1 CREDIT RISK PREMIA Kian-Guan Lim Singapore Management University Prepared for the NUS Institute of Mathematical Sciences Workshop on Computational Finance.

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CREDIT RISK PREMIA

Kian-Guan LimSingapore Management University

Prepared for the NUS Institute of Mathematical Sciences Workshop on Computational Finance

(29 – 30 Aug 2005)

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Ideas

• Defaultable bond pricing

• Recovery method

• Credit spread

• Intensity process

• Affine structures

• Default premia

• Model risk

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Reduced Form Models

• Jarrow and Turnbull (JF, 1995)Jarrow, Lando, & Turnbull (RFS, 1997)RFV (recovery of face value) at TPrice of defaultable bond price under EMM Q

where default time * = inf {s t: firm hits default state}

TT

dssrQt

T

teETt ** 11,

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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

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

TTtp

EeETt

t

TTQt

dssrQt

T

t

*Pr1,

11, **

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

100

1,1,1,

1,1,1,

1,1,1,

1,

,12,11,1

,22,21,2

,12,11,1

ttqttqttq

ttqttqttq

ttqttqttq

ttQ

kkkk

k

k

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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 probabilityAdvantage Using credit rating as an input as in CreditMetrics

of RiskMetricsDisadvantage Misspecification of credit risk with the credit rating

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Hazard rate model – basic idea

duet us

t

s*PrQ

Default arrival time is exponentially distributed with intensity

Under Cox process, “doubly stochastic”

T

tduuQ

t

t

eETtp

TTtpTt

1,

*Pr1,,

where (u) is stochastic

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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

nt

t

duhr

sQt

dshrQt

n

j

dshrQt

s

tuu

nt

tss

jt

tss

ehE

eEecETt

2

1

5.0

,

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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)

TtEhEheTt Qttt

Qtt

rt ,1~

,

TtELE Qttt

Qt ,1

~

L is loss given default

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Duffie & Singleton (RFS, 1999)

TtELhe

TtEhLheTtQttt

r

Qtttt

r

t

t

,1

,11,

TtEeTt Qt

Lhr ttt ,, )(

For small

Hence in continuous time

T

tdsLshsrQ

t eETt)()(

,

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Rt : default-adjusted short rate

LshsrsR

eETtT

tdssRQ

t

)()(where

,

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

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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

T

spread

B

BB

A

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Relation to MC

Under the RFM, for a firm with credit rating i

Defining

i(s) = - ln jk qij(t,t+1) for s(t,t+1]we can recover a Markov Chain structure

Relation to SFMMadan and Unal (RDR, 1996)Defining

(s) = a0+a1Mt+a2(At-Bt)where Mt is macroeconomic variable, and At-Bt are firm specific variable

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Affine Term Structurefor 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

t

tn

t

nxntt

tt

dW

Xba

Xba

dtvuXdX

Xccr

1

11

10

0

0

where

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Advantages

Short rates positive

Tractability

u<0 for mean-reversion in some macroeconomic variables

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Specification of intensity processDuffee (RFS, 1999)

t

tn

t

nxntt

tt

dW

Ydc

Ydc

dtzwXdY

Yddh

1

11

10

0

0

where

Then the default-adjusted rate rt+htL can be expressed in similar form to derive price of defaultable bond

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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

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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

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Vasicek or Ornstein-Uhlenbeck with drift

tnxntt

tt

dWdtzwXdY

Yffh

where

log 10

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

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Extracting and *

From KMV Credit Monitor Distance-to-Default as proxy of default probability

Implying from traded prices of derivatives

Time series

% default prob

Matched pairs , * from same firm and duration

1-3%

3-10%

Q

P

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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)

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Model RiskWrong model or misspecified model can arise out of many possibilities

1. Under-parameterizations in RFM e.g. and 2. Incorrect recovery rate or mode e.g. RT, RFV, RMV,

and timing of recovery at T or *3. 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

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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