CREDIT RISK PREMIA

1. 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 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) Defining (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