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) 1

Ideas

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

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  

E t Q

 

e

 

t T r

 

ds

  1   * 

T

  1   * 

T

     where default time  * = inf {s  t: firm hits default state} 3

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 4

Assuming independence of riskfree spot rate r(s) and default time r.v.  *  

t

,

T

 

E t Q

 

e

 

t T r

 

ds

 

E t Q

  1   * 

T

  1   * 

T

  

p



t

,

T

     1    

t

 * 

T

  JLT (1997) employs a discrete time, time-homogeneous finite state space per period Markov Chain Q to model Pr t (  *>T)

Q



t

,

t

 1        

q

1 , 1

q

2 , 1

q k

 1 , 1  

t t

, ,

t t



t

 0 ,

t

  1 1    1 

q

1 , 2

q

2 , 2

q k

 1 , 2  

t t

, ,  0

t

,

t t t

  1 1    1     

q

1 ,

k q

2 ,

k q k

 1 ,

k

 

t t

, , 

t

1

t

,

t t

   1 1   1        5

T-step transition probability Q(t,T)=Q(t,t+1).Q(t+1,t+2)….Q(T-1,T) If q ik (t,T) is ik th element of Q(t,T), then Pr t (  *>T) = 1- q ik (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 6

Hazard rate model – basic idea

Pr Q 

t

  *  s  

s t

 

e

 

u du

Default arrival time is exponentially distributed with intensity  Under Cox process, “doubly stochastic”    

p

      1    

t

 * 

T

  

p

,      1   

E t Q t T

  

du

    where  (u) is stochastic 7

Lando (RDR, 1998) When recovery  of par only is paid at default time t<  *

cE t Q

 

j

2

n

  1

e

 

t t

 0 .

5

j



r s



h s



ds

  

E t Q



t t

 

n



E t Q

 

h s e

 

t s



r u



h u



du

 

e t t

 

r n s



h s



ds

  8

Recovery – another formulation discrete time approximation  , 

e



r t



h t



E t Q

 

t

  1 

h t

 

E t Q

  

t

  ,

T

   where h s  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)

E t Q

  1 

L t



E t Q

  

t

  ,

T

  L is loss given default 9

Duffie & Singleton (RFS, 1999)

 ,  

e



r t e



r t

 

h t

 1  

h t

 1  

L t L t

  

E t Q

 1    

t h t

   ,  

E t Q T

    

t

  ,

T

   For small   

e

 (

r t



h t L t

)

E t Q

  

t

  ,

T

  Hence in continuous time  

E t Q



e



t T



r

(

s

) 

h

(

s

)

L



ds

  10

R

t

: default-adjusted short rate

 

t

,

T

 

E t Q

 

e

 

t T R ds

  where

R

  

r

(

s

) 

h

(

s

)

L

 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 11

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 spread B BB A T 12

Relation to MC Under the RFM, for a firm with credit rating i Defining  i (s) = - ln  j  k q ij (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) = a 0 +a 1 M t +a 2 (A t -B t ) where M t variable is macroeconomic variable, and A t -B t are firm specific 13

Affine Term Structure

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

r t



c

0 

c

1

X t

where

dX t

 

uX t



v



dt

 

nxn

    

a

1 

b

1 

X t

0  0

a

1 

b n



X t

    

dW t

14

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

Specification of intensity process

Duffee (RFS, 1999)

h t



d

0 

d

1

Y t

where

dY t

 

wX t



z



dt

 

nxn

    

c

1 

d

1 

Y t

0  0

c

1 

d n



Y t

    

dW t

Then the default-adjusted rate r t +h t L can be expressed in similar form to derive price of defaultable bond 16

Comparing physical or empirical intensity process and EMM intensity process Suppose physical g t = e 0 +e 1 Y t And EMM h t = d 0 +d 1 Y t * And both follows square-root diffusion of Y t , Y t * Then h t =  +  g t +u t Another popular form, Berndt et.al.(WP, 2005) and KeWang et.al. (WP, 2005) is log g t = e 0 +e 1 Y t ; log h t = f 0 +f 1 Y t 17

Credit Risk Premia

Difference in processes g t and h t or their transforms provide a measure of default premia Can be translated into defaultable bond prices to measure the credit spread 18

Vasicek or Ornstein-Uhlenbeck with drift log

h t



f

0 

f

1

Y t

where

dY t

 

wX t



z



dt

 

nxn dW t

For which maximum likelihood statistical methods are readily applicable for estimating parameters and for testing the regression relationship 19

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% 1-3% P Time series 20

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) 21

Model Risk

1.

2.

3.

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(g t )-ln(h t ) regression on macroeconomics and other firm specific variables to test for degree of underspecifications – model risk in pricing and in VaR 22

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 23