Enterprise Risk Management Symposium
Bill Pauling, CFA
July 30, 2003

Overview
 Rationale for integrating credit and equity risk
 Possible approaches to integrating credit and equity risk
 Cholesky decomposition
 Transfer functions
 Conclusions
2

Rationale for Integrating Credit and Equity
Risk
 Merton (1974) viewed corporate debt as a risk-free bond plus a short put option on the firm’s equity
 Hence, the value of a firm’s debt and equity are fundamentally linked
 Merton’s model forms the basis of many models commonly used today, including Moody’s KMV
3

Credit and Equity Markets are Connected
15
Equity Return
5
3.3
2.8
-5
-15
-25
-35
-45
Aa Spread
2.3
1.8
1.3
0.8
0.3
Equity Return (LHS) Aa Spread (RHS) 9 per. Mov. Avg. (Equity Return (LHS))
4

36-Month Rolling Correlations to Equity
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Dec
-75
Dec
-76
Dec
-77
Dec
-78
Dec
-79
Dec
-80
Dec
-81
Dec
-82
Dec
-83
Dec
-84
Dec
-85
Dec
-86
Dec
-87
Dec
-88
Dec
-89
Dec
-90
Dec
-91
Dec
-92
Dec
-93
Dec
-94
Dec
-95
Dec
-96
Dec
-97
Dec
-98
Dec
-99
Dec
-00
Dec
-01
Dec
-02
Lehman Govt Index Lehman Credit Index Lehman High Yield Index
5

120-Month Rolling Correlations to Equity
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Dec
-82
Dec
-83
Dec
-84
Dec
-85
Dec
-86
Dec
-87
Dec
-88
Dec
-89
Dec
-90
Dec
-91
Dec
-92
Dec
-93
Dec
-94
Dec
-95
Dec
-96
Dec
-97
Dec
-98
Dec
-99
Dec
-00
Dec
-01
Dec
-02
Lehman Govt Index Lehman Credit Index LehmanHigh Yield Index
6

Historical Correlations
1/73-6/03
7/83-6/03
7/83-6/93
7/93-6/03
Correlations to Equity Returns
Govt Index
0.21
Credit Index
0.35
High Yield Index
0.15
0.30
-0.05
0.28
0.34
0.20
0.50
0.49
0.52
 Over the long-term, the correlation between risky debt and equity is higher than the correlation between government debt and equity
 The correlation between risky bonds and equity appears to be more stable that the correlation between government bonds and equity
7

Cholesky Decomposion
 Method for transforming uncorrelated normal random variables into correlated normal random variables
 Formula for correlating 2 random variables (i.e. 2x2 matrix) error b
 
*
 a

( 1
 
2
) *
 b
 where,

ε a and ε b 
ρ is the correlation between variables a and b
 error b are normally distributed random numbers is a correlated random variable with unit variance
 Can be used to correlate larger matrices
 Can also be used with a covariance matrix
8

Cholesky Decomposition - Example
 Given:
 Random term used to produce the equity market return, Δy~N(0,1)
 Credit spread model:
 s t
 
( s
 s t

1
)
 t
 s t

1
*

*
 z t
 Correlate Δz in credit spread model to Δy in equity model
 Revised credit spread model:
 s t
 
( s
 s t

1
)
 t
 s t

1
*

*


*
 y

( 1
 
2
) *
 z t

9

Transfer Functions
 Useful when two series are believed to be correlated or co-integrated
 Transfer functions are often used in structured economic models to link the economic factors
 Transfer functions can also be used when random terms may not normally distributed (e.g. jump diffusion models)
10

Transfer Functions - Example
 Given
 The periodic equity return r t
, and its long-term average return rbar from jump diffusion equity model
 Credit spread model:
 s t
 
( s
 s t

1
)
 t
 s t

1
*

*
 z t
11

Transfer Functions - Example
 Incorporate the difference between r and rbar in credit spread model to reflect the correlation between the series
 Revised credit spread model:
 s t
 
( s
 s t

1
)
 t
 
( r t
 r )
 t
 s t

1
*

*
 z t
 Revised credit spread model now is correlated to equity model
 The jump diffusion process in the equity model is ‘transferred’ to the credit spread model
 Equity market crashes will be associated with rather large increases in credit spreads
 Negative skewness in equity returns will also be transferred to the credit spread model as positive skewness due to the negative sign of the beta term
12

Conclusions
 Credit and equity risk should be modeled in an integrated fashion
 Cholesky decomposition can be used to reflect the correlation between the random terms in credit and equity models
 Transfer functions can be used to integrate credit and equity models
13

References
 Bevan, Andrew and Garzarelli, Franco, “Corporate Bond Spreads and the Business Cycle: Introducing the GSSPREAD”, The
Journal of Fixed Income , March 2000.
 Dynkin, L., Lindener, P., Phelps, B. and Wu, W., “Equity Market
Impact on Corporate Bond Excess Returns”, Lehman Brothers
Portfolio Strategies, May 7, 2001.
 Kealhofer, Stephen, “Quantifying Credit Risk I: Default Prediction”,
Financial Analysts Journal , January/February 2003.
 Kealhofer, Stephen, “Quantifying Credit Risk II: Debt Valuation”,
Financial Analysts Journal , May/June 2003.
 Merton, Robert, “On the Pricing of Corporate Debt: The Risk
Structure of Interest Rates”, Journal of Finance , Vol. 29, no. 2,
May 1974.
14