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ESSAYS IN PORTFOLIO CREDIT RISK

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MANAGEMENT SCIENCE AND ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Baeho Kim

February 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/bg468jw7546

© 2010 by Baeho Kim. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/bg468jw7546

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Kay Giesecke, Primary Adviser


James Primbs


John Weyant

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

This dissertation considers the measurement and management of portfolio credit risk.

Collateralized debt obligations, which are securities with payoffs that are tied to the

cash flows in a portfolio of defaultable assets such as corporate bonds, play a signif-

icant role in the financial crisis that has spread throughout the world. Insufficient

capital provisioning due to flawed and overly optimistic risk assessments is at the

center of the problem. In the first part of the dissertation, we develop stochastic

methods to measure the risk of positions in collateralized debt obligations and re-

lated instruments tied to an underlying portfolio of defaultable assets. We propose

an adaptive point process model of portfolio default timing, a maximum likelihood

method for estimating point process models that is based on an acceptance/rejection

re-sampling scheme, and statistical tests for model validation. To illustrate these

tools, they are used to estimate the distribution of the profit or loss generated by

positions in multiple tranches of a collateralized debt obligation that references the

CDX High Yield portfolio, and the risk capital required to support these positions.

The second part of the dissertation develops maximum likelihood estimators of the

term structure of systemic risk in the U.S. financial sector, defined as the conditional

probability of failure of a large number of financial institutions. The estimators are

based on a new dynamic hazard model of failure timing that captures the influence

of time-varying macro-economic and sector-specific risk factors on the likelihood of

failures, and the impact of risk spillovers due to contagion or incomplete information

about relevant risk factors. The estimation results, which cover the period January

1987 to December 2008, provide strong evidence for the presence of failure clustering

not caused by variations in the observable explanatory covariates, which include the

iv

trailing return on the S&P 500 index, the lagged slope of the U.S. yield curve, the

default and TED spreads, and other sector-specific variables.

v

Acknowledgments

I would first like to express my immeasurable gratitude to my advisor, Professor Kay

Giesecke. I have been very fortunate to have him as my principal advisor during my

graduate studies. His expertise in credit risk modeling and analysis has been truly

valuable in this dissertation that would have not been possible without him. I am also

indebted to my other members of the reading committee, Professor James A. Primbs

and Professor John Weyant. I would like to thank them for their helpful advices.

Professor Primbs introduced me into the field of financial engineering and I learned a

great deal on the basics of my research from him, which I used extensively throughout

this dissertation. Professor Weyant gave me considerable economic insights, which

helped me in finding a nice topic for my research.

Professor Gerd Infanger has served in my oral exam committee and Professor

Antoine Toussaint has served as the oral exam chair. I would like thank Professor

Infanger for his valuable input as an expert in stochastic optimization and Professor

Toussaint for his comments from his background in mathematical finance. In addi-

tion, I benefited from the guidance of a number of professors at Stanford by taking

invaluable classes, which not only prepared me in progressing my research, but also

increased my academic knowledge in the related field.

It was a real fun to have great friends and colleagues in the MS&E department

at Stanford. I am deeply thankful for helpful discussions with Jack Kim, Xiaowei

Ding, Shahriar Azizpour, Eymen Errais, Matt A.V. Leduc and Mohammad Mousavi

among others. I would like to extend my gratitude to my Korean friends in Corner-

stone Community Church for their great mental support. They include, but are not

limited to, Pastor Hun Sol, Taikyun Shin, Minyong Shin, Sewook Wee, Jihyung Yoo,

vi

Junghyun Kim, Yongkyun Na, Wonyoung Lee, Younggeun Cho and many others. I

know they prayed to give me the courage to move forward and finish my study. I am

also grateful for what I have received from Samsung Scholarship Foundation for their

financial support in most of my study at Stanford.

In addition, I would like to especially thank my parents Myung-hwan Kim and

Young-sook Jeun, and my sister Baeok Kim, for their unconditional love, encourage-

ment and support throughout my entire education, which led to my Ph.D. degree at

Stanford. Moreover, I would like to acknowledge and thank my wife Dawoon Jung.

Without her support, understanding and sacrifice, this dissertation would have been

an impossible task. I dedicate this dissertation to her and my lovely son, Daniel J.

Kim. Last, but not least, I praise for the Lord, especially for having given me the

privilege of my life within his glory.

vii

Contents

Abstract iv

Acknowledgments vi

1 Introduction 1

2 Risk Analysis of CDOs 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Re-sampling based inference . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Preliminaries and problem . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Acceptance/rejection re-sampling . . . . . . . . . . . . . . . . 8

2.2.3 Thinning process specification . . . . . . . . . . . . . . . . . . 10

2.2.4 Likelihood estimators and fitness tests . . . . . . . . . . . . . 12

2.2.5 Portfolios without replacement . . . . . . . . . . . . . . . . . . 14

2.3 An adaptive intensity model . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Re-sampling scenarios . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Intensity specification . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.3 Event and loss simulation . . . . . . . . . . . . . . . . . . . . 23

2.3.4 Likelihood estimators . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.5 Testing in-sample fit . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.6 Testing out-of-sample loss forecasts . . . . . . . . . . . . . . . 28

2.4 Synthetic collateralized debt obligations . . . . . . . . . . . . . . . . 30

2.5 Cash collateralized debt obligations . . . . . . . . . . . . . . . . . . . 35

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

viii

3 Systemic Risk: What Defaults Are Telling Us 44

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Measures of systemic risk . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Statistical methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.1 Economy-wide default timing . . . . . . . . . . . . . . . . . . 52

3.3.2 System-wide default timing . . . . . . . . . . . . . . . . . . . 54

3.3.3 Measures of risk . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Default timing data . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.2 Covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.3 Economy-wide intensity . . . . . . . . . . . . . . . . . . . . . 61

3.4.4 Goodness-of-fit tests . . . . . . . . . . . . . . . . . . . . . . . 63

3.4.5 System-wide intensity . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Systemic risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.1 Risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.2 Forecast evaluation . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6 Sensitivity of systemic risk . . . . . . . . . . . . . . . . . . . . . . . . 74

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Appendix 78

A Risk-neutral tranche loss distributions . . . . . . . . . . . . . . . . . 78

B Cash CDO prioritization schemes . . . . . . . . . . . . . . . . . . . . 80

B1 Uniform prioritization . . . . . . . . . . . . . . . . . . . . . . 81

B2 Fast prioritization . . . . . . . . . . . . . . . . . . . . . . . . . 82

C Cash CDO valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

D Covariate time-series model . . . . . . . . . . . . . . . . . . . . . . . 84

E Default volume model . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Bibliography 89

ix

List of Tables

2.1 Maximum likelihood estimates of the parameters of the intensity λ for

the CDX.HY6 portfolio, along with estimates of asymptotic (A) and

bootstrapping (B) 95% confidence intervals (10K bootstrap samples

were used). The “Median” row indicates the median of the empirical

distribution of the per-path MLEs over all re-sampling paths. The

estimates are based on I = 10K re-sampling scenarios, generated by

Algorithm 1 from the observed defaults in the universe of Moody’s

rated names from 1/1/1970 to 11/7/2008. . . . . . . . . . . . . . . . 25

2.2 Maximum likelihood estimates of the parameters of the portfolio inten-

sity λ for the CDX.HY6, along with estimates of asymptotic (A) and

bootstrapping (B) 95% confidence intervals (10K bootstrap samples

were used). The estimates are based on I = 10K re-sampling scenar-

ios for N , generated by Algorithm 1 from the observed economy-wide

defaults in Moody’s universe of rated names from 1/1/1970 to 3/27/2006. 32

2.3 Fitted annualized par coupon rates and spreads on 3/27/2006 for the

constituent bonds and debt tranches of a 5 year cash CDO referenced

on the CDX.HY6, for each of two standard prioritization schemes. The

principal values are (p1, p2, p3) = (85, 10, 5). The fitting procedure is

described in Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Maximum likelihood estimates (MLE) of economy-wide intensity pa-

rameters, asymptotic standard errors (SE), t-statistics (t-stat), and

Bayes factor statistics (Ψ). . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Goodness-of-fit tests of the economy-wide intensity. . . . . . . . . . . 64

x

3.3 Maximum likelihood estimates of the coefficients β of the covariate pro-

cess X governing the thinning process Z in (3.4), asymptotic standard

errors (SE), t-statistics, p-values, and Bayes factor statistics (Ψ). . . . 66

3.4 Out-of-sample tests of the forecast accuracy of the fitted system-wide

value at risk, for each of several horizons. The period considered is

January 1998 to June 2009. . . . . . . . . . . . . . . . . . . . . . . . 74

A1 Market data from Morgan Stanley and fitting results for 5 year index

and tranche swaps referenced on the CDX.HY6 on 3/27/2006. The

index, (15 − 25%) and (25 − 35%) contracts are quoted in terms of

a running rate S stated in basis points (10−4). For these contracts

the upfront rate G is zero. The (0, 10%) and (10, 15%) tranches are

quoted in terms of an upfront rate G. For these contracts the running

rate S is zero. The values in the column Model are fitted rates based

on model (A4) and 100K replications. We report the minimum value

of the objective function MinObj and the average absolute percentage

error AAPE relative to market mid quotes. . . . . . . . . . . . . . . . 81

A2 Estimates of the parameters of the risk-neutral portfolio intensity λQ,

obtained from market rates of 5 year index and tranche swaps refer-

enced on the CDX.HY6 on 3/27/2006. . . . . . . . . . . . . . . . . . 81

E1 Fitted coefficients of VAR(1) model (D1) as of 12/31/2008. The t-

statistics are shown in parenthesis. . . . . . . . . . . . . . . . . . . . 86

E2 Fitted covariance matrix Σ of the VAR(1) error term εt as of 12/31/2008. 87

xi

List of Figures

2.1 Left panel : Annual default rate, relative to the number of rated names

at the beginning of a year, in the universe of Moody’s rated corporate

issuers in any year between 1970 and 2008, as of 11/7/2008. Source:

Moody’s Default Risk Service. Right panel : Mean annual default rate

for the CDX.HY6 portfolio for I = 10K re-sampling scenarios. . . . . 17

2.2 Left panel : Sample path of the intensity (left scale, solid) and de-

fault process (right scale, dotted) for the adaptive model (2.10) with

θ = (0.25, 0.05, 0.4, 0.8, 2.5), values that are motivated by our estima-

tion results in Section 2.3.4. The reversion level and speed change at

each event. Algorithm 3 is used to generate the paths. Right panel :

Sample path of the intensity and default process for the Hawkes model

dλt = κ(c−λt)dt+δdUt where the jump magnitudes un = 1 so U = N ,

λ0 = 2.5, and κ = 2.5 and c = δ = 1.5 are chosen so that the

expected number of events over 10 years matches that of the model

(2.10), roughly 37. Note that the Hawkes intensity cannot fall below

the global reversion level c. . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Left panel : Fitted mean portfolio intensity λ = I−1∑I

i=1 λθ(ωi) vs.

[5%, 95%] percentiles (boxes), [1%, 99%] percentiles (whiskers) and the

mean number of portfolio defaults in any given year between 1970 and

2008, based on I = 10K re-sampling scenarios for the CDX.HY6. Right

panel : Fitted economy-wide intensity λ/Z∗ vs. economy-wide defaults

between 1970 and 2008, semi-annually. The fitted intensity matches

the time-series fluctuation of observed economy-wide default rates. . . 26

xii

2.4 In-sample fitness tests: empirical distribution of time-scaled, economy-

wide inter-arrival times generated by the fitted (mean) paths of Z∗ and

λ, based on I = 10K re-sampling scenarios for the CDX.HY6. Left

panel : Empirical quantiles of time-scaled inter-arrival times vs. theo-

retical standard exponential quantiles. Right panel : Empirical distri-

bution function (solid) of time-scaled inter-arrival times vs. theoretical

standard exponential distribution function (dotted) along with 1% and

5% bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Out-of-sample test of loss forecasts for a test portfolio with the same

rating composition as the CDX.HY6. We show the [25%, 75%] per-

centiles (box), the [15%, 85%] percentiles (whiskers) and the median

(horizontal line) of the fitted conditional distribution of the incremen-

tal portfolio loss L′τ+1 − L′τ given Fτ for τ varying annually between

1/1/1996 and 1/1/2007, estimated from 100K paths generated by Al-

gorithms 3 and 2, and the realized portfolio loss during [τ, τ + 1]. Left

panel : The test portfolio is selected at the beginning of each period.

Right panel : The test portfolio is selected in 1996. . . . . . . . . . . . 29

2.6 Kernel smoothed conditional distribution of the normalized 5 year cu-

mulative tranche loss for the CDX.HY6 on 3/27/2006, the Series 6

contract inception date, for each of several standard attachment point

pairs. Here, H represents the maturity date 6/27/2011. Left panel :

Actual distribution, estimated using the model and fitting methodol-

ogy developed in Sections 2.2 and 2.3, based on I = 10K re-sampling

scenarios for N , and 100K replications. Right panel : Risk-neutral dis-

tribution, estimated from the market prices paid for 5 year tranche

protection on 3/27/2006 using the method explained in Appendix A,

based on 100K replications. . . . . . . . . . . . . . . . . . . . . . . . 31

xiii

2.7 Kernel smoothed conditional distribution of the normalized cumulative

portfolio loss L′H/C for the CDX.HY6 on 3/27/2006 for horizons H

varying between 3/27/2009 and 3/27/2016. Left panel : Actual distri-

bution, estimated using the model and fitting methodology developed

in Sections 2.2 and 2.3, based on I = 10K re-sampling scenarios for

N , and 100K replications. Right panel : Risk-neutral distribution, es-

timated from the market prices paid for 5 year tranche protection on

3/27/2006 using the method explained in Appendix A, based on 100K

replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.8 Kernel smoothed conditional distribution on 3/27/2006 of the normal-

ized cumulative loss (UH(0, 0.1)−UH(0.1, 0.15))/0.1C associated with

selling equity protection and buying mezzanine protection with match-

ing notional and maturity, along with the distribution of the normalized

cumulative equity loss UH(0, 0.1)/0.1C, for the CDX.HY6. . . . . . . 34

2.9 Cash flow “waterfall” of a sample cash CDO. . . . . . . . . . . . . . . 35

2.10 Kernel smoothed conditional distribution on 3/27/2006 of the nor-

malized discounted loss Ujτ (H, pj, v∗, c∗1, c

∗2)/V jτ (H, pj, v

∗, c∗1, c∗2) for a

5 year cash CDO referenced on the CDX.HY6, whose maturity date H

is 6/27/2011. The initial tranche principals are p1 = 85 for the senior

tranche, p2 = 10 for the junior tranche, and p3 = 5 for the residual eq-

uity tranche. We apply the model and fitting methodology developed

in Sections 2.2 and 2.3, based on I = 10K re-sampling scenarios for

N , and 100K replications. Left panel : Uniform prioritization scheme.

Right panel : Fast prioritization scheme. . . . . . . . . . . . . . . . . . 39


ized discounted profit and loss (V3τ (H, 5, v∗, c∗1, c

∗2)− 5)/5 for the resid-

ual equity tranche of a 5 year cash CDO referenced on the CDX.HY6,

whose maturity date H is 6/27/2011. We apply the model and fitting

methodology developed in Sections 2.2 and 2.3, based on I = 10K

re-sampling scenarios for N , and 100K replications. . . . . . . . . . . 40

xiv


ized discounted profit and loss (Vjτ (H, pj, v∗, c∗1, c

∗2)−pj)/pj for the debt

tranches of a 5 year cash CDO referenced on the CDX.HY6, whose ma-

turity date H is 6/27/2011. We apply the model and fitting method-

ology developed in Sections 2.2 and 2.3, based on I = 10K re-sampling

scenarios for N , and 100K replications. Left panel : Senior tranche with

principal p1 = 85. Right panel : Junior tranche with principal p2 = 10. 41

2.13 99.5% value at risk VaR1τ (0.995, H, 85, v∗, c∗1, c∗2) for the 5 year senior

tranche of a cash CDO referenced on the CDX.HY6 with maturity date

6/27/2011, for τ varying weekly between 3/27/2006 and 11/7/2008,

based on I = 10K re-sampling scenarios for N , and 100K replications.

Left panel : Uniform prioritization scheme. Right panel : Fast prioriti-

zation scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1 Value at risk Vt(α, T ) at level α for a given horizon T . . . . . . . . . . 51

3.2 Default timing and volume data. Left panel : 1-year economy-wide

default rate in the universe of Moody’s rated issuers. Right panel : 1-

year system-wide default rate. The defaults of Lehman Brothers and

Washington Mutual contributed to over 80% of the system-wide default

volume in 2008. Source: Moody’s Default Risk Service. . . . . . . . . 58

3.3 Time-series of explanatory covariates. Left panel : The 1-year lagged

slope of yield curve and the default spread, given by the difference

between Moody’s seasoned Baa-rated and Aaa-rated corporate bond

yields. Right panel : The trailing 1-year returns on the S&P500 index

and the banking and FIRE portfolios. . . . . . . . . . . . . . . . . . . 59

3.4 TED spreads during the sample period, along with significant events. 60

3.5 Fitted economy-wide intensity λ∗. Left panel : Yearly defaults and

fitted intensity. Right panel : Intensity decomposition: fitted baseline

hazard vs. fitted spillover hazard. . . . . . . . . . . . . . . . . . . . . 63

3.6 Left panel : Observed binary response variables Yn and fitted process

Z. Right panel : Power curve for the fitted process Z. . . . . . . . . . 67

xv

3.7 Left panel : Fitted system-wide failure intensity λ, based on the pa-

rameter estimates reported in Tables 3.1 and 3.3. Right panel : Fitted

fraction of λ tied to the spillover hazard term, with default events

indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.8 Fitted conditional distribution (kernel-smoothed) of the system-wide

6-month default rate Dt(t+ 0.5) for conditioning times t varying semi-

annually between 12/31/1997 and 12/31/2008. . . . . . . . . . . . . . 69

3.9 Fitted conditional distribution (kernel-smoothed) of the economy-wide

6-month default rate for conditioning times t varying semi-annually

between 12/31/1997 and 12/31/2008. The default rate is obtained by

normalizing the number of economy-wide defaults during (t, t+ 0.5] by

the total number of firms in the economy at t. . . . . . . . . . . . . . 70

3.10 Left Panel: Fitted value at risk Vt(α, t + 0.5) of the system-wide

default rate, for conditioning times t varying semi-annually between

12/31/1997 and 12/31/2008, versus realized default rate. Right Panel:

Fitted value at risk of the economy-wide default rate versus realized

default rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.11 Left Panel: Term structure of systemic risk on 12/31/2008: fitted value

at risk Vt(α, t + ∆) on 12/31/2008 as a function of ∆. Right Panel:

Fitted conditional probability at t of no failures in the financial system

during (t, t + ∆], for conditioning times t varying quarterly between

12/31/1997 and 12/31/2008, for each of several horizons ∆. . . . . . . 72

3.12 Impact of a default on systemic risk. Left Panel: Absolute change

∆Vt(0.95, t + 1) for conditioning times t varying quarterly between

12/31/1997 and 12/31/2008. Right Panel: Term structure of value at

risk Vt(0.95, t+ ∆) on 12/31/2008, for different scenarios. . . . . . . . 75

D1 Realized vs. VAR(1) predicted time series (monthly) of covariate com-

ponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xvi

E1 Left panel : Empirical default volume distribution vs. fitted generalized

Pareto distribution as of 12/31/2008. Right panel : Empirical quantiles

of the observed default volumes vs. quantiles of realizations of variables

from the fitted Pareto distribution. . . . . . . . . . . . . . . . . . . . 88

xvii

Chapter 1

Introduction

Credit markets in the U.S. have seen a tendency for defaults to cluster due to the

correlation among firms. This correlation comes from either firm-specific sensitivity

to common risk factors or the feedback of an individual event to the aggregate level.

For example, the recent subprime mortgage financial crisis, which has become a global

economic catastrophe, originated with dramatically increased defaults and foreclosure

activities of subprime loans. Despite the impact of the global financial crisis, which

was sparked by the collapse of Lehman Brothers in September 2008, the correlated

default risk is still difficult to measure and its sources are poorly understood, since

financial innovations have enabled risk transfers that were not fully recognized by

financial regulators and institutions.

The first part of the dissertation considers a financial problem of credit investors,

who are exposed to the correlated default risk. The market of credit derivatives has

experienced an impressive growth and one of the most important developments was

the introduction of a collateralized debt obligation (CDO), which is an asset-backed

security and structured credit product whose underlying collateral is usually a port-

folio of defaultable corporate bonds, sovereign bonds, or bank loans. These claims,

in turn, are prioritized by tranches with different levels of seniority in order to allow

a high degree of customization of repackaging risk profiles according to the investors

risk return requirements. However, insufficient capital provisioning with inaccurate

measures of risk transfers due to flawed and overly optimistic risk assessments is at

1

CHAPTER 1. INTRODUCTION 2

the center of the recent financial crisis. To circumvent the limitations of existing

methodologies, we propose new stochastic methods to measure the risk of positions

in collateralized debt obligations and related instruments tied to an underlying port-

folio of defaultable assets under the actual probability measure. Our methodology

provides sophisticated yet practical tools allowing investors to quantify the exposure

associated with their risk positions, and to accurately estimate the amount of risk

capital required to support a position.

The second part of the dissertation extends the scope of ‘portfolio’ and focuses

on the assessment of financial sector-wide systemic risk due to direct and indirect

systemic linkages, including those to the broader economy. An understanding of sys-

temic risk and its sources is of paramount importance, since the cascading failure of

banks and other financial institutions can deprive society of capital and dramatically

increase its cost, which is the most serious consequence of a systemic failure. However,

the effective management of systemic risk in the financial sector is still a key chal-

lenge for regulators and policy makers, since the channels and linkages through which

local financial disturbances can take on systemic characteristics are hard to predict

by their nature. The background of this challenge is the chain of subsequent failures

because of the interconnectedness that leads to risk spillovers for the whole financial

system. For instance, one bank’s default on an obligation to another may adversely

affect that other bank’s ability to meet its obligations to yet other banks and so on

along the financial chain in the banking system. Moreover, the financial system is

composed of intermediaries and the infrastructure of payment, settlement and trad-

ing mechanisms that intertwine banks and other non-bank financial institutions. This

complex network of financial connections is extended through the financing needs of

all economic sectors as well. Hence, a desirable systemic risk measure should be able

to accommodate contagion channeled through trade credit or buyer/supplier relation-

ships in the real sector, and derivatives counterparty relations and interbank lending

arrangements in the financial sector. In this context, we propose a methodology that

develops maximum likelihood estimators of the term structure of systemic risk in the

financial sector, defined as the conditional probability of failure of a large number of

financial institutions, potentially as part of a larger cluster of economy-wide defaults.

CHAPTER 1. INTRODUCTION 3

The estimators are based on a new dynamic hazard model of failure timing that cap-

tures the influence of time-varying macro-economic and sector-specific risk factors

on the likelihood of failures, and the impact of risk spillovers due to contagion or

incomplete information about relevant risk factors. As a modern toolkit for financial

surveillance, the proposed framework provides a metric of potential financial failures

due to both direct and indirect systemic linkages by capturing the statistical implica-

tions of risk spillovers for failure timing without needing to be precise a priori about

the economic mechanisms behind them. It facilitates a macro-prudential supervision

of financial institutions by gauging the vulnerability of the financial system at any

particular time, and thus enhance financial surveillance.

Chapter 2

Risk Analysis of Collateralized

Debt Obligations

Collateralized debt obligations, which are are securities with payoffs that are tied

to the cash flows in a portfolio of defaultable assets such as corporate bonds, play a

significant role in the financial crisis that has spread throughout the world. Insufficient

capital provisioning due to flawed and overly optimistic risk assessments is at the

center of the problem. This chapter develops stochastic methods to measure the

risk of positions in collateralized debt obligations and related instruments tied to an

underlying portfolio of defaultable assets. It proposes an adaptive point process model

of portfolio default timing, a maximum likelihood method for estimating point process

models that is based on an acceptance/rejection re-sampling scheme, and statistical

tests for model validation. To illustrate these tools, they are used to estimate the

distribution of the profit or loss generated by positions in multiple tranches of a

collateralized debt obligation that references the CDX High Yield portfolio, and the

risk capital required to support these positions.

This chapter is joint work with Kay Giesecke.

4

CHAPTER 2. RISK ANALYSIS OF CDOS 5

2.1 Introduction

The financial crisis highlights the need for a holistic, objective and transparent ap-

proach to accurately measuring the risk of investment positions in portfolio credit

derivatives such as collateralized debt obligations (CDOs). Portfolio credit deriva-

tives are securities whose payoffs are tied, often through complex schemes, to the

cash flows in a portfolio of credit instruments such as corporate bonds, loans, or

mortgages. They facilitate the trading of insurance against the default losses in the

portfolio. An investor providing the insurance is exposed to the default risk in the

portfolio.

There is an extensive literature devoted to the valuation and hedging of portfolio

derivatives.1 The basic valuation problem is to estimate the price of default insurance,

i.e., the arbitrage-free value of the portfolio derivative at contract inception. This

value is given by the expected discounted derivative cash flows relative to a risk-

neutral pricing measure. After inception, the derivative position must be marked to

market; that is, the value of the derivative under current market conditions must

be determined. The basic hedging problem is to estimate the sensitivities of the

derivative value to changes of the default risk of the portfolio constituents. These

sensitivities determine the amount of constituent default insurance to be bought or

sold to neutralize the derivative price fluctuations due to changes of the constituent

risks.

The valuation and hedging problems are distinct from the risk analysis problem,

which is to measure the exposure of the derivative investor, who provides default

insurance, to potential payments due to defaults in the portfolio. More precisely, the

goal is to estimate the distribution of the investor’s cumulative cash flows over the

life of the contract. The distribution is taken under the actual measure describing

the empirical likelihood of events, rather than a risk-neutral pricing measure. The

distribution describes the risk/reward profile of a portfolio derivative position, and

1See Arnsdorf & Halperin (2008), Brigo, Pallavicini & Torresetti (2006), Chen & Glasserman(2008), Cont & Minca (2008), Ding, Giesecke & Tomecek (2009), Duffie & Garleanu (2001), Eckner(2009), Errais, Giesecke & Goldberg (2009), Kou & Peng (2009), Longstaff & Rajan (2008), Lopatin& Misirpashaev (2008), Mortensen (2006), Papageorgiou & Sircar (2007) and many others.


is the key to risk management applications. For example, it allows the investor or

regulator to determine the amount of risk capital required to support a position.

The financial crisis indicates the significance of these applications and the problems

associated with the traditional rating based analysis of portfolio credit derivative

positions; see SEC (2008).

The risk analysis problem has been largely ignored in the academic literature.

This chapter provides stochastic methods to address this problem. It makes several

contributions. First, it develops a maximum likelihood approach to estimating point

process models of portfolio default timing from historical default experience. Second,

it devises statistical tests to validate a fitted model. Third, it formulates, fits and

tests an adaptive point process model of portfolio default timing, and demonstrates

the utility of the estimation and validation methods on this model. Fourth, it pro-

vides an exact simulation algorithm for the adaptive model, and uses it to address

important risk management applications. For example, we estimate profit and loss

distributions for positions in multiple tranches of a CDO. These distributions quan-

tify and differentiate the risk exposure of alternative investment positions, and the

impact of complex contract features. They are preferable to agency ratings, which

are often based on the first moment only.

Estimating a stochastic point process model of portfolio default timing under the

actual probability measure presents unique challenges. Most importantly, inference

must be based on historical default timing data, rather than market derivative pricing

data. However, the default history of the reference portfolio underlying the credit

derivative is often unavailable, so direct inference is usually not feasible. We confront

this difficulty by developing an acceptance/rejection re-sampling scheme that allows

us to generate alternative portfolio default histories from the available economy-wide

default timing data. These histories are then used to construct maximum likelihood

estimators for a portfolio point process model that is specified in terms of an intensity

process. A time-scaling argument leads to testable hypotheses for the fit of a model.

The re-sampling approach is predicated on a top-down formulation of the portfolio

point process model. This formulation has become popular in the credit derivatives

pricing literature. Here, the point process intensity is specified without reference to


the portfolio constituents. In our risk analysis setting, the combination of top-down

formulation and re-sampling based inference leads to low-dimensional estimation, val-

idation and prediction problems that are highly tractable and fast to address even

for the large portfolios that are common in practice. Alternative bottom-up formu-

lations in Das, Duffie, Kapadia & Saita (2007a), Delloye, Fermanian & Sbai (2006)

and Duffie, Eckner, Horel & Saita (2009) require the specification and estimation of

default timing models for the individual portfolio constituent securities. They allow

one to incorporate firm-specific data into the estimation, but lead to high-dimensional

computational problems.

To demonstrate the effectiveness of the re-sampling approach and the appropri-

ateness of the top-down formulation, we develop and fit an adaptive intensity model.

This model extends the classical Hawkes (1971) model by including a state-dependent

drift coefficient in the intensity dynamics. The state-dependent drift involves a re-

version level and speed that are proportional to the intensity at the previous event.

While this specification is as tractable as the Hawkes model, it avoids the constraints

imposed by the constant Hawkes reversion level and speed. This helps to better fit

the regime-dependent behavior of empirical default rates. In- and out-of-sample tests

show that our adaptive model captures the clustering in the default arrival data. The

tests indicate that a parsimonious top-down model formulation is also statistically

appropriate.

This rest of this chapter is organized as follows. Section 2.2 develops the re-

sampling approach to point process model estimation and validation. Section 2.3

formulates and analyzes the adaptive point process model, and uses the re-sampling

approach to fit it. Section 2.4 applies the fitted model to the risk analysis of synthetic

CDOs, while Section 2.5 analyzes the risk of cash CDOs. Section 2.6 concludes. There

are several appendices.

2.2 Re-sampling based inference

This section develops a re-sampling approach to estimating a stochastic point process

model of default timing. It also provides a method to validate the estimators.


2.2.1 Preliminaries and problem

The uncertainty in the economy is modeled by a complete probability space (Ω,F , P ),

where P is the actual (statistical) probability measure. The information flow of

investors is described by a right-continuous and complete filtration F = (Ft)t≥0.

Consider a non-explosive counting process N with event stopping times 0 < T1 <

T2 < . . . The Tn represent the ordered default times in a reference portfolio of firms,

with the convention that a defaulted name is replaced with name that has the same

characteristics as the defaulter. A portfolio credit derivative is a security with cash

flows that depend on the financial loss due to default in the reference portfolio.

Suppose N has a strictly positive intensity λ such that∫ t

0λsds < ∞ almost

surely. The intensity represents the conditional mean default rate in the sense that

E(Nt+∆ − Nt | Ft) ≈ λt∆ for small ∆ > 0. This means that N −∫ ·

0λsds is a local

martingale relative to P and F. The process followed by λ determines the distribution

of N . It is the modeling primitive, and is specified without reference to the constituent

firms.

Our goal is to estimate a given model of λ. If the data consist of a path of N ,

then this is a classical statistical problem; see Ogata (1978a), for example. However,

a path of N is rarely available, so direct inference is typically not feasible. Instead,

the data consist of a path of the economy-wide default process N∗ generated by the

default stopping times 0 < T ∗1 < T ∗2 < . . . in the universe of names. We develop a

re-sampling approach to estimating λ from the realization of N∗. The basic idea is to

generate alternative portfolio default histories from N∗, and to estimate λ from these

histories.

2.2.2 Acceptance/rejection re-sampling

We propose to generate paths of N from the realization of N∗ by acceptance/rejection

sampling. Here, we randomly select an event time of N∗ as an event time of N

with a certain conditional probability. The following basic observation provides the

foundation of this mechanism, and the justification of our estimation approach. It

also facilitates the design of tests to evaluate the approach.


Proposition 2.2.1. Let Z∗ be a predictable process with values in [0, 1]. Select an

economy-wide event time T ∗n with probability Z∗T ∗n . If the economy-wide default process

N∗ has intensity λ∗, then the counting process of the selected times has intensity Z∗λ∗.

Proof. Let (U∗n) be a sequence of standard uniform random variables that are indepen-

dent of one another and independent of the (T ∗n). Let π(du, dt) be the random count-

ing measure with mark space [0, 1] associated with the marked point process (T ∗n , U∗n).

Note that N∗t =∫ t

0

∫ 1

0π(du, ds), and that π(du, dt) has intensity λπ(u, t) = λ∗t− for

u ∈ [0, 1], which we choose to be predictable. This means that the process defined

by∫ t

0

∫ 1

0(π(du, ds)− λπ(u, s)duds) is a local martingale. Now for u ∈ [0, 1] define the

process I(u, ·) by I(u, t) = 1u≤Z∗t . Then the counting process N generated by the

selected event times can be written as

Nt =∑n≥1

I(U∗n, T∗n)1T ∗n≤t =

∫ t

0

∫ 1

0

I(u, s)π(du, ds).

Since Z∗ is predictable, I(u, ·) is predictable for u ∈ [0, 1]. The process I(u, ·) is also

bounded for u ∈ [0, 1]. Therefore, a local martingale M is defined by

Mt = Nt −∫ t

0

∫ 1

0

I(u, s)λπ(u, s)du ds.

But the definitions of I(u, s) and λπ(u, s) yield

Mt = Nt −∫ t

0

∫ Z∗s

0

λ∗s− du ds = Nt −∫ t

0

Z∗sλ∗s ds,

which means that N has intensity Z∗λ∗.

Proposition 2.2.1 states that the counting process obtained by thinning N∗ ac-

cording to Z∗ has intensity given by the product of the thinning process Z∗ and the

intensity of N∗. The specification of Z∗ is subject to a mild predictability condition:

the selection probability at T ∗n can only depend on information accumulated up to but

not including time T ∗n . Proposition 2.2.1 is related to the construction of a marked

point process from a Poisson random measure through a state-dependent thinning


Algorithm 1 (Acceptance/Rejection Re-Sampling) Generating a sample pathof the portfolio default process N from the realization of the economy-wide defaultprocess N∗ over the sample period [0, τ ].

1: Initialize m← 0.2: for n = 1 to N∗τ do3: Draw u ∼ U(0, 1).4: if u ≤ Z∗T ∗n then5: Assign Tm+1 ← T ∗n and update m← m+ 1.6: end if7: end for

mechanism in Proposition 3.1 of Glasserman & Merener (2003). It is a generalization

of the classical, state-independent thinning scheme for the Monte Carlo simulation of

time-inhomogeneous Poisson processes proposed by Lewis & Shedler (1979).

We generate paths of the portfolio default process N by thinning N∗ with a fixed

process Z∗, see Algorithm 1. Each of these paths represents an alternative event

sequence, or portfolio default history, assuming a defaulter is replaced with a name

that has the same characteristics as the defaulter. Proposition 2.2.1 implies that the

events in each alternative sequence arrive with intensity λ = Z∗λ∗. This justifies the

estimation of λ from the alternative paths of the portfolio default process N .

It is important to note that this estimation approach does not require the speci-

fication of a model for the economy-wide intensity λ∗. We only need to formulate a

model for the thinning process Z∗, to which we turn next.

2.2.3 Thinning process specification

The random measure argument behind Proposition 3.1 in Giesecke, Goldberg & Ding

(2009) can be adapted to show that Z∗ takes the form

Z∗t (ω) = limε→0

E(Nt+ε −Nt | Ft)E(N∗t+ε −N∗t | Ft)

(ω) (2.1)


in all those points (ω, t) ∈ Ω × (0,∞] where the limit exists.2 The quotient on the

right side of equation (2.1), which is taken to be zero when the denominator vanishes,

represents the conditional probability at time t that the next defaulter is a reference

name, given that a default occurs in the economy by time t+ ε.

We specify Z∗ nonparametrically, guided by formula (2.1). Intuitively, Z∗ must

reflect the relation between the issuer composition of the economy and the issuer

composition of the reference portfolio. We propose to describe the issuer composition

in terms of the credit ratings of the respective constituent issuers. In this case, N∗ is

identified with the default process in the universe of rated issuers.

The use of ratings does not limit the applicability of our specification, because the

reference portfolios of most derivatives consist of rated names only. Moreover, the

rating agencies maintain extensive and accessible data bases that record credit events

including defaults in the universe of rated names, and further attributes associated

with these events, such as recovery rates. The agencies have maintained a high firm

coverage ratio throughout the sectors of the economy, and therefore the universe of

rated names is a reasonable representation of the firms in the economy.

Let [0, τ ] be the sample period, with τ denoting the (current) analysis time. Let

R be the set of rating categories, Xτ (ρ) be the number at time τ of reference firms

with rating ρ ∈ R, and X∗t (ρ) be the number at time t ∈ [0, τ ] of ρ-rated firms in the

universe of rated names. The number of firms X∗(ρ) is an adapted process. It varies

through time because issuers enter and exit the universe of rated names. Exits can be

due to mergers or privatizations, for example. We assume that the thinning process

Z∗ is equal to the predictable projection of the process defined for times 0 < t ≤ τ

by3

Xτ (ρ∗N∗t−+1)

X∗t−(ρ∗N∗t−+1)1X∗t−(ρ∗

N∗t−+1)>0 (2.2)

where ρ∗n ∈ FT ∗n is the rating at the time of default of the nth defaulter in the

2The limit in (2.1) exists and is equal to Z∗t (ω) almost surely with respect to a certain measureon the product space Ω× (0,∞]. See Giesecke et al. (2009) for more details.

3Here and below, if Y is a right-continuous process with left limits, Yt− = lims↑t Ys.


economy.4 We require that Xτ (ρ) ≤ X∗t (ρ) for all t ≤ τ and ρ ∈ R. Since Xτ (ρ) is

a fixed integer prescribed by the portfolio composition and X∗t−(ρ) is predictable for

fixed ρ,

Z∗t =∑ρ∈R∗t−

Xτ (ρ)

X∗t−(ρ)P (ρ∗N∗t−+1 = ρ | Ft−) (2.3)

almost surely, where R∗t is the set of rating categories ρ ∈ R for which X∗t (ρ) >

0. Formula (2.3) suggests to interpret the value Z∗t as the conditional “empirical”

probability that the next defaulter is a reference name. This conditional probability

respects the ratings of the reference names, as P (ρ∗N∗t−+1 = ρ | Ft−) is the conditional

probability that the next defaulter has rating ρ. Our estimator ν∗t (ρ) of this latter

conditional probability is based on the ratings of the defaulters in [0, t). For ρ ∈ R,

it is given by

ν∗t (ρ) =

∑N∗t−n=1 1ρ∗n=ρ + α∑N∗t−

n=1 1ρ∗n∈R∗t− + α|R∗t−|1ρ∈R∗t− (2.4)

where α ∈ (0, 1] is an additive smoothing parameter guaranteeing that ν∗t (ρ) is well-

defined for t < T ∗1 . For α = 0, equation (2.4) defines the empirical rating distribution,

which treats the observations ρ∗1, . . . , ρ∗N∗t−

as independent samples from a common

distribution and ignores all other information contained in Ft−. Our implementation

assumes α = 0.5, a value that can be justified on Bayesian grounds, see Box & Tiao

(1992, pages 34-36).5

2.2.4 Likelihood estimators and fitness tests

Based on a collection of re-sampling paths N(ωi) : i ≤ I of the portfolio default

process N generated by Algorithm 1, we estimate a model λ = λθ, where θ ∈ Θ

4Taking Z∗ to be the predictable projection of the process given by formula (2.2) guarantees thatZ∗ is defined up to indistinguishability; see Dellacherie & Meyer (1982).

5This choice makes (2.4) the so-called expected likelihood estimator. A sensitivity analysis indi-cates that the model parameter estimates reported in Table 2.1 are robust with respect to variationsof α.


is a parameter vector and Θ is the set of admissible parameters. The paths of N

induce intensity paths λθ(ωi) : i ≤ I, θ ∈ Θ. We fit θ by solving the log-likelihood

problem6

supθ∈Θ

∫ τ

0

I∑i=1

(log λθs−(ωi)dNs(ωi)− λθs(ωi)ds). (2.5)

The adequacy of the fitted intensity as a model of portfolio defaults depends on

the effectiveness of the re-sampling procedure and the appropriateness of the para-

metric intensity specification. More precisely, it depends on how well the alternative

re-sampling scenarios generated by Z∗ capture the actual default clustering in the

reference portfolio, and how well the fitted model for λ replicates these clusters. We

require statistical tests to assess this. The following result allows us to design such

tests.

Proposition 2.2.2. Suppose that Z∗t > 0, almost surely. Then the economy-wide

default process N∗ is a standard Poisson process under a change of time defined by

A∗t =

∫ t

0

1

Z∗sλs ds. (2.6)

Proof. Proposition 2.2.1 implies that the compensator A =∫ ·

0λsds to N is absolutely

continuous with respect to the compensator A∗ to N∗, with density Z∗. Because

Z∗t > 0 almost surely, A∗ is also absolutely continuous with respect to A, with density

1/Z∗. It follows that A∗ can be written as

A∗t =

∫ t

0

1

Z∗sdAs =

∫ t

0

1

Z∗sλs ds.

The time change theorem of Meyer (1971) implies the result, since A∗ is continuous

and increases to ∞, almost surely.

Proposition 2.2.2 expresses the compensator A∗ to N∗ in terms of Z∗ and λ, and

6The asymptotic properties of the maximum likelihood estimator of θ are developed by Ogata(1978a). The estimator is shown to be consistent, asymptotically normal, and efficient.


states that this compensator can be used to time-scale N∗ into a standard Poisson

process. We evaluate the joint specification of Z∗ and λ by testing whether the fitted

(mean) paths of these processes generate a realization of A∗ that time-scales the

observed T ∗n into a standard Poisson sequence. The Poisson property can be tested

with a battery of tests.

2.2.5 Portfolios without replacement

The intensity model λ estimated from the re-sampling scenarios generated by Algo-

rithm 1 is based on a portfolio with replacement of defaulters. This is without loss of

generality because we can extend the reach of the fitted model to portfolios without

replacement.

Consider event times Tn of N over some interval [τ,H], where H > τ . The event

times T ′n of the portfolio default process without replacement, N ′, can be obtained

from the Tn by removing event times due to replaced defaulters. This is done by

thinning. To formalize this, let Xt(ρ) be the number of ρ-rated reference names at

time t ≥ τ , assuming defaulters are not replaced. Thus, for fixed ρ, Xt(ρ) ≤ Xτ (ρ)

almost surely for every t ≥ τ . For fixed ρ, the process X(ρ) decreases and vanishes

when all ρ-rated reference names are in default. It suffices to specify Z, the thinning

process for N , at the event times Tn ≥ τ of N . Motivated by formula (2.3), we

suppose that

ZTn =∑ρ∈Rτ

XT−n(ρ)

Xτ (ρ)P (ρn = ρ | FT−n ) (2.7)

where Rτ is the set of rating categories ρ ∈ R for which Xτ (ρ) > 0, and ρn is

the rating of the firm defaulting at Tn. Note that the thinning probability (2.7)

vanishes when all firms in the portfolio are in default. We estimate the conditional

distribution P (ρn = · | FT−n ) by the smoothed empirical distribution ν of the ratings


Algorithm 2 (Replacement Thinning) Generating a sample path of the portfoliodefault process N ′ without replacement from a path of the portfolio default processN with replacement over [τ,H], for a horizon H > τ .

1: Initialize m← 0 as we define T ′0 = τ , and set Y (ρ)← Xτ (ρ) for all ρ ∈ R.2: for n = Nτ + 1 to NH do3: Draw u ∼ U(0, 1).4: if u ≤ ZTn then5: Assign T ′m ← Tn and update m← m+ 1.6: Draw ρm ∼ ν ′, where

ν ′(ρ) =Y (ρ)

Xτ (ρ)ν(ρ)/ZTn , ρ ∈ R. (2.9)

7: Update Y (ρm)← Y (ρm)− 1.8: end if9: end for

of the defaulters in the re-sampling scenarios N(ωi) : i ≤ I, where

ν(ρ) =

∑Ii=1

∑Nτ (ωi)n=1 1ρn(ωi)=ρ + α∑I

i=1

∑Nτ (ωi)n=1 1ρn(ωi)∈Rτ + α|Rτ |

1ρ∈Rτ (2.8)

for an additive smoothing parameter α ∈ [0, 1]; see formula (2.4). This estimator

treats the observations ρn(ωi) of all paths ωi as independent samples from a common

distribution. Algorithm 2 summarizes the steps required to generate N ′ from N .

2.3 An adaptive intensity model

This section demonstrates the effectiveness of the re-sampling approach. It formu-

lates, fits and evaluates an adaptive intensity model for the CDX High Yield portfolio,

which is a standard portfolio that is referenced by a range of credit derivatives. The

fitted intensity model will be used in Sections 2.4 and 2.5 to analyze the risk of such

derivatives.


2.3.1 Re-sampling scenarios

We begin by examining the re-sampling scenarios of the default process of the CDX

High Yield Series 6 portfolio, or CDX.HY6. We adopt the rating system of Moody’s,

a leading rating agency.7 The reference portfolio consists of 2 Baa, 45 Ba, 39 B and

14 C rated firms. The realization of N∗ comes from Moody’s Default Risk Service,

and covers all Moody’s rated issuers between 1/1/1970 and 11/7/2008. We observe

a total of 1447 defaults.

Figure 2.1 shows the annual economy-wide default rate between 1970 and 2008,

along with the mean default rate for the CDX.HY6 portfolio, obtained from Algorithm

1.8 Portfolio defaults cluster heavily, indicating the positive dependence between the

defaults of reference names. The excessive default rate in 1970 is due to an exceptional

cluster of 24 railway defaults on June 21, 1970. Other major event clusters are related

to the 1987 crash and the burst of the internet bubble around 2001. These clusters

translate into substantial fluctuations of the empirical portfolio default rate that the

model portfolio intensity λ must replicate. This calls for special model features.

7We distinguish issuers in terms of their “senior rating,” which is an issuer-level rating gen-erated by Moody’s from ratings of particular debt obligations using its Senior Rating Algorithm,Hamilton (2005). We follow a common convention and subsume the categories Caa and Ca intothe category C. We also subsume the numerical sub-categories Aa1, Aa2, Aa3 into the categoryAa, and similarly for the other numerical sub-categories. Then the set of rating categories isR = Aaa, Aa, A, Baa, Ba, B, C, WR. The category WR indicates a withdrawn rating.

8Each default event in the data base has a time stamp; the resolution is one day. There are dayswith multiple events whose exact timing during the day cannot be established. Thus, the sequenceof raw event dates in the data set is not strictly increasing. Algorithm 1 requires distinct eventtimes, however. To address this issue, we assume that an event date is measured with uniformlydistributed noise. The noise is sampled when the paths of N are generated. We convert a raweconomy-wide event date to a real-valued calendar time equal to 12am on the day of the event,and draw the noise from a uniform distribution on [0, 1/365]. With this choice, the randomizationdoes not alter the original time stamp of an observed event. We have experimented with severalalternative randomization schemes but have found that the model estimation results are insensitiveto the chosen randomization scheme. Further, since the data set allows us to measure the numberof ρ-rated firms in the economy X∗t (ρ) only daily, in formula (2.3) for Z∗t we take X∗t−(ρ) as thenumber of ρ-rated firms on the day prior to the day that contains t.


1970 1975 1980 1985 1990 1995 2000 20050

0.5

1

1.5

2

2.5

3

3.5

4

Pe

rce

nt

1970 1975 1980 1985 1990 1995 2000 20050

2

4

6

8

10

12

Pe

rce

nt

Figure 2.1: Left panel : Annual default rate, relative to the number of rated namesat the beginning of a year, in the universe of Moody’s rated corporate issuers in anyyear between 1970 and 2008, as of 11/7/2008. Source: Moody’s Default Risk Service.Right panel : Mean annual default rate for the CDX.HY6 portfolio for I = 10K re-sampling scenarios.

2.3.2 Intensity specification

Our specification of λ is informed by the results of the empirical analysis in Azizpour

& Giesecke (2008b), which is based on roughly the same historical default data used

here. Using in- and out-of-sample tests, Azizpour & Giesecke (2008b) found that

prediction of economy-wide default activity based on past default timing outperforms

prediction based on exogenous economic covariates. Intuitively, the timing of past

defaults provides information about the timing of future defaults that is statistically

superior to the information contained in exogenous covariates. Further, if the past

default history is the conditioning information set, then the inclusion of additional

economic covariates does not improve economy-wide default forecast performance.

These empirical findings motivate the formulation of a parsimonious portfolio inten-

sity model whose conditioning information set is given by the past default history.

We assume that λt is a function of the path of N over [0, t]. More specifically, we


propose that λ evolves through time according to the equation

dλt = κt(ct − λt)dt+ dJt (2.10)

where λ0 > 0 is the initial intensity value, κt = κλTNt is the decay rate, ct = cλTNt is

the reversion level, and J is a response jump process given by

Jt =∑n≥1

max(γ, δλT−n )1Tn≤t. (2.11)

The quantities κ > 0, c ∈ (0, 1), δ > 0 and γ ≥ 0 are parameters. We denote by

θ the vector (κ, c, δ, γ, λ0). We give a sufficient condition guaranteeing that Nt < ∞almost surely, for all t. This condition relates the reversion level parameter c to the

parameter δ, which controls the magnitude of a jump of the intensity at an event.

Proposition 2.3.1. If c(1 + δ) < 1, then the counting process N is non-explosive.

Proof. It suffices to show that∫ t

0λsds <∞ almost surely for each t > 0. We assume

that γ = 0, without loss of generality. For s ≥ 0 and h(0) > 0, let

h(s) = ch(0) + (1− c)h(0) exp(−κh(0)s).

The function h describes the behavior of the intensity (2.10) between events. For

Tn ≤ t < Tn+1 and h(0) = λTn , we have that λt = h(t− Tn). The inverse h−1 to h is

given by

h−1(u) =1

κh(0)log

(h(0)(1− c)u− ch(0)

)for ch(0) < u ≤ h(0). Consider the first hitting time

ϕ(h(0)) = infs ≥ 0 : h(s) < h(0)/(1 + δ).


For c(1 + δ) < 1, we have

ϕ(h(0)) = h−1(h(0)/(1 + δ)) =1

κh(0)log

((1 + δ)(1− c)1− c(1 + δ)

).

Let T0 = 0. The conditional probability given FTn of the intensity jumping at an

event to a value that exceeds the value taken at the previous event is

P (λTn+1 > λTn | FTn) = P (Tn+1 − Tn < ϕ(λTn) | FTn)

= 1− exp

(−∫ ϕ(λTn )

0

h(s)ds

)= 1− exp

(− c

κlog

((1 + δ)(1− c)1− c(1 + δ)

)− δ

κ(1 + δ)

)=: M,

where M is strictly less than 1 and independent of n = 0, 1, 2, . . . It follows that the

unconditional probability P (λTn+1 > λTn) = M , for any n. Further, an induction

argument can be used to show that for any n and k = 1, 2, . . .

P (λTn+k> λTn+k−1

> · · · > λTn) = Mk.

Now letting

Ckt = ω : Nt(ω) ≥ k and λTn+1(ω) > λTn(ω) for n = Nt(ω)− 1, . . . , Nt(ω)− k

for t > 0 and k = 1, 2, . . ., we have that

P (Ckt ) = P (Nt ≥ k and λTNt > λTNt−1

> · · · > λTNt−k)

≤ P (λTNt+k > λTNt+k−1> · · · > λTNt )

= Mk


independently of t. We then conclude that

P

(∫ t

0

λs(ω)ds <∞)≥ P

(sups∈[0,t]

λs(ω) <∞)

≥ P

( ∞⋂k=1

Ckt

c)(2.12)

= 1− P( ∞⋂k=1

Ckt

)= 1− lim

k→∞P (Ck

t ) (2.13)

≥ 1− limk→∞

Mk

= 1.

Equation (2.13) is due to the fact that Ckt ∈ Ft for all k = 1, 2, . . . and C1

t ⊇ C2t ⊇ · · · .

To justify the inequality (2.12), we argue as follows. For given t > 0 and ω ∈ Ω, define

At(ω) =k ∈ 0, 1, . . . , Nt(ω)− 1 : λTk+1

(ω) ≥ λTk(ω).

Let ω ∈ ⋂∞k=1C

kt c. Then, by definition of Ck

t , either (i) Nt(ω) < ∞, or (ii) there

exists some n <∞ such that λTn+1(ω) < λTn(ω) and |At(ω) \ ATn(ω)| <∞.

In case (i), thanks to the condition |At(ω)| ≤ Nt(ω) <∞, each λTk+1(ω)−λTk(ω) <

∞ for all k ∈ At(ω). Thus, λs(ω) < ∞ for all s ∈ [0, t]. Hence, we conclude that

ω ∈ ω : sups∈[0,t] λs(ω) <∞ in this case.

Now consider case (ii). Note that for any s ∈ [0, t], the condition cλTNs (ω) <

λs(ω) implies that λs(ω) remains at infinity once λTNs (ω) achieves infinity. Hence,

λTn+1(ω) < λTn(ω) implies that λTn(ω) <∞, and we conclude that sups∈[0,Tn] λs(ω) <

∞. Moreover, due to the condition |At(ω)\ATn(ω)| <∞, we conclude that sups∈[Tn,t] λs(ω) <

∞.

Therefore, ⋂∞k=1C

kt c ⊆ ω : sups∈[0,t] λs(ω) <∞ and (2.12) is justified.

The intensity (2.10) follows a piece-wise deterministic process with right-continuous

sample paths. It jumps at an event time Tn. The jump magnitude is random. It is

equal to max(γ, δλT−n ), and depends on the intensity just before the event, which itself


is a function of the event times T1, . . . , Tn−1. The minimum jump size is γ. From Tn

onwards the intensity reverts exponentially to the level cλTn , at rate κλTn . Since the

reversion rate and level are proportional to the value of the intensity at the previous

event, they depend on the times T1, . . . , Tn and change adaptively at each default.

For Tn ≤ t < Tn+1, the behavior of the intensity is described by the FTn-measurable

function

λt = cλTn + (1− c)λTn exp(−κλTn(t− Tn)). (2.14)

The dependence of the reversion level ct, reversion speed κt and jump magnitude

max(γ, δλT−n ) on the path of the counting process N distinguishes our specification

(2.10) from the classical Hawkes (1971) model. The Hawkes intensity follows a piece-

wise deterministic process dλt = κ(c−λt)dt+ δdUt, where U = u1 + · · ·+uN and the

jump magnitudes un are drawn from a fixed distribution on R+. This model is more

rigid than our adaptive model (2.10): it imposes a global, state-independent reversion

level c and speed κ, and the magnitude of a jump in the Hawkes intensity is drawn

independently of past event times. Figure 2.2 contrasts the sample paths of (λ,N)

for the Hawkes model with those for our model (2.10). The paths exhibit different

clustering behavior, with the Hawkes model generating a more regular clustering

pattern. While Azizpour & Giesecke (2008a) found that a variant of the Hawkes

model performs well on the economy-wide default data, we had difficulty fitting this

and several other variants of the Hawkes model to the portfolio default times generated

by the re-sampling mechanism. We found the constant reversion level and speed to

be too restrictive. Our adaptive specification (2.10) relaxes these constraints while

preserving parsimony and computational tractability.

The jumps of the intensity process (2.10) are statistically important. They gener-

ate event correlation: an event increases the likelihood of further events in the near

future. This feature facilitates the replication of the event clusters seen in Figure

2.1, a fact that we establish more formally below. The intensity jumps can also be

motivated in economic terms. They represent the impact of a default on the other

firms, which is channeled through the complex web of contractual relationships in the


0 2 4 6 8 100

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35

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nsi

ty

0

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mb

er

of

De

fau

lts

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Inte

nsi

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mb

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of

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lts

Figure 2.2: Left panel : Sample path of the intensity (left scale, solid) anddefault process (right scale, dotted) for the adaptive model (2.10) with θ =(0.25, 0.05, 0.4, 0.8, 2.5), values that are motivated by our estimation results in Section2.3.4. The reversion level and speed change at each event. Algorithm 3 is used togenerate the paths. Right panel : Sample path of the intensity and default processfor the Hawkes model dλt = κ(c − λt)dt + δdUt where the jump magnitudes un = 1so U = N , λ0 = 2.5, and κ = 2.5 and c = δ = 1.5 are chosen so that the expectednumber of events over 10 years matches that of the model (2.10), roughly 37. Notethat the Hawkes intensity cannot fall below the global reversion level c.

economy. The existence of these feedback phenomena is indicated by the ripple effects

associated with the default of Lehman Brothers on September 15, 2008, and is further

empirically documented in Azizpour & Giesecke (2008a), Jorion & Zhang (2007) and

others. Our jump size specification guarantees that the impact of an event increases

with the default rate prevailing at the event: the weaker the firms the stronger the

impact. An alternative motivation of the intensity jumps is Bayesian learning: an

event reveals information about the values of unobserved event covariates, and this

leads to an update of the intensity, see Collin-Dufresne, Goldstein & Helwege (2009),

Duffie et al. (2009) and others.


Algorithm 3 (Default Time Simulation) Generating a sample path of the port-folio default process N over [τ,H] for a horizon H > τ and the model (2.10).

1: Draw (Nτ , TNτ , λTNτ ) uniformly from (Nτ (ωi), TNτ (ωi), λTNτ (ωi)) : i ≤ I.2: Initialize (n, S)← (Nτ , τ) and λS ← cλTn + (1− c)λTn exp(−κλTn(S − Tn)).3: loop4: Draw E ∼ Exp(λS) and set T ← S + E .5: if T > H then6: Exit loop.7: end if8: λT ← cλTn + (λS − cλTn) exp(−κλTn(T − S))9: Draw u ∼ U(0, 1).

10: if u ≤ λT/λS then11: λT ← λT + max(γ, δλT )12: Assign Tn+1 ← T and update n← n+ 1.13: end if14: Set S ← T and λS ← λT .15: end loop

2.3.3 Event and loss simulation

The piece-wise deterministic dynamics of the intensity (2.10) generate computational

advantages for model calculation. One benefit of this feature is that it allows us to

estimate the distribution of N and related quantities by exact Monte Carlo simulation

of the jump times of N . The inter-arrival times of N can be generated sequentially

by acceptance-rejection sampling from a dominating counting process with intensity

λTNt− ≥ λt−.

The steps are summarized in Algorithm 3. In Step 1, we draw the state at the

analysis time τ from the re-sampling scenarios of N , and the corresponding fitted

intensity scenarios. The rate λS of the dominating Poisson process, from which a

candidate default time is generated in Step 4, is not only redefined at each acceptance,

but also at each rejection of a candidate time. This improves the efficiency of the

algorithm. The acceptance probability is increased and fewer candidate times are

wasted.

Algorithm 3 leads to a trajectory of the default process N for a portfolio with

replacement of defaulters. A trajectory of the corresponding portfolio loss process


L = `1 + · · ·+ `N is obtained by drawing the loss `n at each event time Tn of N . The

distribution µ` of `n is estimated by the empirical distribution of the losses associated

with all defaults in the re-sampling scenarios N(ωi) : i ≤ I.Once a path of N is generated, Algorithm 2 can be used to obtain a path of the

default process N ′ for a portfolio without replacement of defaulters. A trajectory of

the corresponding portfolio loss process L′ = `′1 + · · · + `′N ′ is obtained by drawing

the loss `′n at each event time from µ`.

Algorithm 3 is easy to implement and runs fast. For example, generating 100K

paths of N over 5 years for the fitted model parameters in Table 2.1 takes 1.67 seconds.

Generating 100K paths of N ′ takes, in the same configuration, 2.14 seconds.9

2.3.4 Likelihood estimators

Another advantage of the piece-wise deterministic intensity dynamics (2.10) is that

the log-likelihood function in problem (3.3) takes a closed form that can be computed

exactly. Based on I = 10K re-sampling scenarios, we address the problem (3.3) with

the Nelder-Mead algorithm, which uses only function values. The algorithm is initial-

ized at a set of random parameter values, which are drawn from a uniform distribution

on the parameter space Θ = (0, 2)×(0, 1)×(0, 2)2×(0, 20). For each of 100 randomly

chosen initial parameter sets, the algorithm converges to the optimal parameter value

θ reported in Table 2.1.10 We also provide asymptotic and bootstrapping confidence

intervals. The left panel of Figure 2.3 shows the path of the fitted mean intensity

λ = I−1∑I

i=1 λθ(ωi).

To provide some perspective on the parameter estimates, we employ an alternative

inference procedure. Instead of maximizing the total likelihood (3.3) associated with

9The numerical experiments in this chapter were performed on a desktop PC with an AMDAthlon 1.00 GHz processor and 960 MB of RAM, running Windows XP Professional. The codeswere written in C++ and the compiler used was Microsoft Visual C++ .NET version 7.1.3088. Forrandom number generation, numerical optimization, and numerical root-finding, we used the GNUScientific Library, Version 1.11.

10In the computing environment described in Footnote 9, it takes 0.15 seconds to evaluate thelog-likelihood function for I = 10K and a given parameter set. The full optimization takes 146.88seconds when the initial parameters are set to the mid-points of the parameter space.


Parameter κ c δ γ λ0

MLE 0.254 0.004 0.419 0.810 8.70995% CI (A) (0.252,0.255) (0.004,0.004) (0.417,0.422) (0.806,0.813) (8.566,8.851)95% CI (B) (0.250,0.258) (0.003,0.004) (0.406,0.433) (0.791,0.829) (8.538,8.882)Median 0.263 0.004 0.430 0.779 9.405

Table 2.1: Maximum likelihood estimates of the parameters of the intensity λ forthe CDX.HY6 portfolio, along with estimates of asymptotic (A) and bootstrapping(B) 95% confidence intervals (10K bootstrap samples were used). The “Median” rowindicates the median of the empirical distribution of the per-path MLEs over all re-sampling paths. The estimates are based on I = 10K re-sampling scenarios, generatedby Algorithm 1 from the observed defaults in the universe of Moody’s rated namesfrom 1/1/1970 to 11/7/2008.

all paths of N , we maximize the path log-likelihood

supθ(ωi)∈Θ

∫ τ

0

(log λθ(ωi)s− (ωi)dNs(ωi)− λθ(ωi)s (ωi)ds)

for each i = 1, 2, . . . , I. The last row in Table 2.1 shows the median of the empirical

distribution of the per-path MLE θ(ωi) over all paths ωi. These values are in good

agreement with the MLEs, supporting our total likelihood estimation strategy.

2.3.5 Testing in-sample fit

Above we have stressed the computational advantages of our intensity model (2.10).

In this section and the next, we evaluate the model statistically. We show that the

model fits the default data, and that the fitted model leads to accurate event forecasts.

We evaluate the joint specification of Z∗ and λ based on Proposition 2.2.2.11

The right panel of Figure 2.3 shows the fitted path of the economy-wide intensity

λ∗ = λ/Z∗, which defines the time change. We test the Poisson property of the

time-scaled inter-arrival times W ∗n =

∫ T ∗nT ∗n−1

1Z∗sλsds using a Kolmogorov-Smirnov (KS)

11Proposition 2.2.2 requires the process Z∗ to be strictly positive. The fitted values Z∗t are indeedstrictly positive during our sample period. We approximate the paths of the fitted process Z∗ andthe fitted mean intensity λ on a discrete-time grid with daily spacing. When multiple economy-widedefaults occur on the same day, then the arrival times are spaced equally.


1970 1975 1980 1985 1990 1995 2000 20050

2

4

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8

10

12

14

16

18

20Fitted Mean Portfolio Intensity

1970 1975 1980 1985 1990 1995 2000 20050

10

20

30

40

50

60

70

80

90

100

110Semi−annual DefaultsFitted Economy−wide Intensity

Figure 2.3: Left panel : Fitted mean portfolio intensity λ = I−1∑I

i=1 λθ(ωi) vs.

[5%, 95%] percentiles (boxes), [1%, 99%] percentiles (whiskers) and the mean numberof portfolio defaults in any given year between 1970 and 2008, based on I = 10Kre-sampling scenarios for the CDX.HY6. Right panel : Fitted economy-wide intensityλ/Z∗ vs. economy-wide defaults between 1970 and 2008, semi-annually. The fittedintensity matches the time-series fluctuation of observed economy-wide default rates.

test, a QQ plot, and Prahl’s (1999) test. The KS test addresses the deviation of

the empirical distribution of the (W ∗n) from their theoretical standard exponential

distribution. Prahl’s test is particularly sensitive to large deviations of the (W ∗n)

from their theoretical mean 1. Prahl shows that if the (W ∗n) are independent samples

from a standard exponential distribution, then

M =1

m

∑n:W ∗n<µ

(1− W ∗

n

µ

)(2.15)

is asymptotically normally distributed with mean µM = e−1− 0.189/m and standard

deviation σM = 0.2427/√m, where m = N∗τ is the number of arrivals and µ is the

sample mean of (W ∗n)n≤m. We reject the null hypothesis of a correct joint specification

of Z∗ and λ if the test statistic ∆M = (M−µM)/σM lies outside of the interval (−1, 1).

Figure 2.4 contrasts the empirical distribution of the (W ∗n) with the theoretical

standard exponential distribution. The KS test has a p-value of 0.094, indicating


0 1 2 3 4 5 6 7 8 90

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Th

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s

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

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1

Figure 2.4: In-sample fitness tests: empirical distribution of time-scaled, economy-wide inter-arrival times generated by the fitted (mean) paths of Z∗ and λ, based onI = 10K re-sampling scenarios for the CDX.HY6. Left panel : Empirical quantiles oftime-scaled inter-arrival times vs. theoretical standard exponential quantiles. Rightpanel : Empirical distribution function (solid) of time-scaled inter-arrival times vs.theoretical standard exponential distribution function (dotted) along with 1% and5% bands.

that the deviation of the empirical distribution from the theoretical distribution is not

statistically significant at standard confidence levels. The value of Prahl’s test statistic

∆M = 0.87 leads to the same conclusion. The results of these and other12 tests

suggest that the economy-wide intensity λ∗ generated by (2.3) and (2.10) replicates

the substantial time-series variation of default rates during 1970–2008. This means

that our adaptive intensity model (2.10) captures the clustering of defaults observed

during the sample period.

A benefit of the time-scaling tests is that they can be applied to alternative model

formulations, facilitating a direct comparison of fitting performance. Consider the

bottom-up specification of Das et al. (2007a), which appears to be much richer than

12To detect a possible serial dependence of the W ∗n , we also consider the test statistics SC1 andSC2 described by Lando & Nielsen (2009). These tests check the dependence of the number ofre-scaled events in non-overlapping time bins. Under the null of Poisson arrivals, the event countsare independent. We cannot reject the null for bin sizes 2, 4, 6, 8 and 10 at standard confidencelevels.


ours: they specify firm-level intensity models with conditioning information given by

a set of firm-specific and macro-economic covariates. Das et al. (2007a) find that

the time-scaled, economy-wide times generated by their model deviate significantly

from those of a Poisson process. In particular, they find that their specification does

not completely capture the event clusters in the data. Thus, they reject this model

formulation at standard confidence levels. This may indicate that, relative to our

portfolio-level formulation with conditioning information given by past default timing,

the additional modeling and estimation effort involved in a firm-level intensity model

formulation with a large set of exogenous covariates may not translate into better fits

and forecast performance.

Finally we note that the time-scaling tests can also be used to evaluate the spec-

ification of λ directly on the re-sampling scenarios N(ωi) : i ≤ I. For each path

ωi, we time-scale the portfolio default process N(ωi) with its fitted compensator

A(ωi) =∫ ·

0λs(ωi)ds, and calculate the statistics of the KS test and Prahl’s test for

the time-scaled inter-arrival times Wn =∫ Tn(ωi)

Tn−1(ωi)λθs(ωi)ds, which can be calculated

exactly thanks to formula (2.14). The 95% confidence interval for the p-value of the

KS test is 0.24± 0.01. The 95% confidence interval for Prahl’s test statistic, defined

for (Wn) in analogy to (2.15), is 0.81±0.07. Also these tests indicate that the adaptive

model (2.10) is well-specified.

2.3.6 Testing out-of-sample loss forecasts

Next we assess the out-of-sample event forecast performance of the fitted model (2.10).

We contrast the loss forecast for a test portfolio without replacement with the actual

loss in the test portfolio. The firms in the test portfolio are randomly selected from the

universe of rated issuers, such that the test portfolio has the same rating composition

as the CDX.HY6.13 We apply Algorithms 3 and 2 to obtain an unbiased estimate of

the conditional distribution of the incremental test portfolio loss L′τ+1−L′τ given Fτ .We then contrast this distribution with the actual loss in the test portfolio during

13We do not use the CDX.HY6 itself, since it was formed only in 2006, and we want to test theloss process over a longer time period, starting in 1996. The analysis would suffer from survivorshipbias if we were to take the CDX portfolio itself.


1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 20070

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Figure 2.5: Out-of-sample test of loss forecasts for a test portfolio with the samerating composition as the CDX.HY6. We show the [25%, 75%] percentiles (box), the[15%, 85%] percentiles (whiskers) and the median (horizontal line) of the fitted condi-tional distribution of the incremental portfolio loss L′τ+1 − L′τ given Fτ for τ varyingannually between 1/1/1996 and 1/1/2007, estimated from 100K paths generated byAlgorithms 3 and 2, and the realized portfolio loss during [τ, τ + 1]. Left panel :The test portfolio is selected at the beginning of each period. Right panel : The testportfolio is selected in 1996.

(τ, τ + 1], for yearly analysis times τ between 1/1/1996 and 1/1/2007.

Figure 2.5 shows the results for two settings that represent common situations in

practice. In one setting (left panel), we roll over the test portfolio in each 1 year test

period. That is, for each test period, we select a new test portfolio of 100 issuers at

the beginning of the period, estimate the portfolio loss distribution based on data

available at the beginning of the period, and compare it with the realized portfolio

loss during the forecast period. In the other setting (right panel), we select the test

portfolio only once, in 1996. For each period, we estimate the loss distribution for the

portfolio of names that have survived to the beginning of the period, and compare it

with the realized portfolio loss during the forecast period. This setting is motivated

by the situation of a buy and hold investor. The graphs indicate that the portfolio

loss forecasts are very accurate.


2.4 Synthetic collateralized debt obligations

Having established the fitting and forecast performance of the adaptive model (2.10)

of portfolio default timing, we use it to estimate the exposure of an investor selling

default protection on a tranche of a synthetic CDO. A synthetic CDO is based on

a portfolio of C single-name credit swaps with notional 1 and maturity date H, the

maturity date of the CDO. The constituent credit swaps are referenced on bonds

issued by rated firms. When a bond issuer defaults, the corresponding credit swap

pays the loss associated with the event. A defaulter is not replaced.

A tranche of a synthetic CDO is a swap contract specified by a lower attachment

point K ∈ [0, 1), and an upper attachment point K ∈ (K, 1]. An index swap has

attachment points K = 0 and K = 1. The protection seller agrees to cover all losses

due to default in the reference portfolio, provided these losses exceed a fraction K of

the total notional C of the reference contracts but are not larger than a fraction K

of C. In exchange, the protection buyer pays to the protection seller an upfront fee

at inception, and a quarterly fee, both of which are negotiated at contract inception.

We estimate the risk exposure of the tranche protection seller, which is measured

by the conditional distribution at contract inception of the cumulative tranche loss,

i.e., the portfolio loss allocated to the tranche over the life of the contract. With the

convention that the portfolio loss at contract inception is equal to zero, the cumulative

tranche loss at a post-inception time t ≤ H is given by the “call spread”

Ut(K,K) = (L′t − CK)+ − (L′t − CK)+. (2.16)

The left panel of Figure 2.6 shows the conditional distribution of the normalized 5 year

cumulative tranche loss UH(K,K)/C(K − K) for the CDX.HY6 on 3/27/2006, the

Series 6 contract inception date,14 for each of several standard attachment point pairs.

The maturity date H for Series 6 contracts is 6/27/2011.15 To estimate the tranche

14Every 6 months, the CDX High Yield index portfolio is “rolled.” That is, a new portfolio with anew serial number is formed by replacing names that have defaulted since the last roll, and possiblyother names. The index and tranche swaps we consider are tied to a fixed series.

15Although the actual time to maturity is 5 years and 3 months at contract inception, here andbelow, we follow the market convention of referring to the contract as a “5 year contract.” At an


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Figure 2.6: Kernel smoothed conditional distribution of the normalized 5 year cu-mulative tranche loss for the CDX.HY6 on 3/27/2006, the Series 6 contract inceptiondate, for each of several standard attachment point pairs. Here, H represents the ma-turity date 6/27/2011. Left panel : Actual distribution, estimated using the model andfitting methodology developed in Sections 2.2 and 2.3, based on I = 10K re-samplingscenarios for N , and 100K replications. Right panel : Risk-neutral distribution, esti-mated from the market prices paid for 5 year tranche protection on 3/27/2006 usingthe method explained in Appendix A, based on 100K replications.

loss distribution, we generate default scenarios during 1/1/1970 – 3/27/2006 for the

CDX portfolio from Moody’s default history by the re-sampling Algorithm 1. Next

we estimate the intensity model (2.10) from these scenarios, as described in Section

2.3.4, with parameter estimates reported in Table 2.2. Then we generate event times

during the prediction interval 3/27/2006 – 6/27/2011 from the fitted intensity using

Algorithm 3, which we thin using Algorithm 2 to obtain paths of portfolio default

times and losses without replacement. The trajectories for U(K,K) thus obtained

lead to an unbiased estimate of the desired distribution.

The loss distributions indicate the distinctions between the risk profiles of the

different tranches. The equity tranche, which has attachment points 0 and 10%,

carries the highest exposure among all tranches. For the equity protection seller,

index roll, new “5 year contracts” are issued, and these mature in 5 years and 3 months from theroll date.


Parameter κ c δ γ λ0

MLE 0.260 0.003 0.439 0.856 8.49495% CI (A) (0.259,0.261) (0.003,0.003) (0.437,0.441) (0.853,0.859) (8.368,8.621)95% CI (B) (0.256,0.265) (0.003,0.004) (0.424,0.454) (0.836,0.876) (8.327,8.650)

Table 2.2: Maximum likelihood estimates of the parameters of the portfolio intensityλ for the CDX.HY6, along with estimates of asymptotic (A) and bootstrapping (B)95% confidence intervals (10K bootstrap samples were used). The estimates arebased on I = 10K re-sampling scenarios for N , generated by Algorithm 1 from theobserved economy-wide defaults in Moody’s universe of rated names from 1/1/1970to 3/27/2006.

the probability of trivial losses is roughly 15%, and the probability of losing the

full tranche notional is 20%. For the (10%, 15%)-mezzanine protection seller, the

probabilities are roughly 80% and 10%, respectively. The mezzanine tranche is less

risky, since the equity tranche absorbs the first 10% of the total portfolio loss. For

the (15%, 25%)-senior protection seller, the probabilities are roughly 90% and 5%,

respectively.

To provide some perspective on these numbers, we also estimate the risk-neutral

tranche loss distributions implied by the market prices paid for 5 year tranche protec-

tion on 3/27/2006. The estimation procedure, explained in Appendix A, is based on

the intensity model (2.10) relative to a risk-neutral pricing measure. The risk-neutral

intensity parameters are chosen such that the model prices match the market index

and tranche prices as closely as possible. The fitting errors are small, which indi-

cates that our adaptive intensity model (2.10) performs also well under a risk-neutral

pricing measure.

The right panel of Figure 2.6 shows the risk-neutral distribution of UH(K,K)/C(K−K) for the CDX.HY6, for each of several standard attachment point pairs. For the

equity protection seller, the risk-neutral probability of trivial losses is zero, and the

risk-neutral probability of losing the full tranche notional is 70%. Compare with the

actual probabilities indicated in the left panel of Figure 2.6. For any tranche, the

risk-neutral probability of losing more than any given fraction of the tranche notional

is much higher than the corresponding actual probability. The distinction between


010

2030

40

34

56

78

910

0

0.02

0.04

0.06

0.08

0.1

Loss (%)

Time toMaturity (Years)

010

2030

40

34

56

78

910

0

0.02

0.04

0.06

0.08

0.1

Loss (%)

Time toMaturity (Years)

Figure 2.7: Kernel smoothed conditional distribution of the normalized cumula-tive portfolio loss L′H/C for the CDX.HY6 on 3/27/2006 for horizons H varying be-tween 3/27/2009 and 3/27/2016. Left panel : Actual distribution, estimated using themodel and fitting methodology developed in Sections 2.2 and 2.3, based on I = 10Kre-sampling scenarios for N , and 100K replications. Right panel : Risk-neutral dis-tribution, estimated from the market prices paid for 5 year tranche protection on3/27/2006 using the method explained in Appendix A, based on 100K replications.

the probabilities reflects the risk premium the protection seller requires for bearing

exposure to the correlated corporate default risk in the reference portfolio. Figure

2.7 shows the distributions of the normalized cumulative portfolio loss L′H/C for mul-

tiple horizons H. For all horizons, the risk-neutral distribution has a much fatter

tail than the corresponding actual distribution. In other words, when pricing index

and tranche contracts, the market overestimates the probability of extreme default

scenarios relative to historical default experience.

Our tools can also be used to analyze more complex investment positions. In-

vestors often trade the CDO capital structure, i.e. they simultaneously sell and buy

protection in different tranches. A popular trade used to be the “equity-mezzanine”

trade, in which the investor sells equity protection and hedges the position by buy-

ing mezzanine protection with matching notional and maturity. Figure 2.8 con-

trasts the conditional distribution on 3/27/2006 of the normalized cumulative loss


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Loss (%)

Cu

mu

lativ

e L

oss

Dis

trib

utio

n

Equity−MezzanineEquity Only

Figure 2.8: Kernel smoothed conditional distribution on 3/27/2006 of the normal-ized cumulative loss (UH(0, 0.1)− UH(0.1, 0.15))/0.1C associated with selling equityprotection and buying mezzanine protection with matching notional and maturity,along with the distribution of the normalized cumulative equity loss UH(0, 0.1)/0.1C,for the CDX.HY6.

(UH(0, 0.1) − UH(0.1, 0.15))/0.1C generated by this trade with the conditional dis-

tribution of the normalized cumulative equity loss UH(0, 0.1)/0.1C, both for the

CDX.HY6. While the mezzanine hedge does not alter the probability of trivial losses

in the equity-only position, it does reduce the probability of a total loss of notional

from 20% to virtually zero. This is because the mezzanine protection position gen-

erates cash flows when equity is wiped out. The magnitude of these cash flows is

however capped at 50% of the total equity notional, and this property generates the

point mass at 5% in the loss distribution of the equity-mezzanine position.

We can also measure the risk of positions in tranches referenced on different port-

folios. For example, an investor may sell and buy protection on several of the refer-

ence portfolios in the CDX family, including the High Yield, Investment Grade and

Crossover portfolios. In this case, we estimate default and loss processes for each of

the portfolios based on the realization of the economy-wide default process. We gen-

erate events for each portfolio, and then aggregate the corresponding position losses

as in the equity-mezzanine case.


Figure 2.9: Cash flow “waterfall” of a sample cash CDO.

2.5 Cash collateralized debt obligations

Next we use the adaptive model to estimate the exposure of an investor in a tranche

of a cash CDO. A cash CDO is based on a portfolio of corporate bonds, mortgages

or other credit obligations, which is bought by a special purpose vehicle (SPV) that

finances the purchase by issuing a collection of tranches that may pay coupons. The

interest and recovery cash flows generated by the reference bonds are collected by

the SPV in a reserve account, and are then allocated to the tranches according to a

specified prioritization scheme. Losses due to defaults are absorbed by the tranche

investors, usually in reverse priority order. Figure 2.9 illustrates a typical structure.

The reference portfolio of a cash CDO may be actively managed. In this case, the

asset manager can buy and sell collateral bonds, and replace defaulted obligations.

We analyze a basic unmanaged cash CDO, in which the reference bonds are held

to maturity and defaulted bonds are not replaced. We assume that the C reference

bonds are straight coupon bonds with notional 1, maturity date H equal to the

CDO maturity date, issuance date t0 equal to the CDO inception date, coupon dates

(tm)1≤m≤M with tM = H, and per-period coupon rate v. The total interest income to


the SPV in period m is therefore

W (m) = v(C −N ′tm). (2.17)

When a reference bond defaults, the SPV collects the recovery at the coupon date

following the event. The total recovery cash flow in period m is

K(m) = N ′tm −N′tm−1− (L′tm − L

′tm−1

).

The total cash flows from coupons and recoveries are invested in a reserve account

that earns interest at the risk-free rate r. For period m, they are given by

B(m) =

W (m) +K(m) if 1 ≤ m < M

W (m) +K(m) + C −N ′tm if m = M.

The SPV issues three tranches, of which two are debt tranches that promise to

pay a specified coupon at times (tm). There is one senior debt tranche, represented

by a sinking-fund bond with initial principal p1 and per-period coupon rate c1, and a

junior debt tranche that is represented by a sinking-fund bond with initial principal

p2 and per-period coupon rate c2. The initial principal of the residual equity tranche

is p3 = C − p1 − p2. Each tranche has maturity date H.

For the debt tranches j = 1, 2 and coupon period m, we denote by Fj(m) the

remaining principal. The scheduled interest payment is Fj(m)cj, and the accrued

unpaid interest is Aj(m − 1). If Qj(m), the actual interest paid in period m, is less

than Fj(m)cj, then the difference is accrued at cj to generate an accrued unpaid

interest of

Aj(m) = (1 + cj)Aj(m− 1) + cjFj(m)−Qj(m).

The actual payments to the debt tranches are prioritized. We consider two priori-

tization schemes, the uniform and fast schemes, which were introduced by Duffie &

Garleanu (2001) and which are reviewed in Appendix B for completeness. A pri-

oritization scheme specifies the actual interest payment Qj(m), the pre-payment of

principal Pj(m), and the contractual unpaid reduction in principal Jj(m). The total


Reference Uniform Scheme Fast SchemeBond Senior Junior Senior Junior

Annualized Par Coupon Rate (bp) 811.23 476.58 1299.42 483.38 749.75Par Coupon Spread (bp) 339.20 23.68 846.52 11.29 277.77

Table 2.3: Fitted annualized par coupon rates and spreads on 3/27/2006 forthe constituent bonds and debt tranches of a 5 year cash CDO referenced on theCDX.HY6, for each of two standard prioritization schemes. The principal values are(p1, p2, p3) = (85, 10, 5). The fitting procedure is described in Appendix C.

cash payment in period m is Qj(m) + Pj(m). The remaining principal after interest

payments is

Fj(m) = Fj(m− 1)− Pj(m)− Jj(m) ≥ 0.

The par coupon rate on tranche j is the scheduled coupon rate cj with the property

that the initial market value of the bond is equal to its initial face value Fj(0) = pj.

The residual equity tranche does not make scheduled coupon or principal payments

so c3 = 0. Instead, at maturityH, after the debt tranches have been paid as scheduled,

any remaining funds in the reserve account are allocated to the equity tranche.

We analyze the exposure of a tranche investor for a 5 year cash CDO referenced

on the CDX.HY6 of C = 100 names. This choice of reference portfolio allows us

to compare the results with those for the synthetic CDO. Before we can start this

analysis, we need to price the reference bonds and the debt tranches.16 The first

step is to estimate the par coupon rate of the reference bonds. This is the scheduled

coupon rate v∗ with the property that the initial market value of a reference bond is

equal to its initial face value 1. Given v∗, the second step is to estimate the par coupon

rates c∗j of the debt tranches. Appendix C gives details on these steps, and Table 2.3

reports the annualized par coupon rates and spreads. The par coupon spread is the

difference between the par coupon rate and the (hypothetical) par coupon rate that

would be obtained if the reference bonds were not subject to default risk.

Next we estimate the conditional distribution of the discounted cumulative tranche

16In practice, this is done by the SPV, which then offers the tranche terms to potential investors.We have to perform this task here since we do not have access to actual data for the structure weanalyze.


loss, which is the loss the tranche investor faces during the life of the contract, dis-

counted at the risk-free rate r. At a post-inception time t ≤ H, the discounted

cumulative tranche loss is given by

Ujt(H, pj, v∗, c∗1, c

∗2) = V jt(H, pj, v

∗, c∗1, c∗2)− Vjt(H, pj, v∗, c∗1, c∗2) (2.18)

where Vjt(H, pj, v∗, c∗1, c

∗2) is the present value at time t of the coupon and principal

cash flows actually paid to tranche j over the life of the tranche, and V jt(H, pj, v∗, c∗1, c

∗2)

is the present value of these cash flows assuming no defaults occur during the remain-

ing life. For a debt tranche, V jt(H, pj, v∗, c∗1, c

∗2) represents the present value of all

scheduled coupon and principal payments. For the equity tranche, this quantity rep-

resents the present value of the reserve account value after the debt tranches have

been paid as scheduled.

Note that due to the complex structure of the cash flow prioritization, the tranche

loss Ujt(H, pj, v∗, c∗1, c

∗2) does not admit a simple analytic expression in terms of the

portfolio default count and loss of the reference portfolio. This leaves simulation as

the only feasible tool for analyzing the cash CDO tranche loss, regardless of whether

or not the models of the portfolio default count and loss processes are analytically

tractable.

Figure 2.10 shows the fitted conditional distribution on 3/27/2006 of the normal-

ized loss Ujτ (H, pj, v∗, c∗1, c

∗2)/V jτ (H, pj, v

∗, c∗1, c∗2) for a 5 year structure whose matu-

rity date H is 6/27/2006, for each of several tranches. To estimate that distribution,

we proceed as described in Section 2.4 for the synthetic CDO, and generate paths

of the fitted portfolio default and loss processes. These are then fed into the cash

flow calculator, which computes Vjt(H, pj, v∗, c∗1, c

∗2) for each path. We assume that

coupons are paid quarterly. The risk-free interest rate r is deterministic, and is esti-

mated from Treasury yields for multiple maturities on 3/27/2006, obtained from the

website of the Department of Treasury.

As in the synthetic CDO case, the risk profile of a cash CDO tranche depends on

the degree of over-collateralization, i.e. the location of the attachment points. It also

depends on the prioritization scheme specified in the CDO terms. With the uniform


0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tranche Loss (%)

Cu

mu

lativ

e L

oss

Dis

trib

utio

n

SeniorJuniorResidual

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tranche Loss (%)

Cu

mu

lativ

e L

oss

Dis

trib

utio

n

SeniorJuniorResidual

Figure 2.10: Kernel smoothed conditional distribution on 3/27/2006 of the normal-ized discounted loss Ujτ (H, pj, v

∗, c∗1, c∗2)/V jτ (H, pj, v

∗, c∗1, c∗2) for a 5 year cash CDO

referenced on the CDX.HY6, whose maturity date H is 6/27/2011. The initial trancheprincipals are p1 = 85 for the senior tranche, p2 = 10 for the junior tranche, andp3 = 5 for the residual equity tranche. We apply the model and fitting methodologydeveloped in Sections 2.2 and 2.3, based on I = 10K re-sampling scenarios for N ,and 100K replications. Left panel : Uniform prioritization scheme. Right panel : Fastprioritization scheme.

scheme, the cash flows generated by the reference bonds are allocated sequentially

to the debt tranches according to priority order, and default losses are applied to

all tranches in reverse priority order. With the fast scheme, the cash flows are used

to retire the senior tranche as soon as possible. After the senior tranche is retired,

the cash flows are used to service the junior tranche. After the junior tranche is

retired, any residual cash flows are distributed to the equity tranche. Therefore, the

senior tranche investor is less exposed under the fast scheme: the probability of zero

losses is increased from roughly 95% for the uniform scheme to roughly 98%. This

reduction in risk is reflected by the par coupon spreads reported in Table 2.3. For

the junior investor, the probability of zero losses increases from roughly 82% for the

uniform scheme to roughly 93% for the fast scheme, but the probability of a loss equal

to the present value of all scheduled coupon and principal payments increases from

zero to about 3%. Nevertheless, the junior tranche commands a much smaller spread


−100 −50 0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tranche P&L (%)

Cu

mu

lativ

e D

istr

ibu

tion

Uniform SchemeFast Scheme

Figure 2.11: Kernel smoothed conditional distribution on 3/27/2006 of the nor-malized discounted profit and loss (V3τ (H, 5, v

∗, c∗1, c∗2) − 5)/5 for the residual equity

tranche of a 5 year cash CDO referenced on the CDX.HY6, whose maturity date His 6/27/2011. We apply the model and fitting methodology developed in Sections 2.2and 2.3, based on I = 10K re-sampling scenarios for N , and 100K replications.

under the fast scheme. The risk reduction for the debt tranche investors comes at

the expense of the equity investor: while the probability of zero losses is about 15%

for both schemes, the probability of a loss equal to the present value of the reserve

account value after the debt tranches have been paid as scheduled increases from zero

to roughly 8% for the fast scheme. On the other hand, the equity investor has a

much higher upside under the fast scheme. Figure 2.11 shows the distribution of the

normalized discounted profit and loss (V3τ (H, 5, v∗, c∗1, c

∗2)−5)/5 for an equity tranche

position. For the equity investor, the probability of more than doubling the principal

p3 after discounting is roughly 75% under the fast scheme, while it is only 60% under

the uniform scheme. Figure 2.12 shows that for the debt tranches, the upside is much

more limited, for any scheme. For example, the normalized discounted profit on the

senior tranche can be at most 0.5% under the fast scheme and about 1% under the

uniform scheme.

A standard measure to quantify the risk of a position is value at risk, a quantile


−12 −10 −8 −6 −4 −2 0 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tranche P&L (%)

Cu

mu

lativ

e D

istr

ibu

tion


−100 −50 0 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tranche P&L (%)

Cu

mu

lativ

e D

istr

ibu

tion


Figure 2.12: Kernel smoothed conditional distribution on 3/27/2006 of the normal-ized discounted profit and loss (Vjτ (H, pj, v

∗, c∗1, c∗2)−pj)/pj for the debt tranches of a

5 year cash CDO referenced on the CDX.HY6, whose maturity date H is 6/27/2011.We apply the model and fitting methodology developed in Sections 2.2 and 2.3, basedon I = 10K re-sampling scenarios for N , and 100K replications. Left panel : Seniortranche with principal p1 = 85. Right panel : Junior tranche with principal p2 = 10.

of the position’s loss distribution. We estimate the value at risk at time t ≤ H of a

position in a cash CDO tranche maturing at H, given by

VaRjt(α,H, pj, v∗, c∗1, c

∗2) = inf

x ∈ R : P

(Ujt(H, pj, v

∗, c∗1, c∗2)

V jt(H, pj, v∗, c∗1, c∗2)≤ x

∣∣∣Ft) > α

for some level of confidence α ∈ (0, 1). Figure 2.13 shows the 99.5% value at risk for

the 5 year senior tranche with maturity date H given by 6/27/2011, for analysis times

varying weekly between 3/27/2006 and 11/7/2008, for each of the two prioritization

schemes. Assuming risk capital is allocated according to the value at risk, a value in

the time series represents the amount of risk capital that is needed at that time to

support the tranche position over its remaining life. For both prioritization schemes,

the time series behavior reflects the rising trend in corporate defaults that started in

2008 and that is evidenced in Figure 2.1. The fast scheme requires less risk capital

than the uniform scheme but leads to higher volatility of risk capital.


Mar−06 Aug−06 Jan−07 Jun−07 Nov−07 Apr−08 Sep−0812

13

14

15

16

17

18

19

20

21

22

23

99

.5%

Va

lue

−a

t−R

isk

(%)

Uniform (Senior)

Mar−06 Aug−06 Jan−07 Jun−07 Nov−07 Apr−08 Sep−086

7

8

9

10

11

12

13

14

15

16

99

.5%

Va

lue

−a

t−R

isk

(%)

Fast (Senior)

Figure 2.13: 99.5% value at risk VaR1τ (0.995, H, 85, v∗, c∗1, c∗2) for the 5 year senior

tranche of a cash CDO referenced on the CDX.HY6 with maturity date 6/27/2011, forτ varying weekly between 3/27/2006 and 11/7/2008, based on I = 10K re-samplingscenarios for N , and 100K replications. Left panel : Uniform prioritization scheme.Right panel : Fast prioritization scheme.

2.6 Conclusion

This chapter develops, implements and validates stochastic methods to measure the

risk of investment positions in collateralized debt obligations and related credit deriva-

tives tied to an underlying portfolio of defaultable assets. The ongoing financial crisis

highlights the need for sophisticated yet practical tools allowing potential and ex-

isting investors as well as regulators to quantify the exposure associated with such

positions, and to accurately estimate the amount of risk capital required to support

a position.

The key to address the risk analysis problem is a model of default timing that cap-

tures the default clustering in the underlying portfolio, and a method to estimate the

model parameters based on historical default experience. This chapter contributes

to each of these two sub-problems. It formulates an adaptive, intensity-based point

process model of default timing that performs well according to in- and out-of-sample

tests. Moreover, it develops a maximum likelihood approach to estimating point pro-

cess models. This approach is based on an acceptance/rejection re-sampling scheme


that generates alternative portfolio default histories from the available economy-wide

default process.

The point process model and inference method have potential applications in

other areas dealing with correlated event arrivals, within and beyond financial engi-

neering. Financial engineering examples include the pricing and hedging of securities

exposed to correlated default risk, and order book modeling. Other example areas

are insurance, healthcare, queuing, and reliability.

Chapter 3

Systemic Risk: What Defaults Are

Telling Us

This chapter develops maximum likelihood estimators of the term structure of sys-

temic risk in the U.S. financial sector, defined as the conditional probability of failure

of a large number of financial institutions. The estimators are based on a new dy-

namic hazard model of failure timing that captures the influence of time-varying

macro-economic and sector-specific risk factors on the likelihood of failures, and the

impact of risk spillovers due to contagion or incomplete information about relevant

risk factors. The estimation results, which cover the period January 1987 to Decem-

ber 2008, provide strong evidence for the presence of failure clustering not caused by

variations in the observable explanatory covariates, which include the trailing return

on the S&P 500 index, the lagged slope of the U.S. yield curve, the default and TED

spreads, and other sector-specific variables.

This chapter is joint work with Kay Giesecke.

3.1 Introduction

Systemic risk in the financial sector is difficult to measure, and its sources are poorly

understood. This makes it hard for regulators and policy makers to address it effec-

tively. A major challenge is to take account of the risk spillovers that may be caused

44

CHAPTER 3. SYSTEMIC RISK: WHAT DEFAULTS ARE TELLING US 45

by incomplete information about relevant risk factors or contagion in an increasingly

opaque network of interbank loans, derivative trading relationships, and other links

between firms.

This chapter develops maximum likelihood estimators of the term structure of

systemic risk, defined as the conditional probability of failure of a sufficiently large

fraction of the total population of financial institutions. Its main contribution over

prior work is to incorporate the effects of spillovers when measuring systemic risk. The

estimators provide strong evidence for the presence of risk spillovers in the U.S. finan-

cial system during 1987–2008, after controlling for the exposure of firms to observable

risk factors, a source of systemic risk that has been emphasized in the literature on

bank failure prediction. The part of systemic risk not caused by the variation of the

trailing return of the S&P 500, the lagged slope of the U.S. yield curve, the default

and TED spreads, and other observable variables can be substantial, and tends to

be higher in periods of adverse economic conditions. The results indicate that the

systemic risk in the U.S. financial sector can be much greater than would be esti-

mated under the conventional assumption that bank failure clusters arise only from

exposure to observable risk factors.

Our empirical analysis is based on a new dynamic hazard model of correlated fail-

ure timing that extends the proportional hazards specification used by Das, Duffie,

Kapadia & Saita (2007b), Duffie, Saita & Wang (2007), McDonald & Van de Gucht

(1999) and others to predict non-financial corporate default, and by Brown & Dinc

(2005), Brown & Dinc (2009), Lane, Looney & Wansley (1986), Whalen (1991),

Wheelock & Wilson (2000) and others to forecast bank failures. The distinguishing

feature of our formulation is an additional hazard term that is designed to capture

the statistical implications of risk spillovers within and between the real and finan-

cial sectors. The specification addresses the effects of Bayesian learning at defaults

about unobserved or missing risk factors, a source of spillovers emphasized by Collin-

Dufresne et al. (2009), Duffie et al. (2009) for the real sector, and by Acharya &

Yorulmazer (2008), Aharony & Swary (1983), Cooperman, Lee & Wolfe (1992) and

others for the financial sector. It also covers spillovers channeled through the com-

plex web of derivatives counterparty relations, interbank loans, trade credit chains,


parent-subsidiary relationships, and other links between firms. A traditional propor-

tional hazard formulation ignores spillover effects. It focuses on the dependence of

default timing on observable explanatory variables whose dynamics are assumed to

be unaffected by failure events.

We estimate our extended hazard model using data on economy-wide default ex-

perience in the U.S. for the period January 1987 to December 2008, and a collection of

time-varying explanatory covariates that capture the influence of economic conditions

on failure timing. To address the implications of industrial defaults for bank failures

and vice versa, we develop a two-step maximum likelihood estimation approach. In

contrast to the traditional one-step estimation method, our approach does not treat

the sequence of financial failure events in isolation, but in the context of the sequence

of defaults in the wider economy. It seeks to extract the information contained in in-

dustrial defaults relevant for predicting financial failures, and allows us to capture the

dynamic interaction between the real and financial sectors when measuring systemic

risk.

Rigorous statistical tests demonstrate the in-sample fit of our new hazard model

of correlated failure timing, the significance of the spillover hazard term, and the out-

of-sample predictive power of our fitted measures of systemic risk. For example, the

fitted measures accurately forecast the quantiles of the fraction of failures in the U.S.

financial system during 1998–2009, for each of several confidence levels and forecast

horizons. These tests validate our modeling and estimation approach.

The estimators provide time-series and term-structure perspectives of systemic

risk in the U.S. financial sector, information that is of paramount importance to reg-

ulators and policy makers, and that can be used to implement a macro-prudential

supervision of financial institutions. The estimators indicate that systemic risk in-

creased dramatically during the second half of 2008, and reached unprecedented levels

towards the end of 2008. While the magnitude of economy-wide default risk in 2008

is roughly comparable to the level during the burst of the internet bubble around

2001, we find that the systemic risk during that period is dwarfed by the magnitude

of systemic risk in 2008. During the entire sample period, the failure of a financial

firm is estimated to have a relatively greater impact on systemic risk than the default


of an industrial firm.

The remainder of this introduction discusses the related literature. Section 3.2

introduces our measures of systemic risk and discusses their properties. Section 3.3

develops the statistical methodology. Section 3.4 describes our data, the basic esti-

mation results, and their statistical evaluation. Section 3.5 analyzes the behavior of

systemic risk during 1987–2008, provides risk forecasts for future periods, and evalu-

ates these forecasts. Section 3.6 assesses the impact on systemic risk of the failure of

an industrial firm or a financial institution. Section 3.7 concludes. There are several

appendices.

3.1.1 Related literature

There is a substantial literature on bank failure prediction, which includes Brown &

Dinc (2005), Brown & Dinc (2009), Cole & Wu (2009), Lane et al. (1986), Whalen

(1991), Wheelock & Wilson (2000) and many others. These papers employ traditional

hazard models, in which the timing of bank failures is influenced by observable ex-

planatory covariates, which may be time-varying. They focus on predicting individual

bank failures, and do not directly address the correlation between failures, which is

the driving force behind systemic risk. To incorporate the different sources of this

dependence, we significantly extend the traditional hazard model. Our formulation

assumes that firms are exposed to common, time-varying risk factors. Movements of

these observable factors affect firms across the board, and induce failure clusters. Our

formulation also includes an additional “spillover hazard” term, which seeks to ad-

dress the clustering of failures not due to the variation of observable risk factors. The

spillover hazard depends on the timing of past defaults and the volume of defaulted

debt. It models the influence of past defaults on failure timing, with a role for the

size of a default. This influence can be due to informational spillovers, i.e., Bayesian

learning at defaults about risk factors that are unobserved or missing from the list of

explanatory covariates as in Aharony & Swary (1996), Collin-Dufresne et al. (2009),

Duffie et al. (2009), Giampieri, Davis & Crowder (2005), Giesecke (2004), Koopman,

Lucas & Monteiro (2008) and others. It can also be due to the contagious spread of


distress from one firm to another, as in Azizpour & Giesecke (2008b), Jorion & Zhang

(2007), Lando & Nielsen (2009) and others. Contagion may be channeled through

trade credit or buyer/supplier relationships in the real sector, and derivatives coun-

terparty relations and interbank loans in the financial sector (see Upper & Worms

(2004) and others in this regard). The goal of our dynamic hazard model is to capture

the statistical implications of risk spillovers for failure timing without needing to be

precise a priori about the economic mechanisms behind them.

Our measures of systemic risk are related to alternative measures discussed in the

literature. Adrian & Brunnermeier (2009) propose a family of quantile measures of

systemic risk that are based on the distribution of the change of the market value of

total financial assets of public financial institutions. They estimate these risk mea-

sures from time series of equity returns and balance sheet information using quantile

regressions. Acharya, Pedersen, Philippon & Richardson (2009) develop and esti-

mate expected shortfall measures of systemic risk that are based on the distribution

of market equity returns of financial institutions. Lehar (2005) takes a related struc-

tural perspective, and defines systemic risk in terms of adverse changes in the market

values of several institutions.

We propose quantile measures of systemic risk that are predicated on the condi-

tional distribution of the failure rate in the financial system, and are estimated from

actual failure experience. This is an important difference to Adrian & Brunnermeier

(2009), Acharya et al. (2009) and Lehar (2005), who assess systemic risk in terms of

adverse asset price changes across the financial sector, and use equity price data to

estimate the corresponding risk measures. By tying systemic risk to clustered failures

in the financial sector and focusing on actual failure timing data, our measures tend to

be less susceptible to equity market factors unrelated to systemic risk. Nevertheless,

they incorporate market information through the explanatory variables.

Since they summarize the relevant properties of a conditional distribution, our

risk measures are dynamic, and vary through time with the available information.

They also define a term structure of systemic risk over multiple future periods that

incorporates the dynamics of the explanatory macro-economic and sector-wide vari-

ables. The basic risk measures extend naturally to co-risk measures a la Adrian &


Brunnermeier (2009). A co-risk statistic quantifies the impact on systemic risk of an

adverse event, such as the collapse of an industrial firm or financial institution.

Avesani, Pascual & Li (2006), Bhansali, Gingrich & Longstaff (2008), Chan-Lau

& Gravelle (2005), Huang, Zhou & Zhu (2009) and others estimate alternative risk

measures from market rates of credit derivatives. These measures quantify systemic

risk relative to a risk-neutral pricing measure, and incorporate the risk premia in-

vestors demand for bearing correlated default risk. Our measures are based on actual

failure behavior rather than market prices, and do not reflect risk premia.

Elsinger, Lehar & Summer (2006) develop and estimate a static network model of

the interbank lending market to incorporate spillover phenomena induced by inter-

bank loans when quantifying systemic risk; see also Eisenberg & Noe (2001).

Staum (2009) considers the total premium required to insure all deposits in the

banking system as a measure of systemic risk. A bank’s contribution to this risk

measure is proposed as the bank’s deposit insurance premium.

3.2 Measures of systemic risk

This section discusses our definition of systemic risk, describes a measure to quantify

systemic risk, and examines the basic properties of this measure.

We define systemic risk in the financial sector as the conditional probability of

failure of a sufficiently large fraction of the total population of institutions in the

financial system. This definition targets the scenario of a failure cluster of financial

institutions, potentially as part of a larger cluster of economy-wide defaults. Such

a cluster could be due to a severe macro-economic shock, or a contagious spread of

distress from one institution to another. Financial distress can be propagated through

the informational and contractual relationships within the financial system, or the

relationships between financial institutions and other non-financial firms. Lehman

Brothers is an example of how the collapse of a single institution can induce distress

at multiple other entities.


To provide a quantitative measure of systemic risk, consider the process N count-

ing defaults in the financial system.1 The value Nt represents the number of defaults

in the financial system observed by time t. For a given horizon T , consider the con-

ditional distribution at time t < T of the default rate in the financial system, given

by Dt(T ) = (NT −Nt)/Wt, where Wt denots the number of financial institutions ex-

isting at t. This distribution gives the likelihood of failure by T of any fraction of the

population of financial institutions at t. The right tail of this distribution reflects the

magnitude of systemic risk. To measure this magnitude more precisely, we consider

statistics that summarize the information in the tail of the distribution. A standard

statistic is a quantile of the distribution, or value at risk. The value at risk Vt(α, T )

at level α ∈ (0, 1) is the smallest number x ≥ 0 such that the conditional probability

at t that the default rate Dt(T ) during (t, T ] exceeds x is no larger than (1 − α).

Figure 3.1 illustrates the value at risk.

The value at risk Vt(α, T ) of the financial system is intuitive and easily communi-

cated, relying on the popularity of value at risk in the financial industry. There are

other advantages. As indicated by the notation, Vt(α, T ) depends on the conditioning

time t, and thus changes over time as new information is revealed. This leads to a

dynamic risk measure. The value Vt(α, T ) also depends on the risk horizon T . By

varying T for fixed t we obtain a term structure of systemic risk. Further, as shown

in Section 3.6, Vt(α, T ) extends naturally to a co-risk measure that quantifies the

contribution to systemic risk of a particular event, such as the default of a financial

institution.

The quantification of systemic risk need not be predicated on the value at risk.

Our statistical methodology focuses on the entire conditional distribution of Dt(T ),

so our analysis extends to alternative downside risk measures such as the expected

shortfall or average value at risk, which is defined as the conditional mean of Dt(T )

given Dt(T ) ≥ c, where c is some high level, such as Vt(α, T ). While the value at

risk is silent about the magnitude of the failure rate in excess of Vt(α, T ), expected

shortfall provides more detailed information about the severity of large failure clusters.

1We fix a complete probability space (Ω,F , P ) with an information filtration (Ft)t≥0 that satisfiesthe usual conditions. Here, P denotes the actual (empirical) probability measure.


0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

Dt(T) in Percent

Pro

babi

lity

Vt(α,T)

1 − α

Figure 3.1: Value at risk Vt(α, T ) at level α for a given horizon T .

More generally, our analysis extends to any statistic of the conditional distribution

at t of the system-wide default rate Dt(T ), including the moments and other tail

risk measures. Moreover, our analysis extends to risk measures of the conditional

distribution of the value-weighted default rate, which takes account of the default

volume.

The measures of systemic risk we propose are distinct from the measures discussed

in the literature. The fundamental difference is the underlying distribution. While

we define systemic risk in terms of the distribution of the failure rate in the financial

system, Adrian & Brunnermeier (2009), Acharya et al. (2009) and Lehar (2005) relate

systemic risk to the distribution of the change of the market equity value of financial

institutions. Avesani et al. (2006), Chan-Lau & Gravelle (2005), Huang et al. (2009)

and others define systemic risk in terms of a risk-neutral probability, which reflects

the risk premia investors demand for bearing correlated default risk.


3.3 Statistical methodology

This section develops a likelihood approach to estimating the measures of systemic risk

proposed in Section 3.2. In a first step, we formulate and estimate a new hazard, or

intensity-based, model of economy-wide default timing. In a second step, we extract

the system-wide failure intensity from the economy-wide default intensity. The fitted

system-wide intensity then leads to estimators of our systemic risk measures.

3.3.1 Economy-wide default timing

Consider the process N∗ counting defaults in the economy. The value N∗t is the

number of defaults observed by time t. We suppose that N∗ has hazard rate or

intensity λ∗, which represents the conditional mean default rate in the economy and

is measured in events per year. We assume that the intensity evolves through time

according to the model

λ∗t = exp(β∗X∗t ) +

∫ t

0

e−κ(t−s)dJs (3.1)

where X∗ is a vector of time-varying explanatory covariates specified in Section 3.4.2,

β∗ is a vector of constant parameters, κ is a strictly positive parameter, and

Jt = ν1 + · · ·+ νN∗t (3.2)

where νn = γ+ δmax(0, logD∗n). Here, γ and δ are non-negative parameters, and D∗n

is the default volume, i.e. the total amount of debt outstanding at default of the n-th

defaulter, measured in million dollars.2

The intensity (3.1) is the sum of two terms. The first term, termed baseline hazard,

takes a standard Cox proportional hazards form. It models the influence on default

arrivals of explanatory covariates X∗, and captures the clustering of defaults due to

the exposure of different firms to variations in X∗. The dynamics of the variables X∗

are not affected by default arrivals. The proportional hazards formulation is used by

2We assume that each variable max(0, logD∗n) has finite mean, and that each component of X∗tis finite almost surely. Under these conditions, the process N∗ is non-explosive.


Duffie et al. (2007), McDonald & Van de Gucht (1999) and many others to predict

industrial defaults, and by Brown & Dinc (2005), Brown & Dinc (2009), Cole & Wu

(2009), Lane et al. (1986), Whalen (1991), Wheelock & Wilson (2000) and others to

predict bank failures.

The second term, called spillover hazard, is not present in the traditional propor-

tional hazards formulation. It models the influence of past defaults on current default

rates, which is not captured by the baseline hazard term. At an event, the default

rate jumps, with magnitude given by γ plus δ times the positive part of the logarithm

of the defaulter’s total outstanding debt, which is a proxy of the defaulter’s firm

size.3 Thus, the bigger a defaulter the greater the impact of the event, with minimum

impact governed by γ. After an event, the intensity decays to the baseline hazard,

exponentially at rate κ.

The inclusion of the spillover hazard term is motivated by the results of the em-

pirical analyses of Aharony & Swary (1996), Azizpour & Giesecke (2008b), Collin-

Dufresne et al. (2009), Das et al. (2007b), Duffie et al. (2009), Lando & Nielsen (2009)

and others. For U.S. corporate defaults, these papers found evidence of the presence

of contagion or unobservable or missing explanatory covariates, called frailties. With

contagion, a default increases the likelihood of additional defaults, a process that may

be channeled through trade credit or buyer/supplier relationships in the real sector,

and derivatives counterparty relations and interbank loans in the financial sector.

With frailty, Bayesian updating of the conditional distribution of the unobserved

explanatory variables leads to a jump of the econometrician’s default rate.

The spillover hazard term in (3.1) seeks to capture the statistical implications of

contagion and frailty for failure timing. In particular, it is designed to replicate the

excess default clustering not caused by the variation of the observable covariates X∗

defining the baseline hazard. Owing to the reduced-form nature of the specification,

this formulation does not distinguish the potential sources of the excess clustering.

The inference problem for the default timing model (3.1)–(3.2) is addressed as

follows. Letting θ = (β∗, κ, γ, δ) be the set of parameters of the intensity λ∗ = λ∗(θ),

3For the purposes of our analysis, we found the total amount of debt outstanding at default tobe a better measure of firm size than market capitalization, which was used by Shumway (2001) andothers to predict non-financial corporate default.


Θ be the set of admissible parameters, and [0, t] be the sample period, we solve the

log-likelihood problem

supθ∈Θ

∫ t

0

(log λ∗s−(θ)dN∗s − λ∗s(θ)ds). (3.3)

The calculation of the likelihood function is based on a standard measure change

argument. Since the dynamics of the covariates X∗ are assumed to be unaffected by

default arrivals, the likelihood problem for X∗ can be treated separately from that for

λ∗. Given a trajectory of X∗, the log-likelihood function takes a closed form, allowing

for computational tractability of estimation. Under technical conditions stated in

Ogata (1978b), the maximum likelihood estimator of θ is asymptotically normal and

efficient.

We have experimented with several alternative model formulations, including a

proportional hazards model in which average spillover effects are captured by a co-

variate given by the trailing 1-year default rate, as in Duffie et al. (2009). We have

also tested alternative specifications of the impact variables νn in (3.2). However,

based on the in- and out-of-sample tests described in Section 3.4.3 below, we found

these alternatives to be statistically inferior to the model (3.1)–(3.2).

3.3.2 System-wide default timing

Next we extract from the fitted economy-wide model λ∗ the dynamics of system-wide

defaults, i.e., failures in the financial system. This is based on the following result.

Proposition 3.3.1. There is a (predictable) process Z taking values in the unit in-

terval, such that the intensity λ of system-wide failures is given by λ = λ∗Z.

Proof. The system-wide failure times form a subsequence of the economy-wide default

times. The existence and uniqueness of Z follows from the Radon-Nikodym theorem

applied to the random measures associated with the time-integrals of the intensities

λ∗ and λ. The predictability of Z follows from the predictability of the processes

generated by these time-integrals.


The value Zt is the conditional probability at t that a firm in the financial system

defaults next, given a default in the economy in the next instant. For a precise

statement, see Proposition 3.1 in Giesecke et al. (2009). We formulate and estimate

a parametric model of Z, which then leads to λ via Proposition 3.3.1.

We use probit regression to estimate the process Z from the observed economy-

and system-wide default counting processes N∗ and N , respectively. Letting Yn be

a binary response variable equal to one if the n-th defaulter belongs to the financial

system and 0 otherwise, we obtain a value Yn for each economy-wide default time T ∗n

in the sample. Each Yn is a Bernoulli variable with success probability ZT ∗n , where4

Zt = Zt(β) = Φ(βXt−) (3.4)

and where Φ is the cumulative distribution function of a standard normal variable,

Xt is a vector of time-varying explanatory covariates specified in Section 3.4.2, and

β is a vector of constant parameters. Given observations (Yn)n=1,...,N∗tand (Xs)s≤t

during the sample period [0, t], we estimate β by solving the log-likelihood problem

supβ∈Σ

N∗t∑n=1

[YT ∗n log(ZT ∗n (β)) + (1− YT ∗n ) log(1− ZT ∗n (β))

](3.5)

where Σ is the set of admissible parameters. The maximum likelihood estimator of

β is consistent, asymptotically normal and efficient if the covariance matrix of the

vector of regressors exists and is non-singular. See McCullagh & Nelder (1989) for

details. It can also be shown that the log-likelihood function is globally concave in

β, and therefore a standard numerical optimization routine converges quickly to the

unique maximum.

The two-step approach to estimating λ has a significant advantage over an al-

ternative one-step approach in which λ would be estimated directly based on the

historical default experience in the financial system. The two-step approach allows

us to extract the information contained in the observed default times of non-financial

4We experimented with several alternative link functions, including a logit model. All thesealternatives were found to be statistically inferior to the probit model.


firms, which otherwise would not be utilized in the estimation process. Financial

firms are intertwined with the real sector, so defaults in that sector clearly have an

influence on financial firms, and vice versa. Our estimation approach seeks to capture

this influence. It responds to an argument made by Schwarcz (2008) and many oth-

ers that systemic risk measures should account for the relationship between financial

institutions and industrial firms.

The two-step approach has another, statistical advantage. Failures in the financial

system are relatively rare. The number of economy-wide defaults is much larger,

leading to a greater sample size and more accurate inference.5

3.3.3 Measures of risk

The intensity λ = λ∗Z governs the dynamics of the system-wide default process

N , and hence the measures of systemic risk introduced in Section 3.2. Given the

fitted models of λ∗ and Z, we estimate the entire conditional distribution at t of the

system-wide default rate Dt(T ) by exact Monte Carlo simulation of default times

during (t, T ].6 From the conditional distribution we obtain unbiased estimates of the

value at risk Vt(α, T ) or any other risk measure based on the distribution of Dt(T ) or

related quantities, including the value-weighted default rate.

The risk measure estimates take account of the idiosyncratic and clustered default

risk of financial institutions. They capture several sources of default clustering. Clus-

tering can occur due to the spread of distress within the financial sector, or between

the industrial and financial sectors. It can also occur due to the exposure of institu-

tions to the common risk factors represented by the covariate vector X∗. The risk

measure estimates reflect the time-variation of X∗ and the cross-sectional variation of

the default volume D∗n. As detailed in Appendices D and E, this is based on a vector

autoregressive time-series model of the covariates, and a generalized Pareto model of

the default volume. The importance for industrial default prediction of incorporating

the time-series dynamics of explanatory covariates was emphasized by Duffie et al.

5For our sample period 1987-2008, the number of system-wide failures is 83 while the number ofeconomy-wide defaults is 1193.

6The simulation is based on an acceptance/rejection scheme. Details are available upon request.


(2007).

3.4 Empirical analysis

This section describes the default timing data, the data on explanatory covariates,

our basic estimation results, and their statistical evaluation.

3.4.1 Default timing data

Our sample period is 1/1/1987 to 12/31/2008.7 Data on U.S. corporate default timing

were obtained from Moody’s Default Risk Service. For our purposes, a “default”

is a credit event in any of the following Moody’s default categories: (1) A missed

or delayed disbursement of interest or principal, including delayed payments made

within a grace period; (2) Bankruptcy (Section 77, Chapter 10, Chapter 11, Chapter

7, Prepackaged Chapter 11), administration, legal receivership, or other legal blocks

to the timely payment of interest or principal; (3) A distressed exchange occurs where:

(i) the issuer offers debt holders a new security or package of securities that amount

to a diminished financial obligation; or (ii) the exchange had the apparent purpose of

helping the borrower avoid default. A repeated default by the same issuer is included

in the set of events if it was not within a year of the initial event and the issuer’s rating

was raised above Caa after the initial default. This treatment of repeated defaults is

consistent with that of Moody’s. This leaves us with 1193 economy-wide defaults.

For the purpose of analyzing systemic risk, we take the U.S. financial system to

be the set of firms classified in Moody’s industry category “Banking” or “FIRE”

(Finance, Insurance and Real Estate).8 This set includes commercial and invest-

ment banks, bank holding companies, credit unions, thrifts, investment management,

trading, leasing, mortgage and securities firms, financial guarantors, insurance and

7This period was determined by the availability of data for the covariates specified in Section3.4.2. Default data for the period 1/1/2009 to 6/30/2009 were used for the out-of-sample analysis.

8Moody’s uses several industry classifications. Our analysis is based on the “Moody’s 11” scheme,which specifies 11 industries: 1. Banking, 2. Capital Industries, 3. Consumer Industries, 4. Energyand Environment, 5. FIRE, 6. Media and Publishing, 7. Retail and Distribution, 8. Sovereign andPublic Finance, 9. Technology, 10. Transportation, and 11. Utilities.


1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

2

4

6P

erc

en

t

0

80

160

240

$B

illio

n

Annual U.S. Default Rate (Left Axis)Annual U.S. Default Volume (Right Axis)

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

1

2

3

4

Pe

rce

nt

0

3

6

9

12

$B

illio

n

Annual U.S. Financial Default Rate (Left Axis)Annual U.S. Financial Default Volume (Right Axis) 157.2

Figure 3.2: Default timing and volume data. Left panel : 1-year economy-widedefault rate in the universe of Moody’s rated issuers. Right panel : 1-year system-widedefault rate. The defaults of Lehman Brothers and Washington Mutual contributedto over 80% of the system-wide default volume in 2008. Source: Moody’s DefaultRisk Service.

insurance brokerage firms, and REITs and REOCs. Figure 3.2 shows the 1-year

economy- and system-wide default rates during the sample period, along with default

volume information obtained from Moody’s Default Risk Service.9

3.4.2 Covariates

We examine the influence on systemic risk of two types of macro-economic and sector-

wide variables, which are measured monthly. These include:

(i) The trailing 1-year return on the S&P500 index, obtained from Economagic.

Duffie et al. (2007) found this variable to be a significant predictor of industrial

defaults.

(ii) The 1-year lagged slope of the yield curve, computed as the spread between

10-year and 3-month Treasury constant maturity rates, as a forward-looking

9As explained by Hamilton (2005), the volume reported by Moodys excludes debt obligations thatdo not reflect the fundamental default risk of the obligor such as structured finance transactions,short-term debt (e.g., commercial paper), secured lease obligations, and so forth.


1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009−3

−1

1

3

5

Pe

rce

nt

0

1

2

3

4

Pe

rce

nt

1−year Lagged Slope of Yield Curve (Left Axis)Moody’s Seasoned Baa−Aaa Spread (Right Axis)

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

S&P500BankingFinancialInsuranceReal Estate

Figure 3.3: Time-series of explanatory covariates. Left panel : The 1-year laggedslope of yield curve and the default spread, given by the difference between Moody’sseasoned Baa-rated and Aaa-rated corporate bond yields. Right panel : The trailing1-year returns on the S&P500 index and the banking and FIRE portfolios.

indicator of real economic activity. Estrella & Trubin (2006) found this variable

to have strong predictive power for future recessions. We obtain the H.15 release

of Treasury rates from the website of the Federal Reserve Bank of New York.

(iii) The default spread, defined as the yield differential between Moody’s seasoned

Aaa-rated and Baa-rated corporate bonds. Chen, Collin-Dufresne & Goldstein

(2008) argue that the default spread is a measure of aggregate credit risk that is

largely unaffected by bond market frictions such taxes and liquidity. The data

are obtained from the website of the Federal Reserve Bank of New York. The

left panel of Figure 3.3 shows the time series of the default spread and the slope

of the yield curve.

(iv) The TED (Treasury-Eurodollar) spread, defined as the difference between the 3-

month LIBOR and 3-month Treasury rates, as an indicator of credit risk in the

financial system.10 We obtain the historical LIBOR rates from Economagic.

10An increase of the TED spread is a sign that lenders believe that the risk of default on interbankloans is increasing. In that case, lenders demand a higher rate of interest, or accept lower returns onrisk-free Treasuries. The 3-month LIBOR-OIS (overnight index swap) spread is a similar indicator.


1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Per

cent

Black Monday

Subprime Crisis

EconomicCrisis

in Mexico9.11

Lehman Brothers

Bear Stearns

Ford / GM

Gulf War&

JapaneseAsset Bubble

Collapsed

Internet BubbleBursts

LTCM Crisis

Figure 3.4: TED spreads during the sample period, along with significant events.

Figure 3.4 shows the TED spread during the sample period, with significant

events indicated.

(v) The trailing 1-year returns on banking and FIRE portfolios, as a proxy for

business cycle activity in the financial system. The data were obtained from

the website of Kenneth French.11 The right panel of Figure 3.3 shows the return

series.

(vi) The default ratio (Nt − Nt−h)/(N∗t − N∗t−h + 1), which for fixed h > 0 relates

the number of failures in the financial system during (t − h, t] to one plus the

number of economy-wide defaults during that period. It increases at a failure

in the financial system, and decreases at a default of a non-financial firm.

We have also considered, and rejected for lack of significance in the presence of

the above variables, a number of additional covariates, including the 3-month, 1-year,

10-year, 30-year Treasury rates, the spread between Moody’s Baa rate and the 10

year treasury rate, the monthly VIX, and the 3-month LIBOR rate.

11http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html


3.4.3 Economy-wide intensity

We start by addressing the likelihood problem (3.3) for the economy-wide intensity

(3.1), taking the covariate vector X∗ to include a constant, the trailing return on

the S&P500, the lagged slope of the yield curve, and the default spread. We have

also considered, but rejected for lack of significance in the presence of these variables,

the other covariates discussed in Section 3.4.2. The other covariates are used for the

estimation of the process Z in Section 3.4.5 below.

Table 3.1 reports the parameter estimates, along with estimates of asymptotic

standard errors.12 The intensity is increasing in the default spread, and decreasing

in the trailing return on the S&P 500 and the lagged slope of the yield curve. The

jump of the intensity at a default, measured in events per year, is estimated to be

2.3 plus roughly one half of the logarithm of the default volume, measured in million

dollars. The impact of an event fades away exponentially with time: the fitted half

life is log(2)/6.0592 = 0.1144 years.

To develop some insight into the relative statistical importance of the baseline and

spillover hazard terms for model fit, we take a Bayesian perspective, following Duffie

et al. (2009), Eraker, Johannes & Polson (2003) and others. Specifically, we consider

the Bayes factor, given by the ratio of the likelihood of a benchmark model to the

likelihood of an alternative model, both evaluated at their respective estimators. The

test statistic Ψ is given by twice the natural logarithm of the Bayes factor. According

to Kass & Raftery (1995), a value for Ψ between 2 and 6 provides positive evidence,

a value between 6 and 10 strong evidence, and a value larger than 10 provides very

strong evidence in favor of the benchmark model. Due to the marginal nature of

the likelihoods used for computing Ψ, this criterion does not necessarily favor more

complex models.

We first test our model against an alternative specification that does not include

a spillover hazard term (i.e., a traditional proportional hazards model). When the

covariate set of the alternative model includes a constant, the trailing return on the

12The parameter space Θ = (−5, 5)4× (0, 15)× (0, 5)2. The fmincon routine of Matlab was used tosearch for the optimal parameter set. We performed a search for each of 10 randomly chosen initialparameter sets. Each of these searches converged to the values reported in Table 3.1.


Baseline Hazard Spillover HazardConstant S&P500 Yield Slope Baa-Aaa κ γ δ

MLE 2.3026 −0.4410 −0.2140 0.5092 6.0592 2.3205 0.4781SE 0.0605 0.0524 0.0336 0.0534 0.1108 0.0811 0.0233t-stat 38.04 −8.42 −6.37 9.53 54.71 28.60 20.56

Ψ0.1298 3.0987 1.8310

213.403926.5308

Table 3.1: Maximum likelihood estimates (MLE) of economy-wide intensity parame-ters, asymptotic standard errors (SE), t-statistics (t-stat), and Bayes factor statistics(Ψ).

S&P 500, the lagged slope of the yield curve, and the default spread, then the outcome

of Ψ is 213.4, providing extremely strong evidence in favor of including the spillover

hazard term. When the alternative model is based on an unconstrained covariate

set that includes, in addition to the variables just mentioned, the TED spread, the

trailing 1-year returns of banking, financial, insurance and real-estate portfolios,13

then the outcome of Ψ is 131.4, still providing strong evidence in favor of including

the spillover hazard term. Testing our model against one that does not include the

baseline hazard term, the outcome of Ψ is 26.5, providing very strong evidence in

favor of including the baseline hazard term.

The left panel of Figure 3.5 shows the fitted economy-wide intensity against the

number of economy-wide defaults. The fitted intensity tracks the observed arrivals

well. The right panel of Figure 3.5 graphs the decomposition of the fitted intensity

into baseline and spillover hazards. The time series behavior of the components is

similar. However, during periods of higher than average default activity, the spillover

hazard represents a relatively larger fraction of the total default hazard than the

baseline hazard.

13The parameter estimates are as follows (SE in parentheses): Constant 4.1597 (0.1443), S&P 500−2.1542 (0.3036), Yield Slope −0.2346 (0.0272), Baa-Aaa 0.4612 (0.1370), TED −0.8716 (0.3040),Banking 0.5059 (0.2606), Financial 1.5882 (0.3551), Insurance−0.7845 (0.1548), Real Estate−0.6032(0.1054). The default ratio was found to be insignificant in the presence of these covariates.


1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

20

40

60

80

100

120

140

160

180

200Number of DefaultsFitted Intensity

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

20

40

60

0

60

120

180Fitted Baseline Hazard (Left Axis)Fitted Spillover Hazard (Right Axis)

Figure 3.5: Fitted economy-wide intensity λ∗. Left panel : Yearly defaults and fittedintensity. Right panel : Intensity decomposition: fitted baseline hazard vs. fittedspillover hazard.

3.4.4 Goodness-of-fit tests

We test the fit of the economy-wide intensity model λ∗ to the historical default tim-

ing data. The tests are based on a result of Meyer (1971), which implies that the

default arrivals follow a standard Poisson process under a change of time given by

the cumulative intensity λ∗. Thus, if λ∗ is correctly specified, then the time-scaled

inter-arrival times are independent standard exponential variables.

The properties of the time-scaled arrival times can be analyzed with a battery of

alternative tests. We use a family of tests of the binned arrival time data, following

Das et al. (2007b). For given bin size c, we denote by Un the number of observed

events in the n-th successive time interval lasting for c units of transformed time.

With a total of K bins, the null hypothesis is that the U1, . . . , UK are independent

Poisson variables with mean c. We consider bin sizes c = 2, 4, 6, 8 and 10.

We start with Fisher’s dispersion test. Under the null, W =∑K

n=1(Un− c)2/c has

a chi-squared distribution with K − 1 degrees of freedom. Table 3.2a indicates that

there is no evidence against the null for bin sizes 4 through 10, at standard confidence

levels.


Bin Size Number of Bins χ2 Statistic p-Value2 596 838.50 0.00004 298 332.75 0.07516 198 207.17 0.29568 149 167.38 0.131610 119 125.70 0.2967

(a) Fisher’s Dispersion Test

Mean of Tails Median of TailsBin Size Data Simulation p-Value Data Simulation p-Value

2 3.9694 3.6740 0.0000 4.0000 3.0524 0.00004 6.1739 6.1575 0.3092 6.0000 5.9956 0.04766 8.5676 8.8643 0.6254 8.0000 8.5337 0.54198 11.6667 11.3794 0.2190 11.0000 10.9284 0.091610 14.0313 13.7454 0.2459 13.5000 13.2444 0.2899All - - 0.2799 - - 0.1942

(b) Mean and Median of Default Upper Quartile Tail Test

Bin Size Number of Bins A (tA) B (tB) R2

2 596 2.3634∗ (3.4556) −0.1847∗ (−4.5767) 0.03414 298 4.0348 (0.1321) −0.0121 (−0.2074) 0.00016 198 6.1971 (0.4250) −0.0372 (−0.5203) 0.00148 149 8.8132 (1.1613) −0.1074 (−1.3032) 0.111510 119 10.3584 (0.3650) −0.0378 (−0.4018) 0.0014

(c) Excess Default Autocorrelation Test (t-statistics for A are presented for the test A = c andasterisks indicate significance at the 5% level.)

Table 3.2: Goodness-of-fit tests of the economy-wide intensity.

To examine the extent to which our intensity model captures the clustering of

defaults, we perform an upper tail test developed by Das et al. (2007b). We generate

10,000 data sets by Monte Carlo simulation, each consisting of K iid Poisson random

variables with mean c. The p-value of the test is the fraction of the simulated data

sets whose sample upper-quantile mean (or median) is above the actual sample mean

(or median). The p-values reported in Table 3.2b suggest that there is no significant

deviation of the upper-quartile tails from the theoretical Poisson tails for bin sizes 4

through 10, at standard confidence levels. Furthermore, the null hypothesis cannot

be rejected by the joint test across all bin sizes, at conventional confidence levels.


Finally we test for serial dependence of the Uk. To this end, we estimate an

autoregressive model, given by Uk = A+BUk−1 + εk for coefficients A and B. Under

the null, A = c, B = 0, and the εk are independent, demeaned Poisson random

variables. Table 3.2c shows that the fitted coefficients are not significantly different

from their theoretical values for bin sizes 4 through 10, at standard confidence levels.

The results of these tests suggest that the fitted λ∗ time-scales most arrival times

correctly, indicating a good overall fit of our default timing model (3.1). Additional

experiments suggest that the rejections of the null for bin size 2 are due to events

arriving in very short time intervals. On the time scale of the sample period, which

stretches over 21 years, these are almost simultaneous arrivals. It is difficult to match,

at the same time, the few extremely short inter-arrival times, and the many longer

inter-arrival times that constitute the vast majority of the sample.

3.4.5 System-wide intensity

Next we address the likelihood problem (3.5) for the process Z in (3.4). The value

Zt represents the conditional probability at t that the next defaulter is a financial

firm, given that there is a default in the economy in the next instant. We take the

covariate vector X to include a constant, the 1-year lagged slope of the yield curve,

the TED spread, the trailing 1-year returns of banking and real-estate portfolios, and

the default ratio for h = 1/12.14

Table 3.3 provides the estimates of the coefficient vector β, along with asymptotic

standard errors and t-statistics. A likelihood ratio test indicates that the covariates

are informative. The coefficient linking the trailing 1-year return of the banking port-

folio to the probability Zt is positive, and of unexpected sign by univariate reasoning.

With multiple covariates, however, the sign need not be evidence that a good year in

the banking sector foreshadows a higher fraction of bank defaults.

The time-series behavior of the fitted process Z, shown in the left panel of Figure

3.6, indicates the dramatic increase during the second half of 2008 of the number of

defaults in the financial sector relative to the total number of events in the economy.

14We experimented with different window sizes h, but found h = 1/12 to work best. This windowsize is consistent with the frequency of the observations of the other covariates.


Covariate Coefficient SE t-statistic p-value ΨConstant −2.0873 0.1484 −14.0659 0.0000

Yield Slope 0.1256 0.0585 2.1469 0.0318 4.6502TED Spread 0.3710 0.1506 2.4632 0.0138 5.8223

Banking 0.8952 0.3462 2.5856 0.0097 6.6832Real Estate −0.8073 0.2973 −2.7218 0.0065 7.4439

Default Ratio 1.4171 0.4351 3.2572 0.0011 10.1015Model Fit LR-ratio (χ2) = 36.8117 p-value < 0.0001

Table 3.3: Maximum likelihood estimates of the coefficients β of the covariate processX governing the thinning process Z in (3.4), asymptotic standard errors (SE), t-statistics, p-values, and Bayes factor statistics (Ψ).

To measure how accurately the fitted model of Z distinguishes between economy-

and system-wide events out-of-sample, we construct a power curve, shown in the right

panel of Figure 3.6. The diagonal line represents an uninformative model that sorts

events randomly. The larger the area under the curve (AUC), the more accurate the

model predictions. For our model, the AUC is 0.7076, with 95% confidence interval

given by [0.6433, 0.7719]. The standardized AUC is 6.3283, implying that the area is

statistically greater than 0.5 with p-value less than 0.0001.

3.5 Systemic risk

This section analyzes the behavior of systemic risk during the sample period, provides

risk forecasts for future periods, and evaluates these forecasts.

3.5.1 Risk measures

We start by examining the fitted system-wide intensity λt, which measures the level

of instantaneous systemic risk prevailing at time t. It is calculated as the product of

the economy-wide intensity λ∗t and the thinning variable Zt, as explained in Section

3.3. The time-series behavior of λt, shown in the left panel of Figure 3.7, indicates

that the level of instantaneous systemic risk reached unprecedented levels during the

fall of 2008. The right panel of Figure 3.7 shows the fitted fraction of λt tied to


1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

0.1

0.2

0.3

0.4

0.5

Th

inn

ing

Pro

cess

(Z

)

0

1

Sys

tem

ic I

nd

ica

tor

(Y)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

False Positive Ratio

Tru

e P

osi

tive

Ra

tio

Figure 3.6: Left panel : Observed binary response variables Yn and fitted process Z.Right panel : Power curve for the fitted process Z.

the spillover hazard term, calculated as the fitted ratio of the spillover hazard to the

total default intensity λ∗t . The estimators provide strong evidence for the presence of

failure clustering not caused by variations in the observable explanatory covariates.

The fraction of systemic risk tied to the spillover term can be substantial, and tends

to be higher in periods of adverse economic conditions. Moreover, financial firms

tend to fail when the fitted contribution of spillovers to instantaneous systemic risk

is relatively large.

Next we estimate the conditional distribution at time t of the default rate in the

financial system during the period (t, t + ∆], for given ∆. As indicated in Section

3.3.3, this is done by exact Monte Carlo simulation.15 The estimation is based on

the models for λ∗ and Z, fitted with data observed from 1/1/1987 to t. As explained

above, the estimation takes account of the time-variation of the covariates during the

forecast period, and the cross-sectional variation of the default volume.

Figure 3.8 shows the conditional distribution of the system-wide default rate

Dt(t + 0.5) for conditioning times t varying semi-annually between 12/31/1997 and

12/31/2008, for a 6-month horizon. As explained in Section 3.2, the system-wide

15The estimates are based on 100,000 Monte Carlo replications.


1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

10

20

30

40

50

60

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 20090

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sp

illo

ver

Ha

zard

Ra

tio

Industrial DefaultFinancial Default

Figure 3.7: Left panel : Fitted system-wide failure intensity λ, based on the parameterestimates reported in Tables 3.1 and 3.3. Right panel : Fitted fraction of λ tied to thespillover hazard term, with default events indicated.

default rate is obtained by normalizing the number of system-wide defaults during

(t, t + 0.5] by the number of firms in the financial system at t. The right tail of the

distribution indicates the magnitude of systemic risk. The fatter the tail, the greater

the likelihood that a large fraction of the financial system fails. The time series be-

havior suggests that systemic risk has increased very sharply during the second half

of 2008.

We contrast the system-wide distribution with the distribution of the economy-

wide default rate, shown in Figure 3.9. While the increase of aggregate default risk

in the second half of 2008 is clearly visible, the magnitude of risk is only somewhat

greater than that during the internet bubble. This is in stark contrast to the behavior

of systemic risk shown in Figure 3.8: the systemic risk during the burst of the internet

bubble is dwarfed by the systemic risk prevailing at the end of 2008.

Next we consider the value at risk Vt(α, t + ∆) of the system-wide default rate.

For a given level of confidence α ∈ (0, 1), the conditional probability at time t that

the system-wide default rate Dt(t+∆) exceeds Vt(α, t+∆) is 1−α. For conditioning

times t varying semi-annually between 12/31/1997 and 12/31/2008, the left panel of

Figure 3.10 shows Vt(α, t+ 0.5), along with realized default rates. The right panel of


01

23

45 1998

20002002

20042006

2008

0

0.5

1

1.5

2

Years

Default Rate(Percent)

Figure 3.8: Fitted conditional distribution (kernel-smoothed) of the system-wide 6-month default rate Dt(t+0.5) for conditioning times t varying semi-annually between12/31/1997 and 12/31/2008.

Figure 3.10 plots the value at risk for economy-wide defaults, defined similarly.

The value at risk defines a term structure of systemic risk. To illustrate this, the

left panel of Figure 3.11 plots Vt(α, t + ∆) on 12/31/2008, the end of the sample

period, as a function of ∆, for each of several α.

There are alternative measures of systemic risk that may be of interest. An ex-

ample is the conditional probability at t of no failures in the financial system during

(t, T ]. This measure does not require the choice of a confidence level. The right panel

of Figure 3.11 shows this probability during the sample period, for each of several

horizons ∆.


01

23

45 1998

20002002

20042006

2008

0

0.5

1

1.5

YearsDefault Rate

(Percent)

Figure 3.9: Fitted conditional distribution (kernel-smoothed) of the economy-wide 6-month default rate for conditioning times t varying semi-annually between 12/31/1997and 12/31/2008. The default rate is obtained by normalizing the number of economy-wide defaults during (t, t+ 0.5] by the total number of firms in the economy at t.

3.5.2 Forecast evaluation

We evaluate the out-of-sample forecast accuracy of the fitted value at risk Vt(α, t+∆)

by comparing it to the realized default rate. Our selection of tests is informed by the

results of the test performance analysis in Berkowitz, Christoffersen & Pelletier (2009).

Let n be the number of forecast periods. Further, let n1 ≤ n be the number

of periods for which the corresponding value at risk forecast was violated, i.e. the

number of periods for which the realized default rate was strictly greater than the

fitted value at risk Vt(α, t + ∆). Then, n0 = n − n1 denotes the number of periods

for which the realized rate was less than or equal to the fitted value at risk. We test

whether the actual violation rate n1/n is significantly different than the theoretical

violation rate (1 − α), as in Kupiec (1995). Fixing a level α ∈ (0, 1) and assuming


1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 20090

0.5

1

1.5

2

2.5

3

3.5

4

4.5

De

fau

lt R

ate

(P

erc

en

t)

95% Value−at−Risk99% Value−at−RiskRealized Default Rate

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 20090

0.5

1

1.5

2

2.5

3

3.5

4

4.5

De

fau

lt R

ate

(P

erc

en

t)

95% Value−at−Risk99% Value−at−RiskRealized Default Rate

Figure 3.10: Left Panel: Fitted value at risk Vt(α, t + 0.5) of the system-wide de-fault rate, for conditioning times t varying semi-annually between 12/31/1997 and12/31/2008, versus realized default rate. Right Panel: Fitted value at risk of theeconomy-wide default rate versus realized default rate.

violations are independent of one another, the log-likelihood ratio test statistic

LRUC = −2 log

(αn0(1− α)n1

(n0/n)n0(n1/n)n1

)(3.6)

has, asymptotically, a chi-squared distribution with 1 degree of freedom under the

null hypothesis of the theoretical (1− α) violation rate.16

A test based on the statistic (3.6) does not address the time-series properties

of the sequence of “hit” indicators associated with violations in different periods.

The hit indicator It for the forecast period (t, t + ∆] is equal to 1 if the realized

default rate for the period is greater than the fitted value at risk Vt(α, t+ ∆), and 0

otherwise. A more stringent conditional coverage test with higher power tests whether

the indicators are independent and identically distributed Bernoulli variables with

success probability (1 − α). We consider two alternative tests of this property, a

Markov test due to Christoffersen (1998) and the CAViaR test of Engle & Manganelli

(2004). According to the performance analysis in Berkowitz et al. (2009), the CAViaR

16In case of n1 = 0, we follow the convention 00 = 1 so that the test statistic is well-defined.


0 0.25 0.5 0.75 10

2

4

6

8

10

12

14

∆

Vt(α

,t+

∆)

in P

erc

en

t

α = 95%α = 99%

1998 2000 2002 2004 2006 20080

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pt(D

t(t+

∆)=

0)

∆=1.0∆=0.5∆=0.25

Figure 3.11: Left Panel: Term structure of systemic risk on 12/31/2008: fitted valueat risk Vt(α, t+∆) on 12/31/2008 as a function of ∆. Right Panel: Fitted conditionalprobability at t of no failures in the financial system during (t, t+∆], for conditioningtimes t varying quarterly between 12/31/1997 and 12/31/2008, for each of severalhorizons ∆.

test has particularly high power for the relatively small sample sizes we encounter here,

for both the 99% and 95% levels.

The Markov test of Christoffersen (1998) tests the Bernoulli distribution of the

actual hit indicators and their independence. The test of the Bernoulli property

relies on the statistic (3.6). The independence is tested against an explicit first-order

Markov alternative, with log-likelihood ratio test statistic given by

LRInd = −2 log

((1− π1)n00+n10πn01+n11

1

(1− π01)n00πn0101 (1− π11)n10πn11

11

). (3.7)

Here, nij denotes the number of periods with a state of j following a state of i,

πij = nij/(ni0 + ni1), and π1 = (n01 + n11)/(n00 + n01 + n01 + n11).17 Under the

null of the indicators forming a first-order Markov chain, this statistic has a limiting

chi-squared distribution with 1 degree of freedom. The combined test of the coverage

17In case of n10 + n11 = 0, we suppose π11 = 0 so that the test statistic is well-defined.


ratio and independence is based on the statistic

LRM = LRUC + LRInd,

which has a limiting chi-squared distribution with 2 degrees of freedom.18

The CAViaR test described in Berkowitz et al. (2009), which is based on Engle &

Manganelli (2004), considers a first-order autoregression for the hit indicator:

It = γ + β1It−∆ + β2Vt(α, t+ ∆) + εt (3.8)

where the error term εt has a logistic distribution. We test whether the βi coefficients

are statistically significant and whether P (It = 1) = eγ/(1 + eγ) = 1 − α. Denote

the ith response variable by Yi and the corresponding vector of regressors by Xi, for

i = 1, . . . , n − 1. Also, let πi = eγ+βXi/(1 + eγ+βXi), where (γ, β) is the maximum

likelihood estimator of (γ, (β1, β2)) obtained by logistic regression. Then, under the

null of β1 = β2 = 0 and γ = log(

1−αα

), the log-likelihood ratio test statistic

LRCAViaR = −2 log

(n−1∏i=1

(1− α)Yiα1−Yi

πYii (1− πi)1−Yi

)(3.9)

has a limiting chi-squared distribution with 3 degrees of freedom.

Table 3.4 reports the test results for the system-wide value at risk Vt(α, t + ∆),

for each of several forecast horizons ∆ and confidence levels α.19 None of the null

hypotheses can be rejected at the 10% level. This suggests that the fitted measures

accurately quantify systemic risk, for each of several risk horizons and confidence

levels.


Uncond. Coverage Markov CAViaR∆ Obs. LR p-value LR p-value LR p-value1Y 11 0.3153 0.5744 0.5157 0.7727 2.3203 0.5086

95% 6M 23 2.3595 0.1245 2.3595 0.3074 2.2569 0.5208VaR 3M 45 0.9143 0.3390 0.9609 0.6185 5.5926 0.1332

1M 133 2.6284 0.1050 2.6900 0.2605 4.5851 0.20481Y 11 0.2211 0.6382 0.2211 0.8953 0.2211 0.9741

99% 6M 23 0.4623 0.4965 0.4623 0.7936 0.4422 0.9314VaR 3M 45 0.9045 0.3416 0.9045 0.6362 0.8844 0.8292

1M 133 0.0905 0.7636 0.1057 0.9485 0.6689 0.8805

Table 3.4: Out-of-sample tests of the forecast accuracy of the fitted system-widevalue at risk, for each of several horizons. The period considered is January 1998 toJune 2009.

3.6 Sensitivity of systemic risk

We show how to measure the impact of a hypothetical default event on systemic

risk. This analysis could be useful to regulatory authorities. For example, regulators

could estimate the potential impact on systemic risk of a default of a given financial

institution.

Fix a conditioning time t, horizon ∆ and confidence level α. We consider the

change ∆Vt(α, t+∆) of the value at risk Vt(α, t+∆) at t in response to a default at t,

which measures the event’s impact on systemic risk. To estimate the change, we first

estimate the time t value at risk Vt(α, t+ ∆) based on data up to t. Next we enlarge

the data set by including a hypothetical default event at t, and then re-estimate

Vt(α, t+ ∆) based on the enlarged data set. Finally we calculate ∆Vt(α, t+ ∆) as the

difference between the two risk measure estimates.

The change ∆Vt(α, t + ∆) reflects the influence of the hypothetical event on the

other firms in the financial system and the economy at large, including potential

spillover effects. It depends on the characteristics of the hypothetical event, including

the sector of the defaulter (industrial vs. financial) and the total debt outstanding at

18This ignores the first observation in the hit sequence.19We also use 2009 default data in the tests: we validate the forecasts obtained on 12/31/2008 on

the realized default rates in 2009, which are available for the first 1, 3, and 6 months of 2009.


1998 2000 2002 2004 2006 20080

0.5

1

1.5

2

2.5

Ab

solu

te C

ha

ng

e ∆

Vt(0

.95

,t+

1)

in P

erc

en

t

Industrial DefaultFinancial Default

0 0.25 0.5 0.75 10

2

4

6

8

10

12

∆

Vt(0

.95

,t+

∆)

in P

erc

en

t

OriginalIndustrial DefaultFinancial Default

Figure 3.12: Impact of a default on systemic risk. Left Panel: Absolute change∆Vt(0.95, t + 1) for conditioning times t varying quarterly between 12/31/1997and 12/31/2008. Right Panel: Term structure of value at risk Vt(0.95, t + ∆) on12/31/2008, for different scenarios.

default, which proxies the size of the defaulter.

We calculate the change for each of two hypothetical events, a default of a financial

institution, and a default of an industrial firm. The left panel of Figure 3.12 shows

the absolute change ∆Vt(0.95, t+ 1) for each of the two events, for conditioning times

t varying quarterly between 12/31/1997 and 12/31/2008. The right panel of Figure

3.12 shows the impact of each of these events on term structure of the value at risk

Vt(0.95, t + ∆) on 12/31/2008. The total debt outstanding at default is taken to be

the sample mean of the default volumes observed to the conditioning time, for the

respective firm class.

The failure of a financial institution has a higher impact on systemic risk than the

default of an industrial firm. This means the financial system is more vulnerable to

the collapse of a financial firm. The impact of a financial firm default is also more

volatile during the sample period. If measured on an absolute scale, the impact of

a default has increased dramatically during the second half of 2008, indicating the

vulnerability of the financial system during that period.

The sensitivity analysis can be extended to measure the impact on systemic risk


of a hypothetical adverse shock to the explanatory covariates (risk factors), along the

lines of Avesani et al. (2006), Huang et al. (2009), and others.

3.7 Conclusion

This chapter provides an econometric method for estimating the term structure of

systemic risk over multiple future periods. The maximum likelihood estimators incor-

porate the dependence of failure timing on time-varying macro-economic and sector-

specific risk factors. Unlike traditional estimators, they also capture the impact of

risk spillovers due to contagion or frailty, and the influence of industrial defaults on

financial failures.

Applying our method to data on U.S. firms over 1987 to 2008, we find that the

level and shape of the term structure of systemic risk in the U.S. financial sector

depend on the timing and severity of past financial and industrial failures, as well

as the current values of observable risk factors including the trailing S&P 500 index

return, the lagged slope of the U.S. yield curve, the default and TED spreads, and

other sector-wide variables. We find that the variation of these risk factors generally

has a less significant impact on systemic risk than spillover effects associated with

failures in the past. This highlights the importance of addressing spillovers when

managing systemic risk.

Several topics are left for future research, including the estimation of premia for

systemic risk. The estimates provided in this chapter can be compared to estimates

of the risk-neutral probability of failure of a large number of financial institutions,

obtained from market rates of credit derivatives contracts. This analysis would shed

light on the magnitude of the premia investors demand for bearing systemic risk.

Our econometric method has potential applications in other subject areas requir-

ing estimates of event probabilities in situations where network or information effects

may play a role. These applications include the analysis of market transaction data,

the analysis of purchase-timing behavior of households, the analysis of unemployment

timing, and many others. The extant analyses of these applications in Engle & Rus-

sell (1998), Seetharaman & Chintagunta (2003), and Lancaster (1979), respectively,


employ standard proportional hazard formulations. Our generalized hazard model

could be used to study the implications of network and information effects in these

settings.

Appendix

A Risk-neutral tranche loss distributions

To describe the estimation of the risk-neutral tranche loss distributions discussed in

Section 2.4, it is necessary to explain the arbitrage-free valuation of index and tranche

swaps. These contracts are based on a portfolio of C credit swaps with common

notional 1, common maturity date H and common quarterly premium payment dates

(tm).

In an index swap, the protection seller covers portfolio losses as they occur (default

leg), and the protection buyer pays Sam(C − N ′tm) at each premium date tm, where

S is the swap rate and am is the day count fraction for coupon period m (premium

leg). The value at time t ≤ H of the default leg is given by Dt(H, 0, 1), where

Dt(H,K,K) = EQ

(∫ H

t

exp

(−∫ s

t

rudu

)dUs(K,K)

∣∣∣Ft) . (A1)

Here, EQ(· | Ft) denotes the conditional expectation operator relative to a risk-neutral

pricing measure Q. The value at time t ≤ H of the premium leg is given by

Pt(S) = S∑tm≥t

amEQ

(exp

(−∫ tm

t

rudu

)(C −N ′tm)

∣∣∣Ft) . (A2)

The index rate at t is the solution S = St(H) to the equation Dt(H, 0, 1) = Pt(S).

In a tranche swap with upfront rate G and running rate S, the protection seller

covers tranche losses (2.16) as they occur (default leg) and, for K < 1, the protection

buyer pays GKC at inception and Sam(KC − Utm) at each premium date tm, where

78

APPENDIX 79

K = K −K is the tranche width (premium leg). The value at τ ≤ H of the default

leg is given by (A1). The value at time t ≤ H of the premium leg is given by

Pt(K,K,G, S) = GKC + S∑tm≥t

amEQ

(exp

(−∫ tm

t

rudu

)(KC − Utm)

∣∣∣Ft) .(A3)

For a fixed upfront rate G, the running rate S is the solution S = St(H,K,K,G) to

the equation Dt(H,K,K) = Pt(K,K,G, S). For a fixed rate S, the upfront rate G is

the solution G = Gt(H,K,K, S) to the equation Dt(H,K,K) = Pt(H,K,K,G, S).

The valuation relations are used to estimate the risk-neutral portfolio and tranche

loss distributions from market rates of index and tranche swaps. First we formulate a

parametric model for the risk-neutral dynamics of N and L, the portfolio default and

loss processes with replacement. Our risk-neutral model parallels the model under P .

Suppose that N has risk-neutral intensity λQ with Q-dynamics

dλQt = κQt (cQt − λQt )dt+ dJt (A4)

where λQ0 > 0, κQt = κQλQTNtis the decay rate, cQt = cQλQTNt

is the reversion level, and

J is a response jump process given by

Jt =∑n≥1

max(γQ, δQλQT−n

)1Tn≤t. (A5)

The quantities κQ > 0, cQ ∈ (0, 1), δQ > 0 and γQ ≥ 0 are parameters such that

cQ(1 + δQ) < 1. We let θQ = (κQ, cQ, δQ, γQ, λQ0 ).

Since the Q-dynamics of λQ mirror the P -dynamics of the P -intensity λ, we can

apply the algorithms developed above to estimate the expectations (A1)–(A3) and to

calculate the model index and tranche rates for the model (A4). We first generate

event times Tn of N by Algorithm 3, with λ and its parameters replaced by their risk-

neutral counterparts, and with initial condition (Nτ , TNτ , λQTNτ

) = (0, 0, λQτ ). Then

we apply the replacement Algorithm 2 as stated to generate event times T ′n without

replacement, and the corresponding paths of N ′ and L′ required to estimate the

APPENDIX 80

expectations (A1)–(A3). In this last step, we implicitly assume that the thinning

probability (2.7), the rating distributions (2.8) and (2.9), and the distribution µ` of

the loss at an event are not adjusted when the measure is changed from P to Q.

The risk-free interest rate r is assumed to be deterministic, and is estimated from

Treasury yields for multiple maturities on 3/27/2006, obtained from the website of

the Department of Treasury.

The risk-neutral parameter vector θQ is estimated from a set of market index and

tranche rates by solving the nonlinear optimization problem

minθQ∈ΘQ

∑i

(MarketMid(i)−Model(i, θQ)

MarketAsk(i)−MarketBid(i)

)2

(A6)

where ΘQ = (0, 2)× (0, 1)× (0, 2)2× (0, 20) and the sum ranges over the data points.

Here, MarketMid is the arithmetic average of the observed MarketAsk and MarketBid

quotes. We address the problem (A6) by adapted simulated annealing. The algorithm

is initialized at a set of random parameter values, which are drawn from a uniform

distribution on the parameter space ΘQ. For each of 100 randomly chosen initial

parameter sets, the algorithm converges to the optimal parameter values given in

Table A2. The market data and fitting results for 5 year index and tranche swaps

referenced on the CDX.HY6 on 3/27/2006 are reported in Table A1. The model fits

the data, with an average absolute percentage error of 2.9%.

B Cash CDO prioritization schemes

This appendix describes the prioritization schemes for the cash CDOs analyzed in

this chapter. These schemes were introduced by Duffie & Garleanu (2001).

APPENDIX 81

Contract MarketBid MarketAsk MarketMid ModelIndex 332.88 333.13 333.01 333.080-10% 84.50% 85.00% 84.75% 86.75%10-15% 52.75% 53.75% 53.25% 48.19%15-25% 393.00 403.00 398.00 398.8825-35% 70.00 85.00 77.50 79.43MinObj 33.77AAPE 2.90%

Table A1: Market data from Morgan Stanley and fitting results for 5 year index andtranche swaps referenced on the CDX.HY6 on 3/27/2006. The index, (15 − 25%)and (25 − 35%) contracts are quoted in terms of a running rate S stated in basispoints (10−4). For these contracts the upfront rate G is zero. The (0, 10%) and(10, 15%) tranches are quoted in terms of an upfront rate G. For these contracts therunning rate S is zero. The values in the column Model are fitted rates based onmodel (A4) and 100K replications. We report the minimum value of the objectivefunction MinObj and the average absolute percentage error AAPE relative to marketmid quotes.

Parameter κQ cQ δQ γQ λQτEstimate 0.522 0.333 0.366 1.646 8.586

Table A2: Estimates of the parameters of the risk-neutral portfolio intensity λQ,obtained from market rates of 5 year index and tranche swaps referenced on theCDX.HY6 on 3/27/2006.

B1 Uniform prioritization

The total interest income W (m) from the reference bonds is sequentially distributed

as follows. In period m, the debt tranches receive the coupon payments

Q1(m) = min(1 + c1)A1(m− 1) + c1F1(m),W (m)

Q2(m) = min(1 + c2)A2(m− 1) + c2F2(m),W (m)−Q1(m).

APPENDIX 82

Unpaid reductions in principal from default losses, Jj(m), occur in reverse priority

order, so that the residual equity tranche suffers the reduction

J3(m) = minF3(m− 1), ξ(m),

where

ξ(m) = max0, L′tm − L′tm−1− [W (m)−Q1(m)−Q2(m)]

is the cumulative loss since the previous coupon date minus the collected and undis-

tributed interest income. Then, the debt tranches are reduced in principal by

J2(m) = minF2(m− 1), ξ(m)− J3(m)

J1(m) = minF1(m− 1), ξ(m)− J3(m)− J2(m).

With uniform prioritization there are no early payments of principal, so P1(m) =

P2(m) = 0 for m < M . At maturity, principal and accrued interest are treated

identically, while the remaining reserve is paid in priority order. The payments of

principal at maturity are

P1(M) = minF1(M) + A1(M), R(M)−Q1(M),

P2(M) = minF2(M) + A2(M), R(M)−Q1(M)− P1(M)−Q2(M)

where R(M) is the value of the reserve account at M . The residual equity tranche

receives

D3(M) = R(M)−Q1(M)− P1(M)−Q2(M)− P2(M).

B2 Fast prioritization

The senior tranche collects interest and principal payments as quickly as possible until

maturity or until its remaining principal becomes zero, whichever is first. In period

APPENDIX 83

m, the senior tranche receives interest and principal payments

Q1(m) = minB(m), (1 + c1)A1(m− 1) + c1F1(m)

P1(m) = minF1(m− 1), B(m)−Q1(m)

As long as the senior tranche receives payments the junior tranche accrues coupons.

After the senior tranche has been retired, the junior tranche is allocated interest and

principal until maturity or until its principal is written down, whichever is first:

Q2(k) = min(1 + c2)A2(m− 1) + c2F1(m), B(m)−Q1(m)− P1(m)

P2(k) = minF2(m− 1), B(m)−Q1(m)− P1(m)−Q2(m).

The equity tranche receives any residual cash flows. There are no contractual reduc-

tions in principal.

C Cash CDO valuation

This appendix explains the valuation of the cash CDO reference bonds and debt

tranches. At time t ≤ H, the value of the reference portfolio is

Ot(v) =∑tm≥t

EQ

(exp

(−∫ tm

t

rudu

)B(m, v)

∣∣∣Ft) (C1)

where the notation B(m) = B(m, v) indicates the dependence of the cash flow B(m)

on the coupon rate v of a reference bond. The par coupon rate of a reference bond is

the number v = v∗ such that Ot0(v) = C at the CDO inception date t0.

The par coupon rates for the debt tranches are determined similarly. LetOjt(v∗, c1, c2)

be the value at time t ≤ H of the bond representing tranche j = 1, 2 when the refer-

ence bonds accrue interest at their par coupon rate v∗, and when the coupon rates on

the debt tranches are c1 and c2, respectively. Because of the complexity of the cash

flow “waterfall,” this quantity does not admit a simple analytic expression. The par

coupon rates are given by the pair (c1, c2) = (c∗1, c∗2) such that Ojt0(v

∗, c1, c2) = pj for

APPENDIX 84

j = 1, 2.

For the CDX.HY6 of reference bonds, assuming that the cash CDO is established

on 3/27/06 and has a 5 year maturity, the par coupon rates are estimated as follows.

We first generate a collection of 100K paths of N ′ and L′ under the risk neutral

measure, based on the risk-neutral intensity model (A4) with calibrated parameters

in Table A2, as described in Appendix A. Based on these paths, we can estimate

Ot0(v) for fixed v, and then solve numerically for the par coupon rate v∗. The risk-

free interest rate r is deterministic, and is estimated from Treasury yields for multiple

maturities on 3/27/2006. Given v∗, c1, c2, we can calculate the tranche cash flows for

a given path of (N ′, L′) according to the specified prioritization scheme, and then

estimate Ojt0(v∗, c1, c2). We then solve numerically a system of two equations for

(c∗1, c∗2).

D Covariate time-series model

We formulate a vector autoregressive VAR(1) time-series model for the covariates.

This model incorporates the dynamic relationships between the different variables.

Let Φt denote the (n× 1) vector of covariate values at t. We suppose that

Φt = Π0 + Π1Φt−1 + εt (D1)

where Π0 is an (n×1) vector, Π1 is an (n×n) coefficient matrix and εt is a (n×1) zero

mean vector of error processes that is serially uncorrelated, and has time-invariant

covariance matrix Σ. Table E1 reports the estimators Πi of Πi for i = 0, 1, which

are based on monthly observations of Φt during the sample period. Given the Φt and

the Πi, we recover the corresponding values of εt. From these values, we estimate the

covariance matrix Σ, assuming weak stationarity. The fitted Σ is reported in Table

E2. An analysis of the error series indicates the appropriateness of the model (D1)

for our covariates. Figure D1 visualizes the goodness-of-fit by plotting the predicted

vs. the realized covariates.

APPENDIX 85

1987 1991 1995 1999 2003 20070

0.050.1

T−Bill (3M)

1987 1991 1995 1999 2003 20070

0.050.1

T−Bill (10Y)

1987 1991 1995 1999 2003 20070.05

0.10.15

Moody’s Baa

1987 1991 1995 1999 2003 20070.020.070.12

Moody’s Aaa

1987 1991 1995 1999 2003 20070

0.060.12

LIBOR (3M)

1987 1991 1995 1999 2003 2007−1

01

S&P500

1987 1991 1995 1999 2003 2007−1

01

Banks

1987 1991 1995 1999 2003 2007−1

01

Financial

1987 1991 1995 1999 2003 2007−1

01

Insurance

1987 1991 1995 1999 2003 2007−1

01

Real Estate

Actual Predicted

Figure D1: Realized vs. VAR(1) predicted time series (monthly) of covariate compo-nents.

E Default volume model

We adopt a simple but empirically meaningful model of default volumes. We assume

that each D∗n has a generalized Pareto distribution with shape parameter ξ > 0 and

scale parameter σ > 0. We have

P (D∗n > x) =(

1 + ξx

σ

)− 1ξ

(E1)

for all x ≥ 0. The maximum likelihood estimators of (ξ, σ) are given by (0.5960, 225.8828),

with standard errors (0.0427, 10.9864) as of 12/31/2008. The left panel of Figure E1

contrasts the fitted Pareto distribution with the empirical distribution of default vol-

umes. The right panel of Figure E1 compares the observed default volumes to the

realizations of variables from the fitted Pareto distribution. The plots indicate the

statistical appropriateness of our model.

APPENDIX 86

Con

stan

tT

B(3

M)

TB

(10Y

)B

aaA

aaL

IBO

RS&

P50

0B

anks

Fin

Insu

rR

lEst

TB

(3M

)0.

172

1.00

60.

133

-0.1

280.

031

-0.0

360.

194

-0.2

380.

244

-0.4

990.

200

(1.2

79)

(21.

395)

(2.7

62)

(-1.

776)

(0.3

51)

(-0.

844)

(1.1

70)

(-1.

551)

(1.9

07)

(-2.

872)

(2.6

89)

TB

(10Y

)-0

.179

0.13

30.

925

0.04

70.

037

-0.1

270.

622

-0.2

02-0

.053

-0.1

280.

152

(-1.

127)

(2.3

93)

(16.

299)

(0.5

59)

(0.3

56)

(-2.

564)

(3.1

77)

(-1.

120)

(-0.

352)

(-0.

623)

(1.7

33)

Baa

-0.0

02-0

.018

-0.0

510.

999

0.04

20.

019

0.43

5-0

.249

-0.0

01-0

.043

0.06

4(-

0.01

8)(-

0.37

9)(-

1.04

0)(1

3.57

6)(0

.461

)(0

.445

)(2

.560

)(-

1.59

1)(-

0.00

6)(-

0.24

2)(0

.837

)

Aaa

-0.0

880.

080

-0.0

250.

048

0.98

0-0

.078

0.46

8-0

.270

0.06

0-0

.032

0.07

9(-

0.70

4)(1

.837

)(-

0.56

2)(0

.728

)(1

1.95

8)(-

1.99

8)(3

.050

)(-

1.90

4)(0

.502

)(-

0.20

2)(1

.154

)

LIB

OR

0.14

80.

227

0.10

1-0

.073

0.01

20.

752

0.65

9-0

.564

0.25

5-0

.614

0.25

5(0

.961

)(4

.224

)(1

.835

)(-

0.88

5)(0

.121

)(1

5.59

2)(3

.465

)(-

3.21

2)(1

.737

)(-

3.08

9)(2

.993

)

S&P

500

0.12

60.

020

0.02

3-0

.047

0.01

7-0

.016

0.89

80.

101

-0.0

56-0

.080

-0.0

32(3

.204

)(1

.424

)(1

.648

)(-

2.25

6)(0

.662

)(-

1.29

1)(1

8.48

4)(2

.246

)(-

1.49

9)(-

1.57

3)(-

1.47

7)

Ban

ks0.

057

0.03

10.

010

-0.0

360.

026

-0.0

29-0

.033

0.95

40.

016

-0.0

53-0

.046

(1.0

98)

(1.7

03)

(0.5

52)

(-1.

289)

(0.7

77)

(-1.

812)

(-0.

514)

(16.

264)

(0.3

32)

(-0.

794)

(-1.

607)

Fin

0.12

80.

048

0.00

1-0

.069

0.06

2-0

.044

0.14

70.

111

0.79

5-0

.121

-0.0

37(2

.096

)(2

.261

)(0

.068

)(-

2.11

9)(1

.548

)(-

2.31

3)(1

.952

)(1

.592

)(1

3.65

6)(-

1.53

5)(-

1.10

6)

Insr

0.01

70.

046

-0.0

08-0

.033

0.04

2-0

.039

0.04

00.

016

0.03

40.

801

0.00

3(0

.379

)(2

.987

)(-

0.50

1)(-

1.42

2)(1

.446

)(-

2.82

5)(0

.745

)(0

.331

)(0

.814

)(1

4.21

9)(0

.121

)

RlE

st0.

030

0.05

9-0

.008

-0.0

330.

043

-0.0

580.

055

0.09

2-0

.041

-0.0

670.

921

(0.5

96)

(3.4

10)

(-0.

459)

(-1.

228)

(1.3

09)

(-3.

753)

(0.8

99)

(1.6

24)

(-0.

870)

(-1.

052)

(33.

539)

Tab

leE

1:F

itte

dco

effici

ents

ofV

AR

(1)

model

(D1)

asof

12/3

1/20

08.

Thet-

stat

isti

csar

esh

own

inpar

enth

esis

.

APPENDIX 87

TB

(3M

)T

B(1

0Y)

Baa

Aaa

LIB

OR

S&

P50

0B

anks

Fin

Insu

rR

lEst

TB

(3M

)0.

0424

0.02

200.

0048

0.00

830.

0257

0.00

200.

0006

0.00

360.

0005

0.00

21T

B(1

0Y)

0.02

200.

0589

0.03

960.

0405

0.02

480.

0010

-0.0

009

0.00

16-0

.001

60.

0019

Baa

0.00

480.

0396

0.04

430.

0367

0.02

14-0

.002

1-0

.002

3-0

.003

0-0

.003

10.

0002

Aaa

0.00

830.

0405

0.03

670.

0362

0.01

85-0

.000

9-0

.001

9-0

.001

4-0

.002

30.

0007

LIB

OR

0.02

570.

0248

0.02

140.

0185

0.05

55-0

.000

6-0

.000

80.

0003

-0.0

014

0.00

23S&

P50

00.

0020

0.00

10-0

.002

1-0

.000

9-0

.000

60.

0036

0.00

300.

0038

0.00

240.

0019

Ban

ks

0.00

06-0

.000

9-0

.002

3-0

.001

9-0

.000

80.

0030

0.00

620.

0053

0.00

430.

0031

Fin

0.00

360.

0016

-0.0

030

-0.0

014

0.00

030.

0038

0.00

530.

0087

0.00

400.

0039

Insu

r0.

0005

-0.0

016

-0.0

031

-0.0

023

-0.0

014

0.00

240.

0043

0.00

400.

0045

0.00

24R

lEst

0.00

210.

0019

0.00

020.

0007

0.00

230.

0019

0.00

310.

0039

0.00

240.

0058

Tab

leE

2:F

itte

dco

vari

ance

mat

rix

Σof

the

VA

R(1

)er

ror

term

ε tas

of12

/31/

2008

.

APPENDIX 88

100

101

102

103

104

105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x

)

Empirical DistributionFitted Distribution

−4 −2 0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

8

10

12

14

Empirical Quantiles (Logged)

Sim

ula

ted

Qu

an

tile

s (L

og

ge

d)

Figure E1: Left panel : Empirical default volume distribution vs. fitted generalizedPareto distribution as of 12/31/2008. Right panel : Empirical quantiles of the ob-served default volumes vs. quantiles of realizations of variables from the fitted Paretodistribution.

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