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Chapter 24
Credit Risk
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Credit RatingsIn the S&P rating system, AAA is the bestrating. After that comes AA, A, BBB, BB, B,
CCC, CC, and CThe corresponding Moodys ratings are Aaa,
Aa, A, Baa, Ba, B,Caa, Ca, and C
Bonds with ratings of BBB (or Baa) andabove are considered to be investmentgrade
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Estimating Default Probabil i ties
Alternatives:
use historical data
use credit spreads
use Mertons model
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H istor ical DataHistorical data provided by rating agenciesare also used to estimate the probability of
default
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Cumulative Ave Defaul t Rates (%)(1970-2012,Moodys, Table 24.1, page 545)
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1 2 3 4 5 7 10
Aaa 0.000 0.013 0.013 0.037 0.106 0.247 0.503
Aa 0.022 0.069 0.139 0.256 0.383 0.621 0.922
A 0.063 0.203 0.414 0.625 0.870 1.441 2.480
Baa 0.177 0.495 0.894 1.369 1.877 2.927 4.740
Ba 1.112 3.083 5.424 7.934 10.189 14.117 19.708
B 4.051 9.608 15.216 20.134 24.613 32.747 41.947
Caa-C 16.448 27.867 36.908 44.128 50.366 58.302 69.483
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Interpretation
The table shows the probability of defaultfor companies starting with a particularcredit rating
A company with an initial credit rating ofBaa has a probability of 0.177% ofdefaulting by the end of the first year,0.495% by the end of the second year, andso on
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Do Default Probabil i ties I ncrease
with Time?
For a company that starts with a good credit
rating default probabilities tend to increasewith time
For a company that starts with a poor credit
rating default probabilities tend to decreasewith time
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Conditional vs Unconditional Defaul t
Probabilities (page 545-546)
The conditional default probability is the probability ofdefault for a certain time period conditional on no
earlier defaultThe unconditional default probability is the probabilityof default for a certain time period as seen at timezero
What are the conditional and unconditional defaultprobabilities for a Caa rated company in the thirdyear?
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Hazard RateThe hazard rate (also called default density), l(t), attimet is defined so that l(t)Dtis the conditional default
probability for a short period between tand t+DtIf V(t) is the probability of a company surviving to time t
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 9
tt
dtt
etQ
t
etV
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Recovery Rate
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The recovery rate for a bond is usuallydefined as the price of the bond immediately
after default as a percent of its face valueRecovery rates tend to decrease as defaultrates increase
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Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014
Recovery Rates; Moodys: 1982 to 2012
Class Mean(%)
Senior Secured 51.6
Senior Unsecured 37.0
Senior Subordinated 30.9
Subordinated 31.5
Junior Subordinated 24.7
11
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Using Credit Spreads(Equation 24.2,
page 547)
Supposes(T) is the credit spread for maturity T
Average hazard rate between time zero andtime Tis approximately
whereRis the recovery rate
This estimate is very accurate in mostsituations
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 12
R
Ts
1
)(
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ExplanationLoss rate at time tis l(t)(1R)
If the credit spread is compensation for this
loss rate it should approximately equal
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)1)(( Rt l
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Matching Bond Pr icesFor more accuracy we can work forward intime choosing hazard rates that match bond
pricesThis is another application of the bootstrapmethod
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The Risk-F ree RateThe risk-free rate when credit spreads anddefault probabilities are estimated is usually
assumed to be the LIBOR/swap zero rate (orsometimes 10 bps below the LIBOR/swaprate)
Asset swaps provide a direct estimates of thespread of bond yields over swap rates
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Real World vs Risk-Neutral Default
Probabilities
The default probabilities backed out of bond
prices or credit default swap spreads are risk-neutral default probabilities
The default probabilities backed out of
historical data are real-world defaultprobabilities
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Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014
A ComparisonCalculate 7-year default intensities from theMoodys data, 1970-2012, (These are real
world default probabilities)Use Merrill Lynch data to estimate average 7-year default intensities from bond prices,1996 to 2007 (these are risk-neutral default
intensities)Assume a risk-free rate equal to the 7-yearswap rate minus 10 basis points
17
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Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 18
Data from Moodys and Merrill Lynch
Cumulative 7-year defaultprobability (Moodys: 1970-2012)
Average bond yield spread in bps*(Merrill Lynch: 1996 to June 2007)
Aaa 0.247% 35.74
Aa 0.621% 43.67
A 1.441% 68.68
Baa 2.927% 127.53
Ba 14.117% 280.28
B 32.747% 481.04
Caa 58.302% 1103.70
*The benchmark risk-free rate for calculating spreads is assumed to be theswap rate minus 10 basis points. Bonds are corporate bonds with a life ofapproximately 7 years.
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Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 19
Real World vs Risk Neutral Hazard
Rates(Table 24.4, page 550)
Rating Historical hazard rate
% per annum
Hazard rate from bond
prices2
(% per annum)
Ratio Difference
Aaa 0.04 0.60 17.0 0.56
Aa 0.09 0.73 8.2 0.64
A 0.21 1.15 5.5 0.94
Baa 0.42 2.13 5.0 1.71
Ba 2.27 4.67 2.1 2.50
B 5.67 8.02 1.4 2.35
Caa 12.50 18.39 1.5 5.89
1Calculated as[ln(1-d)]/7where dis the Moodys 7 yr default rate. For example, in thecase of Aaa companies, d=0.00247 and -ln(0.99753)/7=0.0004 or 4bps. For investmentgrade companies the historical hazard rate is approximately d/7.
2 Calculated ass/(1-R) wheresis the bond yield spread andRis the recovery rate(assumed to be 40%).
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Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 20
Average Risk Premiums Earned
By Bond TradersRating Bond Yield
Spread over
Treasuries
(bps)
Spread of risk-free
rate over Treasuries
(bps)1
Spread to
compensate for
historical default
rate (bps)2
Extra Risk
Premium
(bps)
Aaa 78 42 2 34
Aa 86 42 5 39
A 111 42 12 57
Baa 169 42 25 102
Ba 322 42 130 150
B 523 42 340 141
Caa 1146 42 750 323
1Equals average spread of our benchmark risk-free rate overTreasuries.
2Equals historical hazard rate times (1-R) whereRis the recovery rate.For example, in the case of Baa, 25bps is 0.6 times 42bps.
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Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014
Possible Reasons for the Extra Risk
Premium (The third reason is the most important)Corporate bonds are relatively illiquid
The subjective default probabilities of bond traders
may be much higher than the estimates fromMoodys historical data
Bonds do not default independently of eachother. This leads to systematic risk that cannot bediversified away.
Bond returns are highly skewed with limited upside.The non-systematic risk is difficult to diversify awayand may be priced by the market
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Which World Should We Use?We should use risk-neutral estimates forvaluing credit derivatives and estimating the
present value of the cost of defaultWe should use real world estimates forcalculating credit VaR and scenario analysis
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Using Equity Prices: Mertons
Model (page 553-555)
Mertons model regards the equity as an
option on the assets of the firmIn a simple situation the equity value is
max(VTD, 0)
where VTis the value of the firm andD
is thedebt repayment required
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Equity vs. AssetsThe Black-Scholes-Merton option pricingmodel enables the value of the firms equity
today,E0, to be related to the value of itsassets today, V0, and the volatility of itsassets, sV
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 24
E V N d De N d
d V D r T
Td d T
rT
V
V
V
0 0 1 2
10
2
2 1
2
( ) ( )
ln ( ) ( );
where
s
ss
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Volatilities
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 25
s
s sE V VE
E
VV N d V 0 0 1 0 ( )
This equation together with the optionpricing relationship enables V0andsV tobe determined fromE0and sE
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ExampleA companys equity is $3 million and thevolatility of the equity is 80%
The risk-free rate is 5%, the debt is $10million and time to debt maturity is 1 year
Solving the two equations yields V0=12.40
and sv=21.23%The probability of default isN(d2) or 12.7%
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The Implementation of Mertons
ModelChoose time horizon
Calculate cumulative obligations to time horizon. This is
termed by KMV the default point. We denote it byDUse Mertons model to calculate a theoretical probabilityof default
Use historical data or bond data to develop a one-to-one
mapping of theoretical probability into either real-world orrisk-neutral probability of default.
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CVACredit value adjustment (CVA) is the amount bywhich a dealer must reduce the total value oftransactions with a counterparty because ofcounterparty default risk
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The CVA Calculation
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Time 0 t1 t2 t3 t4 tn=T
Default probabilityfor counterparty
q1 q2 q3 q4
qn
PV of expected lossgiven default
v1 v2 v3 v4 vn
n
i
iivq1
CVA
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Calculation of qisDefault probabilities are calculated from creditspreads
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R
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Calculation of vi
sTheviare calculated by simulating the marketvariables underlying the portfolio in a risk-neutral world
If no collateral is posted the loss on aparticular simulation trial during the ithinterval is the PV of (1-R)max(Vi, 0) where Viis the value of the portfolio at the mid point ofthe interval
viis the average of the losses across allsimulation trials
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CollateralIt is usually assumed that the collateral is posted asagreed, and returned as agreed, untilNdays before adefault. TheNdays are referred to as the cure periodor margin period at risk. UsuallyNis 10 or 20.
Suppose that that a portfolio is fully collateralized withno initial margin and its value moves in favor of thedealer during the cure period. Then viis positivebecause
If the portfolio has a positive value to the dealer at the defaulttime, collateral posted by the counterparty is insufficient
If the portfolio has a negative value to the dealer at the defaulttime, excess collateral posted by the dealer will not be returned
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I ncremental CVAResults from Monte Carlo are stored so thatthe incremental impact of a new trade can be
calculated without simulating all the othertrades.
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CVA RiskThe CVA for a counterparty can be regardedas a complex derivative
Increasingly, dealers are managing it like any
other derivative
Two sources of risk:
Changes in counterparty spreads
Changes in market variables underlying theportfolio
34
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Wrong Way/Right Way RiskSimplest assumption is that probability ofdefault qiis independent of net exposure vi.
Wrong-way risk occurs when qiis positivelydependent on viRight-way risk occurs when qiis negatively
dependent on vi
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DVADebit (or debt) value adjustment (DVA) is anestimate of the cost to the counterparty of a
default by the dealerSame formulas apply except that viscounterpartys loss given a dealer default and q
is dealers probability of defaultValue of transactions with counterparty = Nodefault valueCVA + DVA
36
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DVA continued
What happens to the reported value oftransactions as dealers credit spread
increases?
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Credit Risk M itigationNetting
Collateralization
Downgrade triggers
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Simple SituationSuppose portfolio with a counterpartyconsists of a single uncollateralized
transaction that always a positive value to thedealer and provides a payoff at time T
The CVA adjustment has the effect ofmultiplying the value of the transaction bye-s(T)Twheres(T) is the counterpartys creditspread for maturity T
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Example 25.5 (page 560)
A 2-year uncollateralized option sold by anew counterparty to the dealer has a Black-
Scholes-Merton value of $3Assume a 2 year zero coupon bond issued bythe counterparty has a yield of 1.5% greaterthan the risk free rate
If there is no collateral and there are no othertransactions between the parties, value ofoption is 3e-0.0152=2.91
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Uncollateral ized Long Forward with
Counterparty (page 560)For a long forward contract that matures at time Ttheexpected exposure at time t is
whereF0is the forward price today,Kis the delivery price,sis the volatility of the forward price, T is the time tomaturity of the forward contract, and ris the risk-free rate
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)1()(
)()(2/)/ln(
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12
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Example 24.6 (page 561)2 year forward. Current forward price is $1,600 perounce. Two one-year intervals
K = 1,500, s = 20%,R = 0.3, r = 5%
t1=0.5,t2=1.5
Suppose q1=0.02 and q2=0.03
v1 = 92.67 and v2 = 130.65
CVA=0.0292.67+0.03130.65 = 5.77
Value after CVA =(16001500)e-0.052 5.77 = 84.71
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Defaul t CorrelationThe credit default correlation between twocompanies is a measure of their tendency to
default at about the same timeDefault correlation is important in riskmanagement when analyzing the benefits ofcredit risk diversification
It is also important in the valuation of somecredit derivatives, eg a first-to-default CDSand CDO tranches.
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Measurement
There is no generally accepted measure ofdefault correlation
Default correlation is a more complexphenomenon than the correlation betweentwo random variables
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Survival Time CorrelationDefine tias the time to default for company iand Qi(ti)as the cumulative probability
distribution fort
iThe default correlation between companies iandjcan be defined as the correlationbetween tiand tj
But this does not uniquely define the jointprobability distribution of default times
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The Gaussian Copula Model
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-0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
V1V2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U1U2
One-to-onemappings
CorrelationAssumption
V1V2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U1U2
One-to-onemappings
CorrelationAssumption
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Gaussian Copula Model (continued,page 562-563)
Define a one-to-one correspondence between thetime to default, ti, of company i and a variablexiby
Qi(ti) =N(xi ) or xi =N-1[Q(ti)]
whereNis the cumulative normal distributionfunction.
This is a percentile to percentile transformation. Theppercentile point of the Qidistribution is transformedto thep percentile point of thexidistribution. xihas a
standard normal distribution We assume that thexiare multivariate normal. The
default correlation measure, rij between companiesiandj is the correlation betweenxiandxj
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Example of Use of Gaussian Copula (page 563)
Suppose that we wish to simulate the defaults for
ncompanies . For each company the cumulativeprobabilities of default during the next 1, 2, 3, 4,and 5 years are 1%, 3%, 6%, 10%, and 15%,respectively
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Use of Gaussian Copula continuedWe sample from a multivariate normaldistribution (with appropriate correlations) to getthex
i
Critical values ofxiare
N-1(0.01) = -2.33,N-1(0.03) = -1.88,
N-1(0.06) = -1.55,N-1(0.10) = -1.28,
N-1(0.15) = -1.04
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Use of Gaussian Copula continued
When sample for a company is less than-2.33, the company defaults in the first yearWhen sample is between -2.33 and -1.88, the
company defaults in the second yearWhen sample is between -1.88 and -1.55, thecompany defaults in the third yearWhen sample is between -1,55 and -1.28, thecompany defaults in the fourth year
When sample is between -1.28 and -1.04, thecompany defaults during the fifth yearWhen sample is greater than -1.04, there is nodefault during the first five years
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A One-Factor Model for the
Correlation Structure
The correlation betweenxiandxjis aiaj
The ith company defaults by timeTwhenxi
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Credit VaR (page 564-565)
Can be defined analogously to Market Risk
VaRA T-year credit VaR with anX% confidence isthe loss level that we areX% confident willnot be exceeded over
Tyears
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Calculation f rom a Factor-Based
Gaussian Copula Model (equation 24.10, page565)
Consider a large portfolio of loans, each of which hasa probability of Q(T) of defaulting by time T. Suppose
that all pairwise copula correlations are r so that allais are
We areX% certain thatFis less than
N1(1X) = N1(X)
It follows that the VaR is
r
Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014 53
r
r
1
)()(),(
11XNTQN
NTXV
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Example (page 565)A bank has $100 million of retail exposures
1-year probability of default averages 2% and therecovery rate averages 60%
The copula correlation parameter is 0.199.9% worst case default rate is
The one-year 99.9% credit VaR is therefore1000.128(1-0.6) or $5.13 million
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1280101
99901002019990
11
..
).(.).(),.(
NNNV
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CreditMetr ics (page 565-566)
Calculates credit VaR by considering possiblerating transitions
A Gaussian copula model is used to definethe correlation between the ratings transitionsof different companies
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