Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios on Corporate Bonds Stephen...

Structural Models of Credit Risk are Useful:Evidence from Hedge Ratios

on Corporate Bonds

Stephen M. SchaeferLondon Business School

International Financial Research ForumFinancial Risks

New Developments in Structured Products and Credit DerivativesParis, 27-28 March 2008

Structural Models are Useful

Joint work with:

Ilya A. StrebulaevStanford University


Structural Models

• Structural models of credit risk represent default in terms of the value of the firm’s assets (that collateralise the debt): falling short of the face value of the debt at maturity

(Merton model); or hitting a lower threshold representing (e.g.) the point

at which lenders will intervene and liquidate the firm (“second generation” models – e.g., Leland)

• Structural models represent the best framework currently available to analyse “fundamental” value in credit setting

BUT …..


Introduction

• Structural models fail to explain size of yield spreads on corporate bonds e.g. Huang and Huang (2003) - 5 models:

Model Actual Data

A 14-39 123

BBB 39-59 194


What do credit spreads in structural models represent?

• The credit spread in a structural model is approximately:

,

Credit Risk Expected Spread Loss Rate Premium

AnnualRisk Premium Loss-given-default Default

on EquityProbability

D E

• So possible reasons for underestimating spreads: underestimating default probability (or LGD) underestimating hedge ratio (beta) of debt to equity

(or equity risk premium) or .. impact of other variables (liquidity etc.)


Source: Huang & Huang

Yield Spread, Default Loss Rate and Calculated Credit Spread (10-year bonds)

0%

1%

2%

3%

4%

5%

AAA AA A BBB BB B

Credit Rating

Yield Spread

Default Loss Rate

Calculated Credit Spread


• So, is it all bad news?


Structural Models and Default Probabilities

• Actually structural models appear to provide reasonably good (or at least not bad) estimates of default probabilities

Leland (2002), Huang and Huang (2003)

Moody’s KMV


Leland’s Estimates of Default Probabilities

• Leland uses default boundary model with realistic input parameters to calculate default probabilities

• A-rated bonds; asset volatility is 23% (Base case).

• B-rated bonds; asset volatility is 32%

Dotted line is actual. Dotted line is actual.

Source: Leland, H. “Predictions of Expected Default Frequencies in Structural Models of Debt”, Working Paper, Univ. of California, Berkeley, September 2002.


0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 1 2 3 4 5

Horizon

Def

ault

Pro

bab

iity

Leland - Default Prob w. Asset Vol =23%

Moody's Default Prob. BBB 1970-2005

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

0 5 10 15 20

Horizon

Def

ault

Pro

bab

iity

Leland - Default Prob w. Asset Vol =23%

Moody's Default Prob. BBB 1970-2005

Short-Term vs. Long-Term Default Probabilities

• Long-term (7-8 years and longer) default frequencies fit quite well

• Short-term (1-6 years and below) default frequencies are too low


What do we do in this paper?

• Existing research: default probabilities – (partial) success spreads – failure (i.e., so far – no success for variable that

depends on prices)

• This paper: perhaps the risk premium component is underestimated do structural models predict hedge ratios of corporate debt

to equity?

,Credit Default Risk Premium Loss-given-default Spread Probability on EquityD E


Why are hedge ratios important - I?

• Determine risk premia

• In all structural models the bond value is determined as the price of the replicating portfolio in theory portfolio and equity and riskless debt replicates

payoff on bond composition of the replicating portfolio is determined by

the hedge ratios so bond price is determined by the hedge ratios

• If observed hedge ratios consistent with those predicted by models, then that failure of models to predict spreads likely to be due to non credit risk factors


Why are hedge ratios important - II?

• Hedge ratio: measures exposure of debt

value to value of collateralising assets

hedge ratio High => high credit risk

hedge ratio low => low credit risk

0

1

2

3

4

5

6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

value of equity

Va

lue

of

De

bt

value of riskless debt

high hedge ratio (slope) = credit exposure

low hedge ratio (slope) = low credit exposure


Focus of Paper

• Estimate hedge ratio regressions:

• In a world governed by structural models, hedge ratio regressions would produce coefficients j,E close to one

high explanatory power (R2 close to 1) ... but not exactly as a result of (a) non-linearity; (b) discreteness

• We show that (a) and (b) are not important and test hypothesis that j,E = 1

• Consider other systematic factors (a la Collin-Dufresne) and examine their relation to underlying credit risk

,, , , , , ,rf j tj t j j E E j t E t Treas tr h r r


Main Findings - 1

• Simple structural model (Merton, 1974) provides reasonably good estimates of hedge ratios of corporate debt to equity

• BUT returns on corporate bonds also strongly related to: SMB and HML (Fama-French factors) .. But NOT in a way that is

related to exposure to underlying equity and NOT in a way that appears linked to credit exposure

S&P (or VIX) .. but NOT in way that is linked to credit risk

• Thus these factors seem to have significant effects on prices / returns but not via credit risk channel

• Another puzzle: structural (Merton) model fails to explain LOW empirical hedge

ratios of debt to riskless bonds (duration)


Main Findings - 2

• While it is true that structural models underestimate corporate yield spreads

.. if spreads reflected credit risk alone then the sensitivity of bond returns to equity would be higher to be consistent with reasonable estimates of the equity risk premium

in fact .. empirical sensitivities correspond quite well to predictions of simple structural model.


Data• Merrill Lynch: Corporate Master Index and Corporate High

Yield Index covers nearly all corporate bond issues in the U.S. (2114 issuers;

10370 issues)

• Monthly price data from 12.1996–12.2003 (388,000 bond-month observations) final sample satisfies additional standard criteria (only US bonds,

matching with CRSP/COMPUSTAT, no financials, only straight bonds)

• Entire and final sample:

Entire Sample Final Sample

Bonds 10,370 1360

Issuers 2,114 396


Descriptive Statistics: Final Dataset


A Simple Time-Series Hedging Regression

• We run the following regression :

,, ,, ,, j tj t j j rj E Treas tE tr rr

All AAA AA A BBB BB BIntercept 0.02 0.02 0.01 0.00 0.00 0.07 0.7910-year bond (x 100) 49.59 57.28 54.65 53.25 50.33 29.36 -8.70Return on Equity (x 100) 3.79 0.61 1.17 3.16 4.00 8.27 15.22R-squared 0.51 0.66 0.63 0.55 0.48 0.34 0.35Sample size 1370 23 126 620 466 107 26

• Results: : estimated hedge ratios – small (0.006 – 0.04 for IG) but highly

(statistically) significant R2 much less than 100% sensitivity to Treasury returns (“duration”) low


Hedge Ratios

• Are these hedge ratios reasonable? compare with hedge ratios implied by Merton model

• In one-factor structural models the hedge ratio, E, is:

1 1 11 1 1/ E

D E ED D LE E V

where E is the “delta” of equity to the firm’s asset value and L is the debt-to-asset value ratio


Hedge Ratios from the Merton Model

Asset Volatility

Leverage 10 15 20 25 30 40 50

10 0.00 0.00 0.04 0.58 2.53 9.25 17.19

20 0.00 0.03 0.70 2.63 4.80 13.52 21.39

30 0.00 0.86 3.42 4.60 13.60 15.31 29.03

40 0.10 1.42 6.88 10.41 16.44 23.73 31.58

50 0.49 53.19 8.05 11.88 17.50 24.17 29.68

60 2.06 4.47 13.255 16.60 19.53 24.79 34.08

70 3.20 7.98 15.25 14.73 21.79 27.83 29.87


The Volatility of Corporate Assets

2 2 2 2 2

Quasi-market leverage ratio

Book Value of Debt (Compustat items 9 and 34)

Book Value of Debt + Market Value of Equity

Estimated asset volatility

(1 ) 2 (1 )Ajt jt Ejt jt Djt jt jt EDL L L L

, jt

All AAA AA A BBB BB BQuasi-Market Leverage

Mean 0.34 0.10 0.21 0.32 0.37 0.50 0.66Std.Dev. 0.21 0.08 0.19 0.20 0.17 0.23 0.22

Equity VolatilityMean 0.32 0.25 0.29 0.31 0.33 0.42 0.61Std.Dev. 0.13 0.06 0.10 0.11 0.13 0.19 0.19

Estimated Asset VolatilityMean 0.22 0.22 0.22 0.21 0.22 0.23 0.28Std.Dev. 0.08 0.05 0.07 0.08 0.08 0.08 0.08


Testing the Merton Model’s Hedge Ratio Predictions

• Estimates of individual hedge ratios are very noisy: calculate as mean hedge ratio within sub-rating :

,, , , , , ,rf j tj t j j E E j t E t Treas tr h r r


Using the Merton Model to Predict Hedge Ratios

Source: Schaefer / Strebulaev

Average Ratio of Emprical to Theoretical Hedge Ratio

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

All AAA AA A BBB BB B


Explaining Structural Model Spreads with Empirical Betas

,Credit Default Risk Premium Loss-given-default Spread Probability on EquityD E

Ratings: AAA - B

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

0.0% 1.0% 2.0% 3.0% 4.0% 5.0%

S&S Implied Spreads

Hu

ang

an

d H

uan

g S

pre

ads

Ratings: AAA - BBB

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7%

S&S Implied Spreads

Hu

ang

an

d H

uan

g S

pre

ads

• Implies default-boundary structural models produce very similar hedge ratios to Merton


The story so far

• Merton model produces hedge ratios in line with empirical estimates so, structural models appear to capture credit exposure

quite well

• But the R2 are lower than the model predicts equity and risk-free debt should account for large fraction

(80% plus) of debt return variation

BUT we find R2 ~ 50% – 70% for investment grade bonds and R2 ~ 30 – 40% for non-investment grade

• What other factors influence corporate bond returns? “Inside” the model: stochastic interest rates “Outside” the model: other systematic factors


Stochastic Interest Rates

• In Merton model ( effectively, Black-Scholes) riskless interest rates are fixed formally, need model that allows for uncertainty in

riskless rate

• Also, puzzle of low interest rate sensitivity of corporate debt


Low Duration Puzzle: Regressions on riskless bonds only

,, , ,0 j tj t j r Treas tr r


Including Stochastic Interest Rates

• Merton (1974) with affine interest rates (Shimko et. al. (1993), Lando (2004))

• For simplicity, consider one-factor Vasicek model

• Results on hedge ratios unchanged


Other Factors: Running a “kitchen sink” regression

• Hedge ratios for equity and riskless debt are not much changed


Sensitivity to SMB

• Sensitivity to corporate debt returns to SMB: not result of sensitivity of underlying assets to SMB not strongly connected to credit exposure (!!)


Conclusion 1

• Hedge ratios provide a good measure of credit exposure and, in this sense, structural models seem to capture credit exposure better than commonly supposed:

• But do NOT explain level of credit spreads


Conclusion 2

• Understanding identity and role of non credit risk related factors: still incomplete liquidity taxes: ?? imperfect substitution between equity, riskless

bonds and corporate bonds fluctuations in capital allocated to credit risky

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