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![Page 1: 1 CREDIT RISK PREMIA Kian-Guan Lim Singapore Management University Prepared for the NUS Institute of Mathematical Sciences Workshop on Computational Finance.](https://reader036.fdocuments.net/reader036/viewer/2022081602/5517f130550346a2228b45d0/html5/thumbnails/1.jpg)
1
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