credit risk (2)
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TRANSCRIPT
Chapter 22Chapter 22
Credit RiskCredit Risk
資管所 陳竑廷
AgendaAgenda
22.1 Credit Ratings
22.2 Historical Data
22.3 Recovery Rate
22.4 Estimating Default Probabilities from bond price
22.5 Comparison of Default Probability estimates
22.6 Using equity price to estimate Default
Probabilities
• Credit Risk
– Arise from the probability that borrowers and
counterparties in derivatives transactions may
default.
22.1 22.1 Credit RatingsCredit Ratings
• S&P – AAA , AA, A, BBB, BB, B, CCC, CC, C
• Moody– Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C
• Investment grade – Bonds with ratings of BBB (or Baa) and above
best worst
22.2 22.2 Historical DataHistorical Data
• For a company that starts with a good credit rating default probabilities tend to increase with time
• For a company that starts with a poor credit rating default probabilities tend to decrease with time
Default IntensityDefault Intensity
• The unconditional default probability – the probability of default for a certain time period as
seen at time zero
39.717 - 30.494 = 9.223%
• The default intensity (hazard rate)– the probability of default for a certain time period
conditional on no earlier default100 – 30.494 = 69.506%
0.09223 / 0.69506 = 13.27%
ttimetosurvivingcompanytheofyprobabilitcumulativethetV
ttimeatsintensitiedefaultthet
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defaultearliernooncondtional
ttandttimebetweendefaultofyprobabilitthett :)(
et
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tdVt
ttVt
t
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ttVttVttV
tttVttVtV
0
)()(
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)()()]()([
• Q(t) : the probability of default by time t
(22.1) ee
tt
dt
tVtQ
)(
)(
1
1
)(1)(
0
22.3 22.3 Recovery RateRecovery Rate
• Defined as the price of the bond immediately after
default as a percent of its face value
• Moody found the following relationship fitting the
data:
Recovery rate = 59.1% – 8.356 x Default rate
– Significantly negatively correlated with default rates
• Source :– Corporate Default and Recovery Rates, 1920-2006
22.4 22.4 Estimating Default Estimating Default ProbabilitiesProbabilities
• Assumption
– The only reason that a corporate bond sells for less
than a similar risk-free bond is the possibility of
default
• In practice the price of a corporate bond is affected
by its liquidity.
raterecoveryexpectedtheR
yieldbondcorporatetheofspreadthes
yearperintentisydefaultaveragethe
:
:
:
R
s
1 (22.1)
%33.34.01
02.0
200
%40
bps
R
)1(
11)1(
)1(
*1]*1*)1[( )(
Rs
sR
eR
eRes
srr ff
1
1
R
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λ
1-λ
λ
1-λ
fre
1*1*)1(
fre
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Taylor expansion
A more exact calculationA more exact calculation
• Suppose that Face value = $100 , Coupon = 6%
per annum , Last for 5 years
– Corporate bond
• Yield : 7% per annum → $95.34
– Risk-free bond
• Yield : 5% per annum → $104.094
• The expected loss = 104.094 – 95.34 = $ 8.75
Q : the probability of default per year
288.48Q = 8.75
Q = 3.03%
0 1 2 3 4 5
e -0.05 *3.5
22.5 22.5 Comparison of default Comparison of default probability estimatesprobability estimates
• The default probabilities estimated from
historical data are much less than those derived
from bond prices
Historical default intensityHistorical default intensity
The probability of the bond surviving for T years is
(22.1)
))(1ln(1
)(
1)()(
tQt
t
tQ ett
%11.0
]00759.01ln[7
1
)]7(1ln[7
1)7(
Q
Default intensity from bondsDefault intensity from bonds
• A-rated bonds , Merrill Lynch 1996/12 – 2007/10
–The average yield was 5.993%
–The average risk-free rate was 5.289%
–The recovery rate is 40%
%16.14.01
05298.005993.01
R
s (22.2)
0.11*(1-0.4)=0.066
Real World vs. Risk Neutral Real World vs. Risk Neutral Default ProbabilitiesDefault Probabilities
• Risk-neutral default probabilities
– implied from bond yields
– Value credit derivatives or estimate the impact of default risk on
the pricing of instruments
• Real-world default probabilities
– implied from historical data
– Calculate credit VaR and scenario analysis
22.6 22.6 Using equity prices to Using equity prices to estimate default probabilityestimate default probability
• Unfortunately , credit ratings are revised relatively infrequently.
– The equity prices can provide more up-to-date information
Merton’s ModelMerton’s Model
If VT < D , ET = 0 ( default )
If VT > D , ET = VT - D
)0,max( DVE TT
• V0 And σ0 can’t be directly observable.
• But if the company is publicly traded , we can observe E0.
Merton’s model gives the value firm’s equity at time T as
So we regard ET as a function of VT
We write
)0,max( DVE TT
(**))()(
(*))()(
22
11
tdwVdtVdVtdwdtV
dV
tdwEdtEdEtdwdtE
dE
VV
EE
dVV
EdE
Lemma sIto'By
V offunction a is EOther term without dW(t) , so ignore it
Replace dE , dV by (*) (**) respectively
We compare the left hand side of the equation above with that of the right hand side
)(
))(()(
2
21
tdwVV
EdtV
V
E
tdwVdtVV
EtdwEdtE
V
VE
VE
VE
VV
EE
tdWVV
EtdWE
dtVV
EdtE
)()(
and
21
(22.4)
ExampleExample
• Suppose that
E0 = 3 (million) r = 0.05 D = 10
σE = 0.80 T = 1
Solving
then get V0 = 12.40
σ0 = 0.2123 N(-d2) = 12.7%
20
20
0100
21000
),(),(
)N(:),(
)N()N(:),(
VGVFminimize
VdEVG
dDedVEVF
VV
VEV
rTV
Solving
[F(x,y)]2+[G(x,y)]2
=(D2)^2+(E2)^2
F(x,y)=A2*NORMSDIST((LN(A2/10)+(0.05+B2*B2/2))/B2) -10*EXP(-0.05)*NORMSDIST((LN(A2/10)+(0.05+B2*B2/2))/B2-B2)
G(x,y)=NORMSDIST((LN(A7/10)+(0.05+B7*B7/2))/B7)*A7*B7
Excel SolverExcel Solver
• Initial V0 = 12.40 , σ0 = 0.2123
• Initial V0 = 10 , σ0 = 0.1
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Thank youThank you