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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). Ideas. Defaultable bond pricing Recovery method Credit spread Intensity process Affine structures

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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)

  • Defaultable bond pricing
  • Recovery method
  • Credit spread
  • Intensity process
  • Affine structures
  • Default premia
  • Model risk
reduced form models
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}

comparing with structural models or firm value models
Comparing with Structural Models (or Firm Value Models)


  • 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


  • Default event is a surprise; less intuitive than the structural model
assuming independence of riskfree spot rate r s and default time r v
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)

t step transition probability
T-step transition probability


If qik(t,T) is ikth element of Q(t,T), then

Prt(*>T) = 1- qik(t,T)

Q(.,.) is risk-neutral probability


Using credit rating as an input as in CreditMetrics of RiskMetrics


Misspecification of credit risk with the credit rating

hazard rate model basic idea
Hazard rate model – basic idea

Default arrival time is exponentially distributed with intensity 

Under Cox process, “doubly stochastic”

where (u) is stochastic

lando rdr 1998
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

recovery another formulation discrete time approximation
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

duffie singleton rfs 1999
Duffie & Singleton (RFS, 1999)

For small 

Hence in continuous time

r t default adjusted short rate
Rt : default-adjusted short rate


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


Recovery is empirically closer to the RFV approach

credit spreads
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






relation to mc
Relation to MC

Under the RFM, for a firm with credit rating i


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

affine term structure
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]



Short rates positive


u<0 for mean-reversion in some macroeconomic variables

specification of intensity process
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

comparing physical or empirical intensity process and emm intensity process
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, 2005) and KeWang (WP, 2005) is

log gt = e0+e1Yt ; log ht = f0+f1Yt

credit risk premia18
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

vasicek or ornstein uhlenbeck with drift
Vasicek or Ornstein-Uhlenbeck with drift

For which maximum likelihood statistical methods are readily applicable for estimating parameters and for testing the regression relationship

extracting and
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





Time series

  • 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)
model risk
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
  • 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