1. Singapore Management University
CREDIT RISK PREMIA
Kian-Guan Lim
Singapore Management University
Prepared for the NUS Institute of Mathematical Sciences Workshop on Computational Finance
(29 – 30 Aug 2005)

2. Ideas Defaultable bond pricing Recovery method Credit spread
Intensity process
Affine structures
Default premia
Model risk

3. Reduced Form Models Jarrow and Turnbull (JF, 1995)
Jarrow, Lando, & Turnbull (RFS, 1997)
RFV (recovery  of face value) at T
Price of defaultable bond price under EMM Q
where default time
* = inf {s  t: firm hits default state}

4. Comparing with Structural Models (or Firm Value Models)
Advantages
Avoids the problem of unobservable firm variables necessary for structural model; the bankruptcy process is exogenously specified and needs not depend on firm variables
Easy to handle different short rate (instantaneous spot rate) term structure models
Once calibrated, easy to price related credit derivatives
Disadvantage
Default event is a surprise; less intuitive than the structural model

5. Assuming independence of riskfree spot rate r(s) and default time r. v
JLT (1997) employs a discrete time, time-homogeneous finite state space per period Markov Chain Q to model Prt(*>T)

6. T-step transition probability
Q(t,T)=Q(t,t+1).Q(t+1,t+2)….Q(T-1,T)
If qik(t,T) is ikth element of Q(t,T), then
Prt(*>T) = 1- qik(t,T)
Q(.,.) is risk-neutral probability
Advantage
Using credit rating as an input as in CreditMetrics of RiskMetrics
Disadvantage
Misspecification of credit risk with the credit rating

7. Hazard rate model – basic idea
Default arrival time is exponentially distributed with intensity 
Under Cox process, “doubly stochastic”
where (u) is stochastic

8. Lando (RDR, 1998)
When recovery  of par only is paid at default time t<*<T instead of at T
For a n-year coupon bond with 2n coupons

9. Recovery – another formulation discrete time approximation
where hs is the conditional probability at time s of default within (s,s+) under EMM Q given no default by time s
Under RMV (recovery of market value just prior to default)
L is loss given default

10. Duffie & Singleton (RFS, 1999)
For small 
Hence in continuous time

11. Rt : default-adjusted short rate
Advantages
Unlike the RMV approach to recovery, correlation between spot rate and hazard rate or even recovery/loss is straightforward
Easy application as a discounting device
Disadvantage
Recovery is empirically closer to the RFV approach

12. Credit spreads Relation with earlier studies
Given . After obtaining i(t,T),
Per period spot rate is
ln [i(t,T+1)/i(t,T)]-1 B BB A
spread T

13. Relation to MC Relation to SFM
Under the RFM, for a firm with credit rating i
Defining
i(s) = - ln jk qij(t,t+1) for s(t,t+1]
we can recover a Markov Chain structure
Relation to SFM
Madan and Unal (RDR, 1996)
(s) = a0+a1Mt+a2(At-Bt)
where Mt is macroeconomic variable, and At-Bt are firm specific variable

14. Affine Term Structure
for short rate r(t) – square root diffusion model of Xt
Duffie and Kan (MF, 1996), Pearson and Sun (JF, 1994)
(t,T) = exp[a(T-t) + b(T-t)’ Xt]
provided

15. Advantages Short rates positive Tractability
u<0 for mean-reversion in some macroeconomic variables

16. Specification of intensity process
Duffee (RFS, 1999)
Then the default-adjusted rate rt+htL can be expressed in similar form to derive price of defaultable bond

17. Comparing physical or empirical intensity process and EMM intensity process
Suppose physical gt = e0+e1Yt
And EMM ht = d0+d1Yt*
And both follows square-root diffusion of Yt , Yt*
Then ht = +gt+ut
Another popular form, Berndt et.al.(WP, 2005) and KeWang et.al. (WP, 2005) is
log gt = e0+e1Yt ; log ht = f0+f1Yt

18. Credit Risk Premia
Difference in processes gt and ht or their transforms provide a measure of default premia
Can be translated into defaultable bond prices to measure the credit spread

19. Vasicek or Ornstein-Uhlenbeck with drift
For which maximum likelihood statistical methods are readily applicable for estimating parameters and for testing the regression relationship

20. Extracting  and *
From KMV Credit Monitor Distance-to-Default as proxy of default probability
Implying from traded prices of derivatives
Matched pairs , * from same firm and duration
% default prob Q
3-10% P
1-3%
Time series

21. Applications
Using statistical relationship between risk-neutral and physical or empirical measure to infer from traded derivatives empirical risk measures such as VaR given a traded price at any time
Using statistical relationship to estimate EMM in order to price product for market-making or to trade based on market temporary inefficiency or to mark-to-model inventory positions of instruments (assuming no arbitrage is possible even if there is no trade)

22. Model Risk
Wrong model or misspecified model can arise out of many possibilities
Under-parameterizations in RFM e.g.  and 
Incorrect recovery rate  or mode e.g. RT, RFV, RMV, and timing of recovery at T or *
BUT assuming same RFM and same recovery mode, USE ln(gt)-ln(ht) regression on macroeconomics and other firm specific variables to test for degree of underspecifications – model risk in pricing and in VaR

23. Conclusion
Credit Risk is a key area for research in applied risk and structured product industry
Model risk can be significant and is underexplored
RFM provides a regression-based framework to explore model risk implications
Same analyses can be applied to other derivatives using reduced form approach e.g. MBS, CDO