bridging debt and equity markets - moody's analytics

JANUARY 29, 2004

BRIDGING DEBT AND EQUITY MARKETS USING AN OPTION-PRICING MODEL OF CORPORATE DEFAULT MODELINGMETHODOLOGY

ABSTRACT

The option-pricing framework first introduced by Fischer Black, Myron Scholes, and Robert Merton in the 1970s facilitates the development of valuation models to use equity market information to price corporate bonds and credit default swaps. This paper reviews a particular implementation of an option-pricing model developed by Moody’s KMV to relate equity and debt markets. The out-of-sample testing suggests that this structural model can be used effectively in pricing debt when only equity prices are available.

AUTHORS

Deepak Agrawal Navneet Arora Jeffrey Bohn Kehong Wen Bin Zeng

CONTACT

Jeffrey Bohn 415-352-1279 [email protected]

© 2003 Moody’s KMV Company. All rights reserved. Credit Monitor®, EDFCalc®, Private Firm Model®, KMV®, CreditEdge, Portfolio Manager, Portfolio Preprocessor, GCorr, DealAnalyzer, CreditMark, the KMV logo, Moody's RiskCalc, Moody's Financial Analyst, Moody's Risk Advisor, LossCalc, Expected Default Frequency, and EDF are trademarks of MIS Quality Management Corp.

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TABLE OF CONTENTS

BRIDGING DEBT AND EQUITY MARKETS USING AN OPTION-PRICING MODEL OF CORPORATE DEFAULT 3

1 INTRODUCTION .................................................................................................. 5

2 OPTION-PRICING MODEL FOR CORPORATE DEBT AND EQUITY ......................... 5

3 DATA .................................................................................................................. 6 3.1 DATA SOURCES ......................................................................................................... 6 3.2 FILTERS ..................................................................................................................... 6

4 FITTING THE MODEL .......................................................................................... 7 4.1 MODEL PARAMETERIZATION ................................................................................... 7 4.2 OUTLIER CHARACTERIZATION................................................................................. 8

5 OUT-OF-SAMPLE TESTING................................................................................. 9 5.1 OUT-OF-TIME TESTING............................................................................................. 9 5.2 OUT-OF-FITTING-SAMPLE TESTING...................................................................... 12 5.3 OUT-OF-BOND-DATA TESTING .............................................................................. 15

6 CONCLUSION.................................................................................................... 17

APPENDIX A VALUATION MODEL DERIVATION.................................................... 18

REFERENCES.............................................................................................................. 21


1 INTRODUCTION

The corporate debt markets have evolved substantially in recent years. More companies are tapping the capital markets for financing instead of commercial bank loans. The rapid growth of the credit default swap market creates an environment to manage portfolios of corporate debt more efficiently. Despite this growth in the debt markets, the lack of transparency in pricing makes it difficult to mark a portfolio of corporate debt to market on a regular basis. Fortunately, the equity markets provide a transparent, frequent indication of a company’s health.

Using the option-pricing framework, a company’s debt and equity prices can be related in a mathematically consistent way (see (Black and Scholes 1973); (Merton 1974); (Vasicek 1984, 1999) (Bohn 2000a); and (Kealhofer 2003) for detailed discussions of the theoretical foundations of this framework.) This paper reports the results of testing this relationship between debt and equity markets. That is, we use the information from the equity market coupled with systematic factors reflected in the U.S. corporate bond market to fit individual corporate bond spreads. Our working hypothesis in this study is that parameterizing an economic, structural model of debt value across a large number of debt instruments coupled with the relatively better measure of market expectations of firm value reflected in the equity market produces superior calculations of value for a large cross-section of debt-issuing firms despite the poor quality of price data in the credit markets.

Section 1 of this paper reviews the salient aspects of the modeling framework. Section 2 discusses the data used in this study. Section 3 reports the details of the model fitting exercise. Section 4 focuses on the out-of-sample testing done to demonstrate the robustness of the model fits. Section 5 concludes.

2 OPTION-PRICING MODEL FOR CORPORATE DEBT AND EQUITY

The values of debt and equity issued by a firm link back to the underlying value of the issuing firm. Since both debt and equity can be viewed as options written on the issuing firm’s asset value, we can write down formulas relating the market values of debt, equity, and assets. (See (Black and Scholes 1973) and (Merton 1974) for details). If we assume the equity value is a function of the asset value, we can construct a cause and effect model to calculate the default probability associated with debt issued by the firm. In the case of MKMV, we calculate an expected default frequency (EDF) based on a Black-Scholes-Merton-type structural model (we refer to this version as a Vasicek-Kealhofer model which is modified to maximize its predictive power for default; see (Kealhofer 2003a; Kealhofer 2003b) In addition to EDF, the debt value will depend on the debt’s tenor, the expected loss given default (LGD), and the price paid for risk also known as the market risk premium.

The structural model used by MKMV in modeling default risk provides a framework to help understand the various factors driving credit quality in a cause-and-effect way. We use a structural model for valuation because the factors directly affecting credit spreads (or market price of risk) can be analyzed systematically and model estimates are far more easily interpreted. The use of a structural model also overcomes the problem that credit spread data tends to be noisy and does not readily reveal certain structures. These types of data make it difficult to estimate a useful, ad-hoc econometric model.

The key relationship embodied in the structural model discussed in this paper is as follows:

0 (1/ ) ln(1 )TS T LGD CQDF= − ⋅ − ⋅ (1)

where S0 is the option-adjusted spread over the zero-EDF yield curve, T is the tenor measure of a corporate bond, LGD is loss-given-default, and CQDFT is the cumulative risk-neutral default probability which depends on the cumulative EDF (CEDF) as follows (see (Bohn 2000a) for details on the derivation):

1 1/ 2[ ( ) ]T TCQDF N N CEDF RTλ−= + (2)

6

where N denotes the standard cumulative normal distribution function, λ is the market Sharpe ratio – the market price of risk, and R is the square root of the R-squared of the firm issuing the bond, where R-squared measures the degree to which the firm’s asset value is driven by systematic risk factors in the economy.

3 DATA

3.1 DATA SOURCES

We focus on U.S. companies with both publicly traded bonds and equity. The equity data are received from Compustat. The bond data are received from EJV, a subsidiary of Reuters. In addition, we make use of data from Capital Access International to help build filters for the bond data. Credit default swap (CDS) data received from Creditrade are used in the out-of-sample testing. The benchmark yield curve used for calculating spreads is the U.S. swap yield curve as reported by Bloomberg.

Given the noise in corporate bond data, we have developed sophisticated filters to select data that can be trusted to reflect actual value. This filtering step is crucial to finding robust estimates of model parameters. Much of this section will characterize the filters necessary to arrive at a usable sample of bonds.

We begin with data on about 85,000 bonds issued by about 2,900 U.S. companies. After running through our data filters and ensuring that we have both equity and bond data on each company, we end up with 795 U.S. companies in our sample. These companies issued about 3,000 bonds that are tracked by EJV. We analyze daily prices beginning January 1, 1999 and ending January 31, 2002. We end up with 1,203,146 observations of bond prices.

Because the process of transforming the equity prices into actual expected default probabilities is explained in more detail elsewhere, we do not discuss in this paper how the Expected Default Frequencies™ or EDF™ values are calculated (see (Vasicek 1984, 1999) and (Kealhofer 2003a)for details). These EDF values are used as the direct estimate of the actual expected probabilities of default for each of these firms. It is important to note that these EDF values are the result of a calibration to an empirical database of defaulted firms maintained by Moody’s KMV (MKMV). MKMV calculates a term structure of EDF values for each firm for each business day where equity data are available. This term structure of EDF values provides the basis for matching an expected default probability to a particular tenor. We use these EDF values calculated in this modeling exercise. MKMV’s EDF values are available on a subscription basis over a system called CreditEdge™. Currently, 100 financial institutions globally use this system to retrieve expected default probabilities on publicly traded firms.

3.2 FILTERS

Because of the poor quality of bond data, filtering out noisy data becomes a necessary step for robust model parameterization. After exploring the data, we discovered certain types of bonds that had spreads that would be difficult to explain within the context of any reasonable modeling framework that retains economically meaningful interpretation of parameter estimates. For example, bonds issued by foreign subsidiaries of high credit quality parents can trade at substantially wider spreads than suggested by their EDF values. We conjecture that certain aspects of the market for these kinds of bonds produce spreads not easily modeled in our contingent-claims framework. Another example is bonds issued with various embedded options such as calls and puts. The reported pricing for these optionable bonds can, on occasion, exhibit unusually wide spreads (i.e. spreads that exceed what is reasonable for both the firm’s credit quality and the implicit value of the option). Upon further investigation, we determined that the data in a number of these cases were in error. We developed the following filters to effectively remove most of the large, difficult-to-model outliers:

• Remove bonds issued by foreign subsidiaries

• Remove putable, convertible, PIK bonds

• Remove callable bonds that are near being called

• Remove floating rate bonds

• Remove bonds issued by real estate investment trusts (REITs) as identified by EJV industry sector code


• Remove bonds with closing prices that are far from par

• Remove junior debt according to EJV debt type codes

• Remove bonds from issuers with EDF values greater than 18%

• Keep bonds that have amount outstanding greater than USD10,000,000

• Keep bonds with time to maturity between 1 and 10 years

A second level of filtering is used to eliminate bonds that are likely to suffer from lack of liquidity. The prices quoted for these bonds do not reflect the actual credit risk of the issuing firm. The filters are as follows:

• Capital Access International1 transaction filter (the extent to which a particular company does not transact often proxies for illiquidity and these bonds are excluded.)

• Remove bonds with negative OAS or OAS greater than 2000 bps

These filters produce a sample of bond prices driven by credit related affects. The result is a set of model parameters that are economically meaningful and a degree of model fit (to be described later) supporting the usefulness of this model in analyzing the relative value of both traded and untraded corporate bonds. We evaluated the data across a number of dimensions including sector, issuer size, and credit quality and do not observe any unusual biases in the sample.

4 FITTING THE MODEL

4.1 MODEL PARAMETERIZATION

After filtering the corporate bond sample, we are prepared to fit our valuation model that bridges the equity and debt market. We begin with the CEDF estimated out of the equity market. We then use the corporate bond market to estimate spreads for the individual corporate bonds. These spreads are calculated in reference to the zero-EDF curve. We estimate the zero-EDF curve using swap data and high-grade corporate bonds. We then fit a market Sharpe ratio using the sample cross-section of bonds. Recall that the market Sharpe ratio relates to compensation for taking systematic risk. As part of this fitting exercise we use the R-squared estimates from MKMV’s Global Correlation Model. These R-squared values reflect the degree to which an individual firm’s asset value is impacted by changes in systematic risk factors. Finally, we estimate an implicit LGD based on the formulas outlined above.

With just one market-wide parameter, the valuation model performs reasonably well against a well-filtered sample of bond spread data on a daily basis. Additional explanatory power can be exploited on the industry sector level when LGD is taken to be a sector-specific parameter. On the firm level, an additional premium, referred to as size premium, has been helpful in explaining the residual spread differential across names. After analyzing the residuals of the model fit in this manner, we discovered a relationship between residuals and the size of the firm where size is measured as the firm’s total sales. In the end, we estimated the model by assuming that a size premium is added to the premium associated with the firm’s credit risk. This size premium is included as an additive term as follows:

0

1 1/ 2

(1/ ) ln(1 )

[ ( ) ]T

T T

S size premium T LGD CQDF

CQDF N N CEDF RTλ−

= − ⋅ − ⋅

= +

(3)

The size premium is firm-specific and acts as the intercept term on the spread term structure for the firm.

1 This database reflects actual transactions by insurance companies.

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4.2 OUTLIER CHARACTERIZATION

With the extensive set of filters outlined in section 2.2, most of the outliers from the original sample have been well controlled. The absolute and relative levels of fitting errors have been minimized. The remaining outliers can be classified into the following groups:

• Contemporaneous: While the EDF implied spread (EIS) and OAS may diverge on a given date, over time EIS and OAS move with each other without a discernible lead-lag relationship.

• EIS Leading Indicator: EIS leads OAS in the sense that subsequent to the date when the two measures diverge OAS “chases” the EIS.

• LGD Effect: Difficulty in the estimation of LGD is responsible for the deviation between EIS and OAS.

• Subsidiary: Using the parent firm’s EDF value may understate or overstate the actual EDF value for the subsidiary.

• Lagging Liability Data: The lag in obtaining a firm’s financial statement data may cause misalignment between EIS and OAS

• EDF Effect: The EDF level appears to be too low or too high.

• Data Error: Raw OAS data may be wrong.

• Tender Offer: OAS suddenly narrows substantially following a tender offer.

We examined closely a sample of 400 bonds representing the top 20% of issues where the EIS and OAS substantially diverge most from each other. We tracked the behavior of the fitting error of these bonds over a three-year time period within the sample. The following statistics categorize the outliers arising from fitting the model in terms of the types of outliers explained above:

• Contemporaneous 41%

• EIS Leading Indicator 34%

• LGD Effect 17%

• Subsidiary 7%

• Lagging Liability Data 4%

• EDF Effect 4%

• Data Error 2%

• Tender Offer 1%

As we can see from the results, most bond spreads either converge to EIS or move contemporaneously with EIS.


5 OUT-OF-SAMPLE TESTING

5.1 OUT-OF-TIME TESTING

The previous sections reported the strength and unbiased nature of the model’s fit to individual bond spreads for the sample used to estimate the model. This section expands the testing process by analyzing bond data that is different from the sample used to fit the model. The first test looks at the model fits for bond data from dates different than those used to estimate the model parameters.

Four time horizons are analyzed. That is, we analyze the fit to bond data on the following time periods based on previously estimated parameters:

• One business day later

• One business week later

• One month later

• 6 months later

The EIS or fitted OAS is a function of estimated parameters such as the market price of risk (i.e. market Sharpe ratio), a size premium, and an implied LGD. The variables used as input are swap yields, EDF values, and market R-squareds (i.e. the extent to which a custom market index explains the variability of a firm’s asset value).

In this analysis, we test the robustness of the estimated parameters. To address this issue, we calibrate model parameter values on data available on any given date and apply them to data on a following date. If the parameters have been robustly estimated, we would expect the model fits to actual spreads on subsequent dates to still be reasonable, with gradual deterioration in fit as the out-of-sample date is farther away from the fitting date.

Our empirical results are generally consistent with this expectation. Consider the data sample available on July 31, 2001. We estimate the parameters for the model on this date and then choose four subsequent dates to evaluate the quality of estimated spreads using the model parameters estimated on July 31, 2001 together with the EDF values available on the subsequent dates of analysis. We evaluated only the bonds that were available on the date of fitting and the date of evaluation (bonds that might have gone into distress or default in subsequent dates were still included as long as a traded price was available).

We look at the following four dates:

• August 1, 2001

• August 7, 2001

• August 30, 2001

• January 31, 2002

Overall, the model still provides reasonable fits for all the four time horizons, with the correlations between OAS and EIS staying around 82%. The EIS is fairly unbiased even if the time lag is as long as one month. Only when it increases to six months can we identify some underestimation bias of EIS across board. This bias can be mostly explained by the noticeable increase in the implicit market risk premium between July, 2001 and January, 2002. We include graphs for the one month later and six months later results, only. The results for one day later and one week later are even better with much tighter residual distributions.

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FIGURE 1 One month forward spread errors

FIGURE 2 Six month forward spread errors


We repeat this exercise on datasets for each business day from August 1, 1999 to January 31, 2002. Figure 3 reports the aggregate results for the percentage (with respect to yield) errors for the six-month lag. That is, we report the distribution of pricing errors for the entire sample when using parameters estimated on the sample six months prior to the fitting. We do this exercise for each date from August 1, 1999 to January 31, 2002 and aggregate the results into a distribution over the entire sample.

Again, the results demonstrate that EIS generally offers an unbiased model fit to actual bond spreads even with parameters estimated six months earlier; the deviations of EIS from OAS increase somewhat with the length of time lag of the parameter estimation. The aggregate results are better than what we observe for a single day’s data. In particular we do not observe the systematic underestimation of EIS for OAS even for a six-month time horizon. The aggregation of multiple days eliminates idiosyncratic bias that may occur when choosing two particular days for analysis.

Our testing demonstrates that the model’s robustness. Not only are the estimated parameters fairly stable over time, but these parameters also produce reasonably good fits to bond spreads even when the estimated parameters are held constant for as long as six months.

FIGURE 3 Aggregate six month forward spread errors

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5.2 OUT-OF-FITTING-SAMPLE TESTING

While the results on the out-of-time testing are encouraging, potential autocorrelation in the data may render them less convincing. A second test involves dividing the sample into two groups: one group used for fitting and one group used for testing. In this way, we fit the model on one set of data, but test the model on a sample completely different from the one used for training.

Over the three-year sample period, the dataset includes 795 companies. We use all the bond issues from a randomly selected 398 companies as the fitting sample; the rest of the sample is used for testing. Even though we have not deliberately divided the sample to ensure similar characteristics such as number of issues, bond types, seniority of issues, or industries, the fitting sample and testing sample turn out to have similar characteristics along these dimensions.

By following the same procedure as described in the section 3, we use the fitting sample to estimate the model parameters and then apply them to the testing sample to get EIS for every bond issue.

We can look at the testing results in two dimensions. First, we compare the parameters estimated from the fitting sample with those estimated from the testing sample. If the model is robust, the parameters estimated from samples should be similar. Indeed, most model parameters estimated from both sub-samples are strikingly close to each other. Figure 4 is a typical example which contrasts market Sharpe ratios (MSR) over time for the two sub-samples. Over the three-year period, the two sets of MSRs follow each other very closely with an average deviation of 0.025. The average difference between the size premiums for a typical bond issuer (in terms of size) from both subsamples is less than 5 basis points. The comparisons for 29 estimated LGDs generate similar results, as illustrated in Figure 6.

FIGURE 4 Market Sharpe Ratio estimated from two random subsamples


FIGURE 5 Size Premium estimated from two random subsamples

FIGURE 6 LGD estimated from two random subsamples

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Next, we evaluate the model fitting performance by using the fitting-sample estimated parameters on the testing-sample data. Since we have already seen that parameters estimated in both sub-samples are similar, we expect the model fits to be close in the testing sample even if we use estimated parameters from the fitting sample. Figure 7 displays that the correlations between OAS and EIS are around 80% regardless of whether the EIS are obtained using fitting-sample estimated parameters or parameters estimated directly from the testing sample. Figure 8 shows narrow, unbiased distributions of model errors.

FIGURE 7 Correlation between OAS and EIS (computed using fitting vs. testing sample parameters)


FIGURE 8 Spread errors using out of sample parameter estimates

5.3 OUT-OF-BOND-DATA TESTING

The out-of-fitting sample tests demonstrate convincing robustness in this modeling approach. Our final test moves us to using a completely different set of data that can be considered to be out of the universe of bond data. In this case, we use the model estimated with bond data to fit individual premiums in the CDS market.

The CDS data are described in the section 2.1. These data have not been used in any of the fitting exercises described in this paper. The testing includes two parts. First, we estimate the model parameters on the CDS data and compare the CDS-estimated parameters with their values when estimated on the bond data. Second, we take the parameters estimated from the bond sample and use them to determine the EIS for CDS from our CDS dataset. The information on each CDS in our dataset includes the maturity but we do not know the seniority of the underlying reference entity. To carry out our testing, we use the LGD table of the senior, unsecured bonds.

Because we do not have a large sample of CDS data, we focus on the dates for which we have a reasonable number of observations. We do not evaluate the dates of sparsely populated data. In the end we look at data from CDS with a tenor of 5 years for most trading dates in 2001.

The CDS dataset has substantially fewer observations than the bond dataset. The resulting model parameters from the CDS data reflect much more variation than the parameters estimated from the bond data. This behavior is generally expected when the underlying sample is small.

One distinct feature for the CDS data is that while the CDS parameters follow the dynamics for those from the bond data, the MSR values estimated from the CDS data are consistently lower than the MSR values calculated on the same dates from the bond data as illustrated in Figure 9. The size premiums reflected in the CDS data are lower, also. Despite the systematic differences in the other estimated parameters, the LGD estimates from the CDS data are comparable to those estimated from the bond dataset.

16

When we apply the model parameters estimated from the bond data to the CDS data, we expect the EIS will typically over-estimate the CDS premiums. Figure 10 confirms this expectation with lower quality obligors reflecting more of this bias. Despite the slight biases in the model fits for the CDS data, the model performs reasonably well given the completely different nature of the data sources. Given the small number of CDS premiums in the dataset, the parameter estimates are relatively noisier. As more data become available, we will likely develop better estimates of model parameters and see more convergence with results using bond data.

FIGURE 9 Market Sharpe Ratio derived from bond and CDS data


FIGURE 10 CDS vs EIS

6 CONCLUSION

We have outlined a simple, parsimonious model for determining the spreads on corporate bonds using the equity market information contained in the EDF credit measures. Our approach is theoretically well grounded in the option-pricing framework of Black and Scholes (1973) and Merton (1974). We have shown how this model can be implemented in practice on a large but noisy sample of corporate bond data. We find that the estimated parameters have economically intuitive values and also a good degree of stability in out-of-sample and out-of-time tests. The fit of the model is also quite reasonable, both in the fitting sample and in a testing sample. We view these results as evidence of success of our approach in modeling the spreads on credit risky claims.

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APPENDIX A VALUATION MODEL DERIVATION

Because the spread is driven directly by the default probability, the structural model is considered to be a “pure” credit spread model. It can be derived from the Black-Scholes option pricing formula and the Capital Asset Pricing Model (CAPM) applied to asset return.

Consider a zero-coupon bond that matures in T years. The bond has a face value of F. If the bond issuer defaults before or at time T, the bondholder receives a recovery value of (1 – LGD)F. If the issuer does not default, the bondholder gets back F. Let V be the value of the bond at time 0, and let the continuously-compounded default-risk-free discount rate be r. The cumulative probability of default at time T is given by CEDFT and the cumulative risk-neutral default probability is given by CQDFT.

Option pricing theory establishes a relatively straightforward procedure to value any type of contingent claim. The general established principle via no-arbitrage argument is that the value of any risky instrument equals the expected cash flows under the risk-adjusted probability measure, discounted by the risk-free rate. The key for valuation here is to find the correct risk adjustment to the real probability measure so that the so-called risk-neutral probability, usually referred to as the Q-measure, can be defined.

Mathematically, this concept is captured by the following:

1

( )t

Nr t Q

tt

V e E C−

=

= ∑ (4)

where Ct represents the risky cash flow at time t.

Applying the general valuation formula to the specific model described above, the value of the defaultable zero-coupon bond becomes:

( ) ( )

( )

[ (1 ) ]

[1 ]i

rT QT T

rT QT

V e E FI LGD FI

Fe E LGD I

τ τ

τ

−> ≤

−≤

= + −

= −

(5)

where τ denotes the time of default and I is an indicator function.

The first equality follows from the fact that the zero-coupon bond has only one cash flow F and the cash flow depends on whether or not τ is greater than T.

Now assume that the recovery process is independent of the process driving the distance-to-default. The expectation operator in the above equation can now be moved inside of the bracket to obtain:

[1 ( ) ]irT QTV Fe E LGD Q−= − (6)

Here the QT is simply the cumulative risk-neutral default probability CQDFT. Note that without the independence assumption this formula would be considerably more complex and it might become impossible to factor out the risk-neutral default probability.

By definition of the credit spread S0, the following holds true:

0( )r S TV Fe− += (7)

The cumulative, risk-neutral or quasi-probability of default, CQDFT differs from CEDFT in the sense that investors require a higher expected return than the risk-free return on an underlying asset subject to systematic risk. Under the


assumption that the underlying asset process follows a lognormal distribution, the Black-Scholes option pricing argument can be used to derive a closed-form transformation between CQDFT and CEDFT as follows:

1( )T Tr

CQDF N N CEDF Tµ

σ−⎛ ⎞−⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

(8)

If we assume that the market risk premium can be determined by the Capital Asset Pricing Model (CAPM), we can write:

Acov( - ) = ( ),var( )

MM

M

r rr r

rµ µ

⎡ ⎤− −⎢ ⎥

⎣ ⎦

(9)

and thus

cov( - )( ) = A M M

A M A M

r r rr µµσ σ σ σ

⎡ ⎤ ⎡ ⎤−−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(10)

or

= A

rR

µ λσ−

(11)

where the risk-neutral probability CQDFT is a function of the actual probability to time T, CEDFT, the market Sharpe ratio λ, the asset’s correlation with the market R, and the tenor T.

From this derivation, one can see that the market price of risk, or the Sharpe ratio of the market, is closely attached to the systematic risk measure of the asset return. In general, this price of risk reflects the aggregate risk preference of investors in the credit market. In contrast to the potentially rapid change of business risks, this market risk preference is believed to be relatively stable.

REFERENCES


REFERENCES 1. Black, F. and M. Scholes (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy

81: 637-659.

2. Bohn, J. R. (2000a). "A Survey of Contingent-Claims Approaches to Risky Debt Valuation." Journal of Risk Finance 1(3): 53-78.

3. Kealhofer, S. (2003). "Quantifying Credit Risk II: Debt Valuation." Financial Analysts Journal 59(3 (May/June)): 78-92.

4. Kealhofer, S. (2003a). "Quantifying Credit Risk I: Default Prediction." Financial Analysts Journal 59(1 (January/February)): 30-44.

5. Kealhofer, S. (2003b). "Quantifying Credit Risk II: Debt Valuation." Financial Analysts Journal 59(3 (May/June)): 78-92.

6. Merton, R. C. (1974). "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates." Journal of Finance 29: 449-470.

7. Vasicek, O. A. (1984, 1999). "Credit Valuation." KMV Corporation (now Moody's KMV Company) Working Paper.

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