1. Stephen M. Schaefer London Business School
Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios on Corporate Bonds
Stephen M. Schaefer
London Business School
International Financial Research Forum
Financial Risks
New Developments in Structured Products and Credit Derivatives
Paris, March 2008

2. Ilya A. Strebulaev Stanford University Joint work with:
Structural Models are Useful

3. BUT ….. 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 …..
Structural Models are Useful

4. 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
Structural Models are Useful

5. What do credit spreads in structural models represent?
The credit spread in a structural model is approximately:
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.)
Structural Models are Useful

6. Yield Spread, Default Loss Rate and Calculated Credit Spread (10-year bonds)
Source: Huang & Huang
Structural Models are Useful

7. So, is it all bad news?
Structural Models are Useful

8. 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
Structural Models are Useful

9. 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.
Structural Models are Useful

10. 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
Structural Models are Useful

11. 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?
Structural Models are Useful

12. 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
Structural Models are Useful

13. Why are hedge ratios important - II?
low hedge ratio (slope) = low credit exposure
Hedge ratio:
measures exposure of debt value to value of collateralising assets
hedge ratio High => high credit risk
hedge ratio low => low credit risk
high hedge ratio (slope) = credit exposure
Structural Models are Useful

14. Focus of Paper Estimate hedge ratio regressions:
In a world governed by structural models, hedge ratio regressions would produce
coefficients bj,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 bj,E = 1
Consider other systematic factors (a la Collin-Dufresne) and examine their relation to underlying credit risk
Structural Models are Useful

15. 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)
Structural Models are Useful

16. 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.
Structural Models are Useful

17. Data
Merrill Lynch: Corporate Master Index and Corporate High Yield Index
covers nearly all corporate bond issues in the U.S. (2114 issuers; issues)
Monthly price data from – (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
Structural Models are Useful

18. Descriptive Statistics: Final Dataset
Structural Models are Useful

19. A Simple Time-Series Hedging Regression
We run the following regression :
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
Structural Models are Useful

20. Hedge Ratios Are these hedge ratios reasonable?
compare with hedge ratios implied by Merton model
In one-factor structural models the hedge ratio, bE, is:
where DE is the “delta” of equity to the firm’s asset value and L is the debt-to-asset value ratio
Structural Models are Useful

21. Hedge Ratios from the Merton Model
Asset Volatility
Leverage 10 15 20 25 30 40 50
0.00
0.04
0.58
2.53
9.25
17.19
0.03
0.70
2.63
4.80
13.52
21.39
0.86
3.42
4.60
13.60
15.31
29.03
0.10
1.42
6.88
10.41
16.44
23.73
31.58
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
Structural Models are Useful

22. The Volatility of Corporate Assets
Structural Models are Useful

23. Testing the Merton Model’s Hedge Ratio Predictions
Estimates of individual hedge ratios are very noisy: calculate as mean hedge ratio within sub-rating :
Structural Models are Useful

24. Using the Merton Model to Predict Hedge Ratios
Source: Schaefer / Strebulaev
Structural Models are Useful

25. Explaining Structural Model Spreads with Empirical Betas
Implies default-boundary structural models produce very similar hedge ratios to Merton
Structural Models are Useful

26. 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
Structural Models are Useful

27. 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
Structural Models are Useful

28. Low Duration Puzzle: Regressions on riskless bonds only
Structural Models are Useful

29. 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
Structural Models are Useful

30. Other Factors: Running a “kitchen sink” regression
Hedge ratios for equity and riskless debt are not much changed
Structural Models are Useful

31. 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 (!!)
Structural Models are Useful

32. Conclusion 1 But do NOT explain level of credit spreads
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
Structural Models are Useful

33. 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 instruments
Structural Models are Useful