conomic risks into bond pricing. Furthermore, all structural equilibrium models do

not specify how macroeconomic conditions evolve, instead describing output growth

and inflation as exogenously given stochastic processes.

Unlike the above modelling approaches, we explain our findings by postulating a

direct mechanism by which firms are affected by both output growth and inflation,

which, along with interest rates, are endogenously specified jointly in macroeconomic

models. For example, macroeconomic models postulate a VAR that describes the

evolution of output growth, inflation, and interest rates.

9t+i

7T*+1

. Tt+1 .

= A(L)

9t

7T*

. r< .

The VAR jointly specifies how output growth, inflation, and interest rates affect

each other in reduced-form. If we then postulate that output growth is a systematic

component of the firm's cash flow growth, then the firm's bond price, which reflects

the discounted stream of coupon payments made to the bondholder, is affected by

both output and inflation. Output growth affects the firm's cash flow growth, and

the discounting rate is impacted by both output and inflation given by the structural

VAR above that simulates the dynamics of the macroeconomy.

We propose that macroeconomic risks, both output and inflation, directly affect

6

credit spreads in the following manner. Output growth is correlated with firm cashflow

growth as suggested by structural equilibrium models, but output growth and inflation

also enter into the discount rate via the Taylor rule. As seen above, output and

inflation are positively related to the risk-free rate; therefore, an increase in either

quantity increases the discount and reduces asset values. While an increase in inflation

should uniformly increase credit spreads of all ratings and maturities, an increase in

output raises both cashflows and discount rates. Therefore, the impact of an increase

in output should be determined by the net effects on cashflows and the discount

rate. If the bond has little chance of defaulting, then output increases the discount

rate thereby increasing credit spreads for highly-rated bonds. For lower-rated bonds,

output increases also raise cashflow for the firm to pay off debt and offset discount

rate increases. We use this intuitive model of credit risk to explain our results and

propose this as a model to be explored in a future paper.

Each of the mechanisms described above has a different implication for the ef

fect of macroeconomic conditions on credit spreads. After we empirically test the

relationship between macroeconomic variables and credit spreads, we compare our

empirical results with the theoretical predictions of each mechanism to evaluate the

applicability of each in modelling credit risk.

For testing the effects of macroeonomic variables on credit spreads, we use as

controls the following variables as suggested by the structural credit models descibed

above.

1. Short Rate

The short rate represents the risk-neutral drift in structural credit models, as

shown above, and is the basic discount rate faced by all firms in the economy.

As suggested above, the short rate might impact the risk-neutral drift of asset

values, thereby impacting credit spreads.

2. Slope of the Yield Curve

7

The term premium, as captured by the slope of the term structure, might

capture future movements in the short rate. Using logic from Longstaff and

Schwartz (1995), if the short rate is expected to converge to the long rate,

then an increase in the slope of the treasury curve increases expected future

short rates. Other studies have noted that the slope of the term structure also

forecasts future economic output.

3. Equity Returns

Equity returns represent an aggregate measure of leverage in the economy and

also another indicator of business climate. As equity values increase, the firm's

debt-to-equity ratio must necessarily decrease. Furthermore, as equity values

rise, the recovery rate, the amount returned to debtholders upon default of the

firm, should also increase.

4. Market Volatility

Structural models of default imply the firm's debt is equivalent to being short a

put option on the firm's assets. Therefore, the volatility of the firm's assets are

key input into the valuation of risky debt. As suggested by Vassalou and Xing

(2004) , equity volatility can be a good proxy for firm asset volatility under

most normal conditions.

Given the above intuition from structural credit model, we will test the impact of

output and inflation on credit risk, controlling for the above significant factors.

1.3 Data Description

1.3.1 Macroeconomic Data

In this section, we describe the data that we use to proxy for factors of credit risk.

Since our credit risk data series extends from May 1994 to June 2007, we must try to

8

find proxies at the monthly frequency, particularly for macroeconomic series, which

are typically measured quarterly.

The real activity (REAL) factor describes overall economic growth, and we cal

culate the series as the month-on-month percentage change in the non-farm payroll

data. The real activity series calculated in this manner shows a high degree of cor

relation with GDP growth, but is available on a monthly frequency. The inflation

factor (INFL) is the month-on-month percentage change in the personal consumption

expenditure price index. Unlike other forms of inflation measures, such as changes in

the CPI, it is also available on a monthly frequency. Both of these time series are ob

tained from data from the St. Louis Federal Reserve (FRED) Database. Since, for a

given month, both non-farm payroll and personal consumption expenditure informa

tion are announced during the middle of the following month, we lag both series by a

month to reflect when the information was known. This also mitigates the possibility

of simultaneity bias in the coefficients.

The federal funds rate (FFR) factor is the market-traded effective federal funds

rate, taken as the 30-day average of the daily quote. The daily effective federal funds

rate is a weighted average of rates on brokered trades. In addition to the level of

interest rates, as measured by the effective federal funds rates, we include the slope

of the term structure to capture term premia. We measure the slope of the term

structure (SLOPE) as the difference of the constant maturity ten-year and two-year

yields on the Treasury curve, also obtained from the FRED database. These are par

yields, consistent with the par credit spreads that we obtained from Lehman Brothers.

For our measure of the equity market's returns, we use the MKT factors from

Kenneth French's website. The CRSP value-weighted market return (MKT) is the

return of the CRSP value-weighted index in excess of the risk-free rate. We could

also include the other Fama-French factors, including value (HML), size (SMB), and

momentum (UMD), as a robustness check as other factors that explain equity market

returns in future work. Elton, Gruber, et. al, find the Fama-French factors have

9

significant effect in explaining credit spread changes.

We use the VIX index as our proxy for market volatility as it is the fair value of

volatility of S & P 500 index options. Just like the federal funds rate, VIX is similar

to a market price series. As a result, we take the average of the previous month's

VIX values to get an end-of-the-month average measure for VIX. The VIX is often

considered a measure of expected future volatility and represents an easily accessible

forward-looking measure of volatility. Furthermore, the VIX is currently measured as

the fair value of volatility. This method of calculation takes into account the volatility

skew of options, which measures the relative price of insuring against market crashes.

Therefore, it is also a measure of market "fear" or risk aversion.

1.3.2 Credit Spread Data

We use credit spread data obtained from Lehman Brothers for this study. The credit

spread data is calculated from bonds that comprise the Lehman Brothers Credit

Indices. The credit spread quotes have credit ratings from AAA to B and with tenor

from 1 year to 10 years. The credit ratings of the series reflect the average of ratings

of three agencies, S & P, Moody's, and Fitch. We generate monthly credit spreads

quotes from May 1994 to June 2007 by taking the average of credit spread quotes

during throughout the month.

Unlike the data used in previous studies, these data consist of credit spreads

calculated from individual corporate bond prices and aggregated to the credit rating

and maturity level. As a result, this study does not suffer the lack of clarity of results

from using Moody's corporate bond index data. We also reflect more correctly market

perception of credit risk than those studies that use individual corporate bonds, at

the cost of losing information at the firm level.

The Lehman Brothers credit spread data is derived by calculating a survival curve

by fitting the curve to individual bond prices. The price of the bond is related to the

10

survival curve by the following equation:

N C N

PV — 2_^ -JZlibor(ti)Q(ti) + Zubor(tN)Q(tN) + 2_^ RZlibor{ti)(Q(ti-l) ~ Q(ti))

i=l •* i= l

where PV is the bond's dirty price, U and N are coupon times, Q(t) is the survival

probability until time t, C is the coupon rate, / is the coupon frequency, Znbor is the

LIBOR discount rate at time t, and R is the assumed recovery of the bond.

The first two terms reflect the present value of scheduled payments (coupon plus

principal) by discounting by LIBOR and the probability of survival. The third term

accounts for the payment of a recovery payment upon default of the bond, weighted

appropriately by the default probability.

The survival curve Q(t) can be fitted to an individual bond or groups of bonds,

using the exponential spline methodology of Vasicek and Fong. We chose data at the

aggregate credit ratings level.

Based on the fitted survival curves, we can calculate a par-spread term structure

for a hypothetical set of bonds of a particular credit rating or industry class. The

par coupon is defined as the coupon of a hypothetical bond of a given maturity

which would trade at par if evaluated using the group's fitted survival probability

term structure. The par spread is defined by subtracting the fitted par yield of the

risk-free bond of the same maturity from the fitted par coupon of the hypothetical

credit-risky bond.

If the hypothetical bond has frequency / and an integer number of payment peri

ods until maturity t^ = 4 , then the par coupon term structure is defined by finding

the coupons which are the solution to the following fundamental pricing equation.

nvaru \ _ A ~ Q(tN)ZlAb(rr(tN) ~ Rj2i=l(Q(tj-l) ~ Q(U))ZLiber(U)

2-a=l Q\ti)^Libcn-\U)

11

The par yield of the risk-free bond is defined in a similar fashion as

vpar i. \ _ j 1 — ZBasejtN) XBase\lN) — J „ j v , .

The par spread to the base curve, in this case the Treasury curve, can then be

derived by subtracting the par base yields from the par risky coupons of the same

maturities.

sBaL(t) = c^(t)-YZrJt)

The above calculated spread measures the spread of a hypothetical bond of a

particular maturity of a given grouping, whether credit rating or industry class. The

fitting procedure used to derive the survival probability curve reduces the impact of

idiosyncratic risk of the individual bonds and provides a better measure of aggregate

credit risk.

The lack of consistency in empirical facts in other papers about credit spreads

results not only from the different possible econometric properties of the data, but

also from the data used to calculate credit spreads. As detailed above, most em

pirical studies about credit spreads use spreads calculated from credit indices, such

as Moody's corporate bond index and Merrill Lynch's credit indices. Other studies

use individual corporate bond prices to calculate risky yields from spreads can be

calculated. Both data sources have features that make calculated credit spreads not

representative of true credit risk in the market.

The most popular source of credit spread data are the Moody's seasoned corporate

bond indices, which are constructed from an equally weighted sample of yields on 75

to 100 non-financial corporate bonds. The maturity of the bonds that comprise the

indices may be anywhere from 10 to 30 years, making inference of risk premia in

credits of different maturities unclear. In fact, many studies arbitrarily subtract the

10-year Treasury yield from the index yield to calculate credit spread, which leads to

incorrect inference.

12

Furthermore, the Moody's bond indices may have issues with the bonds that com

prise the index and how they are weighted within the index. The Moody's bond index

includes callable bonds with optionality that gets reflected in the average corporate

yield. Additionally, the index must be purged of index rebalancing effects that change

the time series.

Credit spreads derived from individual bond prices have their own idiosyncracies.

Firstly, corporate bonds historically are traded rather infrequently, so many data

sources use matrix pricing, interpolating prices between transaction prices. While this

method completes the data set, it also may not accurately reflect changes in credit

conditions. Secondly, individual bond prices are subject to their own idiosyncracies,

due to unique features in the bond convenants and to market valuation of the issuing

firm. Again, credit spreads derived from bond prices may not reflect aggregate credit

condition, which is the focus of our research.

The features of the data used to measure credit risk may have contributed to the

wide variation of results extant in the literature. As we explain below, the dataset

we propose to use from Lehman Brothers should avoid the idiosyncracies found in

individual bond prices and the induced features found in credit indices. However, as

a robustness test of our empirical findings, we apply the same econometric analysis to

Moody's seasoned Aaa and Baa corporate bond indices in Appendix C. We find that

the results are similar to the results we find for our Lehman Brothers credit spread

series.

1.4 Empirical Properties of Credit Spreads

In this section, we propose a plan to study the impact of macroeconomic conditions

on credit risk empirically. Previous empirical credit risk studies imposed arbitrary

econometric specifications without considering the properties of the time series of

credit spreads when choosing the appropriate econometric model. This may have lead

to inconclusive results regarding the nature of the determinants of credit spreads.

13

We propose to document the time series properties of credit spreads and select

an appropriate economic framework, based on those properties. We find that credit

spreads exhibit persistence and regime shifts through unit root and structural break

tests. Previous studies in the literature account for persistence in spreads empirically

by regressing with first differences in credit spreads. We believe that credit spreads

appear to be nonstationary and cannot be rejected for the presence of a unit root

through conventional econometric tests, because credit spreads exhibit regime shift

ing behavior that violates the stationarity assumption. The standard approach in the

literature of taking first differences to handle persistence removes information from

the time series that can be valuable for studying the properties of credit spreads.

Therefore, we perform our study as a regression in levels with lags of credit spreads

among the regressors to account for persistence in the data, while retaining the infor

mation found in the levels, but not in first differences. Our positive confirmation of

structural breaks via Bai and Perron tests (1998) suggest that we should empirically

test the determinants of credit spreads in the context of a regime shifting model.

1.4.1 Tests of Persistence

To select an appropriate econometric framework in which to study the determinants

of credit spreads, we first examine the existence of persistence or non-stationarity

in credit spreads. Several studies have documented the persistence of interest rates

(Fama, 1976, 1977; Rose, 1998 ) and credit spreads (Pedrosa and Roll, 1998 ), par

ticularly measured on frequencies higher than monthly. We, therefore, test our data

for non-stationarity by looking at the autocorrelation functions for different credit

spreads, as well as applying standard unit root tests to our data.

Tables 1.1 through 1.4 show the autocorrelation function of selected credit spread

series from our dataset. We find strong serial correlation in each credit spread series

across different credit ratings and maturities. When the autocorrelation function is

tested on first difference of the monthly credit spread series, we find low autocorre-

14

lation, although not exactly equal to zero. In tables A.l and A.2, we test each credit

spread series for persistence via unit root tests such as augmented Dickey-Fuller and

Phillips-Perron. We also ran augmented Dickey-Fuller tests, as well, with similar re

sults, so we do not present them here. Conforming with our autocorrelation functions

and previous tests in the literature, we cannot reject the credit spread series for the

presence of a unit root.

Maturity lYr 2Yr 3Yr 5Yr 7Yr

lOYr

Autocorrelation Functions Lag 1 0.94 0.96 0.96 0.96 0.96 0.95

Lag 2 0.90 0.93 0.93 0.93 0.92 0.91

Lag 3 0.85 0.88 0.89 0.88 0.88 0.87

Table 1.1: Autocorrelation function values for single A credit spreads, May 1994 to June 2007

Maturity AAA

AA A

BBB BB

B

Autocorrelation Functions Lag 1 0.94 0.94 0.96 0.97 0.92 0.96

Lag 2 0.89 0.89 0.93 0.94 0.88 0.90

Lag 3 0.84 0.83 0.88 0.91 0.82 0.84

Table 1.2: Autocorrelation function values for 5-year credit spreads, May 1994 to June 2007

15

Maturity lYr 2Yr 3Yr 5Yr 7Yr

lOYr

Autocorrelation Functions Lag 1 -0.06 0.04 0.07 0.05 0.04 0.02

Lag 2 0.09 0.11 0.13 0.09 0.06 0.03

Lag 3 -0.04 -0.06 -0.06 -0.08 -0.07 -0.06

Table 1.3: Autocorrelation function values for first differences in single A credit spreads, May 1994 to June 2007

Maturity AAA

AA A

BBB BB

B

Autocorrelation Lag 1 -0.12 0.02 0.05 0.11 -0.17 0.32

Lag 2 0.10 0.04 0.09 0.17 0.08 0.02

Functions Lag 3 -0.11 -0.14 -0.08 0.06 0.03 -0.03

Table 1.4: Autocorrelation function values for first differences in 5-year credit spreads, May 1994 to June 2007

Although we confirm the presence of strong persistence in the credit spread series,

we have several reasons why we choose not to model credit spreads as a unit root or

take first differences in our econometric tests. Firstly, the standard unit root tests,

such as Dickey-Fuller and Phillips-Perron, have low power, not rejecting those cases

that may not be unit root. Furthermore, we have a relatively short sample of data,

consisting of monthly credit spreads between May 1994 and June 2007, which may

not be sufficient to determine the time series properties of credit spreads. Indeed,

interest rates and credit spreads are often modeled in the theoretical literature as

stationary processes, although they empirically exhibit persistence or close to unit

root behavior.

The inability to reject the presence of a unit root in each of the credit spreads

along with the subsamples of each series suggests we should correct for persistence in

our choice of econometric specification. Unlike the specification used by most of the

16

empirical literature testing first differences in credit spreads, we test the determinants

of credit spreads in levels, including lagged credit spreads to account for persistence.

Therefore, in the structural breaks tests and models we estimate below, we employ

the following specification:

yt = a + y't^p + xti + tt

where yt is a single credit spread series. The vector

xt = [REALt INFLt FFRt SLOPEt MKTt VIXt]

contains changes in real activity (REAL), inflation (INFL), the fed funds rate (FFR),

the slope of the term structure (SLOPE), and VIX. We also include the market excess

return (MKT).

1.4.2 Structural Break Tests and Inference from Markov Regime

Switching

A contingent-claim view implies that the factors of credit risk have a non-linear

impact on credit spreads. A simple way to adapt the traditional, linear econometric

specification, such as the one suggested above, to a non-linear phenomenon would be

to introduce changes of regime. Such regime shifts in the model, though, would only

be warranted if we can document that the model experiences structural breaks.

To determine the prescence of structural breaks in credit spreads, we use a battery

of tests developed in Bai and Perron (1998, 2003) . 2 We summarize the purpose

of each test here and postpone their detailed econometric description to Appendix

A. The Bai and Perron methodology first chooses the exact break dates through

the least-squares principle. In a multiple linear regression model, the partitions are

2 Gauss code to implement structural break tests can be found at Pierre Perron's website

17

chosen by minimizing the sum of square residuals. The partition that minimizes the

sum of square residuals objective are the break dates. There are two types of tests to

determine whether there is a structural change: a supFr(k) test that tests the null

of no breaks versus the alternative of k breaks and the double maximum test that

tests the null of no breaks versus the alternative of an unknown number of breaks.

The method to determine the number of breaks consists of sequentially applying the

supFT(l + l\l) test with / breaks versus the alternative of / + 1 breaks starting with

1 = 1. One concludes for a rejection in favor of a model with (I + 1) breaks if the

overall minimal value of the sum of squared residuals is sufficiently smaller than the

sum of squared residuals from the / breaks model.

To test for the existence, number, and location of structural breaks, we applied the

above Bai and Perron (1998, 2003) structural change econometric procedure to the

common econometric specification given earlier. Given the graphs of different credit

spread series as shown in Figure 1, we choose the maximum number of breaks in the

test to be m = 3. We clearly see two breaks in most of the series in the middle of 1998

and again towards the end of 2002 to the middle of 2003. When applying the Bai and

Perron tests, we also account for potential serial correlation and heteroscedasticity.

The results from the structural break tests show some common results across all

credit ratings and maturities. As shown in Tables 1.5 and 1.6, most of the regressions

show a structural break around the summer of 1998 and again around the end of

2002 and the beginning of 2003, with relatively tight confidence interval within those

periods. Bold numbers denote significance at the 1 percent level. The values of

the supF test statistic which test the null of no break versus the alternative of 1

to 3 breaks are generally significant, with the test statistic almost always significant

for two breaks. Similarly, the values of the UDmax and the WDmax statistics (not

shown in the table) which test for the null of no break versus the alternative of an

unknown number of breaks, are also highly significant at the 1 percent level for all

ratings and maturities. From the strong significance of the above three statistics, we

18

can conclusively state that credit spread do exhibit structural breaks in the period

between May 1994 and June 2007.

With m = 3, the maximum number of breaks, we proceed to estimate the break

dates and their 95% confidence intervals. Under global minimization of the sum of

square residuals, we find the first break date for credit spreads from AAA to BB to be

around August 1998 with relatively tight confidence interval between July 1998 and

February 1999. The second break date for most credit spreads occurs around October

2002, although there is much higher variation in the specific break date, depending

on the credit rating and maturity, with wider confidence intervals between May 2002

and September 2004. The relationship for B credit spreads exhibit different breaks

with the first break date in the series occuring around the beginning of 2000 with the

exception of 5 year and 7 year tenors, which correspond to the late 1998 break date.

The second break date for B credit spreads almost uniformly falls in the beginning of

2004.

The break dates we find in all the credit spread series correspond to key financial

and macroeconomic events, giving further credence to existence of actual breaks in

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Break date 07/1998 03/2003

08/1998 04/2003

08/1998 04/2003

08/1998 04/2003

08/1998 05/2003

08/1998 06/2003

95% C.I. 02/1998 03/2003

07/1998 03/2003

07/1998 07/2002

08/1998 07/2002

08/1998 07/2002

02/1998 07/2002

08/1998 11/2003

11/1998 12/2003

09/1998 05/2003

11/1998 03/2003

11/1998 11/2003

12/1998 04/2004

1 731.13

5442.18

6009.19

25.90

85.04

805.78

supF test 2

7018.15

58233.22

52040.66

214820.02

37406.52

29824.46

3 287281.06

643382.28

309221.33

41417.50

44353.14

525900.69

supF(i+l || i) i = 1 i =2

181.49

579.13

949.42

3239.84

261.73

153.38

65.17

30.54

29.83

30.29

50.41

18.99

Table 1.5: Bai and Perron (1998) structural break test dates and confidence intervals for single A credit spreads

19

Rating AAA

AA

A

BBB

BB

B

Break date 08/1998 02/2003

08/1998 03/2003

08/1998 04/2003

08/1998 04/2003

08/1998 06/2003

02/2000 04/2003

95% C.I. 04/1998 04/2002

01/1998 07/2002

08/1998 07/2002

08/1998 12/2001

04/1998 03/2003

08/1999 03/2003

11/1998 02/2003

11/1998 04/2003

11/1998 03/2003

08/1998 09/2003

12/1998 07/2003

08/2000 02/2004

1 254.12

892.15

25.90

124.57

6.72

38.19

supF test 2

18751.25

55627.44

214820.02

101708.45

14011.47

97037.08

3 46436.70

491577.71

41417.50

751472.02

73961.83

285133.30

supF(i+l || i) i = 1 i =2

35.42 5.77

26.23 22.37

3239.84 30.29

174.02 26.85

43.86 13.45

109.09 8.64

Table 1.6: Bai and Perron (1998) structural break test dates and confidence intervals for 5-year credit spreads

the data. The first break date for most of the series, November 1998, corresponds to

the failure of LTCM around the middle to end of that year. Prior to the failure of the

hedge fund, a series of international financial crises occurred in Asia in late 1997 and

in Russia around the middle of 1998. The impact of these financial crises, followed by

the failure of one of the world's largest hedge funds, may have increased risk aversion

in the markets, causing a widening of credit spreads for the following five or six years.

The second break date in the end of 2002 or the beginning of 2003 may correspond to

the gradual recovery of the markets from September 11 and the corporate accounting

scandals that followed in 2002, coupled with the Federal Reserve's easing of interest

rates. Additionally, global uncertainty regarding the possibility of war in the Mid

dle East may also have been realized, reducing risk aversion and the corresponding

volatility in credit spreads around this time. Single B credits, which behave more like

equity than like debt, may respond more to equity market factors and, as a result,

the first regime shift in our sample occurs around the beginning of 2000 during the

bursting of the dot-com bubble.

When we consider the graph of VIX (Figure 2) over the sample period between

20

May 1994 and June 2007, we also find that VIX experiences a change in mean over

the period corresponding roughly to the breaks found in most of the credit spread

series. VIX remains consistently above the 17% mark in the period between the

middle of 1998 and the middle of 2003. Therefore, we generally describe our credit

spread regimes as high- and low-volatility regimes. The period from the beginning of

our sample until the fall of 1998 (spring of 2000 for single B) and the period from the

spring of 2003 until the end of our sample corresponds to regime 1, the low-volatility

regime. The period between fall 1998 (spring 2000 for single B) and the spring of

2003 corresponds to our regime 2, the high-volatility regime.

1.4.3 Difference from Previous Empirical Studies

As we mentioned in the introduction, the literature contains a number of empirical

studies on the determinants of credit spreads. Unlike our paper, these studies gener

ally arrive at the conclusion that the variation in credit spreads is difficult to explain

with simple factors, but that a common, unknown factor exists that may explain the

majority of the variation. Our results differ as we test credit spreads in levels and

allow for regime shifts.

Studies, including Collin-Dufresne, Goldstein, and Martin , employ ordinary least

squares on first differences in credit spreads as their econometric framework for an

alyzing credit spreads. 4 As we noted above, such studies cannot explain a great

deal of the variation in spreads. This lack of explanatory power may be attributable

to the imposition of an arbitrary econometric framework, testing first differences in

spreads to account for persistence or the exclusion of the structural break properties

we find here in credit spreads.

Other papers, such as Morris, Neal, and Rolph (1998) , also determine unit root

behavior in credit spreads and impose a cointegration framework on credit spreads

4 Other papers in a similar vein include Elton, Gruber, Aggarwal and Mann (2001) , Huang and Kong (2003) , and Alessandrini (1999)

21

with possible aggregate factors. As we stated earlier, we do not believe that credit

spreads are truly unit root in nature, and therefore a cointegration framework may not

be approprate for analyzing determinants of credit spreads. Furthermore, estimating

an error-correction model, as they do, to correct for cointegration does not truly

capture the non-linearity inherent in credit risk.

Our approach differs from existing papers in the literature by testing for the

time series properties of credit spreads and imposing a regime switching model on

a linear model of credit risk. In so doing, we find that simple factors, including

macroeconomic conditions, do explain credit spreads in certain regimes and provides

empirical motivation for theoretical model that link credit risk and macroeconomics.

1.5 Model Estimation

Once we establish the presence and number of structural breaks via the Bai and Per

ron (1998) tests, we should incorporate structural breaks into our model estimation.

Although the structural break tests probabilistically determine the timing of breaks,

we cannot identify the cause for the break. Therefore, we estimate the above linear

specification with a Markov regime switching model where the regimes shift based

on information in all the determinants we chose to study. The coefficients a, (3, and

7 are allowed to vary over some subsamples of the data, determined endogenously

with fixed volatility for the error term. We estimate the regime switching model us

ing Hamilton's (1990) Markov approach with constant transition probabilities. The

model maximizes the likelihood function of the linear model in a number of states

using the EM algorithm of Dempster and Laird (1994).3

In this section, we first estimate the model using OLS regression techniques. Our

results are similar to those found in previous papers of little explanatory power. We

then present our estimation of the model allowing for regime shifts, which increases

3 We estimated the model using the hmarkov_em code from James LeSage's Econometrics toolbox, written in Matlab

22

the explanatory power of the model and the significance of certain factors.

1.5.1 OLS Estimation Results

We first estimate our econometri specification using OLS with no structural breaks.

As we mentioned earlier, previous studies that estimated linear regressions of changes

in spreads on changes in possible factors resulted in low explanatory power and in

significant coefficients for a variety of macroeconomic and firm-specific factors. In

contrast, using our specification in levels with lagged credit spreads, we find a great

deal of explanatory power with R2 around 90% with a few macroeconomic factors

explaining much of the variation. Table 1.7 below shows sample regression results for

single A credit spreads of different maturities, while Table 3b shows sample regression

results for 5-year credit spreads of different ratings. For each maturity or rating, the

first row shows the regression coefficient and the second row shows the t-statistic with

the t-stats significant at the 5% level in bold.

Four common macroeconomic factors seem to explain credit spreads: previous

period credit spreads, real activity, inflation, and VIX. Consistent with the results

of our persistence tests, we find that lagged credit spreads have strongly significant

impact on the current level of credit spreads. Output growth generally has a negative

effect on credit spreads, although it also has a significant and positive effect on AAA

credit spreads. Inflation has a positive effect on credit spreads of all maturities and

ratings, as does VIX, the measure of market volatility and risk aversion.

23

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Adj. R2

0.9172

0.9442

0.9489

0.9471

0.9440

0.9337

Const

-0.0006 -0.7400

-0.0006 -0.8944

-0.0007 -0.9573

-0.0005 -0.6172

-0.0002

-0.2239

0.0001 0.1364

cs^ 0.8059

20.5299

0.8613 27.0646

0.8542

27.0828

0.8484 26.2869

0.8444

26.0583

0.8344 24.4374

REAL

-0.0380 -2.9959

-0.0255 -2.3485

-0.0291 -2.6000

-0.0343 -2.7695

-0.0339 -2.7674

-0.0312 -2.6294

INPL

0.0263 1.9783

0.0259 2.0429

0.0306 2.3709

0.0365 2.6423

0.0374

2.7393

0.0376 2.8047

FFR 0.0109 1.1151

0.0050 0.5794

0.0058 0.6588

0.0046 0.4962

0.0023 0.2557

0.0019 0.2140

SLOPE

-0.0289 -1.3693

-0.0284 -1.5214

-0.0316 -1.6860

-0.0423 -2.0765

-0.0512

-2.5214

-0.0561 -2.8227

MKT 0.0003 0.1881

-0.0008

-0.4988

-0.0006 -0.3977

-0.0002 -0.1052

-0.0004

-0.2238

-0.0006 -0.3709

VIX 0.0080

4.9879

0.0073 5.0122

0.0081 5.3877

0.0087 5.5214

0.0085 5.5743

0.0079 5.3160

Table 1.7: OLS regression coefficients for single A credit spreads on regressors

Rating

AAA

AA

A

BBB

BB

B

Adj. R2

0.9032

0.9183

0.9471

0.9615

0.8839

0.9340

Const

0.0004

0.6438

-0.0002 -0.2438

-0.0005 -0.6172

-0.0006

-0.5477

-0.0065 -1.5809

-0.0233 -2.2378

CSt^

0.8138

20.6120

0.8161 22.5977

0.8484 26.2869

0.8702

29.3963

0.7637 17.0164

0.8508 22.2116

REAL

0.0156 2.3954

-0.0245 -2.3646

-0.0343 -2.7695

-0.0535 -3.0759

-0.1663 -2.7529

-0.3153 -2.0596

INFL

0.0199 2.8117

0.0336 2.6176

0.0365 2.6423

0.0361 2.8813

0.1955 2.5010

0.3869 1.9919

FFR -0.0038 -0.5312

-0.0013 -0.1541

0.0046 0.4962

0.0093

0.7138

0.0790 1.4665

0.2872 2.0663

SLOPE

-0.0372

-2.3119

-0.0437 -2.2713

-0.0423 -2.0765

-0.0527

-1.8973

0.0278 0.2452

0.1006 0.3691

MKT 0.0001

0.0820

-0.0003 -0.1984

-0.0002 -0.1052

0.0005 0.2001

-0.0029 -0.3087

-0.0285 -1.2741

VIX 0.0055

4.3548

0.0078 5.3469

0.0087 5.5214

0.0117 5.2629

0.0427 4.9171

0.0875 3.9030

Table 1.8: OLS regression coefficients for 5-year credit spreads on regressors

24

1.5.2 Markov Regime-Switching Results

Our intent in this paper is to attribute some of the variation in credit spreads to

macroeconomic factors through an appropriate, yet parsimonious econometric model.

Based on the results presented above and the graph of different credit spreads, we esti

mate the general specification presented above with a 2-state Markov regime switching

model. While we do not pre-specify the determinants of the states, we find that they

usually correspond to the high- and low-volatility regimes discussed earlier in our

structural break results. Tables 1.9 and 1.10 present our Markov regime switching

estimation results. In each row corresponding to a credit rating/maturity pair, the

first two rows present the coefficients and t-statistics for the low volatility regime,

and the second two rows present the corresponding coefficients and t-statistics for the

high-volatility regime.

By estimating the relationship between credit spread changes and the macroeco

nomic and financial determinants as a regime-switching model, we find that impact of

macroeconomic variables have significant impact on in the high volatility regime. In

the low volatility regime, it seems that only the previous level of credit spreads is rel

evant for the current level of spreads, consistent with the persistence we cannot reject

in the credit spread series. While Collin-Dufresne, Goldstein, and Martin (2001) and

others find low explanatory power and conclude the existence of a common, unidenti

fied systematic factor, we find that, by estimating a model taking non-linearities into

account, that certain common factors explain credit spreads over certain periods in

the sample

Furthermore, the smoothed probabilities of the Markov regime-switching models

for the different states for each model correspond to the break dates we find through

the global minimization procedure. Figure A.3 shows smoothed probabiities for the

regime-switching model for 5-year A credit spreads. The smoothed probability for

regime 2 (high volatility) increases around the end of 1998 and falls around 2003.

This gives further evidence for the existence of structural breaks, particularly during

25

the periods we find through the Bai and Perron (1998) tests.

In the high volatility regime, the coefficients of the regression on macroeconomic

factors, such as real activity and inflation, are consistent with the signs we find under

OLS. We find the inflation is positively related to credit spreads across all maturities

and ratings classes, although mostly in the high volatility regime. Real activity or

output also affect spreads only in the high volatility regime, but its effect varies with

credit rating. For AAA and AA bonds, an increase in output also increases credit

spreads. For all other credit spreads, particularly lower than investment grade bonds,

output is negatively correlated with credit spreads.

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. R2

0.9598

0.9783

0.9832

0.9840

0.9819

0.9708

Const 0.0008 1.4659

-0.0011 -0.1853

0.0000 0.8565

-0.0033 -6.5662

0.0007 1.5904

-0.0019 -3.8528

0.0002 0.2194

-0.0078 -0.8457

0.0003 0.6377

-0.0069 -2.7133

-0.0009 -1.3659 0.0000

-0.0036

CSt-i 0.9500

21.9782 0.3628

3.8695

0.8923 35.5516

0.7833 6.5453

0.8530 25.3093

0.7440 5.5315

0.9892 9.8172 0.3569 1.7984

0.9678 21.2988

0.3572 4.3204

1.0415 8.8436 0.4389

4.6397

REAL 0.0066 0.4103

-0.1931 -2.7541

-0.0108 -1.1287 -0.0424 3.1243

-0.0216 -2.1349 -0.0221

-2.0736

0.0128 0.3227

-0.1134 -2.8605

0.0064 0.4540

-0.1100 -3.2511

0.0169 0.6125

-0.1176 -3.5198

INFL -0.0116 -1.1512 0.1093

2.5765

0.0142 3.5285 0.0644

1.9755

0.0130 1.7369 0.0332

3.7207

-0.0079 -0.6706 0.0637

2.1150

-0.0025 -0.3853 0.2762

5.7875

0.0104 0.5797 0.0903

3.4604

FFR -0.0145

-2.1267 0.0488 0.7578

0.0034 0.9323 0.0245 0.8565

0.0012 0.2026 0.0120 0.4810

-0.0085 -1.4099 0.0830 1.5016

-0.0078 -1.8441 0.0722

3.3233

-0.0043 -0.3808 0.0475 1.1596

SLOPE -0.0430

-2.7907 -0.0885 -0.4723

-0.0153 -2.5890 -0.0049 -0.3403

-0.0273 -2.4790 -0.0205 -0.6321

-0.0220 -1.2697 -0.0039 -0.0831

-0.0238 -1.8675 -0.0577

-1.9741

-0.0133 -0.6654 -0.0935

-4.4543

MKT -0.0004 -0.2350 0.0008 0.1127

-0.0004 -0.3432 0.0039 1.2111

-0.0009 -0.6608 -0.0026 -0.9063

0.0004 0.1538

-0.0002 -0.0446

-0.0002 -0.1875 0.0023 0.6040

-0.0012 -0.5090 0.0009 0.2824

VIX 0.0026 1.7996 0.0222

3.9386

0.0017 1.6078 0.0197

6.0123

0.0021 1.9439 0.0203

9.6121

0.0016 1.3355 0.0347

3.6194

0.0020 1.8758 0.0349

7.4425

0.0023 1.3876 0.0205

3.7003

Table 1.9: Markov regime switching model coefficients for single A credit spreads on regressors

26

Rating AAA

AA

A

BBB

BB

B

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. R2

0.9553

0.9732

0.9840

0.9901

0.9524

0.9709

Const 0.0010 0.0003

-0.0010 0.0006

-0.0014 -3.1226

0.0008 8.4425

0.0002 0.2194

-0.0078 -0.8457

-0.0007 -2.5062 0.0078 4.9930

-0.0245 -2.1327 0.0014 0.4979

-0.1954 -9.5151 -0.0100 -0.8703

CS,_i 0:8556

5.9474 0.7819

22.6640

0.8477 9.1639 0.8038

30.2014

0.9892 9.8172 0.3569

3.7984

1.0122 43.7808

0.4776 7.9258

0.8275 5.9326 0.7558

12.2385

0.9759 5.4522 0.9329

20.8274

REAL 0.0121 0.9272 0.0240

3.2327

0.0124 0.7992

-0.0214 -2.7477

0.0128 0.3227

-0.1134 -2.8605

0.0138 0.9823

-0.4388 -12.2073

-0.4160 -3.1525 -0.0841

-4.3992

-0.6050 -0.7562 -1.0260

-3.2106

INFL 0.0250 1.9572 0.0160

2.3349

0.0110 0.6229 0.0246

3.4463

-0.0079 -0.6706 0.0637

2.1150

0.0002 0.0225 0.0548

2.6205

0.4566 3.2595 0.0777

3.6482

3.5731 4.0761 1.1537 0.7742

FFR -0.0015 -0.3529 -0.0223 -1.4887

-0.0085 -0.6612 -0.0065 -1.4782

-0.0085 -1.4099 0.0830 1.5016

-0.0059 -0.9087 -0.0113 -0.4658

0.2120 0.6424 0.0165 1.5225

1.2886 2.0295 0.1007 0.7984

SLOPE -0.0226

-2.4924 -0.0792

-3.2184

-0.0296 -1.1994 -0.0404

-4.2189

-0.0220 -1.2697 -0.0039 -0.0831

-0.0226 -1.8616 -0.5926

-8.7880

-0.0210 -0.0570 0.0007 0.0138

1.0215 1.1842

-0.0838 -0.3556

MKT -0.0011 -0.8552 -0.0003 -0.0968

0.0002 0.0683

-0.0003 -0.2593

0.0004 0.1538

-0.0002 -0.0446

0.0020 1.0679

-0.0001 -0.0103

-0.0529 -3.0180 -0.0011 -0.1427

-0.3860 -4.4033

0.0138 0.7372

VIX 0.0022

2.0973 0.0154

6.2167

0.0181 8.5420 0.0036

3.4920

0.0016 1.3355 0.0347

3.6194

0.0037 2.8265 0.0390

10.6517

0.1042 6.8452 0.0185

3.2424

0.0346 1.2830 0.4443

5.7224

Table 1.10: Markov regime switching model coefficients for 5-year credit spreads on regressors

1.5.3 Robustness Check with First Differences

Although we find strong evidence for the impact of output growth and inflation on

credit spreads, even while controlling for persistence, the large R2 and significant

coefficients may be due to persistence in the credit spread time series. To check the

robustness of our results, we test the impact of our macroeconomic factors on credit

spread changes. This approach is akin to assuming that credit spreads are, in fact,

unit root, a notion that we reject from our theoretical knowledge of interest rates and

credit spreads.

We employ the following specification:

Ayt = a + x't(3 + et

27

where yt is a single credit spread series. Running a regression in first differences

in credit spreads removes persistence and avoids spurious regression estimates. The

vector xt = [AREALt AINFLt AFFRt ASLOPEt MKT AVIXt] contains changes

in real activity (REAL), inflation (INFL), the fed funds rate (FFR), the slope of the

term structure (SLOPE), and VIX. We also include the market excess return (MKT).

All changes in the regression are taken as the one-month difference in the variable.

Like many previous studies, we find that OLS estimation cannot capture the

impact of macroeconomic factors on credit spread changes. Our estimation, not shown

here, of a simple linear model results in low explanatory power and insignificance of

most factors, except VIX, similar to the results in the the literature.

Estimating the specification in differences using Markov regime switching confirms

the results we obtain from the regression in levels. We find that, in certain regimes,

changes in output growth and inflation do impact changes in credit spreads. The

regression exhibits structural breaks at the same points as the specification in levels.

Estimating the model with two regimes also increases the explanatory power of the

model. Furthermore, changes in output growth have a negative effect and inflation

have a positive effect on changes in credit spreads.

28

Rating AAA

AA

A

BBB

BB

B

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. R2

0.5436

0.6218

0.6748

0.6057

0.4884

0.5990

Const 0.0000 0.1450 0.0001 0.5987

0.0000 -0.5237 0.0003 0.6509

0.0000 -0.5119 0.0002 0.9337

-0.0001 -1.2306 -0.0013 0.0000

-0.0009 -2.7047

0.0048 2.8037

-0.0002 -0.2976 -0.0035 -1.0803

REAL 0.0040 0.1222 0.0355

2.5260

-0.0298 -0.7897 -0.4750 -1.5799

-0.0330 -0.8897 -0.4447

-3.5807

-0.0355 -0.6618 -0.6541

-5.7734

-0.2297 -0.9751 -2.2798

-3.0210

-0.7676 -1.6403 -4.8168

-3.8020

INFL -0.0040 -0.2016 0.3965

3.7518

0.0235 0.9331 0.2359

2.3599

0.0126 0.4723 0.0452

2.7224

-0.0096 -0.2605 0.0148

3.0567

0.0780 0.5092 1.4439 1.7933

-0.3338 -1.0352 4.5114

3.0835

A F F R -0.0531 -1.8547 0.4780

6.0867

-0.0160 -0.4237 0.4870

5.4227

-0.0294 -0.8826 0.4414

5.9739

0.0147 0.3051 0.2335

2.1430

-0.9489 -1.0524 0.5736

2.7679

0.7321 1.7232 0.9718 0.4574

A SLOPE -0.0258 -0.6684 0.1205 0.5596

-0.0532 -1.2326 0.6602

2.4753

-0.0322 -0.7511 0.4545

2.5419

0.0017 0.0280

-1.2083 0.0000

-0.2605 -0.8940 2.5843

2.6166

0.3730 0.6155

-3.3120 -1.0403

MKT 0.0001 0.0617

-0.0077 -3.4580

0.0002 0.1451

-0.0056 -3.2154

0.0007 0.5452

-0.0051 -1.9713

0.0036 1.8546

-0.0292 3.2349

-0.0003 -0.0311 -0.0776

-2.7101

0.0158 0.9160

-0.2871 -4.3243

A V1X 0.0004 0.2878 0.0200

5.7959

-0.0007 -0.4036 0.0264

7.3939

-0.0033 -1.4848 0.0298

10.3452

-0.0010 -0.5366 0.0746

4.0230

0.0081 0.8111 0.1266

4.5887

-0.0185 -0.8095 0.5951

10.6494

Table 1.11: Markov regime switching model coefficients for first differences in 5-year credit spreads on regressors

1.6 Discussion of Results

1.6.1 Regime Switching in Credit Spreads

As shown previously, the time series of credit spreads tested here between May 1994

and June 2007 exhibit regime shifts at least twice in the sample. These results are

consistent with new models of credit risk and with empirical evidence

Both Hackbarth, Miao, and Morellec (2006) and David (2007) incorporate regime

switching into their theoretical models of credit risk and, in so doing, relate macroeco-

nomic conditions with credit spreads. Hackbarth, Miao, and Morellec (2006) observe

that firms adapt their default and financing policies to the position of the economy in

the business cycle phase. Based on this concept, they find that credit spreads change

29

with aggregate economic shocks, due to changes in the firms' behaviors with regards

to their capital structures. Hackbarth, Miao, and Morellec (2006) and Chen (2007)

use the ideas that firms change capital structure under different macroeconomic con

ditions to answer many puzzles extant in both the theoretical and empirical credit

risk literature.

David (2007), on the other hand, specifies that current inflation and output signal

future conditions and affect asset valuations. Asset valuation ratios and volatilities

vary over time as investors update their beliefs about the hidden states of fundamen

tals and reasses the prospects of future real growth in fundamentals. He, therefore,

models the regime shifts as an exogenous change in fundamentals.

Among the empirical literature, Alessandrini (1999) observes that credit spreads

are more sensitive to macroeconomic risks during recessions than during expansions.

Although he tests credit spread factors ony with OLS, his observation of different

behaviors in subsamples provides some empirical evidence for regime shifts.

While we arrive at the same conclusions as the authors mentioned above, our

rationale about the cause of regime shifting in credit spreads is different. We do find,

based on the Bai and Perron (1998) tests and Markov regime-switching models, that

most credit spreads change regime once around the summer/fall of 1998 and again

around the spring of 2003. Single B credits, however, exhibit their first regime change

around the early part of 2000, reflecting that they behave more like equities. Based

on such evidence, we believe that credit spreads do exhibit regime shifts not based

on macroeconomic fundamentals, but rather based on market risk aversion.

We base this on the similarities in behavior between credit spreads of all ratings

and the VIX, the measure of the fair value of volatility in the market. VIX, measured

as the fair value of volatility of S&P 500 index options, reflects not only the present

level of expected volatility in the market, but also is a measure of risk aversion as

it incorporates information about the index volatility skew. The volatility smile on

index options reflects how risk averse investors are, because it incorporate information

30

from out-of-the-money puts, which investors purchase and bid up to protect against

market downturns.

The key regime shift dates of fall 1998, spring 2000, and spring 2003 are marked

not by revelations about macroeconomic fundamentals, but rather exogenous asset

market events. Risk aversion and credit spreads appear to have jumped in the fall

of 1998 during the Russian default and the collapse of LTCM and again during the

2000 market crash. Neither events had direct connection with macroeconomic funda

mentals. In the case of the collapse of LTCM, macroeconomic fundamentals remained

strong and asset valuations recovered quickly; yet, credit spreads remained high there

after. The moderation of credit spreads during the early part of 2003 also had little

basis in macroeconomic events or even asset valuations. That date seems to reflect

the quick resolution of the military conflict in Iraq, after which credit spreads seemed

to decline continuously until the end of our sample.

Furthermore, we find that the impact of many of the factors we have chosen for

credit spreads seem only to have impact in the high-volatility state. This may explain

lack of explanatory power in other work when testing for credit risk factors with first

differences, particularly with a simple linear regression.

1.6.2 The Impact of Macroeconomic Variables

As we observed earlier, inflation in the high volatility regime increases credit spreads

across all ratings and maturities. Increased output in the high volatility regime, on

the other hand, increases spreads for higher rated bonds and decreases spreads for

lower rated bonds. This result contradicts the predictions of most structural credit

models and confirms the predictions of structural equilibrium models.

As mentioned previously, structural models of default take the risk-free rate as

given and not determined endogenously. In such a model, an increase in output and

in inflation, which increases the risk-free rate via the Taylor rule, would cause the

risk-neutral drift of the firm value process to increase. This, in turn, reduces credit

31

spreads as the firm hits its default boundary less frequently. While this prediction

matches what we find empirically with lower-rated firms, it does not conform with

our results for an increase in output for AAA and AA rated firms nor does it predict

the correct sign for the sensitivity of credit spreads with inflation.

Structural equilibrium models which embed credit risk within a general equilib

rium framework allow the firm value and risk-free rate to be determined endogenously.

Therefore, papers like Hackbarth, Miao, and Morellec (2006) and David (2007) pre

dict that increased output should decrease credit spreads. Again, this result matches

what we find for lower-rated firms and contradicts what we find for higher-quality

firms. Furthermore, David (2007) finds that inflation increases asset volatilities and

reduces valuations, thereby increasing spreads. This is consistent with the effect we

find for inflation for all classes of credit spreads.

To gain better intuition of the possible drivers behind our empirical results, we

propose the following intuition. Suppose the payout of the bond at each period is

git) = cX{t < T)X(t <r) + FS(t - T)X{t < r) + uF8{t - r)X(t < T)

where c is the coupon, T is the maturity, K(t) is the firm's cash flow at time t, F is

the face value of the bond, r represents the first time Kit) < c, and UJ represents the

recovery rate when the firm defaults. The function x(.) is the indicator function, and

the function 5(.) is the Dirac delta function.

The price of the bond at time 0 is therefore

P(0, T) = EQ

where Q represents the risk-neutral measure.

When realized macroeconomic output or real activity increases, it has two effects

on the bond pricing equation. If we assume that an individual firm's cash flow growth

is directly correlated with GDP growth, then, as GDP growth increases, market par-

g(t)

i + rty

32

ticipants would expect that future cash flows of the firm would increase, thereby

increasing the probability of the firm meeting its coupon obligation in every period

until maturity. However, the imposition of a Taylor rule would also imply the mon

etary policy agency would increase interest rates, thereby increasing the discount

factor, and reducing future cash flows. As a result, the impact of increasing real ac

tivity on firms differs by credit rating, as shown in the empirical results. Firms with

higher credit ratings already are generally able to meet their coupon obligations, so

that cash flow impact of increased GDP growth is minimal. They, however, are im

pacted more by an increase in interest rates, which lowers the price of their bonds and

subsequently increase credit spreads. In contrast, firms with lower credit quality see

an increase in their cash flows and their ability to meet debt obligations with GDP

growth increases. This positive effect offsets the discounting effects of the Taylor rule

and reduces the spreads on lower-rated bonds. Furthermore, the recovery rate can be

assumed to be positively correlated with GDP growth, contributing to lower spreads

for firms that default more often. Bondholders can expect to recover more of their

investment in case of default when macroeconomic conditions are good.

The effects described here parallel the "good beta, bad beta" theory, proposed

by Campbell and Vuolteenaho (2004) . They separate the beta of a stock with the

market portfolio into two components, one reflecting news about the market's future

cash flows and one reflecting news about the market's future discount rate. We

propose a similar effect with credit spreads here through this simple bond-pricing

equation. The change in GDP growth reflects information about future cash flows

and discount rates for both the macroeconomy and the firm. Firms with higher

credit ratings have greater sensitivity to discount rate information or "bad beta" and

perform like large-cap or growth stocks. Similarly, firms with lower credit ratings

have greater sensitivity to cash flow information or "good beta" and perform like

small-cap or value stocks.

The effects of an increase in inflation are far clearer and reflected in the strong

33

positive relationship between inflation and credit spreads in the results. In the context

of the bond-pricing equation above, inflation increases interest rates via the Taylor

rule, which, in turn, reduces discounted expectations of future cash flows.

1.7 Conclusions and Future Research

In this paper, we intend to study the impact of macroeconomic variables on credit

spreads as a measure of credit risk. First, we study the time series properties of credit

spreads as proxied by a relatively new dataset from Lehman Brothers. This dataset

avoids the pitfalls associated with other time series used to calculate credit spreads

and, therefore, provides a cleaner series for econometric inference. We find that credit

spreads between May 1994 and June 2007 exhibit regime shifts around key market

events, including the collapse of LTCM, the NASDAQ market collapse of early 2000,

and the end of the military phase of the war in Iraq in 2003.

Based on the above time series properties of credit spreads, we test the explanatory

of a set of factors, including output and inflation, on credit spreads in a Markov

regime-switching linear model. We find that such a model explains up to 70 % of

the variations in credit spreads and, in the high-volatility regime between 1998 and

2003, a number of factors are significant in the explanation of changes in credit

spreads. Based on a simple bond-pricing equation and the Taylor rule, we explain

the significant effects of output and inflation on credit spreads in terms of the future

cash flow and discount rate effects.

The results presented in this paper help resolve some of the ambiguity found in

previous empirical studies of credit spreads. The conclusive test of regime shifts in

the series may help explain the assertion in Collin-Dufresne, Goldstein, and Martin

(2001) of the existence of an unknown common factor affecting all credit spreads. Fur

thermore, the cash flow/discount rate explanation of the impact of macroeconomic

variables on credit spreads could be extended in a theoretical model. Currently, theo

retical models only consider the firm's capital structure behavior in isolation without

34

macroeconomic effects. Our future work will focus only explaining the behavior of

credit spreads in the face of changing macroeconomic conditions by embedding a

model of credit risk within a monetary economy model.

35

Chapter 2

Credit Spreads in a New

Keynesian Macro Model

2.1 Introduction

Valuation of credit risk is central to corporate financing decisions. Most of the models

in the credit risk literature, following Merton (1974), value the firm's risky debt as

a short put option on the firm's assets. This structural approach takes as given the

dynamics of the risk-free rate and the dynamics of firm value with default occuring

when the firm's asset value falls below a pre-specified boundary. Corporate bonds

are, therefore, valued as claims on the firm's asset value contingent on default oc

curence. This approach, however successful in characterizing firm behavior, ignores

how aggregate economic conditions affect the dynamics of the firm value and risk-free

interest rate processes.

The above structural credit approach also does not address several empirical ob

servations about credit spreads, a measure of credit risk. Fama and French (1989)

were among the first to document that credit spreads widen when economic conditions

are weak. Collin-Dufresne, Goldstein, and Martin (2001) and Elton, et. al, (2001)

both document that firm-specific default risk factors " account for a surprisingly small

36

fraction" of credit spreads. Alessandrini (1999) finds greater sensitivity of credit risk

to aggregate and firm-specific factors during recessions than during expansions. As

suggested by Collin-Dufresne, et. al (2001), the literature "suggests the need for fur

ther work on the interaction between market risk and credit risk - that is, general

equilibrium models embedding default risk."

Two recent empirical papers directly verify the impact of macroeconomic con

ditions on credit spreads. In a previous paper, we examine the direct relationship

between credit spread changes and macroeconomic conditions, as summarized by

output and inflation, in a linear model. We find that increasing inflation raises credit

spreads uniformly, while greater output growth raises higher-rated credit spreads and

lowers the spreads on lower-rated bonds. When the linear model is then estimated

with two possible regimes, we find that output growth and inflation impact credit

spreads in the manner described earlier in periods of high volatility or risk aversion in

the market. This result concurs with an earlier study done by Wu and Zhang (2008)

who arrive at the same conclusion when valuing credit spreads in a no-arbitrage term

structure model.

Recently, several structural equilibrium models have been developed in the liter

ature to explain the connection between macroeconomic conditions and credit risk.

1 Such structural equilibrium models explicitly link macroeconomic dynamics with

the firm value or cash flow process to generate macroeconomic implications for credit

risk. Hackbarth, Miao, and Morellec (2006) and Chen (2007) , for example, suggest

firms adjust their capital structure optimally in response to changes in aggregate out

put and find counter-cyclicality of credit spreads with economic growth as a result.

David (2007) prices corporate debt in a model where expected earnings growth rates

and expected inflation are unobservable, but follow a Markov regime switching pro

cess. Tang and Yan (2006) link macroeconomic output to the cash flow process of

the firm, which is then used to price risky debt in a contigent claims fashion. Al-

1 This terminology "structural equilibrium" was first used by Bhamra, Kuehn, and Sterbulaev to describe equilibrium models that embed credit risk

37

though each of these studies link macroeconomic conditions to the firm's default, all

these studies model the economy with exogenous stochastic processes, rather than

in a structural framework. Furthermore, with the exception of David (2007), most

structural equilibrium models can only show how output affects credit risk, ignoring

inflation altogether. Therefore, these models do not depict a true picture of how

credit risk respond to aggregate macroeconomic risk.

This paper rectifies this gap in the present literature by employing a simple new

Keynesian macroeconomic model and linking the time series of output, inflation,

and interest rates generated by the model to the firm's cash flow process. Real

aggregate output growth comprises part of the systematic component of the firm's

overall real cash flow growth. The firm issues a bond with finite maturity and a

fixed coupon to be serviced each period by the cash flow until maturity. Default

occurs when the firm's cash flow cannot meet its coupon obligation. Upon default,

the bondholders receive a percentage of the principal loaned called the recovery rate,

which is positively correlated with the prevailing output growth of the economy. The

risky bond is thus valued as a contingent claim on the firm's discounted cash flows.

Inflation, as well as output growth, impacts the discounting of the bond's cash flow,

because new Keynesian models impose a Taylor rule to relate the risk-free rate and

the term structure to macroeconomic conditions.

While previous structural credit models consider macroeconomic conditions as

exogenous and existing structural equilibrium models exogenously specify the dy

namics of the macroeconomic condition, we explicitly consider the dynamics of the

macroeconomy in a model where output growth, inflation, and the risk-free rate are

determined endogenously. Treating macroeconomic variables in reduced form, as is

done by other papers, may mask the determinants and dynamics of credit spreads.

We also model the firm's cash flow rather than its asset values, because it allows a

period-by-period view of the impact of default and liquidity risk.

We first simulate a standard new Keynesian macroeconomic model, such as those

38

postulated by Woodford (2003) and King (2000), using calibrated parameters that

match the features of historical US data. The results of this macroeconomic model

match the observed empirical correlations of output, inflation, and the risk-free term

structure. We then relate the time series properties of the marginal firm's cash flow

growth to the generated output growth of the macroeconomy, simulating a number

of possible cash flow growth paths for the firm. To define the initial credit risk of the

firm, we calibrate the realized default probability along simulation paths to actual

default probabilities provided in Huang and Huang (2003) and Leland (2004) , thereby

choosing initial coverage ratio for each credit rating class. We then price risky bonds

and calculate credit spreads based on simulated discounted cash flow paths for each

time period in the macroeconomy. This procedure generates a time series of credit

spreads for different credit ratings contemporaneous with the output, inflation, and

term structures generated by the macroeconomic model.

After simulating the model, we test the model-generated credit spreads by re

gressing credit spread levels on contemporaneous output growth and inflation. Since

inflation is positively related to the nominal term structure used in discounting via

the Taylor rule and no-arbitrage properties of the macroeconomic model, we expect

that an increase in inflation should increase future discount rates, thereby reducing

bond prices and increasing credit spreads. Output growth positively affects firm-level

cash growth and the nominal term structure; therefore, its effect depends upon the

credit rating of the considered firm. A firm with high credit rating and low initial

probability of default will have a negligible effect on ability to meet its coupon obli

gations, but will have its coupon payments discounted more when output growth

increases. This, in turn, will lower its bond price and raise its spread to the Treasury

rate. Conversely, a firm with low credit rating will lower its probability of default

upon an increase of output growth to counteract the rise in the discount rate. As a

result, its bond price should rise and its credit spread should fall.

To model the impact of macroeconomic conditions on credit spreads, we make

39

a necessary deviation from the traditional New Keynesian macroeconomic model.

Unlike most micro-founded macroeconomic models, we assume that industries, not

firms, are the optimizing productive agents in the economy. Firms are passive units in

our model that produce as much as is demanded by the particular industry in which

they reside and hire the amount of labor necessary for that output amount. While

this assumption may differ from the common practice in the literature, it allows us to

model the impact of the macroeconomy on the credit risk of firms without the more

intractable problem of modeling the impact of firm default on the macroeconomy. If

we model the macroeconomy with firms as optimizing agents that default, then the

macroeconomy would not have complete markets and would be much more intractable

to model using standard numerical techniques. Furthermore, this assumption allows

us to model the direct impact of the macroeconomy on credit risk, a feature not found

in either the macroeconomics or credit risk literature to date.

This model not only generates realistic interactions between macroeconomic vari

ables and credit spreads, but also generates credit spread properties observed in data

which are not generated by existing structural credit models. For example, the credit

spread volatility puzzle, as termed in the literature, is the inability of structural credit

models to generate the high levels of credit spread volatility observed empirically.

The model we propose generates larger credit spread volatility by correlating states

in which defaults are more prevalent with macroeconomically "bad" states, times of

high marginal utility and low recovery rates. In such periods, both the probability of

default and the recovery rate upon default fluctuate with the state of the economy,

thereby generating greater fluctations in credit spreads. Therefore, this approach

taken in this paper should be able to match higher levels of credit spread volatility

found in the data.

Another counterfactual result generated by many structual credit models that

incorporate asset pricing models are procyclical default probabilities. For example,

Chen, Collin-Dufresne, and Goldstein (2005) attempt to explain the equity premium

40

puzzle and credit spread puzzle jointly from asset pricing models, such as Camp

bell and Cochrane (1999) and Bansal and Yaron (2004) . While they find that the

larger time-varying risk premia generate credit spreads larger than those generated

by existing structural models, they also find that their approach generates procyclical

default probabilities without the imposition of a counter-cyclical default boundary.

In contrast, because our model separates the pricing of corporate bonds from the

default condition of the firm, we can generate counter-cyclical default probabilities

observed in reality. Our model can allow for alternative preference specification that

can generate larger credit spreads, while maintaining procyclical default probabilities.

The connection between credit spreads and macroeconomic conditions is of great

interest beyond the need to explain the empirical properties of credit spreads. While

the finance literature focuses on asset pricing issues such as the size and volatility

of credit spreads, the macroeconomics literature has focused on how the availability

of credit affects the transmission of monetary policy and its impact on the macroe-

conomy. Bernanke, Gertler, and Gilchrist (2000) model a framework in which en

dogenous developments in the credit markets amplify and propagate shocks in the

macroeconomy, including monetary policy. Stiglitz and Greenwald (2003) suggest,

in their book, that the entire purpose of monetary policy is to control the supply of

credit and, therefore, liquidity in the economy, not the money supply as is conven

tionally believed. Regardless of the interpretation of the purpose of monetary policy,

this paper provides a framework in which to study the relationship between monetary

policy and credit.

The remainder of this paper is as follows. Section 2 explores a simple, intuitive

mechanism of how credit spreads are related to macroeconomic conditions. Section 3

describes the theoretical setup of the full model. Section 4 describes the calibration

coefficients for the macroeconomic sub-model, the calibration approach for the credit

part of the model, and the resulting simulation. Section 5 describes the results gen

erated by simulation of the model and compare with empirical observations. Section

41

6 describes some comparative statics of the model results and credit spread impulse

responses generated by the model. Section 7 concludes the papers and discusses areas

of future research.

2.2 An Intuitive Framework

Bhamra, Kuehn, and Strebulaev (2007) observe that the price of a one-period risky

bond B can be written as the payoff P of the bond divided by the risk-free rate

r plus the credit spread s as in equation 2.1. Alternatively, in a contingent claims

setting, the price of the bond B is the price of a risk-free bond — times the Arrow-

Debreu probability of survival qp plus the recovery value of the bond A times the

Arrow-Debreu price of default as in equation 2.2.

£ = — (2.1)

B = -(l-qD) + qDA (2.2) r

Setting the two bond pricing equations equal to each other, they find that credit

spread on default risky debt can be written as

where r is the risk-free rate, I is the loss ratio of the bond which gives the proportional

value loss if default occurs, and q^ is the Arrow-Debreu security price, which pays

out 1 unit of consumption upon default. A more detailed derivation of this result is

presented in Appendix A. Furthermore, they derive that the Arrow-Debreu security

price can be further decomposed into three factors

qD = TKpD (2.4)

42

where pr> is the actual default probability, T is the discounting for the time value of

money, and 1Z is an adjustment for risk.

In the context of our model, we can relate each of the terms in the credit spread

equation to output growth and inflation, which define our macroeconomy. Via the

Taylor rule, output growth and inflation directly impact the risk-free rate, as the mon

etary policy agency sets risk-free rates as a linear function of the two macroeconomic

conditions. Again, via the Taylor rule, output growth and inflation impact T, the

time adjustment, because future discounting is performed through the term structure

of interest rates, which, in turn, is affected by the Taylor rule through no-arbitrage

relationships.

While both the risk-free rate and the discount factor positively relate output

growth and inflation to credit spreads in a linear context, higher output growth also

lower the loss given default and the probability of default in the context of our model.

2.3 The Model

The model we present here consists of two parts: the macroeconomy and individual

firms. The macroeconomic model we employ is a standard new Keynesian macroe

conomic model with pricing rigidities. Households optimize their consumption and

labor decisions, subject to a period-by-period budget constraint. Industries exhibit

monopolistic competition in the goods that they produce. Furthermore, they exhibit

Calvo (1983) type pricing rigidities that allow them to optimize their product prices

only at random times. Industries, when they are allowed to, optimize prices subject

to meeting all demand. This pricing rigidity induces inflation with effects on real

variables into the model. Finally, a monetary policy agency sets nominal one-period

rates, according to a contemporaneous Taylor rule, which makes both discount rates

depend on economic output growth and inflation.

The resulting output growth, inflation, and risk-free term structure that evolve

from the macroeconomic model are then inputs into pricing risky corporate debt. We

43

assume that the optimizing productive agents in the macroeconomic model summa

rized above are industries that set product prices and employee wages. In our model,

the marginal firm is a measure-zero productive unit that takes the price of goods and

wages as given by the industry to which they belong, rather than as the output of an

optimization. In this construct, the default of the firm does not impact the industries'

ability to meet aggregate consumption. Output growth affects the cash flow of the

marginal firm, while the current term structure of interest rates includes information

about future output growth and inflation risks. The marginal firm uses its cash flow

to meet its coupon obligations, and the price of a risky bond is the sum of discounted

coupon payments plus either the principal or fraction of the principal recovered upon

default, discounted appropriately. We then can calculate credit spreads from the

difference between the yields of risky bonds and their risk-free counterparts.

2.3.1 The Macroeconomy

To model the macroeconomy explicitly, we employ a standard new Keynesian macroe

conomic model that includes a monetary policy rule and nominal price rigidities to

generate inflation and monetary policy with real effects. The macroeconomic litera

ture has many papers on this type of model, including Woodford (2003), Rotemberg

and Woodford (1997), Lubik and Schorfheide (2004) , Ravenna and Seppala (2006)

, and others. This type of model is standard in the macroeconomics literature and

allows us to link real features in the model and in empirical studies with credit spreads.

Households

The economy consists of a continuum of infinitely lived households, indexed by

j E [0,1]. Consumers demand differentiated consumption goods, choosing from a

continuum of goods, indexed by z E [0,1]. Therefore, C\{z) indicates consumption

from household j at time t of the good produced by firm z. As we shall explain

further, an economy with differentiated goods allows optimizing producers to set the

44

price of these goods differently and at different time, thereby causing inflation with

real effects. Households' preferences over the basket of differentiated goods are defined

by the aggregator:

d = Jo

dz ,9> 1 (2.5)

where 9 is the elasticity of demand.

Household j chooses (C^+i, N(+i, B{+j) where Nt denotes the labor supply, and

Bt are bond holdings to maximize a power utility function with disutility of labor.

Ut = Et Y,(3i{u(Cl+i,Dt)-v(N^t+i)) i-Q

= Et Y,?[ (c?+i)1~7A i=0 1 - 7

t i v j , t+i

1+V

In addition to choosing period-by-period consumption C°t+i) the agent also chooses

hours of labor N^+i in a particular industry for a particular good. His hours con

tributed to labor detract from his overall utility.

Household j maximizes the above utility subject to the aggregator in (2.5) and

the budget constraint

C{(z)Pt(z)dz = WtN> + E> - pt(B> - BU) (2.6)

where Wt is the nominal wage rate, n t is the share of the agent's profit from firms, pt

is a vector of asset prices, and Bt is the agent's holding of the corresponding assets

at time t. Therefore, Pt(Bj — B3t_x) represents the nominal value of household j ' s net

asset holdings.

Since asset markets are assumed to be complete, we do not need to designate

individual assets, but we can assume that every agent holds the complete market

through shares of a index fund that includes all assets in the market, including risk-

free bonds and firm shares. As stated in Woodford (2003), the above specification

need not refer to the quantity held of some specific asset. Since we assume complete

markets, households must be able to hold a wide selection of instruments with state-

45

contingent returns. Furthermore, there exists a set of spanning assets with returns

that can span any possible state-contingent returns. Therefore, we do not need to

refer to any particular asset, but know the household can achieve any state-contingent

return it wants.

The choice of capital structure, the composition of a firm's market securities, is de

termined by the individual firms' initial capital structure and not by market demand.

Consuming agents simply invest in the market portfolio and make no individual se

curity selection decisions.

The household j determines its demand for individual good z among the differen

tiated goods with the following condition:

ci(z) = Pt{z)

CI (2.7)

where Pt is the associated price index measuring the minimum expenditure on differ

entiated goods that will buy a unit of the consumption index

Pt.= Pt{zf-edz (2-

Since all households solve an identical optimization problem and face the same

aggregate variables, we can omit the index j in the above optimization problem. The

other optimal conditions for the individual are

MUCt = ^ - = Et A d

(2.9)

~P~t ~ MUCt (2.10)

Et P^Rt l MUCt

where MUCt is the marginal utility of consumption.

(2.11)

46

Industry Price Sett ing and Symmetr ic Equil ibrium Solution

Different industries produce differentiated goods, indexed by z, and optimize linear

production by controlling labor choosing the wage. The production function of the

industry producing good z is Yt(z) — AtNt(z) where Nt(z) is the labor allocated to

the production of differentiated good z and At is an aggregate productivity shock.

Each industry maximizes its profit function

Ut(z) = Pt(z)Yt(z) - Wt(z)Nt(z)

Every industry has a fixed capital stock. Furthermore, each industry hires as much

labor as is necessary in each period and the labor stock is common to all industries.

Industries take the wage demanded as given as an input and hire as much labor as

necessary to optimize the production function.

Industries have monopolistic competition in their particular good, but can only

set prices at particular periods in time. In each period, an industry will be able to

adjust its price with constant probability (1 — 9p), regardless of past history.

The problem of the industry setting the price at time t consists of choosing Pt(z)

to maximize their expected discounted stream of profits, as follows:

Et £0W Mua t+i

j = 0 MUC*

Pt(z)

Pt Yt t,t+i\

t+i

MC™Y (z) —5 rt,t+i{z)

subject to

Yt,t+i(z) = Pt(z)

Pt t+i Y, t+i

where Ytjt+i(z) is the industry's demand for its output at time t + i condition on the

prices set at time t, Pt(z). This optimization constraint is given by market clearance,

the supply of a good equal to its demand (Yt = Ct).

Solving the model with a symmetric equilibrium implies that C\ = Ct and MUC\ —

MUCt- Given that all industries are able to purchase the same labor service bundle

47

and are charged the same aggregate wage, they face the same marginal cost. The lin

ear production technology ensures that the marginal cost is equal across industries,

whether or not they update prices, regardless of the level of production.

Industries are heterogeneous, because only a fraction (1— 6P) of firms can optimally

choose the price charged at time t. In equilibrium, each producer that chooses a new

price Pt(z) in period t will choose the same new price Pt(z) and the same level of

output. The dynamics of the consumption-based price index will follow

Pt = [ < V t i ' + (i - epWz)1-*] ^ (2.12)

We assume that, within each industry, there exists a continuum of firms that take

prices and wages as set by the industry and hire a measure-zero portion of the labor re

quired by the total industry. Since firms are measure zero in their particular industry

and the entire economy, an individual firm's behavior does not impact the aggregate

economy. By abstracting firm behavior from the aggregate behavior of the indus

try, we isolate the firm's default and, as a result, the price of its default-risky bonds

without the firm default affect the industry's or economy's output. Furthermore, a

default is an unhedgeable event and, as a result, the solution to the equilibrium in the

macroeconomy becomes much more complex as financial markets are now incomplete.

In this model, we deviate from the standard specification of traditional micro-

founded macroeconomic models that assume that firms are optimizing agents that

decide product prices and wages. Instead, we assume that infinitessimally small,

passive firms comprise industries that perform the actual optimizing decision over

prices and wages. We are only interested in the impact of macroeconomic conditions

on credit spreads as a measure of credit risk. Therefore, we abstract default able

firms from the industries that help set the market equilibrium. A firm produces an

infinitesimally small portion of the output for its industry and the economy. If it

should default, another firm in the industry can replace its productive capacity and

thereby not affect the equilibrium solution. Furthermore, since the firm does not

48

affect the equilibrium, we can specify its cash flow behavior arbitrarily as the rest of

the firms in the industry will close the economy to produce the equilibrium solution.

Monetary Policy

The monetary policy authority follows a Taylor rule, as follows:

'l + Rt,t+l\ , / l+7Tt \ , , / Yt loS ( -, , n .<? ) = u* loS I i , ) + ^y l o§ y l + Rss J - b V l + W " \Yss

Furthermore, we assume the central bank assigns positive weight to an interest

rate smooth objective so that the domestic short-term interest rate at time t is set

according to

(1 + Rttl) = [(1 + i W ] 1 _ X [ l + fl*-i,i]x (2-13)

The nominal discount rate changes based on the deviations of current output

and inflation from steady-state trend. The Taylor rule and no-arbitrage that applies

through the stochastic discount factor link macroeconomic conditions, output and

inflation, to asset prices. The consumer Euler equation, 2.11, provides the connection

between the monetary policy agency rate-setting rule and the consumer's investment

decisions. Other rates of return, riskless or risky, are tied to monetary policy via the

pricing kernel.

While this is the standard approach used in the no-arbitrage term structure lit

erature, I apply this approach to the evaluation of credit risk. This model does not

include all the possible factors for credit spreads, but suggests how macroeconomic

conditions in particular affect credit spreads.

Reduced-Form Macro Model

Several papers in the macroeconomics literature, including Woodford (2003) and

Goodfried and King (1997), solve new Keynesian macroeconomic model with a first-

order log-linear approximation, resulting in a linear state space model of the following

49

form:

AEt Vt+i

h+i = B

Vt

h + Cet (2.14)

where y is a vector of non pre-determined variables and k is a vector of pre-determined

variables. We can further decompose the state-space representation into

yt = Dkt + Fet

kt+i = Gkt + Het

This representation arises from linearizing the first-order optimality conditions of

the consumer and firm problems described above. In the case of the model above,

the vector y contains [Y it R] where Y is output, IT is inflation, R is the one-period

nominal interest rate, and x represent the log deviation from steady state of the

variable x.

The resulting state-space model can generate a time series of real output growth,

inflation, and real interest rates that are structurally related to each other via the

dynamics suggested by the above macroeconomic model. The real and nominal term

structures are related to all three variables by the Taylor rule and the no-arbitrage

imposition of the stochastic discount factor. If the systematic portion of a firm's

real cash flow growth, then the price of a firm's risky bonds and the resulting credit

spreads are also functions of output growth, as well as the interest rate and inflation

via the term structure used to discount the firm's cash flow.

We solve the New Keynesian macroeconomic model describes above using Dynare++,

a C+-(-/Matlab package that takes the first-order conditions of the model and per

forms simulations to solve the model.2

2 We thank Juha Seppala for his code, which was used as a template to solve the macroeconomic model using Dynare++ as described in this paper.

50

2.3.2 Risk-Free and Risky Yields

Risk-Free Term Structure

In this section, we use the results of the macroeconomic model to calculate prices and

yields for the real risk-free term structure.

Using the marginal utility of consumption described earlier, the real stochastic

discount factor is given by re-arranging the Euler condition.

_ MUCt+l qt+1 ~ P^fucT

The price of an n-period zero-coupon real bond is

Pn,t = Et

The yields for real bonds are,

n &+j b'=i

= -Etfe+lPn-l.t+l]

1

rn,t = --log(pbnit)

The Firm and Its Risky Bonds

Now that we have characterized the entire economy including the behavior of house

holds and industries, we now focus on the specifics of firm behavior necessary to

generate risky bond prices. As we assumed before, firms are passive and receive

prices and wages as given from the optimal choices of their respective industry. The

marginal firm has an initial capital structure, consisting of some amount of financing

coming from the issue of a risky bond.

Furthermore, we assume that the marginal firm has real cash flow that is a measure

zero portion of the aggregate output and whose growth is given by

gK{t) = Q9t + & + P<?K€? + ( W l - A f (2-15)

51

where gt represents the growth in aggregate output, gx(t) is the growth of the

marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggre

gate output growth, £t is the mean of firm-specific cash flow growth, p is the correla

tion of output growth and firm-specific cash flow growth, <TK is volatility of firm cash

flow growth, and ef, ef ~ N(0,1) independent of each other. The sensitivity of firm

growth to economic growth,

g = cov(gt,g?)/var(gt) = p —

can be thought of as the cash flow beta in Campbell and Vuolteenaho (2004). The

variable oc represents the unconditional volatility of consumption growth.

We base this approach on arguments presented in Longstaff and Piazzesi (2004)

and Tang and Yan (2006) . Longstaff and Piazzesi (2004) suggest that modeling

corporate cash flows, which are more volatile than aggregate output, may resolve the

equity premium puzzle. Cash flows vary with aggregate consumption, but are more

volatile. They, therefore, believe that corporate cash flows are the primitive process

that should be modeled to resolve puzzles in asset pricing. We base our expression for

firm-level cash flow growth on Tang and Yan (2006), who assume that firm-level cash

flow growth has economy-wide and firm-specific components. This assumed form for

cash flow growth allows for a direct correlation with aggregate output while allowing

for a higher degree of volatility.

Similar to Bakshi and Chen (1997), we assume that the rest of the industry and

economy follow a process that allows the sum of the individual firm's cash flow and the

output of the rest of the industry and the economy sum to total aggregate industry

and economy-wide output. Therefore, the assumption of the form of the marginal

firm's cash flow process does not change the path of aggregate output and does not

contradict the equilibrium given by the New Keynesian macro model described earlier.

Our model abstracts away from the firm's capital structure decisions. Implicitly,

therefore, we assume that the Modigliani-Miller theorem holds such that a firm's

52

financing decisions do not impact its value and, therefore, its cash flow available

to stock and bond investors. We assume this approach to identify clearly the rela

tionships between macroeconomic conditions and credit risk without introducing the

complexities of optimal capital structure issues, addressed in other papers such as

Hackbarth, Miao, and Morellec (2006) and Chen (2007). In fact, these papers take

the opposite approach and consider abstract specifications of the macroeconomy to

analyze the impact of optimal capital structure decisions on a firm's credit risk. Like

these models, we could introduce the complexity of optimal capital structure decisions

into our model and show how changes in capital structure affect the firm's cash flow

and credit risk.

Furthermore, the cash flow specification we have chosen above is not limited to

just firms, but can also explain the behavior of any agent in the economy. With

this approach, we hope to describe the sensitivity of any kind of credit risk, not just

corporate credit risk, to macroeconomic conditions, although most of the literature

and empirical observations made by us and others in literature usually pertain to

corporate credit.

Once we define the firm's cash flow, we can also value any contingent claim written

on the cash flow. If the marginal firm issues a bond with face value F and coupon c,

then, in each period, the firm's real payout to the defaultable bond is

kB(t) = ci(t < T)i(t <T) + FS(t - T)i(t <T)+ u(ih)F5(t - r)i(t < T) (2.16)

where r = inf(£ : K(t) < c) is the first passage time of default, l is the indicator

function, and S(t — r ) is the Kronecker delta.

The value us represents the percentage of the face value of the debt that can be

reclaimed upon default. The recovery rate also varies positively with the economy in

the following fashion:

ut = w(gt) =a + bgt

53

We choose to vary the recovery rate with aggregate economic output growth,

due to a number of empirical observations about recovery after default. Altman

and Kishore (1996) show that recovery rates are time-varying, while Collin-Dufresne,

Goldstein, and Martin (2001) suggest that "even if the probability of default remains

constant for a firm, changes in credit spreads can occur due to changes in the expected

recovery rate. The expected recovery rate should be a function of the overall business

climate." Shleifer and Vishny (1992) find that a firm's liquidation value is lower when

its competitors are experiencing cash flow problems. Other empirical papers in the

literature with similar conclusions include Thorburn (2000), Gupton and Stein (2002),

Altman et al. (2002) and Acharya et al (2003). Acharya et al. (2003) also show

that default risk and the recovery rate are only weakly linked by the same factors.

We then assume that they are essentially independent, which makes our simulation

easier. Furthermore, since the actual empirical correlation is slightly positive, the

independence assumption decreases expected loss and biases our model-generated

credit spreads downward slightly.

The value of the risky debt at t = 0 is

The rates r(0, t) represents the real term structure generated by the macro model.

To adjust the firm's cash flow growth to the risk neutral measure, which simplifies

bond pricing, we must reduce the drift of the firm cash flow growth process by its

market price of risk. Since the market price of risk for an economy is simply the

product of relative risk aversion and the volatility of consumption growth, the above

macroeconomy produces a constant market price of risk, since power utility admits

a constant relative risk aversion 7 and the first-order approximation we use to solve

the model admits only a constant volatility of consumption growth equal to the

unconditional volatility of consumption growth of the sample. Therefore, under the

54

risk-neutral measure Q, the cash flow dynamics are

9x{t) = ggt + & - 'yp(rC(7K + pcr^ef + cr^V1 - A f (2.17)

where uc? is the volatility of consumption growth. We explore the issue of calculating

the market price of risk to adjust to represent the firm's cash flow process under the

Q measure in Appendix B.

The risk-free term structure results from the macroeconomic model; therefore we

must adjust the firm's cash flow growth to the risk-neutral measure.

The yield-to-maturity of the risky bond is the solution Y to the equation

DV = ± + (F-±)(I-Y)T

The yield-to-maturity R of a risk-free bond with the same payment structure

The resulting credit yield spread is, therefore, just Y — R. The assumption of this

form for the bond payout and recovery rate allow for current economic conditions as

represented by current output growth and inflation to impact the firm's cash flow and

how its bond's coupon payments are discounted.

2.4 Model Calibration and Simulation

2.4.1 The Macroeconomic Model

To calibrate the new Keynesian macroeconomy in the first part of the model, we

choose calibrated parameters and the estimation technique found in Ravenna and

Seppala (2006). Using parameters chosen from related papers in the monetary policy

literature, they estimate the standard new Keynesian monetary policy model using

55

Dynare++ to match historical correlations of output and inflation as observed from

1952 to present. Table 1 lists the parameters we employ for the macroeconomic

model.3

The preference and technology exogenous shocks follow an AR(1) process:

logZt = (1 - Pz)logZ + pzlogZt^ + ef, ef ~ N(Q, a\) (2.18)

where Z is the steady state value of the variable

This calibration produces correlations between model output, inflation, and real

interest rates that match historical empirical correlations. This match between model

and empirical correlations between macroceonomic conditions and risk-free and risky

yields is one of the primary goals of this paper and differentiates this model from

others in the credit literature that specify output growth and inflation processes

exogenously, not structurally. Table 2.2 compares the model's second moments and

correlations with output with U.S. post-war sample data. The model matches the

3 The literature on parametrizing New Keynesian models to match historical data is extensive. These include Ravenna and Seppala (2006), Ravenna (2006), Christiano, Eichenbaum, and Evans (2005), Ireland (2001), Woodford (2003), and Rabanal and Rubio-Ramirez (2005)

Symbol

7 £ P 9P

it 9

UJy

U*

X Pa

Pd

<?a 4

Description Relative risk aversion

Inverse of labor supply elasticity Discount factor

Price stickiness probability Steady state inflation (quarterly)

Demand elasticity Taylor coefficient (output)

Taylor coefficient (inflation) Monetary policy smoothing parameter Autocorrelation of technology shock Autocorrelation of production shock

Volatility of technology shock Volatility of production shock

Value 2.5 0.5 0.99 0.75 0.75 11

0.01 1.5 0

0.9 0.95

0.0035 0.008

Table 2 .1 : Macroeconomic model coefficients based on Ravenna and Seppala (2006), calibrated to historical correlations of output, inflation, and interest rates

56

volatility and correlations of output and inflation, but increases the volatility and

output correlation of interest rates. We can improve the fit of the model to historical

data by including a number of modifications, including habit persistence, higher-order

approximation, and sticky wages. The model, however, provides a basic framework

linking macroeconomic conditions jointly and allowing us to test their behavior versus

model-generated credit spreads. Table 2.3 shows means, standard deviations, and

correlations for output growth, inflation, and real interest rates.

2.4.2 Calibrating and Simulating Credit Spreads

Once we estimate the macroeconomic model, we simulate the firm's cash flow process

with equation 2.15. We follow Longstaff and Piazzesi (2004) to choose parameters for

the firm's cash flow process. Longstaff and Piazzesi (2004) note that corporate cash

flows are highly volatile, strongly procyclical, and could answer part of the equity

premium puzzle by generating higher volatility for corporate cash flow that underlie

firm-issued securities. They find that the correlation of firm's cash flow with total

output growth to be around p ~ 0.6 — 0.7, while firm cash flow growth has volatility

around 20%.

For each quarter t, we assume that output growth, inflation, and the real term

structure for the given quarter are fixed and that bondholders use the fixed macroe

conomic conditions to forecast firm cash flows and price bonds. We choose 10,000

Variable Yt

7T*

Rt n

Standard Deviation Model 2.02 3.28 4.94 4.59

US Data 1.59 3.00 2.82 2.32

Correlation with Output Model

1.00 0.21 0.32 0.30

US Data 1.00 0.19 0.17 -0.13

Table 2.2: Selected variable volatilities and correlations, model vs. historical 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt

is real GDP, irt is annualized CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms.

57

Variables 9t

r3

r 12 11 11

r120 11

Mean Std. Dev 0.01 0.89 2.97 3.28 2.75 4.59 3.26 3.90 3.83 2.13 4.00 0.34

Correlation coefficients 9t 7T* r* rf rf r\w

1.00 0.13 0.64 0.64 0.65 0.66 0.13 1.00 0.27 0.24 0.19 0.18 0.64 0.27 1.00 1.00 0.99 0.99 0.64 0.24 1.00 1.00 1.00 0.99 0.65 0.19 0.99 1.00 1.00 1.00 0.66 0.18 0.99 0.99 1.00 1.00

Table 2.3: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates

paths for cash flow growth each quarter and calculate simulated firm cash flows and

payments to bondholders. The average of the sum of payments to bondholders dis

counted by the prevailing term structure gives us the bond price, from which we can

calculate risky yields and spreads.

After we generate a realistic set of time series for output growth, inflation, and

interest rates, we calibrate the model to generate credit spreads to match historical

default probabilities. Taking the parameters from above, we can fix the coupon of the

bond issued by the firm to a value, 5%, and calibrate inital cash flow K0 for each credit

rating category to match historical default probabilities in simulation. Huang and

Huang (2003) provide the following default probabilities for different credit ratings.

The calibration of initial cash flow to historical default probabilities to generate

different credit ratings categories uses the assumption that the rating agencies adopt

the "through the business cycle" approach to assign firms to a rating category. The

representative firm we model is assigned its rating based on its average probability of

Symbol

P £

&K

a b

Description Correlation of macro growth and firm c.f. growth

Mean of firm-specific cash flow growth Standard deviation of firm cash flow growth

Recovery rate constant Recovery rate amplifier

Value 0.6 0%

20% 51.31%

2.5

Table 2.4: Firm-specific cash flow parameters, based on Longstaff and Piazzesi (2004) and recovery rate in Huang and Huang (2003), assuming average of zero economic growth and recovery rate of 51.31% for all bond ratings and maturities

58

Rating

AAA AA

A BBB

BB B

Cumulative default prob. (%) 1 yr. 0.00 0.03 0.10 0.12 1.29 6.47

4 yrs. 0.04 0.23 0.35 1.24 8.51

23.32

10 yrs. 0.77 0.99 1.55 4.39

20.63 43.91

Recovery rate

(%) 51.31 51.31 51.31 51.31 51.31 51.31

Table 2.5: Moody's default probabilities and recovery rates (Huang and Huang, 2003)

default throughout the business cycle we model, rather than on the basis of current

conditions. This assumption allows us to vary the firm's default probability and

credit risk within the same rating category. Again, we consider our firm to be a

representative firm of a particular ratings category throughout the business cycle.

After generating the initial credit spread results based on the values for firm

specific growth mean and variance, we can then perform a sensitivity analysis of the

results by varying the firm-specific properties of cash flow growth. Finally, we conduct

a comparative static analysis to gauge the effects of firm specific characteristics on

credit spread levels and volatilities generated by the model and their relationship with

output growth and inflation.

2.5 Results

In this section, we discuss the properties of the credit spreads generated by the model

we describe above. First, we describe the empirical properties of the credit spread

time series generated by the model. We then discuss the contemporaneous relation

ship between credit spreads and model-generated macroeconomic conditions, namely


59

2.5.1 Properties of Model-Generated Credit Spreads and De

fault Probabilities

Our model produces credit spreads and volatilities of the order of magnitude found

in the data and comparable with other structural credit models. Unfortunately, the

credit spread levels produced by the model are lower than those found in the data

and in structural models. The inability to produce the levels seen in data could be

due to the lack of large premia in the model. Chen, Collin-Dufresne, and Goldstein

(2005) and others have suggested that the equity risk premium puzzle and the credit

spreads level puzzle are linked and that an asset pricing model that captures one also

captures the other.

Furthermore, it has been well documented that asset pricing models based on

production economies have greater difficulty in explaining asset pricing anomalies

than exchange economies. The macroeconomic model that we employ does not have

elements that could solve the equity premium puzzle, such as a time-varying risk aver

sion parameter. Although the credit spreads generated by model depend upon firm

cash flows that are more volatile than consumption growth in the model, Longstaff

and Piazzesi (2004) suggest that this may not entirely resolve the equity premium

puzzle and, additionally, the credit spreads level puzzle.

In constrast with the credit spread level results, the model generates credit spread

volatilities closer to empirical values than other structural credit models. Chen,

Collin-Dufresne, and Goldstein (2005) state structural credit models generate credit

spreads whose volatilities are too low relative to actual data. Most models gener

ate a volatility of the spread between BBB and AAA of around 35 bps per annum,

while the actual volatility is around 75 bps per annum for the entire history of the

Moody's corporate bond index and 56 bps for the post-war period. Our model gener

ates a BBB-AAA spread volatility of around 52 bps, closer than other models to the

full sample volatility and close to the post-war period value. In general, the model

generates volatilities close to the credit spread volatilities observed in the data.

60

Horizon and Maturity 4-year A

4-year BBB 4-year B

10-year A 10-year BBB

10-year B

Model 2.0 6.7

121.0 3.4 8.4

103.0

Historical 96.0

158.0 470.0 123.0 194.0 470.0

TY 13.2 56.6

240.5 48.1

109.6 318.5

LS 7.5

25.4 406.0

14.5 38.6

341.9

LT CDG 9.9

- 31.1 - 435.3

38.5 22.5 59.5 52.3

408.4 371.6

Table 2.6: Average credit spread levels, model generated vs. historical and literature. Historical data taken from Huang and Huang (2003). Comparable models including Tang and Yan (2006) (TY), Longstaff and Schwartz (1995) (LS), Leland and Toft (1996) (LT), and Collin-Dufresne and Goldstein (2005) (CDG)

Our simulation methodology, in addition to providing us with credit spreads, also

finds forward-looking default probabilities. Many previous credit risk models, partic

ularly those that link consumption growth with credit spreads, generate procyclical

default probabilities, an obviously counterfactual property. As shown in Table B.l,

we regress model-generated default probabilities on contemporaneous output growth

and inflation and find a strongly significant countercyclical relationship of default

probabilities and output growth, but no connection with inflation. Both regression co

efficients conform with empirical data that default probabilities rise as output growth

falls, but have inflation has little impact on actual defaults.

2.5.2 Macroeconomic Factors of Credit Spreads: Contempo

raneous Relationships

Given the model developed above, we expect inflation to be positively related to credit

spreads, as an increase in inflation should increase the discount rate, depressing bond

prices and increasing yield spreads. Furthermore, we expect higher output growth to

have a positive impact on credit spreads of AAA and AA and a negative effect on

credit spreads of A rating or below. For the higher rated bonds, increased output

growth should increase the discount rate applied to coupon payments, but should not

substantially decrease the probability of default for the bond. Therefore, the bond

price should be depressed and the credit spread should rise. For the lower rated

61

bonds, the increase in output growth should increase the firm's cash flow, helping

it meet cash flow obligations and reducing the probability of default. This, in turn,

should counteract the effect of an increase in the discount rate and should decrease

the credit spread.

The model produces results close to our intuition and what we observe empirically.

To evaluate the impact of output growth and inflation on credit spread level, we

perform regression on with the following specification:

st = a + f3sst_i + (3ggt + (3vixt (2.19)

As seen in Table B.2, inflation varies positively with credit spread levels, as (3n > 0

for all ratings and maturities. This result is consistent with our empirical findings in

our earlier research. Furthermore, it matches our intuition that higher inflation raises

the risk-free real rate demanded by investors, which correspondingly reduces demands

for risky securities or raise the yield required of risky debt securities to compensate

investors to purchase them.

The coefficient on output growth, however, does not exactly match what we find in

our previous empirical study. Output growth varies negatively with credit spreads of

all maturites and ratings in the model. However, while output growth has a positive

coefficient for higher-rated credit spreads in the empirical data, the model generates

higher-rated credit spreads that vary negatively with output growth, although not

significantly.

2.6 Impulse Response Functions and Comparative

Statics

In addition to replicating the empirical sensitivities of credit risk to macroeconomic

conditions, the model we present here also has several features that could be im-

62

portant for those interested in the behavior of credit markets, particularly monetary

policymakers. In this section, we present impulse response functions of the various

credit spread series of different ratings and maturities to shocks in the macroecon-

omy. Unlike standard credit risk models found in the literature, our ablity to connect

a New Keynesiam macroeconomic model with a model of credit risk allows us to

test the impact of shocks in the macroeconomy on credit spreads. Furthermore, we

also present comparative statics of the model, varying the calibrated parameters to

represent different macroeconomic and firm conditions, including different monetary

policy rules and firm characteristics.

2.6.1 Impulse Response Functions

The macroeconomic model employed in this paper has three sources of uncertainty:

technology, preference, and monetary policy shocks. Our proposed link between the

macroeconomic and credit risk models allows not only to test the sensitivity of credit

spreads to macroeconomic conditions, but also to determine the impact of macroe

conomic shocks on credit risk. We do not show our impulse responses of the credit

spreads in our model to preference shocks, as the results are not entirely clear and

are of too small a magnitude. Figures B.l and B.4 show the responses of macroeco

nomic conditions, namely consumption growth, inflation, and the short-term interest

rate, to an adverse technology shock and to a positive monetary policy shock of 1%,

respectively.

Constructing the impulse response to credit spreads is not as simple as construct

ing the impulse responses to macroeconomic variables, because bond pricing as a con

tingent claim is essentially non-linear. To construct credit spread impulse response,

we first find the impulse response of the individual macroeconomic series. We then

convert the impulse responses of the individual series, which are in log deviations,

back to actual time series of consumption growth, inflation, and interest rates, which

are then used to calculate risky bond spreads. We then calculate bond spreads for

63

each credit category of bond assuming that the macroeconomy is at the steady state

and subtract these spreads from the spreads given by the macroeconomic time series

derived from the impulse response functions.

As seen in figures B.2-B.4, credit spreads respond strongly to an adverse technol

ogy shock in the economy. If we interpret a negative technology shock as a drastic

loss of productivity of almost 3% per quarter, say during a time of war, the impulse

responses for all credit spreads increase dramatically in initial response and decline

over the next couple of years, but do not return to their steady-state levels.

Figures B.6-B.8 shows the response of credit spreads to a positive monetary policy

shock. A positive monetary policy shock reduces consumption growth and inflation,

but they more quickly return to their steady-state levels. A positive monetary policy

shock also increases in credit spreads, which also recover to their original levels within

12 quarters, if not earlier.

2.6.2 Correlation with Output Growth, p

The coefficient p and the resulting sensitivity g determine how connected firm cash

flow growth is with the aggregate macroeconomy. The correlation with the macroe

conomy plays an important role in determining credit spreads. The size of the cor

relation p can proxy for different industries, as well, based on elasticity of demand

faced by the industry. For example, a low correlation with output can correspond to

the utility industry, while a high correlation could correspond to the technology or

financial sector.

Tables B.3 and B.4 show the results of regressing current credit spreads on lagged

spreads and current macroeconomic conditions as described in equation 2.19. Not

surprisingly, the greater the correlation of firm cash flow growth with total output

growth, the larger and more significant the regression coefficients in the regression of

credit spreads on output growth and inflation. Furthermore, the explanatory power

of regression, as measured by the adjusted R2, also increases with increasing p.

64

2.6.3 Idiosyncratic Cash Flow Growth and Volatility, (£, OK)

The firm's cash flow growth process has two components: a systematic component

that varies with the macroeconomy and an idiosyncratic component that is firm-

specific. As we shall demonstrate, the firm's idiosyncratic component also affects

the impact of macroeconomic factors on its credit spread. As we stated earlier, we

assume that firms in the economy obey the Modigliani-Miller theorem, which implies

that their capital structure decisions do not impact the firm's value or cash flow

processes. The idiosyncratic components of the firm's cash flow (£, OK) could be used

in future research to link the firm's optimal capital structure decision to the firm's

cash flow process.

The mean of the firm's idiosyncratic cash flow growth affects its default probability,

as a firm with greater cash flow growth will not default as frequently. Table B.5

show the regression test results for credit spreads generated by firms with mean

idiosyncratic growth of -2% per annum, while table B.6 show results for firms with

idiosyncratic growth of 2% per annum. We notice that output growth and inflation

have greater explanatory power for firms with negative idiosyncratic growth. The

coefficient on output growth, /3g, is larger in magnitude for all ratings, while /3n is

larger for higher-rated credits and smaller for lower rated credits.

A firm with greater idiosyncratic cash flow volatility will have greater variability

of its cash flow and, correspondingly, will default with greater probability and have

credit spreads that are more sensitive to macroeconomic conditions. The impact of

idiosyncratic cash flow volatility on credit spreads and their sensitivity to macroeco

nomic conditions is clearly seen in Tables B.7 and B.8. The larger the idiosyncratic

volatility, the greater the explanatory power of macroeconomic variables and the

larger the magnitude and significance of the regression coefficients on output growth

and inflation.

65

2.6.4 Relative Risk Aversion, 7

Several recent papers in the term structure and credit risk literature have found

evidence of regime switching behavior in the time series of credit spreads. As we

mentioned earlier, Alessandrini (1999) finds varying sensitivity to macroeconomic

conditions during recessions and expansions. In our previous empirical study, we

showed that credit spreads exhibit regime switching with respect to macroeconomic

factors and that regime switches occur during specific, identifiable financial market

events, such as the failure of LTCM and the cessation of hostilities in the Iraq War in

the beginning of 2004. This evidence points to changes in risk aversion changing the

sensitivity of credit risk to macroeconomic conditions. Although the model we employ

in this paper only allows for a constant risk aversion, we re-ran the simulations with

different relative risk aversion parameters to illustrate the change in sensitivity.

As seen in tables B.9 and B.10, increasing risk aversion increases the sensitivity

of credit spreads to macroeconomic conditions. This provides us an impetus for

employing a model with changing risk aversion or curvature of the utility function that

generates time varying risk premia and changing sensitivity to the macroeconomy.

2.6.5 Alternative Monetary Policy, (x^y)

In addition to changing the firm characteristics or the preferences in the economy,

this model is also useful for testing the impact of changes in the monetary policy rule.

Variable Yt

TTi

Rt

n


US Data 1.59 3.00 2.82 2.32


1.00 0.13 0.08 -0.06

US Data 1.00 0.19 0.17 -0.13

Table 2.7: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, nt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Coefficient of relative risk aversion is 10

66

Variables gt (quarterly) 7rt (quarterly)

r 3

r 12

' t r120 11

Mean Std. Dev 0.00 0.85 2.66 3.32 2.32 4.79 3.56 2.51 3.89 0.77 3.95 0.46

Correlation coefficients gt 7rt rf r]2 rt

60 r\20

1.00 0.06 -0.10 -0.14 -0.09 -0.02 0.06 1.00 -0.95 -0.93 -0.73 -0.59

-0.10 -0.95 1.00 0.98 0.87 0.75 -0.14 -0.93 0.98 1.00 0.90 0.79 -0.09 -0.73 0.87 0.90 1.00 0.97 -0.02 -0.59 0.75 0.79 0.97 1.00

Table 2.8: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Relative risk aversion is 10.

Variable Yt

n Rt rt


US Data 1.59 3.00 2.82 2.32


1.00 0.37 0.06 -0.27

US Data 1.00 0.19 0.17 -0.13

Table 2.9: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, wt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Coefficient of relative risk aversion is 25

Variables gt (quarterly) nt (quarterly)

r 3

r 12

-60 ' t

r120 11

Mean Std. Dev 0.00 0.78 2.64 3.52 3.91 3.79 3.96 2.47 4.00 0.41 4.02 0.25

Correlation coefficients gt TTt rf r\2 rf° r t

120

1.00 0.06 -0.10 -0.14 -0.09 -0.02 0.06 1.00 -0.95 -0.93 -0.73 -0.59

-0.10 -0.95 1.00 0.98 0.87 0.75 -0.14 -0.93 0.98 1.00 0.90 0.79 -0.09 -0.73 0.87 0.90 1.00 0.97 -0.02 -0.59 0.75 0.79 0.97 1.00

Table 2.10: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Relative risk aversion is 25.

67

We believe this is the first model that allows a direct connection between policy rules

used by the monetary policy agency and credit risk, as measured by credit spreads.

While this is just a first pass attempt at such a model, it provides the natural building

block for policy analysis.

As shown in Tables B.ll and B.12, increasing the smoothing of the monetary

policy function or increasing the Taylor coefficient on output make the impact of

output growth on credit spreads stronger, while reducing the impact of inflation

changes on credit spreads.

2.7 Conclusion

In this paper, we explore a model that combines realistic macroeconomic dynamics

with credit risk. The model generates realistic dynamics of macroeconomic variables

using a simple New Keynesian model and empirically observed correlations of credit

spreads with macroeconomic risk, namely output growth and inflation, using Monte

Carlo simulations.

The first contribution of our paper is methodological. We show how to connect a

New Keynesian model that generates realistic dynamics for macroeconomic variables

with an asset pricing model. Previous papers on the relationship between macroeco

nomic and credit risk exogenously specify either the behavior of the macroeconomy,

such as Hackbarth, Miao, and Morellec (2006) and Chen (2007), or how macroeco-

Variable Yt

Kt

Rt rt


US Data 1.59 3.00 2.82 2.32


1.00 0.38 -0.08 -0.28

US Data 1.00 0.19 0.17 -0.13

Table 2.11: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, irt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Degree of monetary policy smoothing is 0.9.

68


r 3

r 12 ' t

' t „120 11

Mean Std. Dev 0.00 3.08 2.98 3.25 3.81 3.71 3.87 2.46 3.95 0.76 3.98 0.44

Correlation coefficients gt irt rf r\2 r®° rt

120

1.00 0.22 -0.20 -0.19 -0.11 -0.07 0.22 1.00 -0.97 -0.94 -0.71 -0.58

-0.20 -0.97 1.00 1.00 0.86 0.77 -0.19 -0.94 1.00 1.00 0.91 0.82 -0.11 -0.71 0.86 0.91 1.00 0.99 -0.07 -0.58 0.77 0.82 0.99 1.00

Table 2.12: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Degree of monetary policy smoothing is 0.9.

Variable Yt

nt

Rt n


US Data 1.59 3.00 2.82 2.32


1.00 -0.52 -0.46 0.36

US Data 1.00 0.19 0.17 -0.13

Table 2.13: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, nt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Taylor coefficient of output is 0.1.

Variables gt (quarterly) TTt (quarterly)

r 3

r 12 11

rm 11

-120 11

Mean Std. Dev 0.00 2.76 3.11 4.59 3.91 1.48 3.93 0.86 3.97 0.48 3.99 0.31

Correlation coefficients gt 7Ti rf r\2 rf° r]20

1.00 -0.16 0.15 0.23 0.20 0.19 -0.16 1.00 -0.56 -0.82 -0.88 -0.90 0.15 -0.56 1.00 0.85 0.72 0.69 0.23 -0.82 0.85 1.00 0.97 0.96 0.20 -0.88 0.72 0.97 1.00 1.00 0.19 -0.90 0.69 0.96 1.00 1.00

Table 2.14: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Taylor coefficient of output is 0.1.

69

nomic conditions impact credit risk, as in Bernanke, Gertler, and Gilchrist (1996).

We structurally specify both the macroeconomy and credit risk in our model in the

simplest fashion possible. To connect macroeconomic dynamics with credit risk, we

assume that industries, not firms, are the optimizing agents in the macroeconomy,

while firms are passive in their optimization behavior. This assumption allows us to

investigate the unidirectional impact of macroeconomic conditions on credit risk. This

approach also allows a method to analyze the impact of macroeconomic conditions

and different forms of monetary policy on credit risk.

Our key result is that credit spreads generated by our model exhibit a negative

relationship with output growth and a positive relationship with inflation that is

found in the data. Furthermore, we can generate credit spreads that display higher

volatility than what results from existing structual credit models and closer to what

is observed in data. The model-generated spreads also exhibit a negative relationship

with future model-generated output growth and inflation, which matches empirical

observations in other papers.

Our new methodology also allows us to study credit spreads in a new way, namely

with macroeconomic shocks and with alternatively macroeconomic specifications,

such as different monetary policy rules. This approach is valuable for policymak

ers who would like to study the impact of their policies on financial markets. If we

follow the arguments of Stiglitz and Greenwald in their book, monetary policy is a

method for the government to increase or decrease the amount of credit in the econ

omy, not just the money supply. The model we present here can be used to simulate

and verify their proposition.

We have quite a few avenues of research to improve upon the model and match

more empirical observations on credit spreads. First, the credit risk literature has

focused much effort on trying to explain the observed size of credit spreads, which

cannot be replicated by standard structural credit risk models that consider the im

pact of the capital structure of the firm. Some recent papers that employ " structural

70

equilibrium" models try to explain the level of credit spreads by incorporating varia

tion of default risk with the macroeconomy, as we do in this paper. However, we do

not incorporate any features in our economy that could explain the credit spread puz

zle. As shown by Chen, Collin-Dufresne, and Goldstein (2005), any model that can

explain the size of credit spreads should also be able to explain the equity premium

puzzle. We do not incorporate any such feature that could explain either puzzle,

such as time-varying risk premia. In a later paper, we intend to propose a model

with preferences defined by habit persistence, an approach which has been shown to

resolve the equity premium puzzle.

Chapter 3

Credit Spreads in a New

Keynesian Macro Model with

Habit Persistence

3.1 Introduction

In our previous research, we established an empirical link between macroeconomic

conditions, such as output growth and inflation, and credit risk, as measured by corpo

rate credit spreads. Credit spreads generally increase with increasing inflation, while

increasing output growth causes higher-rated credit spreads to widen and lower-rated

credit spreads to narrow. Furthermore, the magnitude of the effect of macroeconomic

conditions on credit spreads is higher in periods of uncertainty in financial markets,

such as during the crash of LTCM or during the lead-up to the second Gulf War.

In periods of relative financial calm, credit spreads, just like interest rates, strongly

behave like a random walk with little or no variation arising from macroeconomic

conditions.

Upon empirically establishing the properties of credit spreads and their link to

the macroeconomy, we built a model linking a new Keynesian macroeconomic model

71

72

with a model of credit risk. By assuming a unidirectional connection between the

macroeconomy and the growth of the marginal firm, we show that prices on contingent

claims on the firm's cash flow, such as corporate bonds, vary with output growth and

inflation. With this model, we show that corporate credit spreads vary positively

with inflation and negatively with output growth, even when controlling for lagged

credit spreads. Furthermore, this model generates a high volatility for credit spreads

closer to what is observed in the data than other credit risk models, although the

model does not generate a high enough level of credit spreads to match empirical

observations.

Our previous model also does not duplicate the regime shifting behavior that we

found empirically with the relationship between macroeconoic conditions and credit

spreads. We notice in our empirical work that the relationship between credit spreads

and macroeconomic conditions becomes more significant during periods of financial

uncertainty. To duplicate this empirical observation and generate credit spreads that

match empirically observed levels, we propose explicitly allowing modeling agents'

preferences in the macroeconomic model to be time-varying. To model preferences,

we use an internal habit persistence specification, as first described by Constantinides

(1990) , where agents' utility is based on how much current consumption differs from

consumption in the previous period.

Chen, Collin-Dufresne, and Goldstein (2005) and Bhamra, Kuehn, and Strebulaev

(2007) both postulate a connection between the equity premium puzzle, the inability

of existing asset pricing models to explain the level and volatility of historical equity

returns, and the credit spreads level puzzle , the inability of structural credit mod

els based on current asset pricing techniques to explain the historical level of credit

spreads. Chen, Collin-Dufresne, and Goldstein (2006) find that preference specifica

tions that resolve the equity premium puzzle, such as those presented in Campbell

and Cochrane (1999) and Bansal and Yaron (2004), also resolve the credit spread

puzzle, but generate pro-cyclical default probabilities. As shown in our previous

73

paper, by design, our model ensures counter-cyclical default probabilities as is empir

ically observed, while generating higher credit spreads with time-varying sensitivities

to the macroeconomy. For our preference specification, we chose an internal habit

persistence model, as it allows for time-varying elasticity of utility with respect to

consumption, while maintaining relative risk aversion at levels that are empirically

realistic.

The rest of this paper is organized as follows. Section 2 summarizes the model we

proposed in our previous paper with the addition of time-varying preference modeled

with internal habit persistence. Section 3 describes our calibration and simulation

methodology, Section 4 discusses our simulation results, and Section 5 provides our

discussion and conclusion.

3.2 The Model

In this section, we summarize the model we employ to link macroeconomic conditions

and credit spreads. Since, with the exception of the preference specification, the

model is the same as our previous paper, we relegate specific details of the model

specification to our earlier paper.

The model consists of two parts: the macroeconomy and individual firms. The

macroeconomic model we employ is a standard new Keynesian macroeconomic model

with pricing rigidities. Households optimize their consumption over which they have

time-varying preferences and labor decisions, subject to a period-by-period budget

constraint. Industries exhibit monopolistic competition in the goods that they pro

duce. Furthermore, they exhibit Calvo (1983) type pricing rigidities that allow them

to optimize their product prices only at random times. Industries, when they are

allowed to, optimize prices subject to meeting all demand. This pricing rigidity in

duces inflation with effects on real variables into the model. Finally, a monetary policy

agency sets nominal one-period rates, according to a contemporaneous Taylor rule,

which makes discount rate depend on both economic output growth and inflation.

74

The resulting output growth, inflation, and risk-free term structure that evolve

from the macroeconomic model are then inputs into pricing risky corporate debt. We

assume that the optimizing productive agents in the macroeconomic model summa

rized above are industries that set product prices and employee wages. In our model,

the marginal firm is a measure-zero productive unit that takes the price of goods and

wages as given by the industry to which they belong, rather than as the output of an

optimization. In this construct, the default of the firm does not impact the industries'

ability to meet aggregate consumption. Output growth affects the cash flow of the

marginal firm, while the current term structure of interest rates includes information

about future output growth and inflation risks. The marginal firm uses its cash flow

to meet its coupon obligations, and the price of a risky bond are discounted coupon

payment plus either the principal or fraction of the principal recovered upon default,

discounted appropriately. We then can calculate credit spreads from the difference of

the yields of risky bonds and their risk-free counterparts. Since households' prefer

ences vary with time, so does the pricing kernel and the sensitivity of credit spreads

to macroeconomic conditions.

3.2.1 The Macro economy

To model the macroeconomy explicitly, we employ a standard new Keynesian macroe

conomic model that includes a monetary policy rule and nominal price rigidities to

generate inflation and monetary policy with real effects. The macroeconomic litera

ture has many papers on this type of model, including Woodford (2003), Rotemberg

and Woodford (1997), Lubik and Schorfheide (2004) , Ravenna and Seppala (2006),

and others. This type of model is standard in the macroeconomics literature and al

lows us to link real features in the model and in empirical studies with credit spreads.

75

Households

The economy consists of a continuum of infinitely lived households, indexed by

j G [0,1]. Consumers demand differentiated consumption goods, choosing from a

continuum of goods, indexed by z G [0,1]. Therefore, C{{z) indicates consumption

from household j at time t of the good produced by firm z. Households' preferences

over the basket of differentiated goods are defined by the aggregator:

C\ = C\{zp-dz ,9> 1 (3.1;

where 9 is the elasticity of demand.

Household j chooses (CJ+i, N

3+i, Bj+i)i_Q where Nt denotes the labor supply, and

Bt are bond holdings to maximize an internal habit persistent utility function with

disutility of labor.

Ut = Et f2^nci+i,Dt)-v(^t+i)) i=0

Et

i=0

n t+i ^ci+i_x \ l - 7

-A 7 l+ri

In addition to choosing period-by-period consumption C3t+i, the agent also chooses

hours of labor N3+i in a particular industry for a particular good. His hours con

tributed to labor detract from his overall utility.

Household j maximizes the above utility subject to the aggregator in (3.1) and

the budget constraint

Jo C3

t(z)Pt(z)dz = WtNi + Ul -pt(Bi - BU) (3.2)

where Wt is the nominal wage rate, Tlt is the share of the agent's profit from firms, pt

is a vector of asset prices, and Bt is the agent's holding of the corresponding assets

at time t. Therefore, Pt{B3t — B3

t_^) represents the nominal value of household j ' s

net asset holdings.We assume that asset markets are complete and that agents invest

only in the market portfolio to hedge their consumption risk.

76

The household j determines its demand for individual good z among the differen

tiated goods with the following condition:

ci(z) = Pt(z) Pt

CI (3.3)

where Pt is the associated price index measuring the minimum expenditure on differ

entiated goods that will buy a unit of the consumption index

Pt f, Pt(zf-edz (3.4)

Since all households solve an identical optimization problem and face the same

aggregate variables, we can omit the index j in the above optimization problem. The

other optimal conditions for the individual are

MUCt dU

dCt

= Et ci (3.5)

Wt _ IW

~P ~ MUCt

(3.6)

E, 'p^^Rt = l MUCt

where MUCt is the marginal utility of consumption.

(3.7)

Industry Price Sett ing and Symmetr ic Equilibrium Solution

Different industries produce differentiated goods, indexed by z, and optimize linear

production by controlling labor choosing the wage. The production function of the

industry producing good z is Yt(z) = AtNt(z) where A^(^) is the labor allocated to

the production of differentiated good z and At is an aggregate productivity shock.

77

Each industry maximizes its profit function

Ut(z) = Pt{z)Yt(z) - Wt(z)Nt(z)

Every industry has a fixed capital stock. Furthermore, each industry hires as much

labor as is necessary in each period and the labor stock is common to all industries.

Industries take the wage demanded as given as an input and hire as much labor as

necessary to optimize the production function.

Industries have monopolistic competition in their particular good, but can only

set prices at particular periods in time. In each period, an industry will be able to

adjust its price with constant probability (1 — 9P), regardless of past history.

The problem of the industry setting the price at time t consists of choosing Pt(z)

to maximize their expected discounted stream of profits, as follows:

E, Y,w) MUG t+i

i=0 MUCt

Pt{z)y , MC?+i ~T> rt,t+i\z) ^

t L *t+i t+i Yt,t+i(z)

subject to

Yt,tu{z) = Pt(*)' R t+i

Yt+i

where Yt}t+i(z) is the industry's demand for its output at time t + i condition on the

prices set at time t, Pt(z). This optimization constraint is given by market clearance,

the supply of a good equal to its demand (Yt = Ct).

Solving the model with a symmetric equilibrium implies that C\ = Ct and MUC\ =

MUCt- Given that all industries are able to purchase the same labor service bundle

and are charged the same aggregate wage, they face the same marginal cost. The lin

ear production technology ensures that the marginal cost is equal across industries,

whether or not they update prices, regardless of the level of production.

Industries are heterogeneous, because only a fraction (1—9P) of firms can optimally

choose the price charged at time t. In equilibrium, each producer that chooses a new

78

price Pt(z) in period t will choose the same new price Pt(z) and the same level of

output. The dynamics of the consumption-based price index will follow

Pt = [OpPil? + (1 - Op)Pt(zf-e} ^ (3.8)

We assume that, within each industry, there exists a continuum of firms that take

prices and wages as set by the industry and hire a measure-zero portion of the labor re

quired by the total industry. Since firms are measure zero in their particular industry

and the entire economy, an individual firm's behavior does not impact the aggregate

economy. By abstracting firm behavior from the aggregate behavior of the indus

try, we isolate the firm's default and, as a result, the price of its default-risky bonds

without the firm default affect the industry's or economy's output. Furthermore, a

default is an unhedgeable event and, as a result, the solution to the equilibrium in the

macroeconomy becomes much more complex as financial markets are now incomplete.

In this model, we are only interested in the impact of macroeconomic conditions

on credit spreads as a measure of credit risk. Therefore, we abstract defaultable

firms from the industries that help set the market equilibrium. A firm produces an

infinitesimally small portion of the output for its industry and the economy. If it

should default, another firm in the industry can replace its productive capacity and

thereby not affect the equilibrium solution. Furthermore, since the firm does not

affect the equilibrium, we can specify its cash flow behavior independently as the

rest of the firms in the industry will close the economy to produce the equilibrium

solution.

Monetary Policy

The monetary policy authority follows a Taylor rule, as follows:

log (TT^1) = <"•log (ITS) + •+log Gl)

79

Furthermore, we assume the central bank assigns positive weight to an interest

rate smooth objective so that the domestic short-term interest rate at time t is set

according to

(1 + i?t;1) = [(1 + Rt,t+i}^x[l + Rt-i,i]x (3.9)

The nominal discount rate changes based on the deviations of current output

and inflation from steady-state trend. The Taylor rule and no-arbitrage that applies

through the stochastic discount factor link macroeconomic conditions, output and

inflation, to asset prices. The consumer Euler equation, 3.7, provides the connection

between the monetary policy agency rate-setting rule and the consumer's investment

decisions. Other rates of return, riskless or risky, are tied to monetary policy via the

pricing kernel.

While this is the standard approach used in the no-arbitrage term structure liter

ature, we apply this approach to the evaluation of credit risk. This model does not

include all the possible factors for credit spreads, but suggests how macroeconomic

conditions in particular affect credit spreads.

Reduced-Form Macro Model

Several papers in the macroeconomics literature, including Woodford (2003) and

Goodfried and King (1997), solve new Keynesian macroeconomic model with a first-

order log-linear approximation, resulting in a linear state space model of the following

form:

AEt Vt+i

. kt+1 .

= B Vt

_kt_ + Cet (3.10)

where y is a vector of non pre-determined variables and A; is a vector of pre-determined

variables. We can further decompose the state-space representation into

yt = Dkt + Fet

80

kt+1 = Gkt + Het

This representation arises from linearizing the first-order optimality conditions of

the consumer and firm problems described above. In the case of the model above,

the vector y contains [Y n R] where Y is output, n is inflation, R is the one-period

nominal interest rate, and x represent the log deviation from steady state of the

variable x.

The resulting state-space model can generate a time series of real output growth,

inflation, and real interest rates that are structurally related to each other via the

dynamics suggested by the above macroeconomic model. The real and nominal term

structures are related to all three variables by the Taylor rule and the no-arbitrage

imposition of the stochastic discount factor. If the systematic portion of a firm's real

cash flow growth is a function of economy-wide output growth, then the price of a

firm's risky bonds and the resulting credit spreads are also functions of output growth,

as well as the interest rate and inflation via the term structure used to discount the

firm's cash flow.

We solve the New Keynesian macroeconomic model describes above using Dynare-|—f-,

a C++/Matlab package that takes the first-order conditions of the model and per

forms simulations to solve the model.2

In addition to simplifying the solution of the macroeconomy, the loglinear ap

proximation we derive above is useful for obtaining the market price of risk when

simulating to calculate bond prices and yields. As stated in Lettau and Uhlig (2002),

if all the relevant random variables in an economy are lognormal and the pricing kernel

exhibits loglinear behavior, then the market price of risk has a closed-form solution

based on the preference specification. The loglinear approximation of the solution to

the macroeconomy allows us to make the above assumptions when we evaluate bond

yields. Boldrin, Christiano, and Fisher (1997) and Lettau (2003) use this approach

2 We thank Juha Seppala for his code, which was used as a template to solve the macroeconomic model using Dynare++ as described in this paper.

81

when studying the asset pricing implications of habit persistence for exchange and

real business cycle economies, respectively.

3.2.2 Risk-Free and Risky Yields

Risk-Free Term Structure

In this section, we use the results of the macroeconomic model to calculate prices and

yields for the real risk-free term structure.

Using the marginal utility of consumption described earlier, the real stochastic

discount factor is given by re-arranging the Euler condition.

_ „MUCt+l qt+1 ~ P~MUC7

The price of an n-period zero-coupon real bond is

The yields for real bonds are,

nq^ = Et[qt+iPb„-h t+u

rn,t = --fog(pbn,t)

The Firm and Its Risky Bonds

Now that we have characterized the entire economy including the behavior of house

holds and industries, we now focus on the specifics of firm behavior necessary to

generate risky bond prices. As we assumed before, firms are passive and receive

prices and wages as given from the optimal choices of their respective industry. The

marginal firm has an initial capital structure, consisting of some amount of financing

coming from the issue of a risky bond.

Furthermore, we assume that the marginal firm has real cash flow that is a measure

82

zero portion of the aggregate output and whose growth is given by

9K(t) = ggt + & + paKef + oK^J\ - p2ef

where gt represents the growth in aggregate output, gxit) is the growth of the

marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggre

gate output growth, £t is the mean of firm-specific cash flow growth, p is the correla

tion of output growth and firm-specific cash flow growth, &K ls volatility of firm cash

flow growth, and ef, ef ~ iV(0,1) independent of each other. The sensitivity of firm

growth to economic growth,

Q = cov(gt, g?)/var(gt) = p—

can be thought of as the cash flow beta in Campbell and Vuolteenaho (2004). The

variable oc represents the unconditional volatility of consumption growth.

Similar to Bakshi and Chen (1997) , we assume that the rest of the industry and

economy follow a process that allows the sum of the individual firm's cash flow and the

output of the rest of the industry and the economy sum to total aggregate industry

and economy-wide output. Therefore, the assumption of the form of the marginal

firm's cash flow process does not change the path of aggregate output and does not

contradict the equilibrium given by the New Keynesian macro model described earlier.

Our model abstracts away from the firm's capital structure decisions. Implicitly,

therefore, we assume that the Modigliani-Miller theorem holds such that a firm's

financing decisions do not impact its value and, therefore, its cash flow available

to stock and bond investors. We assume this approach to identify clearly the rela

tionships between macroeconomic conditions and credit risk without introducing the

complexities of optimal capital structure issues, addressed in other papers such as

Hackbarth, Miao, and Morellec (2006) and Chen (2007).

Furthermore, the cash flow specification we have chosen above is not limited to

83

just firms, but can also explain the behavior of any agent in the economy. With

this approach, we hope to describe the sensitivity of any kind of credit risk, not just

corporate credit risk, to macroeconomic conditions, although most of the literature

and empirical observations made by us and others in literature usually pertain to

corporate credit.

Once we define the firm's cash flow, we can also value any contingent claim written

on the cash flow. If the marginal firm issues a bond with face value F and coupon c,

then, in each period, the firm's real payout to the defaultable bond is

kB(t) = ci(t < T)i(t <T) + F5(t - T)i(t < r) + Lo(^t)F5{t - r ) i ( t < T) (3.11)

where r = inf(£ : K{t) < c) is the first passage time of default, 1 is the indicator

function, and S(t — T) is the Kronecker delta.

The value UJ represents the percentage of the face value of the debt that can be

reclaimed upon default. The recovery rate also varies positively with the economy in

the following fashion:

ut = u(gt) = a + bgt

We choose to vary the recovery rate with aggregate economic output growth,

due to a number of empirical observations about recovery after default. Altman

and Kishore (1996) show that recovery rates are time-varying, while Collin-Dufresne,

Goldstein, and Martin (2001) suggest that "even if the probability of default remains

constant for a firm, changes in credit spreads can occur due to changes in the expected

recovery rate. The expected recovery rate should be a function of the overall business

climate." Shleifer and Vishny (1992) find that a firm's liquidation value is lower when

its competitors are experiencing cash flow problems. Other empirical papers in the

literature with similar conclusions include Thorburn (2000), Gupton and Stein (2002),

Altman et al. (2002) and Acharya et al (2003). Acharya et al. (2003) also show

that default risk and the recovery rate are only weakly linked by the same factors.

84

We then assume that they are essentially independent, which makes our simulation

easier. Furthermore, since the actual empirical correlation is slightly positive, the

independence assumption decreases expected loss and biases our model-generated

credit spreads downward slightly.

The value of the risky debt at t = 0 is

DV = y r kB(t)

^(l+r(0,t))*\

The rates r(0,t) represents the real term structure generated by the macro model.

To adjust the firm's cash flow growth to the risk neutral measure, which simplifies

bond pricing, we must reduce the drift of the firm cash flow growth process by its

market price of risk. Under the risk-neutral measure Q, the cash flow dynamics are

gK(t) = Q9t + & - n7ecP°c°K + (XTK<$ + °KV1 - A f (3.12)

where oc is the volatility of consumption growth and r^hec reflects the elasticity of

the pricing kernel to innovations in consumption growth. Under power utility, this

elasticity is just the coefficient of risk aversion, which corresponds to 7. However,

under a utility function that does not exhibit time separability, a series of assumptions,

including our loglinear assumption as described earlier, makes the above expression

a first-order approximation to the market price of risk, which is sufficient for our

purposes. Furthermore, since our agents' prefences are dependent on the previous

period's consumption, the elasticity ?7™ec is time-varying, which, in turn, makes our

market price of risk also time-varying. This is the key effect that we obtain by using

an internal habit persistence specification for our utility function, as it allows us to

have time-varying sensitivities to macroeconomic conditions not found in our previous

model and it allows for the possibility of larger risk premia than allowed by a power

utility specification, thereby resulting in higher credit spreads.

The risk-free term structure results from the macroeconomic model; therefore we

85

must adjust the firm's cash flow growth to the risk-neutral measure.

The yield-to-maturity of the risky bond is the solution Y to the equation

OV = ^ + (F~P)(l~Yf

The yield-to-maturity R of a risk-free bond with the same payment structure is

given by

The resulting credit yield spread is, therefore, just Y — R. The assumption of this

form for the bond payout and recovery rate allow for current economic conditions as

represented by current output growth and inflation to impact the firm's cash flow and

how its bond's coupon payments are discounted.

3.3 Model Calibration and Simulation

3.3.1 The Macroeconomic Model

To calibrate the new Keynesian macroeconomy in the first part of the model, we

choose calibrated parameters and the estimation technique found in Ravenna and

Seppala (2006). Using parameters chosen from related papers in the monetary policy

literature, they estimate the standard new Keynesian monetary policy model using

Dynare++ to match historical correlations of output and inflation as observed from

1952 to present. Table 1 lists the parameters we employ for the macroeconomic

model.3

The preference and technology exogenous shocks follow an AR(1) process:

logZt = (1 - pz)logZ + pzlogZt^ + ef, ef ~ N(Q, a\) (3.13)

3 The literature on parametrizing New Keynesian models to match historical data is extensive. These include Ravenna and Seppala (2006), Ravenna (2006), Christiano, Eichenbaum, and Evans (2005), Ireland (2001), Woodford (2003), and Rabanal and Rubio-Ramirez (2005)

86

where 2 is the steady state value of the variable

This calibration produces correlations between model output, inflation, and real

interest rates that match historical empirical correlations. This match between model

and empirical correlations between macroceonomic conditions and risk-free and risky

yields is one of the primary goals of this paper and differentiates this model from

others in the credit literature that specify output growth and inflation processes

exogenously, not structurally. The habit perisistence coefficient, ip, is the standard

value used by Constantinides (1990) and other similar papers. Table 3.2 compares

the model's second moments and correlations with output with U.S. post-war sample

data. The model matches the volatility and correlations of output and inflation,

but increases the volatility and output correlation of interest rates. Table 3.3 shows

means, standard deviations, and correlations for output growth, inflation, and real

interest rates.

Symbol

7 1> £ P 8p

TX

e LVy

UJn

X Pa

Pd

%

4

Description Relative risk aversion

Habit persistence coefficient Inverse of labor supply elasticity

Discount factor Price stickiness probability

Steady state inflation (quarterly) Demand elasticity

Taylor coefficient (output) Taylor coefficient (inflation)

Monetary policy smoothing parameter Autocorrelation of technology shock Autocorrelation of production shock

Volatility of technology shock Volatility of production shock

Value 2.5 0.8 0.5

0.99 0.75 0.75 11

0.01 1.5 0

0.9 0.95

0.0035 0.008

Table 3.1: Macroeconomic model coefficients. Values based on Ravenna and Seppala (2006), calibrated to historical correlations of output, inflation, and interest rates

87

Variable Yt

nt

Rt n


US Data 1.59 3.00 2.82 2.32

Correlation with Output Model 1.00 0.13 0.08 -0.06

US Data 1.00 0.19 0.17 -0.13

Table 3.2: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, irt

is annualized CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms.


r 3

r 12

„60 ' t

-120 11

Mean Std. Dev 0.00 0.78 2.66 3.32 2.30 3.74 3.56 2.50 3.89 0.77 3.94 0.45

Correlation coefficients gt nt r\ r]2 rf° r]20

1.00 0.06 -0.10 -0.14 -0.09 -0.02 0.06 1.00 -0.94 -0.92 -0.73 -0.59

-0.10 -0.94 1.00 1.00 0.88 0.76 -0.14 -0.92 1.00 1.00 0.90 0.79 -0.09 -0.73 0.88 0.90 1.00 0.98 -0.02 -0.59 0.76 0.79 0.98 1.00

Table 3.3: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates

3.3.2 Calibrating and Simulating Credit Spreads

Once we estimate the macroeconomic model, we simulate the firm's cash flow process

with equation 3.3. We follow Longstaff and Piazzesi (2004) to choose parameters for

the firm's cash flow process. Longstaff and Piazzesi (2004) note that corporate cash

flows are highly volatile, strongly procyclical, and could answer part of the equity

premium puzzle by generating higher volatility for corporate cash flow that underlie

firm-issued securities. They find that the correlation of firm's cash flow with total

output growth to be around p fa 0.6 — 0.7, while firm cash flow growth has volatility

around 20%.

For each quarter t, we assume that output growth, inflation, and the real term

structure for the given quarter are fixed and that bondholders use the fixed macroe

conomic conditions to forecast firm cash flows and price bonds. We choose 10,000

paths for cash flow growth each quarter and calculate simulated firm cash flows and

88

payments to bondholders. The average of the sum of payments to bondholders dis

counted by the prevailing term structure gives us the bond price, from which we can

calculate risky yields and spreads.

After we generate a realistic set of time series for output growth, inflation, and

interest rates, we calibrate the model to generate credit spreads to match historical

default probabilities. Taking the parameters from above, we can fix the coupon of the

bond issued by the firm to a value, 5%, and calibrate inital cash flow KQ for each credit

rating category to match historical default probabilities in simulation. Huang and

Huang (2003) provide the following default probabilities for different credit ratings.

The calibration of initial cash flow to historical default probabilities to generate

different credit ratings categories uses the assumption that the rating agencies adopt

the "through the business cycle" approach to assign firms to a rating category. The

representative firm we model is assigned its rating based on its average probability of

default throughout the business cycle we model, rather than on the basis of current

conditions. This assumption allows us to vary the firm's default probability and

credit risk within the same rating category. Again, we consider our firm to be a

representative firm of a particular ratings category throughout the business cycle.

After generating the initial credit spread results based on the values for firm

specific growth mean and variance, we can then perform a sensitivity analysis of the

results by varying the firm-specific properties of cash flow growth. Finally, we conduct

a comparative static analysis to gauge the effects of firm specific characteristics on

credit spread levels and volatilities generated by the model and their relationship with

Symbol

P £

OK

a b

Description Correlation of macro growth and firm c.f. growth

Mean of firm-specific cash flow growth Standard deviation of firm cash flow growth

Recovery rate constant Recovery rate amplifier

Value 0.6 0%

20% 51.31%

2.5

Table 3.4: Firm-specific cash flow parameters, based on Longstaff and Piazzesi (2004) and recovery rate in Huang and Huang (2003), assuming average of zero economic growth and recovery rate of 51.31% for all bond ratings and maturities

89

Rating

AAA AA

A BBB

BB B

Cumulative default prob. (%) 1 yr. 0.00 0.03 0.10 0.12 1.29 6.47

4 yrs. 0.04 0.23 0.35 1.24 8.51

23.32

10 yrs. 0.77 0.99 1.55 4.39

20.63 43.91

Recovery rate

(%) 51.31 51.31 51.31 51.31 51.31 51.31

Table 3.5: Moody's default probabilities and recovery rates (Huang and Huang, 2003)


3.4 Results

In this section, we discuss the properties of the credit spreads generated by the model

we describe above. First, we describe the empirical properties of the credit spread

time series generated by the model. We then discuss the contemporaneous relation

ship between credit spreads and model-generated macroeconomic conditions, namely


3.4.1 Properties of Model-Generated Credit Spreads and De

fault Probabilities

Our model produces credit spreads and volatilities of the order of magnitude found

in the data and comparable with other structural credit models. Unfortunately, the

credit spread levels produced by original model are lower than those found in the

data and in structural models. When internal habit persistence is the preference

specification of the model, we produce substantially higher credit spreads on par with

other structural credit models in the literature, but not as high the levels observed in

the data. The inability to produce the levels seen in data could be due to the reduced

90

consumption growth volatility that naturally occurs with higher elasticity of income

substitution that an internal habit persistent model generates. Chen, Collin-Dufresne,

and Goldstein (2005) and others have suggested that the equity risk premium puzzle

and the credit spreads level puzzle are linked and that an asset pricing model that

captures one also captures the other. Therefore, any model that does not produce

high elasticity of income substitution with low relative risk aversion for agents in the

economy cannot generate high enough equity or credit risk premia, because either the

volatility of consumption growth or the elasticity of the pricing kernel to consumption

growth innovations are not large enough to justify a large maximal Sharpe ratio.

In constrast with the credit spread level results, the model generates credit spread

volatilities closer to empirical values than other structural credit models. Chen,

Collin-Dufresne, and Goldstein (2005) state structural credit models generate credit

spreads whose volatilities are too low relative to actual data. Most models generate

a volatility of the spread between BBB and AAA of around 35 bps per annum, while

the actual volatility is around 75 bps per annum for the entire history of the Moody's

corporate bond index and 56 bps for the post-war period. Our model with power

utility generates a BBB-AAA spread volatility of around 52 bps and generates an

annual volatility for the BBB-AAA spread of 64 bps, close enough with the actual

observed values.

Our simulation methodology, in addition to providing us with credit spreads, also

finds forward-looking default probabilities. Many previous credit risk models, partic-

Horizon and Maturity 4-year A

4-year BBB 4-year B

10-year A 10-year BBB

10-year B

Power 2.0 6.7

121.0 3.4 8.4

103.0

Habit 25.2 86.5

142.0 44.7 75.6 217

Historical 96.0

158.0 470.0 123.0 194.0 470.0

TY 13.2 56.6

240.5 48.1

109.6 318.5

LS 7.5

25.4 406.0

14.5 38.6

341.9

LT CDG 9.9

- 31.1 - 435.3

38.5 22.5 59.5 52.3

408.4 371.6

Table 3.6: Average credit spread levels. Historical data taken from Huang and Huang (2003). Comparable models including Tang and Yan (2006) (TY), Longstaff and Schwartz (1995) (LS), Leland and Toft (1996) (LT), and Collin-Dufresne and Goldstein (2005) (CDG)

91

ularly those that link consumption growth with credit spreads, generate procyclical

default probabilities, an obviously counterfactual property. As shown in Table C.l,

we regress model-generated default probabilities on contemporaneous output growth

and inflation and find a strongly significant countercyclical relationship of default

probabilities and output growth, but no connection with inflation. Both regression co

efficients conform with empirical data that default probabilities rise as output growth

falls, but have inflation has little impact on actual defaults.

3.4.2 Macroeconomic Factors of Credit Spreads: Contempo

raneous Relationships

Given the results of our previous model, we expect inflation to be positively related

to credit spreads, as an increase in inflation should increase the discount rate, de

pressing bond prices and increasing yield spreads. Furthermore, we expect higher

output growth to have a positive impact on credit spreads of AAA and AA and a

negative effect on credit spreads of A rating or below. For the higher rated bonds,

increased output growth should increase the discount rate applied to coupon pay

ments, but should not substantially decrease the probability of default for the bond.

Therefore, the bond price should be depressed and the credit spread should rise. For

the lower rated bonds, the increase in output growth should increase the firm's cash

flow, helping it meet cash flow obligations and reducing the probability of default.

This, in turn, should counteract the effect of an increase in the discount rate and

should decrease the credit spread.

The model produces results close to our intuition and what is seen empirically. To

evaluate the impact of output growth and inflation on credit spread level, we perform

regression on with the following specification:

st = a + psst-i + Pggt + /3n7Tt (3.14)

92

As seen in Table C.2, inflation varies positively with credit spread levels, as /3n > 0.

This result is generally consistent with our empirical findings in our earlier research.

Furthermore, it matches our intuition that higher inflation raises the risk-free real rate

demanded by investors, which correspondingly reduces demands for risky securities

or raise the yield required of risky debt securities to compensate investors to purchase

them.

The coefficient on output growth, however, does not exactly match what we find in

our previous empirical study. Output growth varies negatively with credit spreads of

all maturites and ratings in the model. However, while output growth has a positive

coefficient for higher-rated credit spreads in the empirical data, the model generates

higher-rated credit spreads that vary negatively with output growth, although not

significantly or with large magnitude. This result is present in both the power utility

and habit persistence versions of our model.

3.4.3 Macro economic Factors of Credit Spreads: Different

Regimes

In addition to testing the magnitude and volatility of credit spreads generated by

the model as well the relationship with macroeconomic conditions, we also wanted to

produce credit spreads whose relationship with macroeconomic variables has different

sensitivities in different periods. In particular, we developed a model to replicate

the phenomenon that macroeconomic conditions primarily impact credit spreads in

periods of financial uncertainty. To acheive this objective, we chose a preference

specification with time variation, namely internal habit persistence.

With habit persistence, the elasticity of the pricing kernel with consumption

growth innovation is time-varying, in the range of 10 to 22. In contrast, under power

utility, the elasticity of the pricing kernel is equal to the relative risk aversion of 2.5.

To test if our model produces different regimes, we divided our time series of credit

spreads into regimes of high and low pricing kernel elasticity and performed the above

93

regression exclusively under the regimes.

As seen in Tables C.3 and C.4, credit spreads demonstrate little sensitivity and

cannot be explained by macroeconomic conditions under low elasticity regimes. Under

regimes of high pricing kernel elasticity, credit spreads are related to output growth

in a manner consistent with the whole sample. This is exactly the behavior exhibited

empirically in our credit spread data. Inflation impacts credit spreads positively and

significantly in the high elasticity regime only for BBB-B spreads, but seems not

to affect AAA-A rated spreads. Although we can replicate the qualitative signs of

the relationships between credit spreads and macroeconomic conditions, we cannot

exactly replicate the perisistence of credit spreads, particularly in the low elasticity

regime, where empirically the coefficients on lagged credit spreads are close to one.

3.5 Conclusion

In this paper, we present a variation of the macroeconomic-credit model that we pre

sented earlier with agents' utility specified by an internal habit specification. We chose

this specification for the utility function, because it is time-varying and generates a

time-varying elasticity of the pricing kernel to consumption innovations. The time-

varying elasticity allows us to replicate the changing sensitivities of credit spreads to

macroeconomic conditions and generates higher levels and volatilities of credit spreads

than our previous model. As stated in other research, the size of credit spreads and

the equity premium are related and, therefore, any model specification, such as habit

persistence, that helps to generate an empirically-plausible equity premium also can

generate realistic credit spreads.

The credit spreads generated by the model presented in this paper do conform

with the empirical properties observed in the data better than our previous model.

The model-generated credit spreads demonstrate greater sensitivity to macroeconomic

conditions during periods of lower elasticity of the pricing kernel to consumption, a

feature that we found in our empirical results. Furthermore, we generate larger credit

94

spreads on the order of those found in other structural credit models and the volatility

of the credit spreads we find are close to those observed empirically and much higher

than those generated by any other credit risk model.

While this model comes closer to matching empirical observations about credit

spreads and their relationship with macroeconomic variables, there are further av

enues we can explore. In this and our previous papers, we abstracted away from the

optimal capital structure decisions the firm can make to take advantage of macroeco

nomic conditions. Optimal capital structure decisions are a relatively new avenue in

the credit risk literature and maybe, we can incorporate such a feature in our model

of the macroeconomy with credit risk.

Bibliography

[1] V.V. Acharya, S.T. Bharath, and A. Srinivasan. Understanding the recovery

rates on defaulted securities. Working paper, LBS, Michigan, and Georgia, 2003.

[2] Fabio Alessandrini. Credit risk, interest rate risk, and the business cycle. The

Journal of Fixed Income, 9(2):2177-2207, Dec 1999.

[3] E. Altman, B. Brady, A. Resti, and A. Sironi. The link between default and

recovery rates: implications for credit risk models nad procyclicality. Working

paper, NYU, 2002.

[4] E. Altman and V.M. Kishore. Almost everything you wanted to know about

recoveries on defaulted bonds. Financial Analysts Journal, 526:57-64, 1996.

[5] Jushan Bai and Pierre Perron. Estimating and testing linear models with mul

tiple structural changes. Econometrica, 66:47-78, 1998.

[6] Jushan Bai and Pierre Perron. Computation and analysis of multiple structural

change models. Journal of Applied Econometrics, 18:1-22, 2003.

[7] G. Bakshi and Z. Chen. An alternative valuation model for contingent claims.

Journal of Financial Economics, 44:123-165, 1997.

[8] Ravi Bansal and Amir Yaron. Risks for the long run: A potential resolution of

asset pricing puzzles. The Journal of Finance, 59(4): 1481-1509, August 2004.

95

96

[9] Ben Bernanke, Mark Gertler, and Simon Gilchrist. The financial accelerator

and the flight to quality. The Review of Economics and Statstics, 78(1):1—15,

February 1996.

[10] Ben Bernanke, Mark Gertler, and Simon Gilchrist. The financial accelerator in

a quantitative business cycle framework. Handbook of Macroeconomics, pages

1341-93, 2000.

[11] Harjoat Bhamra, Lars-Alexander Kuehn, and Ilya Strebulaev. The levered equity

risk premium and credit spreads: A unified framework. University of British

Columbia and Stanford University, Oct 2007.

[12] M. Boldrin, L. Christiano, and J.D.M. Fisher. Habit persistence and asset returns

in an exchange economy. Macroeconomic Dynamics, pages 312-332, 1997.

[13] Guillermo Calvo. Staggered pricing in a utility-maximizing framework. Journal

of Monetary Economics, 12(3):383-98, 1983.

[14] John Campbell and Tuomo Vuolteenaho. Bad beta, good beta. The American

Economic Review, 94(5):1249-1275, December 2004.

[15] J.Y. Campbell and J.H. Cochrane. By force of habit: A consumption-based

explanation of aggregate stock market behavior. Journal of Political Economy,

107:205-251, 1999.

[16] Hui Chen. Macroeconomic conditions and the puzzles of credit spreads and

capital structure. University of Chicago GSB Working Paper, Jan 2007.

[17] Long Chen, Pierre Collin-Dufresne, and Robert S. Goldstein. On the relation

between the credit spread puzzle and the equity premium puzzle. Presented at

the Second Credit Risk Conference, London, England, May, 2005, Mar 2006.

97

[18] Pierre Collin-Dufresne, Robert S. Goldstein, and J. Spencer Martin. The deter

minants of credit spread changes. The Journal of Finance, 56(6):2177-2207, Dec

2001.

[19] George Constantinides. Habit formation: A resolution of the equity premium

puzzle. Journal of Political Economy, 98:519-543, 1990.

[20] Alexander David. Inflation uncertainty, asset valuations, and the credit spread

puzzles. Forthcoming: Review of Financial Studies, 2007.

[21] Edwin Elton, Martin Gruber, Deepak Agarwal, and Christopher Mann. Ex

plaining the rate spread on corporate bonds. Journal of Finance, 41(l):247-278,

February 2001.

[22] Paul Glasserman. Monte Carlo Methods in Financial Engineering. Springer

Verlag, 2004.

[23] Dirk Hackbarth, Jianjun Miao, and Erwan Morellec. Capital structure, credit

risk, and macroeconomic conditions. Journal of Financial Economics, 82(3) :519-

550, December 2006.

[24] J. Huang and W. Kong. Explaining credit spread changes: New evidence from

option-adjusted bond indexes. The Journal of Derivatives, pages 30-42, 2003.

[25] Hayne Leland. Predictions of default probabilities in structural models of debt.

Journal of Investment Management, pages 5-20, 2004.

[26] M. Lettau and H. Uhlig. The sharpe ratio and preferences: A parametric ap

proach. Macroeconomic Dynamics, pages 242-265, 2002.

[27] Martin Lettau. Inspecting the mechanism: Closed-form solutions for asset prices

in real business cycle models. The Economic Journal, pages 550-575, July 2003.

98

[28] George Li. Time-varying risk aversion and asset prices. Journal of Banking and

Finance, 31(1), Jan 2007.

[29] Francis Longstaff and Monika Piazzesi. Corporate earnings and the equity pre

mium. Journal of Financial Economics, 74:401-421, 2004.

[30] Francis Longstaff and Eduardo Schwartz. A simple approach to valuing risky

fixed and floating rate debt. Journal of Finance, 50:789-819, 1995.

[31] Thomas Lubik and Frank Schorfheide. Testing for indeterminacy: An application

to u.s. monetary policy. American Economic Review, 94(1):190-217, 2004.

[32] R. Mashal and M. Naldi. Default-adjusted credit curves and bond analytics

on lehmanlive. Lehman Brothers Fixed Income Research Quantitative Credit

Strategies, September 2005.

[33] R. C. Merton. On the pricing of corporate debt: The risk structure of interest

rates. Journal of Finance, pages 449-470, 1974.

[34] C. Morris, R. Neal, and D. Rolph. Credit spreads and interest rates: A cointe-

gration approach. Federal Reserve Bank of Kansas City, 1998.

[35] M. Pedrosa and R. Roll. Systematic risk in corporate bond credit spreads. Jour

nal of Fixed Income, Dec 1998.

[36] Federico Ravenna and Juha Seppala. Monetary policy and the rejections of the

expectations hypothesis. Working paper, Aug 2006.

[37] Andrew Rose. Is the real interest rate stable. Journal of Finance, 43:1095-112,

1988.

[38] A. Shleifer and R. Vishny. Liquidation values and debt capacity: A market

equilibrium approach. The Journal of Finance, 47:1343-1366, 1992.

99

[39] Joseph Stiglitz and Bruce Greenwald. Towards a New Paradigm in Monetary

Economics. Cambridge University Press, 2003.

[40] Dragon Yongjun Tang and Hong Yan. Macroeconomic conditions, firm charac

teristics, and credit spreads. Journal of Financial Services Research, 29:311-344,

Apr 2006.

[41] Maria Vassalou and Yuhang Xing. Default risk in equity returns. Journal of

Finance, 59(2):831-868, Aug 2004.

[42] Jing zhi Huang and Ming Huang. How much of the corporate-treasury yield

spread is due to credit risk? Penn State University and Stanford University,

May 2003.

Appendix A

Chap. 1 Tables and Figures

A. l Unit Root Tests

Period Full Sample

May 1994 - Aug 1998 Sep 1998 - Mar 2003 Apr 2003 - Jun 2007

1 Yr -2.0148 -0.5959 -2.3081

-4.9418

2 Yr -1.7000 -1.8881 -1.9823

-3.2817

3 Yr -1.7026 -0.4323 -1.5556

-7.6593

5 Yr -1.7162 -0.0039 -1.6578

-9.0780

7Yr -1.7424 -0.2186 -1.7961

-5.1038

10 Yr -1.8222 -1.0940 -1.8936

-3.8597

Table A . l : Phillips-Perron test results for single A credit spreads

Period Full Sample

May 1994 - Aug 1998 Sep 1998 - Mar 2003 Apr 2003 - Jun 2007

AAA -2.1264 1.7604

-2.4991 -4.1803

AA -2.0849 1.7577

-2.5629 -2.7661

A -1.7162 -0.0039 -1.6578

-9.0780

BBB -1.6050 -0.3675 -0.8276

-10.8433

BB -2.1860 -1.2582 -1.3030

-6.0400

B -2.3163 -1.1323 -2.3888 -2.2181

Table A.2: Phillips-Perron test results for 5-year credit spreads

100

101

A.2 Structural Break Tests

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Break date

08/1998

03/2003

08/1998 03/2003

08/1998 03/2003

08/1998 02/2003

08/1998

02/2003

08/1998 02/2003

95% C.I.

05/1998

12/2001

05/1998 12/2001

05/1998 12/2001

04/1998 04/2002

08/1998 02/2003

04/1998 01/2003

02/1999

05/2003

02/1999 06/2003

01/1999 05/2003

11/1998

02/2003

10/1998 04/2003

01/1999 06/2003

1 621.31

71.49

112.23

254.12

2395.04

121.52

supF test

2

8780.78

23562.32

45254.88

18751.25

31290.68

26640.68

3 24795.86

82791.26

68309.23

46436.70

111695.89

77551.85

supF(i+l || i)

i = 1 i =2

21.62 12.34

75.45 24.00

36.45 3.61

35.42 5.77

42.77 12.50

114.63 1.51

Table A.3 : Bai and Perron (1998) structural break test dates and confidence intervals for AAA credit spreads

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Break date

08/1998 01/2003

08/1998 01/2003

08/1998

01/2003

08/1998 03/2003

08/1998 03/2003

08/1998 12/2002

95% CI.

04/1998 01/2003

11/1997 01/2003

07/1998 07/2002

01/1998 07/2002

08/1998 07/2002

01/1998 06/2002

03/1999 05/2003

09/1998 12/2003

12/1998

03/2003

11/1998 04/2003

05/1999 03/2003

01/1999 07/2003

1 573.33

648.87

63.87

892.15

909.38

228.60

supF test 2

52403.16

18723.58

94448.78

55627.44

155117.04

8319.77

3 476589.82

21686.91

52793.13

491577.71

149751.62

68649.57

supF(i+l || i)

i = 1 i =2

337.05 35.24

121.83 82.62

586.09 57.57

26.23 22.37

61.44 10.03

162.81 5.93

Table A.4: Bai and Perron (1998) structural break test dates and confidence intervals for AA credit spreads

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Break date

07/1998 03/2003

08/1998

04/2003

08/1998

04/2003

08/1998 04/2003

08/1998

05/2003

08/1998

06/2003

95% C.I.

02/1998 03/2003

07/1998

03/2003

07/1998 07/2002

08/1998 07/2002

08/1998 07/2002

02/1998

07/2002

08/1998 11/2003

11/1998

12/2003

09/1998 05/2003

11/1998 03/2003

11/1998 11/2003

12/1998 04/2004

1 731.13

5442.18

6009.19

25.90

85.04

805.78

supF test 2

7018.15

58233.22

52040.66

214820.02

37406.52

29824.46

3 287281.06

643382.28

309221.33

41417.50

44353.14

525900.69

supF(i+l || i) i = 1 i =2

181.49 65.17

579.13 30.54

949.42 29.83

3239.84 30.29

261.73 50.41

153.38 18.99

Table A.5: Bai and Perron (1998) structural break test dates and confidence intervals for A credit spreads

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Break date

08/1998 03/2003

08/1998 04/2003

08/1998 04/2003

08/1998 04/2003

08/1998 04/2003

08/1998 04/2003

95% C.I.

05/1998 03/2003

08/1998 09/2002

01/1998 12/2001

08/1998

12/2001

07/1998 12/2001

01/1998 12/2001

11/1998 04/2004

12/1999 08/2003

11/1998 09/2003

08/1998

09/2003

11/1998 07/2003

10/1998 08/2003

1 58.92

707.29

357.01

124.57

132.79

326.35

supF test 2

30619.86

21319.92

39080.97

101708.45

387344.74

44555.87

3 364333.35

1959487.32

620366.28

751472.02

680318.31

636413.12

supF(i+l || i) i = 1 i=2

362.44 18.28

342.34 117.16

366.58 45.95

174.02 26.85

60.13 37.34

4808.33 14.90

Table A.6: Bai and Perron (1998) structural break test dates and confidence intervals for BBB credit spreads

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Break date

08/1998 03/2003

08/1998 03/2003

08/1998

03/2003

08/1998 06/2003

08/1998 06/2003

08/1998 06/2003

95% C.I.

05/1998 11/2002

11/1997 08/2002

01/1998

01/2003

04/1998 03/2003

04/1998 03/2003

04/1998 03/2003

12/1998 04/2003

03/1999 04/2003

11/1998 01/2004

12/1998 07/2003

12/1998 08/2003

12/1998 09/2003

1 27.20

5.90

4.29

6.72

6.93

18.72

supF test 2

1441.19

415.51

91.02

14011.47

8339.88

7662.66

3 18042.79

28912.14

81660.83

73961.83

47409.48

51620.18

supF(i+l || i)

i = 1 i =2

386.04

456.46

167.89

43.86

69.59

159.97

11.87

14.64

13.48

13.45

19.35

80.46

Table A.7: Bai and Perron (1998) structural break test dates and confidence intervals for BB credit spreads

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Break date

02/2000 03/2003

02/2000 03/2003

03/2000 03/2003

02/2000 04/2003

03/2000

03/2003

03/2000

03/2003

95% C.I. 03/1998 09/2002

08/1998

01/2003

10/1999 09/2003

08/1999 03/2003

05/1998

03/2003

05/1998

03/2003

08/2000 03/2003

08/2000

09/2003

08/2000 09/2002

08/2000 02/2004

04/2000 12/2003

04/2000

12/2003

1 73.09

90.40

84.10

38.19

10.69

4.90

supF test 2

332.36

460.47

292.82

97037.08

435.67

650.68

3 310189.92

54992.04

27833.24

285133.30

42795.27

69522.23

supF(i+l 1 i) i = 1 i =2

96.72 16.38

107.03 75.21

110.51 15.14

109.09 8.64

95.94 22.57

212.14 58.14

Table A.8: Bai and Perron (1998) structural break test dates and confidence intervals for B credit spreads

A.3 OLS Est imation

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Adj. R2

0.7550

0.8887

0.9071

0.9032

0.8978

0.8824

Const

0.0014

2.3841

0.0006

1.2992

0.0004

0.7573

0.0004

0.6438

0.0004

0.7111

0.0003

0.4540

C$-i 0.6651

12.1792

0.8257

21.4695

0.8313

22.3918

0.8138

20.6120

0.8107

20.3787

0.8249 20.3688

REAL

0.0117

2.8748

0.0109

1.7985

0.0134

1.9827

0.0156

2.3954

0.0120

2.2785

0.0133

1.5467

INFL

0.0078

1.9672

0.0093

2.1251

0.0131

2.4450

0.0199

2.8117

0.0203

2.8145

0.0178 2.5655

FFR -0.0076

-1.1236

-0.0069

-1.2458

-0.0048

-0.8107

-0.0038

-0.5312

-0.0036

-0.4854

-0.0013

-0.1757

SLOPE

-0.0517

-3.2341

-0.0316

-2.5422

-0.0304

-2.2919

-0.0372

-2.3119

-0.0407

-2.4817

-0.0354

-2.1603

MKT 0.0013

1.0491

-0.0001

-0.1058

0.0000

-0.0371

0.0001

0.0820

-0.0003

-0.2444

-0.0007

-0.5518

VIX 0.0044

4.1365

0.0038

3.9214

0.0044

4.1716

0.0055

4.3548

0.0055

4.3248

0.0047 3.6345

Table A.9: OLS regression coefficients for AAA credit spreads on regressors

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Adj. R2

0.8511

0.9099

0.9196

0.9183

0.9150

0.9021

Const

-0.0001

-0.1259

-0.0003

-0.5091

-0.0004

-0.6204

-0.0002

-0.2438

0.0001

0.1589

0.0002

0.3553

CSt-i 0.7561

16.7965

0.8340

23.6269

0.8258

23.6394

0.8161

22.5977

0.8056

22.0602

0.7979

21.0342

REAL

-0.0235

-2.3555

-0.0138

-1.6259

-0.0177

-1.9726

-0.0245

-2.3646

-0.0252

-2.4467

-0.0209

-2.1214

INFL

0.0227

1.9754

0.0229

2.0989

0.0277

2.4121

0.0336

2.6176

0.0345

2.7109

0.0336

2.6868

FFR

0.0038

0.4544

-0.0004

-0.0603

0.0002

0.0269

-0.0013

-0.1541

-0.0034

-0.4042

-0.0032

-0.3903

SLOPE

-0.0388

-2.0052

-0.0299

-1.8182

-0.0326

-1.9224

-0.0437

-2.2713

-0.0527

-2.7395

-0.0548

-2.9206

MKT

0.0007

0.4670

-0.0006

-0.4767

-0.0006

-0.4150

-0.0003

-0.1984

-0.0007

-0.4351

-0.0010

-0.6923

VIX

0.0066

4.7967

0.0061

4.8545

0.0070

5.2339

0.0078

5.3469

0.0079

5.4891

0.0073

5.1939

Table A. 10: OLS regression coefficients for AA credit spreads on regressors

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Adj. R2

0.9172

0.9442

0.9489

0.9471

0.9440

0.9337

Const

-0.0006 -0.7400

-0.0006 -0.8944

-0.0007

-0.9573

-0.0005 -0.6172

-0.0002 -0.2239

0.0001 0.1364

C$_! 0.8059

20.5299

0.8613 27.0646

0.8542

27.0828

0.8484 26.2869

0.8444 26.0583

0.8344 24.4374

REAL

-0.0380 -2.9959

-0.0255 -2.3485

-0.0291 -2.6000

-0.0343 -2.7695

-0.0339 -2.7674

-0.0312 -2.6294

INFL

0.0263 1.9783

0.0259 2.0429

0.0306 2.3709

0.0365 2.6423

0.0374 2.7393

0.0376 2.8047

FFR 0.0109 1.1151

0.0050 0.5794

0.0058

0.6588

0.0046 0.4962

0.0023 0.2557

0.0019 0.2140

SLOPE

-0.0289 -1.3693

-0.0284 -1.5214

-0.0316

-1.6860

-0.0423 -2.0765

-0.0512 -2.5214

-0.0561 -2.8227

MKT 0.0003 0.1881

-0.0008 -0.4988

-0.0006 -0.3977

-0.0002 -0.1052

-0.0004 -0.2238

-0.0006

-0.3709

VIX 0.0080

4.9879

0.0073 5.0122

0.0081 5.3877

0.0087 5.5214

0.0085 5.5743

0.0079

5.3160

Table A . l l : OLS regression coefficients for A credit spreads on regressors

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Adj. R2

0.9584

0.9686

0.9677

0.9615

0.9541

0.9421

Const

-0.0004

-0.4045

-0.0005

-0.5427

-0.0006

-0.6580

-0.0006

-0.5477

-0.0003

-0.3151

-0.0002

-0.1387

CSt-!

0.8685 28.0947

0.8959

33.8695

0.8867

32.4830

0.8702

29.3963

0.8575

27.3248

0.8414

24.7578

REAL

-0.0484

-3.1536

-0.0386

-2.7803

-0.0434

-2.9137

-0.0535

-3.0759

-0.0584

-3.1382

-0.0617

-3.1608

INFL

0.0376

2.0558

0.0390

2.2114

0.0252

2.5095

0.0361

2.8813

0.0416

2.0139

0.0471

2.1400

FFR 0.0106 0.9021

0.0065

0.6022

0.0082

0.7117

0.0093

0.7138

0.0086

0.6145

0.0099

0.6615

SLOPE

-0.0374

-1.4983

-0.0402

-1.7397

-0.0426

-1.7506

-0.0527

-1.8973

-0.0634

-2.1226

-0.0710

-2.2540

MKT -0.0003

-0.1394

-0.0009

-0.4565

-0.0004

-0.2120

0.0005

0.2001

0.0006

0.2520

0.0008

0.3017

VIX 0.0093

4.7948

0.0090

4.9194

0.0101

5.1454

0.0117

5.2629

0.0124

5.2272

0.0125

5.0414

Table A. 12: OLS regression coefficients for BBB credit spreads on regressors

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Adj. R2

0.8691

0.8778

0.8810

0.8839

0.8795

0.8670

Const

-0.0093

-2.0465

-0.0084

-1.9243

-0.0079

-1.8377

-0.0065

-1.5809

-0.0055

-1.3510

-0.0045

-1.1217

CSt-i

0.7005

13.7510

0.7322

15.1626

0.7445

15.8632

0.7637

17.0164

0.7758

17.6438

0.7842

17.8634

REAL

-0.2227

-3.3544

-0.1941

-3.0523

-0.1836

-2.9401

-0.1663

-2.7529

-0.1472

-2.5002

-0.1248

-2.1595

INFL

0.1893

2.2590

0.1905

2.3358

0.1952

2.4229

0.1955

2.5010

0.1898

2.4624

0.1825

2.3678

FFR 0.1365

2.2850

0.1124

1.9589

0.1009

1.7943

0.0790

1.4665

0.0632

1.1958

0.0501

0.9503

SLOPE

0.1443

1.1401

0.1036

0.8511

0.0771

0.6484

0.0278

0.2452

-0.0034

-0.0302

-0.0257

-0.2327

MKT 0.0061

0.5960

0.0027

0.2716

0.0007

0.0748

-0.0029

-0.3087

-0.0067

-0.7275

-0.0112

-1.2249

VIX 0.0486

5.0875

0.0460

4.9785

0.0452

4.9735

0.0427

4.9171

0.0403

4.7624

0.0374

4.4798

Table A. 13: OLS regression coefficients for BB credit spreads on regressors

110

Maturity

lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Adj. R2

0.9273

0.9298

0.9313

0.9340

0.9351

0.9320

Const

-0.0302

-2.6025

-0.0279

-2.4778

-0.0264

-2.3989

-0.0233

-2.2378

-0.0207

-2.0749

-0.0181

-1.8651

CSt-t

0.8294

20.6372

0.8376

21.2238

0.8418

21.5378

0.8508

22.2116

0.8583

22.6387

0.8623

22.3690

REAL

-0.3472

-2.0907

-0.3357

-2.0702

-0.3325

-2.0861

-0.3153

-2.0596

-0.2897

-1.9683

-0.2615

-1.8268

INFL

0.3887

1.9763

0.3877

1.9876

0.3916

1.9773

0.3869

1.9919

0.3736

1.9925

0.3609

1.9629

FFR 0.3774

2.4663

0.3449

2.3175

0.3252

2.2319

0.2872

2.0663

0.2559

1.9085

0.2295

1.7384

SLOPE

0.2953

0.9727

0.2305

0.7818

0.1826

0.6344

0.1006

0.3691

0.0430

0.1650

-0.0008

-0.0031

MKT -0.0282

-1.1374

-0.0282

-1.1668

-0.0277

-1.1756

-0.0285

-1.2741

-0.0319

-1.4932

-0.0388

-1.8744

VIX 0.1010

4.0329

0.0969

3.9853

0.0942

3.9718

0.0875

3.9030

0.0806

3.7754

0.0728

3.5367

Table A. 14: OLS regression coefficients for B credit spreads on regressors

I l l

A.4 Markov Regime Switching Estimation

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. R2

0.8894

0.9602

0.9671

0.9553

0.9566

0.9639

Const 0.0025

7.9982 -0.0029 -1.7836

0.0001 1.1150 0.0005 1.7195

0.0005 7.6962 0.0002 0.5196

0.0010 0.0003

-0.0010 0.0006

0.0012 6.5477 -0.0006 -0.6020

0.0019 15.3284

-0.0005 -1.6927

CSt-i 1.1198

3.9219 0.5761

11.3321

0.8498 21.1861

0.8157 12.8333

0.9234 11.6369

0.8237 26.1342

0.8556 5.9474 0.7819

22.6640

0.8771 8.1834 0.7538

20.0207

0.8887 14.0252

0.6329 16.7866

REAL -0.0225 -1.5718 0.0327

4.3360

0.0091 1.3386 0.0153

2.9762

-0.0093 -1.4038 0.0100

3.0688

0.0121 0.9272 0.0240

3.2327

-0.0098 -0.4154 0.0237

3.0762

-0.0088 -0.4896 0.0296 1.2958

INFL 0.0220 1.2940 0.0050

2.6135

0.0097 1.0805 0.0154

2.3403

-0.0070 -0.5616 0.0152

2.6446

0.0250 1.9572 0.0160

2.3349

0.0159 0.6398 0.0258

2.0594

0.0268 0.0190 0.0136

2.5739

FFR 0.0077 0.4461

-0.0109 -1.7967

-0.0028 -0.6710 -0.0169

-2.9459

-0.0023 -0.6120 -0.0201

-2.5313

-0.0015 -0.3529 -0.0223 -1.4887

-0.0229 -1.5089 -0.0022 -0.4846

-0.0042 -0.3646 0.0020 0.4498

SLOPE 0.0246 0.7986

-0.0684 -6.3532

-0.0060 -0.7138 -0.0593

-5.2681

-0.0228 -2.9895 -0.0477

-3.1520

-0.0226 -2.4924 -0.0792

-3.2184

-0.0929 -2.9754 -0.0253

-2.4727

-0.0688 -3.6838 -0.0190 -1.9332

MKT -0.0015 -1.3144 0.0106

2.1064

-0.0016 -1.4556 -0.0003 -0.1401

-0.0008 -0.8716 -0.0002 -0.1327

-0.0011 -0.8552 -0.0003 -0.0968

-0.0016 -1.3246 -0.0011 -0.4042

-0.0016 -1.6284 0.0006 0.2925

VIX 0.0026

2.2135 0.0065

2.2532

0.0005 0.5322 0.0094

7.6377

0.0016 1.9284 0.0097

5.8266

0.0022 2.0973 0.0154

6.2167

0.0025 2.3820 0.0150

4.7064

0.0022 2.3613 0.0138

7.3022

Table A. 15: Markov regime switching model coefficients for AAA credit spreads on regressors

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. R2

0.9338

0.9722

0.9762

0.9732

0.9727

0.9734

Const 0.0012

2.8723 -0.0021 -0.7730

-0.0013 -5.4356

0.0006 0.7685

0.0003 3.0878 -0.0009 -1.8527

-0.0014 -3.1226

0.0008 8.4425

-0.0012 -2.8656

0.0013 10.4628

-0.0005 -1.7077 0.0022

23.1075

CS-i 0.9560

14.4920 0.3500

3.5754

0.9877 4.3576 0.8357

32.8941

0.9955 8.1220 0.8364

32.7081

0.8477 9.1639 0.8038

30.2014

0.8287 10.9921

0.7653 26.4742

0.7756 13.0908

0.7086 23.9550

REAL 0.0148 1.0409

-0.0860 -3.0903

0.0159 1.0273

-0.0086 -1.3449

0.0225 1.3792

-0.0224 -3.8872

0.0124 0.7992

-0.0214 -2.7477

0.0140 0.9497

-0.0253 -3.2789

0.0210 1.3621

-0.0346 -4.1309

INFL -0.0181

-2.1135 0.0961

2.7951

-0.0015 -0.1351 0.0141

2.9825

-0.0082 -0.4340 0.0209

3.5042

0.0110 0.6229 0.0246

3.4463

0.0122 0.6958 0.0260

3.6418

-0.0036 -0.2817 0.0141

1.9756

FFR -0.0181

-3.0849 0.0391

2.0731

-0.0126 -1.7947 -0.0044 -1.1682

-0.0285 -2.0894 -0.0015 -0.3930

-0.0085 -0.6612 -0.0065 -1.4782

-0.0087 -0.7245 -0.0100

-2.2690

-0.0037 -0.2909 -0.0081 -1.8508

SLOPE -0.0376

-2.6048 -0.0394 -0.7437

-0.0228 -1.3292 -0.0271

-3.1797

-0.0395 -1.6970 -0.0234

-3.0083

-0.0296 -1.1994 -0.0404

-4.2189

-0.0383 -1.6115 -0.0535

-5.5537

-0.0860 -3.9962 -0.0506

-4.9623

MKT 0.0005 0.1907 0.0028 0.9424

0.0043 1.1407 0.0000 0.0425

-0.0001 -0.1109 0.0004 0.1780

0.0002 0.0683

-0.0003 -0.2593

-0.0005 -0.2089 -0.0009 -0.8000

-0.0022 -1.1248 -0.0011 -1.0015

VIX 0.0010 0.7188 0.0174

4.7025

0.0018 0.3428 0.0128

5.3356

0.0026 2.9392 0.0151

6.2740

0.0181 8.5420 0.0036

3.4920

0.0180 9.0463 0.0044

4.1405

0.0167 9.5984 0.0034

3.1755

Table A. 16: Markov regime switching model coefficients for A A credit spreads on regressors

113

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. _R2

0.9598

0.9783

0.9832

0.9840

0.9819

0.9708

Const 0.0008 1.4659

-0.0011 -0.1853

0.0000 0.8565

-0.0033 -6.5662

0.0007 1.5904

-0.0019 -3.8528

0.0002 0.2194

-0.0078 -0.8457

0.0003 0.6377

-0.0069 -2.7133

-0.0009 -1.3659 0.0000

-0.0036

CSt-i 0.9500

21.9782 0.3628

3.8695

0.8923 35.5516

0.7833 6.5453

0.8530 25.3093

0.7440 5.5315

0.9892 9.8172 0.3569

3.7984

0.9678 21.2988

0.3572 4.3204

1.0415 8.8436 0.4389

4.6397

REAL 0.0066 0.4103

-0.1931 -2.7541

-0.0108 -1.1287 -0.0424 3.1243

-0.0216 -2.1349 -0.0221

-2.0736

0.0128 0.3227

-0.1134 -2.8605

0.0064 0.4540

-0.1100 -3.2511

0.0169 0.6125

-0.1176 -3.5198

INFL -0.0116 -1.1512 0.1093

2.5765

0.0142 3.5285 0.0644

1.9755

0.0130 1.7369 0.0332

3.7207

-0.0079 -0.6706 0.0637

2.1150

-0.0025 -0.3853 0.2762

5.7875

0.0104 0.5797 0.0903

3.4604

FFR -0.0145

-2.1267 0.0488 0.7578

0.0034 0.9323 0.0245 0.8565

0.0012 0.2026 0.0120 0.4810

-0.0085 -1.4099 0.0830 1.5016

-0.0078 -1.8441 0.0722

3.3233

-0.0043 -0.3808 0.0475 1.1596

SLOPE -0.0430

-2.7907 -0.0885 -0.4723

-0.0153 -2.5890 -0.0049 -0.3403

-0.0273 -2.4790 -0.0205 -0.6321

-0.0220 -1.2697 -0.0039 -0.0831

-0.0238 -1.8675 -0.0577

-1.9741

-0.0133 -0.6654 -0.0935

-4.4543

MKT -0.0004 -0.2350 0.0008 0.1127

-0.0004 -0.3432 0.0039 1.2111

-0.0009 -0.6608 -0.0026 -0.9063

0.0004 0.1538

-0.0002 -0.0446

-0.0002 -0.1875 0.0023 0.6040

-0.0012 -0.5090 0.0009 0.2824

VIX 0.0026 1.7996 0.0222

3.9386

0.0017 1.6078 0.0197

6.0123

0.0021 1.9439 0.0203

9.6121

0.0016 1.3355 0.0347

3.6194

0.0020 1.8758 0.0349

7.4425

0.0023 1.3876 0.0205

3.7003

Table A. 17: Markov regime switching model coefficients for A credit spreads on regressors

114

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. i?2

0.9876

0.9911

0.9898

0.9901

0.9830

0.9777

Const -0.0015 0.2752 0.0004 0.9857

-0.0021 0.3568

-0.0031 -0.2411

-0.0003 -1.8454 0.0033 0.9247

-0.0007 -2.5062

0.0078 4.9930

0.0012 0.7352 0.0005 1.0146

-0.0006 -1.0553 0.0055

3.9644

CSt-! 0.9538 1.0194 0.8579

33.9610

0.8972 12.3747

0.8947 43.5943

0.9931 43.2413

0.5439 19.6351

1.0122 43.7808

0.4776 7.9258

0.8822 12.4711

0.8455 31.4550

0.9305 27.6513

0.6311 23.1531

REAL 0.0018 0.9847

-0.0333 -18.1722

-0.0086 -1.1078 -0.0188

-3.5268

-0.0113 0.8726

-0.2128 4.5682

0.0138 0.9823

-0.4388 -12.2073

-0.0666 -1.0307 -0.0410

-2.7487

-0.0139 -0.9089 -0.3841

-7.2695

INFL 0.0051 0.3627 0.0154

3.1742

0.0107 0.6731 0.0195

2.4215

0.0050 0.7839 0.0365

4.0265

0.0002 0.0225 0.0548

2.6205

0.0128 0.3545 0.0352

3.1739

0.0065 0.2724 0.0405

3.7747

FFR -0.0193 -1.4338 0.0129 1.6753

-0.0011 -0.0966 0.0070 1.1910

0.0036 0.9945 0.0271 0.1348

-0.0059 -0.9087 -0.0113 -0.4658

-0.0592 -2.1824

0.0088 1.1333

0.0037 0.5831 0.0659

2.0650

SLOPE -0.0521 -1.7906 -0.0147 -0.9946

-0.0293 -1.3063 -0.0248

-1.9677

-0.0068 -0.6089 -0.2327 1.4233

-0.0226 -1.8616 -0.5926

-8.7880

-0.2035 -2.7955 -0.0443

-2.5577

-0.0424 -2.5815 -0.3530

-3.7149

MKT 0.0062

2.6384 -0.0031 -1.7597

0.0044 1.7566

-0.0012 -0.7740

-0.0006 -0.7564 -0.0042 0.8721

0.0020 1.0679

-0.0001 -0.0103

-0.0093 -1.7321 -0.0002 -0.0826

-0.0060 -3.0029

0.0342 7.3967

VIX 0.0188 1.1938 0.0177

3.5255

0.0214 11.0257

0.0263 2.1442

0.0018 0.9237 0.0305

5.7563

0.0037 2.8265 0.0390

10.6517

0.0055 3.5730 0.0305

6.5524

0.0051 3.2772 0.0245

5.0412

Table A. 18: Markov regime switching model coefficients for BBB credit spreads on regressors

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. R2

0.9476

0.9512

0.9454

0.9524

0.9412

0.9328

Const -0.0401

-2.2612 0.0015 0.3599

0.0056 0.2414 0.0016 0.3734

-0.0371 -1.3756 0.0009 0.2127

-0.0245 -2.1327 0.0014 0.4979

-0.0297 -1.6550 -0.0005 -0.1137

-0.0260 -2.0754

0.0000 -0.0041

csw 0.8237 8.3246 0.7094

13.8365

0.8518 6.2826 0.7269

15.6094

1.0869 6.7507 0.7801

18.8823

0.8275 5.9326 0.7558

12.2385

1.0801 6.8815 0.7859

17.4791

1.0785 7.3088 0.7845

18.6815

REAL -0.7694

-3.4635 -0.1216

-2.3234

-1.4828 -3.3945 -0.1088

-2.2472

-0.2495 -0.8627 -0.0773

-3.5709

-0.4160 -3.1525 -0.0841

-4.3992

-0.2009 -1.1252 -0.0707

-3.3885

-0.2072 -1.2000 -0.0546

-3.0977

INFL 1.1970

2.9732 0.0220

3.3119

0.1729 0.4817 0.0264

4.3404

0.5794 1.3678 0.0513

2.6393

0.4566 3.2595 0.0777

3.6482

0.3657 1.3404 0.1143

2.3544

0.3016 1.4596 0.1210

2.5505

FFR 0.3479

3.4162 0.0410 0.7590

0.4804 2.9543 0.0312 0.5676

0.2812 1.1181 0.0115 0.2162

0.2120 0.6424 0.0165 1.5225

0.2359 0.9345 0.0201 0.3506

0.2232 1.1983 0.0103 0.1960

SLOPE -0.4858 0.0001 0.0332 0.3116

-0.8282 -1.5572 0.0219 0.2063

0.1824 0.2433

-0.0100 -0.0929

-0.0210 -0.0570 0.0007 0.0138

0.1801 0.3359

-0.0076 -0.0641

0.1331 0.3120

-0.0205 -0.1875

MKT -0.0083 -0.3423 0.0053 0.6972

-0.0147 -0.6124 0.0032 0.4049

-0.0534 -1.8196 0.0018 0.2357

-0.0529 -3.0180 -0.0011 -0.1427

-0.0391 -1.3977 -0.0017 -0.2158

-0.0382 -1.4085 -0.0066 -0.8390

VIX 0.1257

3.1378 0.0220

2.6260

0.0348 0.5910 0.0219

2.4211

0.0935 2.5380 0.0203

2.4378

0.1042 6.8452 0.0185

3.2424

0.0800 3.5152 0.0207

2.4057

0.0725 3.6144 0.0194

2.4031

Table A. 19: Markov regime switching model coefficients for BB credit spreads on regressors

116

Maturity lYr

2Yr

3Yr

5Yr

7Yr

lOYr

Regime 1

2

1

2

1

2

1

2

1

2

1

2

Adj. R2

0.9611

0.9618

0.9687

0.9709

0.9824

0.9698

Const -0.0253

-2.3286 -0.0843 -1.0675

-0.0212 -2.0132 -0.1067 -1.1466

-0.0124 -1.3282 -0.3438

-25.5459

-0.1954 -9.5151 -0.0100 -0.8703

-0.0096 -1.0593 -0.0092 -1.1265

-0.0089 -0.9056 -0.0650 -0.6691

CSi_i 0.8979

21.9418 0.3132

3.0986

0.9113 21.1380

0.2731 2.5599

0.9408 24.5209

0.4795 6.2758

0.9759 5.4522 0.9329

20.8274

0.9644 20.8050

0.6459 6.9774

0.9206 16.8215

0.6491 4.5444

REAL -0.0501 -0.3604 -1.8640 3.0567

-0.0591 -0.4096 -1.3088

-3.5667

-0.0822 -0.5936 -1.5633

-2.7628

-0.6050 -0.7562 -1.0260

-3.2106

0.0934 0.7560

-0.9915 -4.2432

-0.0250 -0.1718 -1.3424

-2.7937

INFL 0.3624 1.8715 0.9902

2.9415

0.3159 1.6842 1.3848

3.2254

0.2086 1.2413 0.6188

2.6402

3.5731 4.0761 1.1537 0.7742

0.1347 0.8245 0.6175

3.0476

0.1582 0.8673 0.6915

2.8570

FFR 0.2915

2.0751 1.3484

2.0404

0.2526 1.8128 1.2990 1.1634

0.1712 1.4262 5.2149

10.5672

1.2886 2.0295 0.1007 0.7984

0.0135 0.1131 2.3582

12.3867

0.0833 0.5957 1.4455 1.0891

SLOPE 0.1520 0.5732 2.6881 0.0023

0.0732 0.2830 3.2497 1.1338

-0.1116 -0.5060 10.8152 5.8239

1.0215 1.1842

-0.0838 -0.3556

-0.0291 -0.1565 -4.0918

-6.9775

-0.0263 -0.1187 0.7971 0.2578

MKT -0.0249 -1.1715 0.1615 0.0001

-0.0234 -1.1004 0.1393

2.1321

-0.0130 -0.7045 0.0003 0.0067

-0.3860 -4.4033

0.0138 0.7372

0.0199 1.4898

-0.0119 -0.2004

0.0189 1.1669

-0.3371 -6.9086

VIX 0.0541

2.5139 0.3309

2.9654

0.0471 2.1072 0.3811

3.5278

0.0322 1.6731 0.4379

6.2480

0.0346 1.2830 0.4443

5.7224

0.0338 1.3850 0.1923

3.0058

0.0332 1.2953 0.1941

2.8636

Table A.20: Markov regime switching model coefficients for B credit spreads on regressors

117

A.5 Robustness Check: Tests with Moody's Data

Full Sample May 1994 to Aug 1998 Sep 1998 to Mar 2003 Apr 2003 to Jun 2008

Aaa -1.73861 -0.62319 -2.10791

-2.74863

Baa -1.59253 -0.51291 -0.63282

-3.90358

Table A.21: Phillips-Perron test results for Moody's Corporate Credit Spread series

Ratings

Aaa

Baa

Adj. i?2

0.9443

0.9643

Const CSt-i

•0.0012 0.8657

-1.3030 22.7058

0.0014 0.8638

1.2205 23.9641

REAL INFL

-0.0211 0.0256

-1.5395 2.5044

-0.0513 0.0149

•3,0558 2.8452

FFR SLOPE

0.0227 0.0232

1.9442 0.9158

0.0110 -0.0221

0.9149 -0.8479

MKT VIX

-0.0049 0.0083

•2.3914 3.7454

-0.0049 0.0090

•2.2737 3.9728

Table A.22: OLS model coefficients for Moody's corporate credit spread series

Rating

Aaa

Baa

Regime

1

2

1

2

Adj. #

0.9732

0.9837

Const C$i-\

-0.0003 0.9574

-0.3461 17.9843

-0.0049 0.3773

-1.6940 2.3676

0.0011 0.9104

0.8123 37.6073

0.0002 0.8439

0.7645 2,1435

REAL INFL FFR SLOPE

-0.0232 -0.0029 0.0219 0.0227

-1.3041 -0.3056 2.8222 1.3478

-0.0290 0.1864 0.0625 0.0887

•2.5929 3,8772 1.2291 0 . 1 3

-0.0235 0.0034 0.0153 0.0115

-1.4174 0.2316 1.4196 0.4743

-0.1002 0.0544 0.0125 -0.0710

-7.4412 2,7083 0.4323 -1.3018

MKT VK

-0.0038 0.0008

-2,0678 0.3148

-0.0174 0.0343

•2.5619 6,8289

-0.0037 0.0005

-1.6102 0.4048

-0.0021 0.0214

-0.5384 7,5534

Table A.23: Markov regime-switching model coefficients for Moody's corporate credit spread series

118

A.6 Figures

Lehman 5-Year Credit Spread Data, 5/1994 - 6/2007

•AAA

-AA

A

BBB

-BB

•B

- Regime Shift

993-D1-31 1995-10-28 1998-07-24 2001-04-19 2004-01-14 2006-10-10 2009-07-06

Date

Figure A.l: 5-year credit spreads and possible break dates, May 1994 to June 2007

Volatility Factor

50.00% -J

45.00% -

40.00% -

35.00%

30.00%

0)

=j 25.00%

20.00% -

15.00% -

10.00%

5.00%

0.00%

•VIX

1993-01-31 1995-10-28 1998-07-24 2001-0419 2004-01-14 2006-10-10 2009-07-06

Date

Figure A.2: VIX, May 1994 to June 2007 to

o

Markov Smoothed Probabilities -5 Year A Credit Spreads

60% A

40<M

fl»N

1994-06 1995-10 1997-03 1998-07 1999-12 2001-04 2002-09 2004-01 2005-05 2006-10

Date

Figure A.3: Smoothed regime probabilities, 5 year A credit spreads

Appendix B

Chap. 2 Proofs, Tables, and

Figures

B.l Credit Spread Expression

As stated earlier, the price of a one-period default-risky bond can be expressed as

B = -^~ (B.l)

where B is the risky bond price, c is the face value it pays upon maturity, r is the risk-free rate, and s is the credit spread, representing the extra yield demanded by investors for a default-risky bond over the risk-free rate.

The default-risky bond can also be priced as a contingent claim, contingent on the default event. Along the path with no default, the bond simply returns the face value c, while, along the path with default, the bond returns rA where r again is the risk-free rate and A is the current recovery value of the bond upon default. The Arrow-Debreu price of the default event is qo and, accordingly, the A-D price of survival is (1 — g£>). The bond price is, therefore, the sum of the payoffs along different events paths discounted by the risk-free rate or

B = -(l-qD) + qDA (B.2)

Setting the two bond pricing expressions equal and performing some algebra, we find

122

123

that the expression of the spread in terms of the bond pricing parameters is

r W £ -A) , N

s = qDK\ >—- B.3 c(l - qD) + rqDA

We define the loss given default as percentage of the value loss upon default. From our definition of the payoffs under different states, the loss given default / is

- — A I = r—r- (B.4)

Substituting this expression into the expression for credit spreads B.3, we obtain

s = rT^ (B-5)

B.2 The Market Price of Risk and the Risk-Neutral

Measure

To price a contingent claim, which is how we evaluate a corporate bond to determine its spread, we must evaluate the expectation of its contingent payment at a future date under the risk-neutral measure, an approach underlying most of continuous-time finance. Since the corporate bond we are pricing depends upon the firm's cash flow, which we model as a primitive process, we must simply determine the expectation of future firm cash flows under the risk-neutral or Q measure. Pricing under the risk-neutral measure simplifies to this relationship:

y(0) = e-rTEq[V(T)}

To evaluate the price of a bond, we must simply therefore evaluate its payoff along different possible paths or scenario and discount back by the the risk-free rate of the appropriate maturity under the risk-neutral measure. We must then, therefore, determine how to model the price process V(t) under the risk-neutral measure to simulate its paths. The risk-neutral measure is an equivalent martingale measure under which every price process discounted by the price of a risk-free bond is a martingale.

B(0) L5(T)J

where B{T) is the payoff of a risk-free bond and B(0) = 1.

124

If we assume that the price process V(t) is

dV{t)

V(t) = fj,(t)dt + a(t)dW{t)

under the real measure, then, under some technical conditions, V(t) has the following representation under the risk-neutral measure by Girsanov's Theorem

^=IM(t)dt + *(t)dW*(t)

where dW(t) = dWq{t) - fj,(t)dt

The term //(£) that adjust the general price process to the risk-neutral measure is called the market price of risk.

Since we have an economy with explicit explicit preference specifications, we can directly specify the market price of risk, which adjust the price process to the risk-neutral measure. As derived by Lettau and Uhlig (2002), the market price of risk in the case of a power utility function is simply the relative risk aversion coefficient, 7. Therefore, in our model, we try to price the firm's cash flow, which under the real measure, is postulated to be

9K (t) = Q9t + 6 + P^Kef + aKy/l - p2e

where gt represents the growth in aggregate output, <?#•(£) is the growth of the marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggregate output growth, £f is the mean of firm-specific cash flow growth, p is the correlation of output growth and firm-specific cash flow growth, &K is volatility of firm cash flow growth, and ef, ef ~ iV(0,1) independent of each other.

Under the risk-neutral measure, the cash flow growth process is

9K{t) = Q9t + & - Ipo-c^K + PO-RC? + o-K^l-p2ef (B.6)

where ac is the volatility of consumption growth.

After simulating the firm's cash flow growth process under the risk-neutral measure, we can then price contingent claims on the firm's cash flow by taking an expectation along different simulated paths and discounting by the risk-free rate of the appropriate tenor.

B.3 Regression Test Coefficients

125

Ratings

A A A

AA

A

BBB

BB

B

Maturity

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.017142

0.018305

0.090266

0.054131

0.050117

0.113329

0.084525

0.069319

0.158236

0.087802

0.156368

0.283167

0.275245

0.467394

0.617847

0.559428

0.712586

0.773477

Const

2.87E-06 2.196326 0.000295 2.215974

0.006769 7.610134

0.000315

5.251523 0.001887 4.811875 0.009657 9.214268 0.000946 7.499122

0.003163 6.140926 0.016463

12.31004 0.001032 7.7634

0.011618 12.0407

0.042143 21.51603 0.012779 21.52676

0.082723

39.71867 0.203372 69.58464

0.067078 54.96241

0.230885 95.81524 0.435189

157.4966

9t -0.00146 -13.3039 -0.1533 -13.7501

-2.33965

-31.4085 -0.12024

-23.9199 -0.75406 -22.9641

-3.12569 -35.6131 -0.32034 -30.3242

-1.17552

-27.2491 -4.83134

-43.1397 -0.34462

-30.9539 -3.46299 -42.8569 -10.2466 -62.4687 -3.04663

-61.2861

-16.2265 -93.0344

-30.8633 -126.1 -11.433 -111.866 -31.5019 -156.108 -42.3299

-182.933

7T*

3.47E-05 1.166342 0.003752

1.240606

0.026477

1.310343

0.002017

1.479549 0.011683

1.311676 0.029843 1.253482 0.004366 1.523528 0.016349

1.39712

0.036307

1.195129 0.004484

1.484781 0.030112 1.373804 0.060309 1.355437 0.022057 1.635694

0.073162

1.546388 0.04901

0.738193 0.042788 1.543376 0.032217 0.588549 -0.07928

-1.263

Table B . l : Regression of forward-looking default probabilities on contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.042197

0.031394

0.223458

0.112457

0.120064

0.256394

0.16354

0.164427

0.300232

0.166849

0.289734

0.383985

0.346449

0.507206

0.639084

0.533304

0.69762

0.821846

Const 0.00000 1.44194 0.00001 1.37358 0.00005

2.98259 0.00003

2.78614 0.00005

2.16652 0.00007

3.57753 0.00009

3.53017 0.00008

2.48007 0.00013

4.91325 0.00010

3.63704 0.00024

4.27772 0.00038

9.56764 0.00106

7.75085 0.00220

16.95086 0.00264

35.36535 0.00631

18.72564 0.00802

44.01298 0.00851

87.37827

S t - l

0.16272 16.54662

0.12945 13.09460

0.39391 43.67327

0.26189 27.42157

0.28919 30.47670

0.41306 46.49036

0.31540 33.78856

0.33998 36.60994

0.42304 48.46917

0.31735 34.03920

0.42759 48.89528

0.39948 47.15600

0.41283 48.08396

0.38381 48.17375

0.28667 38.93279

0.40109 51.32028

0.27398 39.50623

0.14947 25.92743

9t -0.00024

-11.42325 -0.00703

-11.24655 -0.03296

-21.44806 -0.01817

-18.10075 -0.03241

-16.27113 -0.04298

-23.98976 -0.04640

-21.45246 -0.04879

-18.47166 -0.06616

-28.84779 -0.04984

-21.78441 -0.13251

-26.98801 -0.14639

-41.53342 -0.42470

-35.88039 -0.65933

-56.88010 -0.55032

-82.89121 -1.73796

-58.20048 -1.50395

-94.12992 -1.10617

-141.36248

V"i

0.00001 1.45641 0.00025 1.47748 0.00116

2.85037 0.00059

2.20844 0.00111

2.06982 0.00148

3.12276 0.00151

2.61880 0.00172

2.43838 0.00214

3.56899 0.00160

2.62878 0.00457

3.55034 0.00397

4.44414 0.01359

4.46775 0.01650

5.88127 0.00961

6.39214 0.04699

6.54332 0.02429

6.88394 0.00853

5.27901

Table B.2: Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.00205

0.005215

0.018033

0.115046

0.022935

0.033709

0.151231

0.037002

0.083682

0.169208

0.115569

0.23907

0.348856

0.410843

0.555764

0.542525

0.649196

0.821513

Const

0.00000 1.30401 0.00000 0.58801 0.00000 1.08265 0.00000

3.07643 0.00000 1.26535 0.00000 1.37305 0.00001

4.01071 0.00000 1.52154 0.00001 1.84012 0.00001

4.18434 0.00002

2.50528 0.00005

3.34738 0.00026

9.22445 0.00058

9.10471 0.00119

22.22070 0.00298

22.00356 0.00444

33.27960 0.00712

84.34375

St-l

-0.00231 -0.23130 -0.00064 -0.06400 0.08833

8.88632 0.25518

26.70615 0.10209

10.28920 0.13756

13.93095 0.28985

30.80124 0.14300

14.49859 0.23815

24.67370 0.30882

33.08964 0.27378

28.74265 0.40094

44.75989 0.39719

46.12140 0.43575

52.53087 0.35417

45.59808 0.38599

49.50554 0.33064

45.74079 0.16553

28.73276

9t 0.00000

-4 .83856 -0.00001

-7 .49212 -0.00130

-9.72376 -0.00083

-19.38710 -0.00052

-10.62438 -0.00265

-11.28071 -0.00345

-22.24494 -0.00142

-11.87845 -0.00701

-14.36614 -0.00378

-23.14955 -0.01308

-17.32220 -0.02972

-23.05116 -0.09320

-37.75482 -0.22774

-40.88427 -0.31207

-65.41077 -0.72261

-60.73231 -0.94893

-80.39608 -0.95196

-139.31380

7T*

0.00000 -0.40240 0.00000 1.40593 0.00005 1.38410 0.00003

2.60731 0.00002 1.46330 0.00009 1.47950 0.00011

2.69304 0.00005 1.52520 0.00024 1.84547 0.00012

2.83554 0.00044

2.15005 0.00104

3.06087 0.00284

4.49945 0.00718

5.09333 0.00711

6.30167 0.01896

6.66997 0.01927

7.18517 0.00846

5.98888

Table B.3: Correlation of firm cash flow growth with output growth, rho, is 0.4. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.03706

0.178215

0.328705

0.162892

0.281831

0.349687

0.218963

0.309826

0.389495

0.224123

0.383778

0.478969

0.407213

0.576666

0.684716

0.57112

0.727916

0.820649

Const 0.00001 1.50606 0.00011 2.44078 0.00027

7.28379 0.00024

3.11894 0.00027

4.07312 0.00034

8.48324 0.00047

3.91118 0.00039

4.98127 0.00050

10.87913 0.00049

3.97566 0.00097

8.91332 0.00104

17.59516 0.00284

9.05460 0.00451

25.99404 0.00395

45.58344 0.01175

22.18200 0.01108

53.12692 0.00943

89.12216

St-l

0.14023 14.21548

0.35791 38.84084

0.39793 45.84481

0.32303 34.64891

0.42539 48.46188

0.39196 45.50186

0.36877 40.60484

0.43022 49.57514

0.38028 44.82841

0.37262 41.13031

0.41232 48.80464

0.34596 42.38748

0.43388 52.18846

0.32681 42.49994

0.24286 34.28013

0.37913 49.57889

0.23543 35.03147

0.14109 24.39737

9t -0.00783

-12.19978 -0.06866

-18.59650 -0.11300

-35.26578 -0.13160

-20.42194 -0.15099

-26.11425 -0.13487

-38.36801 -0.24658

-24.06060 -0.19556

-29.28937 -0.17859

-44.04745 -0.25897

-24.37895 -0.38363

-40.20744 -0.29972

-57.13953 -1.11616

-40.66535 -1.09401

-70.57957 -0.71689

-94.52897 -3.06588

-64.92492 -1.87109

-104.03788 -1.19648

-141.90552

TTi

0.00029 1.67373 0.00243 2.46751 0.00314

3.80118 0.00442

2.57208 0.00514

3.38523 0.00359

3.99809 0.00830

3.06130 0.00639

3.66922 0.00451

4.41329 0.00875

3.11448 0.01084

4.47569 0.00688

5.40310 0.03536

5.09020 0.02259

6.23805 0.01016

6.06564 0.07498

6.72432 0.02485

6.36905 0.00844

4.84481

Table B.4: Correlation of firm cash flow growth with output growth, rho, is 0.8. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. i?2

0.041555

0.082425

0.301174

0.156296

0.209678

0.32404

0.215053

0.254157

0.367097

0.220571

0.366856

0.477391

0.414962

0.605007

0.741343

0.605567

0.789841

0.889473

Const 0.00000 1.69631 0.00003

1.85038 0.00013

4.99933 0.00008

3.44792 0.00011

2.93524 0.00018

6.14901 0.00021

4.42583 0.00017

3.59985 0.00032

8.62122 0.00022

4.54415 0.00057

7.03907 0.00085

16.50101 0.00210

10.53445 0.00441

28.25220 0.00469

58.27485 0.01156

26.98781 0.01349

70.59951 0.01225

134.57722

St-l

0.14842 15.07128

0.23611 24.44701

0.42217 48.37649

0.30802 32.88615

0.38083 41.92696

0.41711 48.16728

0.35447 38.85938

0.41009 46.11571

0.40349 47.30912

0.35838 39.38913

0.43513 51.39920

0.36131 44.36626

0.42130 50.69514

0.33002 43.90691

0.21094 31.84955

0.36719 49.18374

0.19851 32.42671

0.08116 17.14338

9t -0.00092

-12.85009 -0.02020

-14.30154 -0.06582

-29.08230 -0.04262

-21.07209 -0.06985

-21.01349 -0.08487

-32.59950 -0.09993

-25.15819 -0.09999

-24.15418 -0.12593

-39.20007 -0.10638

-25.52579 -0.25091

-35.82302 -0.25359

-55.58539 -0.75046

-42.84129 -1.03156

-74.06814 -0.75406

-109.47683 -2.67383

-70.53909 -1.97651

-124.26737 -1.25083

-190.59571

7T*

0.00003 1.67477 0.00069

1.82211 0.00211

3.57256 0.00139

2.57360 0.00247

2.79765 0.00257

3.82024 0.00327

3.11777 0.00350

3.20251 0.00349

4.26610 0.00345

3.13532 0.00791

4.40072 0.00605

5.44081 0.02327

5.27708 0.02119

6.58140 0.00888

5.99604 0.06303

7.15286 0.02125

6.35095 0.00491

3.75175

Table B.5: Mean of idiosyncratic firm cash flow growth is -2% per annum. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. i?2

0.018654

0.013163

0.1416

0.085417

0.061109

0.173962

0.125152

0.092612

0.233257

0.12767

0.211995

0.320636

0.293772

0.429421

0.544614

0.471549

0.610856

0.743263

Const 0.00000 1.83249 0.00000

0.99151 0.00003

2.04210 0.00001

2.39705 0.00002

1.67098 0.00004

2.37455 0.00004

2.90217 0.00004

1.93542 0.00006

3.11309 0.00005

3.03247 0.00012

2.99401 0.00018

5.85804 0.00056

6.10317 0.00113

10.68685 0.00150

22.62192 0.00359

13.73548 0.00472

28.67569 0.00564

57.75592

St-l

0.07496 7.53689 0.06719

6.74536 0.32203

34.34316 0.22824

23.61995 0.19865

20.36621 0.35377

38.31167 0.27719

29.19013 0.25144

26.15718 0.40066

44.62824 0.27907

29.41557 0.38208

42.10649 0.42022

48.49730 0.39817

45.27415 0.41811

50.64848 0.33656

42.85196 0.41842

51.79978 0.33146

44.32775 0.21803

33.05142

9t -0.00007

-11.00451 -0.00213

-9.09225 -0.01760

-16.39968 -0.00789

-16.01982 -0.01431

-13.14509 -0.02330

-18.37644 -0.02210

-18.89270 -0.02356

-14.89360 -0.03614

-22.11405 -0.02425

-19.16329 -0.07305

-21.23237 -0.08482

-31.82255 -0.24485

-31.02896 -0.41640

-44.77182 -0.38863

-65.64723 -1.14118

-49.45279 -1.09896

-74.74829 -0.91379

-109.25395

Kt

0.00000 0.86339 0.00008

1.31240 0.00063

2.18608 0.00025

1.90935 0.00049

1.68388 0.00084

2.46587 0.00074

2.37603 0.00081

1.90978 0.00128

2.95013 0.00078

2.30785 0.00257

2.80818 0.00262

3.79275 0.00794

3.86547 0.01192

5.11799 0.00846

6.03434 0.03371

5.91873 0.02268

6.68659 0.01123

6.24319

Table B.6: Mean of idiosyncratic firm cash flow growth is 2% per annum. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

131

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0

0

0

0

0

0

0

0

0.001837

-0.00028

0.00286

0.014154

0.021283

0.152485

0.38981

0.180878

0.458793

0.73678

Const 0.00000 0.06459 0.00000

-0.52709 0.00000 1.54456 0.00000 0.06459 0.00000

-0.52709 0.00000 1.54456 0.00000 0.06459 0.00000

-0.52709 0.00000

-0.10960 0.00000 0.06459 0.00000 0.20189 0.00000 0.98028 0.00000 1.20921 0.00007

2.38173 0.00041

9.93457 0.00011

3.73934 0.00147

12.85370 0.00496

56.58320

St-l

-0.00846 -0.84555 -0.01557 -1.55721 0.00523 0.52244

-0.00846 -0.84555 -0.01557 -1.55721 0.00523 0.52244

-0.00846 -0.84555 -0.01557 -1.55721 -0.00169 -0.16910 -0.00846 -0.84555 0.00952 0.95218 0.07587

7.62187 0.09200

9.26453 0.32783

35.09913 0.39748

47.02724 0.33151

35.79998 0.40584

49.78090 0.21755

32.66451

9t 0.00000 0.66683 0.00000

-0.84454 0.00000 1.20864 0.00000 0.66683 0.00000

-0.84454 0.00000 1.20864 0.00000 0.66683 0.00000

-0.84454 -0.00001

-4.65627 0.00000 0.66683

-0.00001 -5.56618

-0.00259 -8.91175

-0.00025 -10.82439

-0.04365 -17.82018

-0.15290 -42.40106

-0.05705 -22.47793

-0.49840 -49.28117

-0.81050 -107.73689

TTi

0.00000 -0.09152 0.00000 0.78720 0.00000

-0.90215 0.00000

-0.09152 0.00000 0.78720 0.00000

-0.90215 0.00000

-0.09152 0.00000 0.78720 0.00000 1.46822 0.00000

-0.09152 0.00000 1.30273 0.00010 1.24774 0.00001 1.65889 0.00153

2.33230 0.00411

4.51298 0.00188

2.78540 0.01353

5.41124 0.00992

6.11942

Table B.7: Standard deviation of idiosyncratic firm cash flow growth is 10% per annum. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

.4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.348073

0.422699

0.516314

0.482626

0.489582

0.536023

0.523152

0.513814

0.569234

0.526489

0.584488

0.635125

0.644904

0.725423

0.770465

0.750356

0.824682

0.874056

Const 0.00105

7.74355 0.00104

10.26658 0.00117

19.91324 0.00398

14.48589 0.00191

15.37876 0.00134

21.90000 0.00576

17.78214 0.00233

17.59787 0.00168

25.59159 0.00593

18.07008 0.00390

25.38290 0.00258

34.72173 0.01576

32.92125 0.00948

50.43988 0.00614

66.89640 0.03369

56.23516 0.01672

83.85174 0.01168

117.69324

St-l

0.41458 48.33843

0.42010 50.73904

0.34661 43.42116

0.41630 51.83482

0.39196 48.76824

0.33794 42.79496

0.40465 51.48360

0.38003 47.85739

0.32320 41.75364

0.40395 51.49714

0.34331 45.05496

0.28901 39.12642

0.34449 47.55924

0.25419 37.82228

0.19079 30.06758

0.26759 41.32909

0.17923 31.41637

0.11431 22.87314

9t -0.42364

-35.88283 -0.39156

-43.80790 -0.32157

-61.37375 -1.23515

-50.90975 -0.59895

-54.09577 -0.35356

-64.48437 -1.62756

-56.67173 -0.68445

-58.01202 -0.41059

-69.90363 -1.66240

-57.13202 -0.96238

-70.09128 -0.54161

-82.08076 -3.29695

-78.35083 -1.65623

-101.67278 -0.91989

-119.16066 -5.34419

-105.94939 -2.24021

-139.17448 -1.29049

-173.95372

7T<

0.01358 4.47569 0.01129

5.03185 0.00728

5.79590 0.03599

6.03739 0.01546

5.72259 0.00776

5.95455 0.04479

6.45656 0.01698

5.96411 0.00845

6.14832 0.04545

6.47561 0.02084

6.49765 0.00956

6.38605 0.07087

7.38659 0.02420

6.82404 0.00956

5.84455 0.08224

7.55939 0.02103

6.31661 0.00709

4.75041

Table B.8: Standard deviation of idiosyncratic firm cash flow growth is 40%. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.04082

0.030041

0.258087

0.109594

0.117622

0.290317

0.159822

0.161451

0.331832

0.163044

0.284537

0.412879

0.338857

0.494887

0.66363

0.520911

0.682627

0.835956

Const 0.00000 1.93416 0.00001

1.84743 0.00007

3.97320 0.00004

3.55139 0.00006

2.84411 0.00010

4.77133 0.00011

4.47410 0.00010

3.28674 0.00017

6.51569 0.00012

4.59833 0.00030

5.59189 0.00049

12.29130 0.00123

9.51106 0.00241

19.45447 0.00315

43.14372 0.00691

21.48139 0.00831

47.36447 0.00947

102.00524

St-l

0.16324 16.59210

0.12995 13.13896

0.43744 49.56118

0.26311 27.51942

0.29025 30.56029

0.45369 52.19500

0.31704 33.91441

0.34132 36.71132

0.45892 53.69133

0.31904 34.16918

0.43011 49.06537

0.42621 51.24858

0.41630 48.29875

0.38952 48.45847

0.29929 41.68548

0.40664 51.55205

0.28029 39.70693

0.15539 28.04139

9t -0.00023

-10.80949 -0.00664

-10.64041 -0.03169

-20.54264 -0.01724

-17.18257 -0.03065

-15.39770 -0.04205

-23.12716 -0.04405

-20.36310 -0.04613

-17.46725 -0.06633

-28.13484 -0.04734

-20.68125 -0.12553

-25.51270 -0.15263

-41.44285 -0.40484

-34.03017 -0.63418

-53.97214 -0.58006

-84.32254 -1.67119

-55.15725 -1.46791

-89.30982 -1.12385

-145.57633

n 0.00001 1.00212 0.00017

1.04969 0.00078

1.93390 0.00040

1.52901 0.00077

1.46058 0.00096

2.03675 0.00100

1.76177 0.00117

1.69403 0.00132

2.19889 0.00105

1.75301 0.00285

2.24900 0.00248

2.73385 0.00831

2.77328 0.00968

3.48647 0.00483

3.23740 0.02740

3.85266 0.01369

3.88780 0.00144

0.94689

Table B.9: Coefficient of relative risk aversion is 10. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.050654

0.066978

0.290813

0.176079

0.184416

0.317123

0.2378

0.232991

0.353546

0.239485

0.353737

0.445028

0.419083

0.568108

0.701435

0.590364

0.751557

0.861449

Const 0.00000 2.26172 0.00002

2.10770 0.00010

4.77161 0.00005

4.11692 0.00008

3.20022 0.00013

5.77575 0.00014

5.17776 0.00013

3.80689 0.00023

7.95154 0.00015

5.23310 0.00042

6.89298 0.00065

14.97173 0.00166

11.41434 0.00351

25.94451 0.00390

51.63829 0.00948

26.91699 0.01139

63.83647 0.01084

119.04969

St-l

0.18215 18.58946

0.21845 22.47525

0.45415 52.26369

0.34720 37.53992

0.37282 40.64128

0.45978 53.50097

0.39774 44.31802

0.41497 46.37385

0.45124 53.15288

0.39786 44.35807

0.47029 55.71749

0.41088 50.02530

0.45913 55.82789

0.37796 49.24574

0.27102 39.18422

0.41034 54.72055

0.24393 37.62282

0.12884 24.93113

9t -0.00033

-11.84106 -0.00856

-12.16389 -0.04179

-23.13698 -0.02241

-19.41962 -0.03936

-17.32304 -0.05528

-26.06877 -0.05699

-22.96021 -0.05954

-19.78898 -0.08633

-31.81687 -0.06153

-23.20281 -0.16591

-29.56496 -0.19088

-46.53301 -0.52075

-38.59808 -0.82028

-63.62671 -0.65730

-93.65968 -2.09670

-63.10480 -1.73787

-107.19658 -1.18050

-162.52765

VTi

0.00001 0.97215 0.00023

1.22064 0.00095

2.03961 0.00052

1.70730 0.00102

1.70700 0.00115

2.11950 0.00127

1.96916 0.00150

1.91431 0.00159

2.32285 0.00140

2.03427 0.00361

2.53074 0.00296

2.97533 0.01051

3.14794 0.01099

3.71943 0.00444

2.97875 0.03069

4.04240 0.01088

3.24395 0.00025

0.17445

Table B.10: Coefficient of relative risk aversion is 25. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

135

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4Yr

10 Yr

Adj. i?2

0.005629

0.010807

0.080781

0.032977

0.03693

0.101378

0.056161

0.051961

0.1447

0.057537

0.120477

0.260458

0.180275

0.360645

0.543398

0.332238

0.539713

0.575723

Const 0.00000 1.10086 0.00003

1.40878 0.00024

6.84047 0.00010

3.74530 0.00017

3.58998 0.00035

8.13129 0.00030

5.60783 0.00029

4.75523 0.00061

10.87625 0.00033

5.73514 0.00118

9.46569 0.00175

19.09981 0.00424

14.62709 0.00983

27.13682 0.00916

46.91798 0.02404

26.49322 0.02866

46.88648 0.02622

53.62860

S t - l

0.00006 0.00585

-0.00016 -0.01652 0.01098 1.14512 0.00041 0.04176 0.00037 0.03753 0.01451 1.53168 0.00164 0.16878 0.00187 0.19251 0.01636 1.77064 0.00188 0.19351 0.01087 1.15886 0.01540 1.78637 0.00725 0.79962 0.01266 1.57624 0.00312 0.45732 0.00937 1.14293 0.00446 0.65089 0.00573 0.87201

9t -0.00024

-7.70920 -0.00518

-10.56861 -0.02523

-29.21094 -0.01233

-18.34578 -0.02217

-19.47344 -0.03408

-33.05715 -0.03149

-24.12512 -0.03473

-23.19860 -0.05438

-40.41228 -0.03380

-24.42821 -0.10954

-36.40920 -0.12523

-57.98687 -0.31835

-45.99658 -0.61633

-73.25055 -0.45951

-105.97595 -1.45442

-68.93054 -1.42887

-105.31179 -1.20943

-113.30136

7T<

0.00002 0.65732 0.00053 1.13786 0.00102 1.24637 0.00080 1.25160 0.00166 1.53826 0.00128 1.31229 0.00168 1.35531 0.00217 1.53007 0.00174 1.35916 0.00175 1.33449 0.00392 1.37471 0.00119 0.58082 0.00776 1.18154 0.00224 0.27996

-0.00371 -0.89472 0.01714 0.85395

-0.00575 -0.44375 -0.00404 -0.39583

Table B . l l : Coefficient of monetary policy smoothing is 0.9. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

8 ^ CD P rt cT

Co I CL tO CO ' '

g H P P & • %

O ST o >-i

5. " St O 2 CD P 2-so a

O c+

c i— e-t- CO

era o

P CD

CO

EC P

§• 8 , o

CD

CD P a co o

o cp

P

P cm era

CD

O O

,0£Z

~J CD

O 4^ Cn i—1

00 -J

I

M O • 39

1

#». to

o b to -a oo o

I—1

o

i-i

p Cn cn m oo 4^ oo

o .023

Cn O

o b o oo o 00

i H .225

4^ 00

O

b o o 1—' CD

on oo 7361

o CO

o 4^ bO ai CD 4 -

HJ

O Or • 892.

* » •

-i

o b 4^ C35 00 -J

4^

i-i

p Cn CO Cn to I—'

CO

o .026

co C5

o b o to CD OO

i h—» .496

Oi 03

p b o o 4^ o

CO h-»

• 185;

CO -J

p Cn CO to o o

1

CJ CD • 33

1

14 ^ i

O Cn i—1

CD oo o

tt)

1—'

tf p CO

to CD -a 4^ -a

o .024

K Cn

p b o 4^ oo Ci

i i—> .590

oi H

I

O b o -a i—* CO

at CO 978!

o h-»

O I—1

4^ 4 > oo oo

M

o -J 553;

CO to

o b to 00 CD to

h-»

o

p cn 4^ to CD Cn co

o

800'

4^ oo

p b o o co CD

i

p 4^ oo

o OS

O

b o o o oo

00 © 764

#>. o

o bo 4^ oo co 4^

i -J to • 799

to ^

p i—1

co to -J Oi

4^

i-S

O

co Cn to 00 -a Oi

o .009

05 o

p b o Oi 00 Cn

i

o b -a 4^ CO to

p b o o -Q 4^

H-» 00 181

CO -1

p to CD Cn Cn to

, ^ P

£86

CO co

1

p b ~a co to to

cd td

I—1

tf p 1—»

oo co oo o Cn

O .004

Cn O

O

b o to 05

-a

i

o .348

CO - i

p b o co o 4^

to to • 023

-* to

p Cn 4^ CO CO 4^

, CN 00 •

9S0

~ j

cn

o to oo -a -a - j

H

o

tf p to Cn 05 CO 00 I—1

O b o i—» - j OS

o b o 4^ Oi CD

i

O h- 1

OO oo oo CO

p b o o 4 to

M to • 351i

OS 00

p Cn 00 O 4^ -a

i CO ^1 139

00

o 1

p Cn

co -a CD O

4^

p t—'

to CO to CO co

p b o CO Cn

O

b O Cn 4^ to

i

p I— 1

to to CO I—1

1

p b o i—1 o -J

00 613

co 00

i

o b to Cn 00 O

I

to Cn 8

74

00

o i

i — i

b co h — i

o OS

Cd Cd Cd

i—1

tf o b C35 CO to -<l •f^

o

000'

^ o 1

o b o o to Ol

1

o b co H-1

co

1

p b o o oo -J

M 00 838:

to <i

K^

b 05 to I—1

Cn

! ^ M •

908

00 CO

1

p h-'

oo M Cn i—1

i—» O

tf O I—1

•1^ oo Cn CD -~a

o

000'

ai oo

p b o CD -a oo

i

p b 1 — '

a> K

o b o o i—1 C5

<I 2871

~* CO

i

p b H CO O Cn

I

to oo 821;

CO hi

1

o bo Cn oo oo o

rf^ < I-i

o b Cn 4^ I—» 4^ 4^

O

000'

oo oo t

o b o o h-' 00

1

o b oo Cn co

i

p b o o oo H

00

002

00 M

O

b 1—J

Cn Cn -a

, to ^ 879

on 00

i

o CO C75 to co oo

>

H

O

b Cn 00 -J Oi 00

o

ooo-

co -CI

o b o o H Cn

i

o '.033

- j

oo

p b o o -a CO

M © 888

i—1 CO

i—i

b CD to o oo

• 00

^ 003

CO 05

I

p Cn -J GO 00 oo

I—1

O

i-i

O i—1

O Cn co i—1

CD

O

000'

4^ I—1

p b w O to co

i

o b CO oo to CD

O

b o o 00 co

on 965

00 M

i

O I—1

o -J as co

i M CO 9

22

00 05

O CD 4^ to Oi Cn

4^

*

O

b CO oo to oo oo

o

000'

to co i

o b o H o 03

1

o .023

co Oi

o b o o OJ Oi

p 108:

to CO

1

o b Cn OJ Cn co

i M co 1

58

to CO

i

I—1

O oo Cn I—>

> >

h-'

^

O

b co Cn 4^

O

000'

h-'

#=-1

p b o o Cn Oi

i

O (.012

oo as

i

p b o o 4^ Cn

CO 376

H^ 00

p ^ CD Cn h-»

to

to co

026

to -1

1

p ^ Cn -Q Cn CO

i—1

o

i-i1

O

b oo CO to Cn CO

O

000'

to co

p b o - j Cn CO

i

o b to oo i—»

o

1

p b o o 4^ co

00 024

00 l-»

1

p b Oi oo Cn Cn

! M O

9S2

to 00

1 I— 1

b CO Cn 4^ i—1

4^

i-i

O

b o co CD oo 00

o .000

o Cn

i

o b o o a> oo

i

o b o Cn O Cn

©

b o o co CO

to bo M I— 1

o oo i

o b oo OJ o -a

i KI 3

20

cn CO

i

H

t—»

co 4^ -a o

> > >

i—1

tf o b o 4^ CO oo Cn

o

000'

o o

o b o o co OJ

1

o b o o to 4^

i

O b o o o to

Rati

p TO cn

vlatur

>

S3 to

Cons

c-t-

C/i I

£

co 0 5

137

B.4 Impulse Response Functions

138

Cons, growth (quarterly)

10 11 12

Figure B.l: Impulse response of macroeconomic conditions to adverse technology/productivity shock

139

AAA1 Yr M l Yr A1 Yr

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12

BBB 1 Yr BB1 Yr B1 Yr

2 4 6 8 10 1 2 4 6 8 10 1: 2 4 6 8 10 12

Figure B.2: Impulse response of 1 year credit spreads to adverse technology/productivity shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters

140

AAA 4 Yr AA4 Yr A4 Yr

0.1

2 4 6 8 10 12 2 4 6 8 10 12 0.05

2 4 6 8 10 12

BBB 4 Yr BB4 Yr B4 Yr

0.1

0.05 0.05 0.05 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12


141

AAA 10 Yr AA10 Yr A 10 Yr 0.07

0.06

0.05

0.04 0.04 0.04 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12

BBB 10 Yr BB 10 Yr 0.08 r-—•—• 1 0.08

B10Yr

0.07

0.06

0.05

0.07

0.06

0.05

0.08

0.07

0.06

0.05 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12


142

-0.005 h

Cons, growth (quarterly)

6 7

Inflation

5 6 7 8

Interest Rate

12

10 11 12

Figure B.5: Impulse response of macroeconomic conditions to positive monetary policy shock

143

1 0 i \AA1 Yr x10"4AA1 Yr x K f A I Y r

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12

x 10-£BB 1 Yr BB 1 Yr 0.03, , 0.06

0.02

B1 Yr

0.01

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12

Figure B.6: Impulse response of 1 year credit spreads to positive monetary policy shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters

144

<I0"&AA4 Yr AA4 Yr A4 Yr

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12

BBB4 Yr BB4 Yr B4 Yr

0.02

0.01

0.04

0.02

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12


145

A M 10 Yr AA10 Yr 0.02

0.015

0.01

0.005

0.02

0.015

0.01

0.005

2 4 6 8 10 12

0.02

0.015

0.01

0.005

n

A1 0 Yr

•

•

2 4 6 8 10 12 2 4 6 8 10 12

BBB 10 Yr 0.02

0.015

0.01

0.005

0.02

0.015

0.01

0.005

n

BB 10 Yr

\

\

\

B10Yr 0.02

0.015

0.01

0.005

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12


Appendix C

Chap. 3 Proofs and Tables

C.l Market Price of Risk

To price a contingent claim, which is how we evaluate a corporate bond to determine its spread, we must evaluate the expectation of its contingent payment at a future date under the risk-neutral measure, an approach underlying most of continuous-time finance. Since the corporate bond we are pricing depends upon the firm's cash flow, which we model as a primitive process, we must simply determine the expectation of future firm cash flows under the risk-neutral or Q measure. Pricing under the risk-neutral measure simplifies to this relationship:

V(0) = e-rTEq[V(T)]

To evaluate the price of a bond, we must simply therefore evaluate its payoff along different possible paths or scenario and discount back by the the risk-free rate of the appropriate maturity under the risk-neutral measure. We must then, therefore, determine how to model the price process V(t) under the risk-neutral measure to simulate its paths. The risk-neutral measure is an equivalent martingale measure under which every price process discounted by the price of a risk-free bond is a martingale.

v(o) -YM- E®\viT)] V{0) ~ B(0) ~ E [B(T)1

where B{T) is the payoff of a risk-free bond and B(Q) = 1.

146

147

If we assume that the price process V(t) is

dV(t)

V(t) fi(t)dt + a(t)dW(t)

under the real measure, then, under some technical conditions, V(t) has the following representation under the risk-neutral measure by Girsanov's Theorem

^ | = »{t)dt + a(t)dW®(t)

where dW(t) = dWQ(t) - n(t)dt

The term p,(t) that adjust the general price process to the risk-neutral measure is called the market price of risk.

Since we have an economy with explicit explicit preference specifications, we can directly specify the market price of risk, which adjust the price process to the risk-neutral measure. As derived by Lettau and Uhlig (2002), the market price of risk in the case of a power utility function is simply the relative risk aversion coefficient, 7. However, under internal habit persistence, the market price of risk actually equals the elasticity of the pricing kernel with respect to consumption innovations. Lettau and Uhlig (2002) derive this to be, under the assumption loglinearity of the pricing kernel and lognormality of all relevant random variables,

7 1 + f3b2e~9(l+^

where g is the steady-state growth rate of consumption, x is the steady-state ratio of habit to consumption, and b is the coefficient of the previous period consumption in the habit specification in the utility function.

In our model, we try to price the firm's cash flow, which under the real measure, is postulated to be

9K (t) = Q9t + it + P°K<^ + VKV1 - P 2 e f

where gt represents the growth in aggregate output, gx(t) is the growth of the marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggregate output growth, £t is the mean of firm-specific cash flow growth, p is the correlation of output growth and firm-specific cash flow growth, GK is volatility of firm cash flow growth, and ef, ef ~ iV(0,1) independent of each other.

Under the risk-neutral measure, the cash flow growth process is

9K(t) = ggt + 6 - r,™cpacaK + paKef + aKyJ\ - p2ef (C.l)

148

where oc is the volatility of consumption growth and r\^'ic reflects the elasticity of the pricing kernel to innovations in consumption growth.

After simulating the firm's cash flow growth process under the risk-neutral measure, we can then price contingent claims on the firm's cash flow by taking an expectation along different simulated paths and discounting by the risk-free rate of the appropriate tenor.

C.2 Regression Test Coefficients

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.017909

0.015756

0.076977

0.056171

0.043155

0.098818

0.084729

0.060301

0.142128

0.087829

0.142407

0.265857

0.270307

0.452774

0.607937

0.549468

0.703733

0.770879

Const 0.00000 2.92947 0.00020 2.52844 0.00589

7.68930 0.00027

6.41854 0.00152

5.03862 0.00860

9.40121 0.00083

8.71071 0.00267

6.41985 0.01514

12.66543 0.00090

8.93561 0.01063

12.58538 0.04057

22.39260 0.01226

23.71492 0.08210

41.90582 0.20593

73.58938 0.06670

58.88270 0.23354

101.13211 0.44026

167.27936

9t -0.00081

-13.61381 -0.09058

-12.76875 -1.95698

-28.92104 -0.08987

-24.45237 -0.56715

-21.30579 -2.67709

-33.14090 -0.25573

-30.46624 -0.93345

-25.38658 -4.29882

-40.70675 -0.27678

-31.06896 -3.04031

-40.76029 -9.62422

-60.14299 -2.77638

-60.83382 -15.72249

-90.86594 -30.72044

-124.30211 -11.03344

-110.29335 -31.37281

-153.82886 -42.52863

-182.96928

V£

0.00001 0.79860 0.00180 0.95240 0.02010 1.11469 0.00109 1.10834 0.00802 1.13037 0.02157 1.00204 0.00273 1.22104 0.01143 1.16644 0.02622 0.93171 0.00293 1.23392 0.02229 1.12126 0.04806 1.12701 0.01545 1.27008 0.05617 1.21810 0.01305 0.19822 0.02905 1.08968 0.00172 0.03165

-0.09658 -1.55922

Table C . l : Regression of forward-looking default probabilities on contemporaneous and inflation.

Ratings AAA

AA

A

BBB

BB

B

Maturity 1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

1 Yr

4 Yr

10 Yr

Adj. R2

0.052106

0.058085

0.276128

0.170283

0.170067

0.305571

0.218936

0.215351

0.341609

0.228244

0.339787

0.426486

0.402389

0.546754

0.677164

0.573513

0.729194

0.843842

Const 0.00000 2.19369 0.00001 2.03647 0.00008

4.40955 0.00005

3.84921 0.00007

3.01850 0.00012

5.29124 0.00013

4.85192 0.00011

3.57511 0.00020

7.23153 0.00014

4.91535 0.00037

6.31281 0.00056

13.47681 0.00147

10.59223 0.00305

23.29991 0.00343

46.10380 0.00834

24.63871 0.01012

57.16553 0.00992

107.00139

St-l

0.18739 19.14339

0.20174 20.67070

0.44695 51.07106

0.34539 37.27266

0.35906 38.86341

0.45841 53.09303

0.38263 42.22967

0.40039 44.33780

0.45415 53.27235

0.39195 43.47920

0.46879 55.20774

0.41838 50.54996

0.45779 55.20953

0.38963 50.13558

0.28857 40.69552

0.41922 55.30654

0.26290 39.36740

0.14660 26.99181

9t -0.00028

-11.65271 -0.00759

-11.67716 -0.03749

-21.96590 -0.01993

-18.60756 -0.03534

-16 .57117 -0.04931

-24.65577 -0.05168

-21.99283 -0.05364

-18.90440 -0.07692

-29.98419 -0.05509

-22.23368 -0.14849

-27.98201 -0.17104

-43.75068 -0.47234

-36.86107 -0.74620

-59.95466 -0.61174

-87.59751 -1.92260

-60.14085 -1.63056

-100.12294 -1.14648

-150.49763

n 0.00001 0.94375 0.00019 1.13147 0.00088

1.98859 0.00047

1.66194 0.00092

1.64286 0.00106

2.06793 0.00119

1.94560 0.00137

1.84723 0.00146

2.24778 0.00126

1.96034 0.00331

2.44134 0.00271

2.84074 0.00956

2.99692 0.01054

3.65421 0.00471

3.13618 0.02975

4.03531 0.01177

3.44929 0.00107

3.71545

Table C.2: Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

151

Ratings Maturity AAA lYr

4Yr

lOYr

AA lYr

4Yr

lOYr

A lYr

4Yr

lOYr

Regime L

H

L

H

L

H

L

H

L

H

L

H

L

H

L

H

L

H

Adj. R2

0.00027

0.04671

0.00012

0.04527

0.00003

0.14728

-0.00027

0.10216

0.00012

0.10108

0.00003

0.16086

0.00192

0.12524

0.00012

0.12108

0.00003

0.18138

Const 0.00000 0.99873 0.00000

-0.98186 0.00000

-1.53393 -0.00002 -1.30283 0.00000 0.30677

-0.00013 -2.44587

0.00000 1.36392

-0.00005 -1.67418 0.00000

-1.53393 -0.00013

-2.01004 0.00000 0.30677

-0.00014 -2.35655

0.00000 5.33145

-0.00011 -1.54820 0.00000

-1.53393 -0.00018

-2.18717 0.00000 0.30677

-0.00014 -1.80489

St-l

0.00000 1.34868 0.17874

12.27943 0.00000

-1.47666 4.61410

11.83308 0.00000

-1.43791 19.44407

18.11847 -0.00129 -0.07219 10.60683

16.20919 0.00000

-1.47666 22.21483

16.96320 0.00000

-1.43791 22.00478

17.44482 -0.01015 -0.21111 24.31079

16.81542 0.00000

-1.47666 31.28069

17.77467 0.00000

-1.43791 24.91307

15.48597

9t 0.00000

-0.81007 -0.00044

-6.46414 0.00000 1.50965

-0.01219 -6.74934

0.00000 -1.39023 -0.08934

-17.97782 0.00000

-1.40274 -0.03944

-13.01610 0.00000 1.50965

-0.07222 -11.90935

0.00000 -1.39023 -0.11968

-20.49019 -0.00001

-3.67655 -0.10756

-16.06608 0.00000 1.50965

-0.11707 -14.36590

0.00000 -1.39023 -0.18381

-24.67431

7T*

0.00000 -0.08745 0.00000

-1.24933 0.00000

-0.00346 0.00004 0.11633 0.00000

-0.47214 0.00009 0.08786 0.00000 0.85517 0.00010 0.16828 0.00000

-0.00346 0.00007 0.05839 0.00000

-0.47214 0.00017 0.14362 0.00000

-0.60373 0.00032 0.23402 0.00000

-0.00346 0.00014 0.08395 0.00000

-0.47214 0.00047 0.31097

Table C.3: Regression of AAA-A credit spreads with regime switching on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

152

Ratings Maturity BBB lYr

4Yr

lOYr

BB lYr

4Yr

lOYr

B lYr

4Yr

lOYr

Regime L

H

L

H

L

H

L

H

L

H

L

H

L

H

L

H

L

H

Adj. R2

0.00147

0.12802

0.00012

0.17955

0.00003

0.23011

0.10526

0.21008

0.07805

0.27889

0.06888

0.30247

0.17336

0.27655

0.12617

0.30513

0.13992

0.31605

Const 0.00000

5.55824 -0.00011 -1.50533 0.00000

-1.53393 -0.00030 -1.83553 0.00000 0.30677 0.00018 1.57091 0.00002

41.30838 0.00074 1.93776 0.00001

31.47698 0.00361

10.08167 0.00001

28.87154 0.00607

32.24364 0.00073

58.37372 0.01317

14.12400 0.00057

44.09497 0.01906

44.16364 0.00193

47.31271 0.01954

126.88663

S t_l

-0.01113 -0.20056 25.88789

16.92799 0.00000

-1.47666 58.40557

17.42646 0.00000

-1.43791 25.24195

10.55921 1.39768 0.89913

132.36116 16.55827

0.57350 1.06485

57.72695 7.74413 0.26828 0.22812

12.72014 3.24371 34.25463 0.86412

238.21676 12.26141

34.73198 0.85715

27.26243 3.03125

127.16300 0.99215

12.05100 3.75519

9t -0.00001

-3.22549 -0.11590

-16.36686 0.00000 1.50965

-0.35596 -22.93621

0.00000 -1.39023 -0.36685

-33.14091 -0.00118

-25.01402 -1.01158

-27.32880 -0.00035

-21.20930 -1.36461

-39.53406 -0.00071

-19.89198 -0.78914

-43.45833 -0.04007

-33.38108 -3.35693

-37.31459 -0.03400

-27.70629 -1.82386

-43.79424 -0.11410

-29.39506 -0.66480

-44.73724

VTi

0.00000 -0.97650 0.00027 0.18778 0.00000

-0.00346 0.00110 0.34905 0.00000

-0.47214 0.00246 1.09309 0.00001 1.01728 0.00542 0.72128 0.00000 0.48806 0.01376

1.96325 0.00000 0.46611 0.00854

2.31718 0.00045

2.68753 0.02789

4.52740 0.00031 1.14986 0.02181

4.58012 0.00106 1.23075 0.00453

3.50176

Table C.4: Regression of BBB-B credit spreads with regime switching on one-quarter lagged credit spreads and contemporaneous output growth and inflation.

essays on macroeconomics and credit risk · sudarshan p. gururaj the first chapter of this...

Documents