Credit Risk

Chapter 22

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Credit Ratings

In the S&P rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, and CCC

The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B, and Caa

Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade”

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Historical Data

Historical data provided by rating agencies are also used to estimate the probability of default

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Cumulative Ave Default Rates (%) (1970-2006, Moody’s, Table 22.1, page 490)

1 2 3 4 5 7 10 Aaa 0.000 0.000 0.000 0.026 0.099 0.251 0.521

Aa 0.008 0.019 0.042 0.106 0.177 0.343 0.522

A 0.021 0.095 0.220 0.344 0.472 0.759 1.287

Baa 0.181 0.506 0.930 1.434 1.938 2.959 4.637

Ba 1.205 3.219 5.568 7.958 10.215 14.005 19.118

B 5.236 11.296 17.043 22.054 26.794 34.771 43.343

Caa-C 19.476 30.494 39.717 46.904 52.622 59.938 69.178

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Interpretation

The table shows the probability of default for companies starting with a particular credit rating

A company with an initial credit rating of Baa has a probability of 0.181% of defaulting by the end of the first year, 0.506% by the end of the second year, and so on

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Do Default Probabilities Increase with Time?

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

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Default Intensities vs Unconditional Default Probabilities

The default intensity (also called hazard rate) is the probability of default for a certain time period conditional on no earlier default

The unconditional default probability is the probability of default for a certain time period as seen at time zero

What are the default intensities and unconditional default probabilities for a Caa or below rate company in the third year? [ （ 39.717%-30.494%)/(1-30.494%)]

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Define V(t) as cumulative probability of the company surviving to time t.

t

(1-V(t+ t))- (1-V(t))V(t)

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Recovery Rate

The recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face value

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Recovery Rates(Moody’s: 1982 to 2006, Table 22.2, page 491)

Class Mean(%)

Senior Secured 54.44

Senior Unsecured 38.39

Senior Subordinated 32.85

Subordinated 31.61

Junior Subordinated 24.47

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Estimating Default Probabilities

Alternatives: Use Bond Prices Use CDS spreads Use Historical Data Use Merton’s Model

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Using Bond Prices

Average default intensity(h) over life of bond is approximately

where s is the spread of the bond’s yield over the risk-free rate and R is the recovery rate

显然这样估计出来的是风险中性违约概率。

1

sh

R

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More Exact Calculation

Assume that a five year corporate bond pays a coupon of 6% per annum (semiannually). The yield is 7% with continuous compounding and the yield on a similar risk-free bond is 5% (with continuous compounding)

Price of risk-free bond is 104.09; price of corporate bond is 95.34; expected loss from defaults is 8.75

Suppose that the probability of default is Q per year and that defaults always happen half way through a year (immediately before a coupon payment).

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Calculations

Time

(yrs)

Def

Prob

Recovery Amount

Risk-free Value

LGD Discount Factor

PV of Exp Loss

0.5 Q 40 106.73 66.73 0.9753 65.08Q

1.5 Q 40 105.97 65.97 0.9277 61.20Q

2.5 Q 40 105.17 65.17 0.8825 57.52Q

3.5 Q 40 104.34 64.34 0.8395 54.01Q

4.5 Q 40 103.46 63.46 0.7985 50.67Q

Total 288.48Q

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Calculations continued

We set 288.48Q = 8.75 to get Q = 3.03% This analysis can be extended to allow

defaults to take place more frequently With several bonds we can use more

parameters to describe the default probability distribution

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The Risk-Free Rate

The risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/swap zero rate (or sometimes 10 bps below the LIBOR/swap rate)

To get direct estimates of the spread of bond yields over swap rates we can look at asset swaps

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18

19

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 default probabilities

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A Comparison

Calculate 7-year default intensities from the Moody’s data (These are real world default probabilities)

Use Merrill Lynch data to estimate average 7-year default intensities from bond prices (these are risk-neutral default intensities)

Assume a risk-free rate equal to the 7-year swap rate minus 10 basis point

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Real World vs Risk Neutral Default Probabilities, 7 year averages (Table 22.4, page 495)

Rating Real-world default probability per yr (% per annum)

Risk-neutral default probability per yr (% per year)

Ratio Difference

Aaa 0.04 0.60 16.7 0.56 Aa 0.05 0.74 14.6 0.68 A 0.11 1.16 10.5 1.04 Baa 0.43 2.13 5.0 1.71 Ba 2.16 4.67 2.2 2.54 B 6.10 7.97 1.3 1.98 Caa-C 13.07 18.16 1.4 5.50

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The calculation of default intensities using historical data are based on equation(22.1) and table 22.1. From equation (22.1),we have

Where is the average default intensity(or hazard rate)by time t and Q(t) is the cumulative probability of default by time t. The value of Q(7) are taken directly from Table 22.1.Consider, for example, an A-rated company. The value of Q(7) is 0.00759.The average 7-year default intensity is therefore

(7) 1/ 7 ln(0.99241) 0.0011

(7) 1/ 7 ln[1 (7)]Q

( )t

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The calculations using bond prices are based on equation(22.2)and bond yields published by Merrill Lynch. The results shown are averages between December 1996 and Oct. 2007.The recovery rate is assumed to be 40% and, the risk-free interest rate is assumed to be the 7-year swap rate minus 10 basis points. Foe example, for A-rated bonds the average Merrill Lynch yield was 5.993%. The average swap rate was 5.398%, so that the average risk-free rate was 5.298%. This gives the average 7-year default probability as

0.05993 0.052980.0116

1 0.4

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Risk Premiums Earned By Bond Traders (Table 22.5, page 496)

Rating Bond Yield Spread over Treasuries

(bps)

Spread of risk-free rate used by market

over Treasuries (bps)

Spread to compensate for

default rate in the real world (bps)

Extra Risk Premium

(bps)

Aaa 78 42 2 34 Aa 87 42 4 42 A 112 42 7 63 Baa 170 42 26 102 Ba 323 42 129 151 B 521 42 366 112 Caa 1132 42 784 305

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Possible Reasons for These Results

Corporate bonds are relatively illiquid The subjective default probabilities of bond

traders may be much higher than the estimates from Moody’s historical data

Bonds do not default independently of each other. This leads to systematic risk that cannot be diversified away.

Bond returns are highly skewed with limited upside. The non-systematic risk is difficult to diversify away and may be priced by the market

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Which World Should We Use?

We should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost of default

We should use real world estimates for calculating credit VaR and scenario analysis

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Merton’s Model

Merton’s model regards the equity as an option on the assets of the firm

In a simple situation the equity value is

max(VT -D, 0)

where VT is the value of the firm and D is the debt repayment required

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Equity vs. Assets

An option pricing model enables the value of the firm’s equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, V

E V N d De N d

dV D r T

Td d T

rT

V

V

V

0 0 1 2

10

2

2 1

2

( ) ( )

ln ( ) ( );

where

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Volatilities

E V VEE

VV N d V0 0 1 0 ( )

This equation together with the option pricing relationship enables V0 andV to be determined from E0 and E

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Example

A company’s equity is $3 million and the volatility of the equity is 80%

The risk-free rate is 5%, the debt is $10 million and time to debt maturity is 1 year

Solving the two equations yields V0=12.40 and v=21.23%

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Example continued

The probability of default is N(-d2) or 12.7% The market value of the debt is V0-E0= 9.40 The present value of the promised payment

is 9.51 The expected loss is about

(9.51-9.4)/9.51=1.2% The recovery rate is 91%

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The Implementation of Merton’s Model (e.g. Moody’s KMV) Moody 公司则利用股票可视为公司资产期权这一思想计算

出风险中性世界的违约距离（如图 1 所示），之后再利用其拥有的海量历史违约数据库，建立起风险中性违约距离与现实世界违约率之间的对应关系，从而得到预期违约频率（ Expected Default Frequency, EDF ），作为违约概率的预测指标。

图 2 就是 Moody 公司用这种方法计算出来的贝尔斯登预期违约频率时间序列。从图上可以看出，在 2008 年 3 月14 日贝尔斯登被摩根大通接管前后，其预期违约频率最高飙升到 80 ％左右。可见，从股票价格中提炼出来的违约概率具有很强的信息功能。

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从期权价格中可以推导出风险中性违约概率

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从期权价格中可以推导出风险中性违约概率

运用上述方法，我们就可根据 2008 年 3 月 14 日贝尔斯登将于 2008 年 3 月 22 日到期的期权价格，计算出贝尔斯登的风险中性违约概率和公司价值的概率分布（如图 13 所示）。贝尔斯登于2008 年 3 月 14 日被摩根大通接管。图 13 显示，市场对贝尔斯登一周后的命运产生巨大分歧，公司价值大涨大跌的概率远远大于小幅变动的概率，这样的分布与正常情况的分布有天壤之别。可见期权价格可以让我们清楚地看出市场在非常时期对未来的特殊看法。

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风险中性违约概率 风险中性违约概率虽然不同于现实概率，但其变化

可以反映现实世界违约概率的变化。在金融危机时期，它可能比信用违约互换（ CDS ）的价差能更敏感地反映出违约概率的变化（如图 14 所示）。在贝尔斯登于 2008 年 3 月 14 日被接管前后，根据上述方法计算出来的风险中性概率每天的变化比 CDS 的价差更敏感。这是因为在金融危机期间，金融机构自身的信用度大幅降低，造成在场外（ OTC ）市场交易的 CDS 交易量急剧萎缩，价差大幅扩大，信号失真。 CDS 价差所隐含的信息将在本文第五点讨论。

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Credit Risk in Derivatives Transactions (page 491-493)

Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability

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General Result

Assume that default probability is independent of the value of the derivative

Consider times t1, t2,…tn and default probability is qi at time ti. The value of the contract at time ti is fi and the recovery rate is R

The loss from defaults at time ti is qi(1-R)E[max(fi,0)].

Defining ui=qi(1-R) and vi as the value of a derivative that provides a payoff of max(fi,0) at time ti, the cost of defaults is

n

iiivu

1

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If Contract Is Always a Liability (equation 22.9)

maturity this for yieldfree-risk the is and derivative the of seller theby

issued bonds coupon zero on yieldthe is value. free-default the is

and derivative the of value actual the is time at payoff a provides derivative where

yyff

T

eff Tyy

*0

*0

)(0

*0

.

*

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Credit Risk Mitigation

Netting Collateralization Downgrade triggers

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Default Correlation

The credit default correlation between two companies is a measure of their tendency to default at about the same time

Default correlation is important in risk management when analyzing the benefits of credit risk diversification

It is also important in the valuation of some credit derivatives, eg a first-to-default CDS and CDO tranches.

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Correlation Correlation measures the degree to which the

probability of one event happening moves in sync with the probability of another event happening.

In terms of default correlation, Zero correlation means that the default of one

company has no bearing on the default of another company – the companies are completely independent of each other.

Perfect positive correlation means that if one company defaults, the other will automatically follow suit.

Perfect negative correlation means that if one company defaults the other one will certainly not.

Factors of Default Correlation

macroeconomic environment: good economy = low number of defaults

same industry or geographic area: companies can be similarly or inversely affected by an external event

credit contagion: connections between companies can cause a ripple effect

Binomial Correlation Measure (page 508)

uses a set of rules to change time-to-default values into either the value 1 or 0, so that if a company defaults within time T, it is assigned the variable 1, and otherwise it is assigned a 0.

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Binomial Correlation continued

Denote Qi(T) as the probability that company i will default between time zero and time T, and Pij(T) as the probability that both i and j will default. The default correlation measure is

])()(][)()([

)()()()(

22 TQTQTQTQ

TQTQTPT

jjii

jiijij

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David Li Statistician who moved over to business Worked at a credit derivative market in 1997 and

knew about the need to measure default correlation

Colleagues in actuarial science working on solution for death correlation, a function called the copula

“Default is like the death of a company, so we should model this the same way as we model human life” (Li)

(Whitehouse, Salmon)

Survival Time Correlation

Define ti as the time to default for company i and Qi(ti) as the cumulative probability distribution for ti

If the probability distributions of t1 and t2 were normal, we could assume the joint probability distribution is bivariate normal. But they are not.

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Gaussian Copula Model Define a one-to-one correspondence between the

time to default, ti, of company i and a variable xi byQi(ti ) = N(xi ) or xi = N-1[Qi (ti)]

where N is the cumulative normal distribution function.

This is a “percentile to percentile” transformation. The p percentile point of the Qi distribution is transformed to the p percentile point of the xi distribution. xi has a standard normal distribution

Then we assume that xi are multivariate normal. The default correlation between ti and tj is measured as the correlation between xi and xj .This is referred to as the copula correlation.

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Marginal Probability Distributions

Copula

Latin word that means “to fasten or fit” Bridge between marginal distributions and a

joint distribution. In the case of death correlation, the

marginal distribution is made up of probabilities of time until death for one person, and joint distribution shows the probability of two people dying in close succession.

(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/copula_17.gif)

Joint distribution

Marginal distributions

Skylar’s Theorem (1959)

If you have a joint distribution function along with marginal distribution functions, then there exists a copula function that links them; if the marginal distributions are continuous, then the copula is unique.

(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/copula_14.gif)

Gaussian copula

Assumes that if the marginal probability distributions are normal, then the joint probability distribution will also be normal.

( ( ), ( )) [ , ; ] ( )

where C is the copula function,

is the cumulative bivariate normal

probability distribution function.

i i j j i j ij ijC Q t Q t M x x P T

M

Copula Correlation Number Specifies the shape of the multivariate distribution

Zero correlation = circular Positive or negative correlation = ellipse

The correlation number is always independent of the marginals (Hull 507).

Assumptions one-to-one relationship between asset correlation and

default correlation based on the definition of default as an asset falling below a certain value.

Correlation number is always positive (Li 11-12). The correlation number is an extremely important factor in

this model because it determines the information you get out of the model.

(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/gmdistribution_fit1.gif)

Marginal distributions(default probability distributions from market data)

+

Correlation number(estimated by asset correlation)

+

Choice of copula(Gaussian / normal copula)

=

Fully defined multivariate distribution of the probability of defaulting within time T

Binomial vs Gaussian Copula Measures

The measures can be calculated from each other

function ondistributiy probabilitnormal bivariate cumulative the is where

that so

M

TQTQTQTQ

TQTQxxMT

xxMTP

jjii

jiijjiij

ijjiij

])()(][)()([

)()(];,[)(

];,[)(

22

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Comparison

The correlation number depends on the way it is defined.

Suppose T = 1, Qi(T) = Qj(T) = 0.01, a value of ij equal to 0.2 corresponds to a value of ij(T) equal to 0.024.

In general ij(T) < ij and ij(T) is an increasing function of T

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Example of Use of Gaussian Copula

Suppose that we wish to simulate the defaults for n companies . For each company the cumulative probabilities 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 continued

We sample from a multivariate normal distribution to get the xi

Critical values of xi are

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 year

When sample is between -2.33 and -1.88, the company defaults in the second year

When sample is between -1.88 and -1.55, the company defaults in the third year

When sample is between -1,55 and -1.28, the company defaults in the fourth year

When sample is between -1.28 and -1.04, the company defaults during the fifth year

When sample is greater than -1.04, there is no default during the first five years

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A One-Factor Model for the Correlation Structure

The correlation between xi and xj is aiaj

The ith company defaults by time T when xi < N-1[Qi(T)] or

Conditional on M the probability of this is

1

2

[ ( )]

1i i

i

i

N Q T a MZ

a

iiii ZaMax 21

2

1

1

)()(

i

iii

a

MaTQNNMTQ

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A One-Factor Model for the Correlation Structure

Suppose that for all i and the common correlation is ,so that for all i.Then

( ) ( )iQ T Q T

ia

1( ( ))( ) ( )

1

N Q T MQ T M N

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Credit VaR

Can be defined analogously to Market Risk VaR

A T-year credit VaR with an X% confidence is the loss level that we are X% confident will not be exceeded over T years

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Calculation from a Factor-Based Gaussian Copula Model

Consider a large portfolio of loans, each of which has a probability of Q(T) of defaulting by time T. Suppose that all pairwise copula correlations are so that all ai’s are

We are X% certain that M is greater than N-1(1−X) = −N-1(X)

It follows that the VaR is

1

)()(),(

11 XNTQNNTXV

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CreditMetrics

Calculates credit VaR by considering possible rating transitions

A Gaussian copula model is used to define the correlation between the ratings transitions of different companies

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Future of the Model

Schönbucher and Schubert devised a method that allows the default correlation to be dynamic (2001) (Meneguzzo and Vecchiato 41)

Fit of different copulas, like Student-t copula (Meneguzzo and Vecchiato 41, Jabbour et al. 32)

Implied correlation from new credit derivative indices (Jabbour et al. 43)