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

:)(

:)(

defaultearliernooncondtional

ttandttimebetweendefaultofyprobabilitthett :)(

et

dtV

tVtdt

tdVt

ttVt

t

tVttV

ttVttVttV

tttVttVtV

0

)()(

)()()(

)()()()(

)()()()(

)()()]()([

• 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

1

λ

1-λ

λ

1-λ

fre

1*1*)1(

fre

R*1*)1(

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