Introduction to Credit Risk

Credit Risk - Definitions

Credit risk - the risk of an economic loss from the failure of

a counterparty to fulfill its contractual obligations.

Credit Exposure (CE) or Exposure at Default (EAD) – the

economic value of the claim on the counter party at time of

default.

Recovery Rate (RR) – the payment ratio given default

Loss Given Default (LGD) – the fractional loss to default,

which is equal to 1 - RR

Measuring Credit Risk-Distribution of loss

Definitions

bi - a “bernoulli” random variable that take the value of 1 if

default occurs and 0 otherwise, with probability of pi.

CEi - the credit exposure at the time of default.

fi - the recovery rate (RR)

(1-fi) – the loss given default (LDG)

N – number of instruments

Measuring Credit Risk-Distribution of loss

The distribution of losses due to credit risk can be described

as:

Assuming the only random variable is bi:

)1(ECCLN

1ii ii f

~~b~

)1(CEE[CL]N

1ii ii fp

Joint Events

The CL distribution depends on the correlation between the

default events.

When the defaults events are uncorrelated:

When the defaults events are perfectly correlated

p(B)p(A)p(A&B)

p(A)p(A)1p(A)A)p(Bp(A&B) |

Joint Events

For Instance, p(A)=p(B)=1%

In the uncorrelated case:

In the perfectly correlated case:

%.... 01000010010010p(B)p(A)p(A&B)

%.| 1010p(A)A)p(Bp(A&B)

Joint Events

When <1:

p(B)p(A)p(A&B) BA

p(A)1p(A)σA

p(B)p(A)p(B)]1p(B)[p(A)]1p(A)[p(A&B)

Joint Events

Consider the pervious example and assume the =0.5:

09949.001.0101.0σ BA

%...... 50005050010010099499050

p(B)p(A)p(B)]1p(B)[p(A)]1p(A)[p(A&B)2

Credit VaR

Consider a portfolio of $100M composed of 3 bonds A, B and

C with the following default probabilities and CE:

BondCE ($M)Default Prob.

A250.05

B300.10

C450.20

For simplicity, assume: 1. Exposures are constant;

2. The recovery rates are zero; 3. The default events are

independent

Credit VarDefaultL ($M)P(L)C. Prob.E(L)=Lp(L)2=(L-(EL))2p(L)

None00.6840.68400120.8

A250.03600.72000.9004.97

B300.07600.79602.28021.32

C450.17100.96707.695172.38

A&B550.00400.97100.2206.97

A&C700.00900.98000.63028.99

B&C750.01900.99001.42572.45

A&B&C1000.00101.00000.1007.53

Sum13.25434.7

6840)2.01()1.01()05.01(p(None) .0360)2.01()1.01(05.0p(onlyA) .

0040)2.01(1.005.0B)p(A .&

Credit VaR

25134520301025050

CEpL)p(LE(CL)N

1iii

n

1iii

....

9207434CL

7434)p(LE(L))(Lp(CL) i2

n

1iii

2

..)(

.

Credit VaR

With a confidence level of 95% the VaR is $45M

The unexpected loss is:

$31.75M13.2545

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0-25-30-45-55-70-75-100

Loss ($M)

Fre

qu

ency

Unexpected Loss

Expected Loss

Credit Diversification

A portfolio of loan is less risky than single loans

Consider different alternatives for $100M loan portfolio:

One loan of $100M

10 loans each for $10M

100 loans each for $1M

1,000 loan each for $0.1M

Assume a fixed default probability of 1% for all loans and

are independence across loans

Credit Diversification

In the first case:M1$10001.0EL

M10$100)01.01(01.0σ

0%

20%

40%

60%

80%

100%

0-10-20-30-40-50-60-70-80-90-100

Credit Diversification

In the second case:

M1$EL M3σ

0%

20%

40%

60%

80%

100%

0-10-20-30-40-50-60-70-80-90

Credit Diversification

In the third case:

M1$EL M1σ

0%

20%

40%

60%

80%

100%

0-10-20-30-40-50-60-70-80-90-100

Credit Diversification

In the last case:

M1$EL M30σ .

0%

20%

40%

60%

80%

100%

0-1-2-3-4-5-6-7-8-9-10

This reflects the Central Limit Theory by which the distribution of

the sum of independent variables tends to normal distribution.

Credit Diversification

The loans diversification does not effect the expected loss

but decreases the variance.

With N independent defaults events with the same

probability of p, we have:

100pN

100NppLE(CL)

N

1ii

22N

1i

2i

2

N

100)p1(p

N

100)p1(NpL)p1(p(CL)

N

100)p1(p(CL)

Credit Diversification

In reality, there is some correlation between the defaults

events, which are all affected by the general state of the

economy:

many more defaults occur in a recession than in

expansion.

In this case the distribution will lose its asymmetry more

slowly.

The solution for this is to limit the exposure to a particular

sectors – defaults are more correlated among sectors than

across sectors.

Historical Default Rates

Cumulative default rate measure the total frequency of

default at any time between the starting date and year T.

According to the S&P experience - from 10,000 BBB rated

firms, there where 36 defaults over one year, and 96 defaults

over 2 years.

Based on the Cumulative default rate one can derives the

marginal default rate, which is the frequency of default during

year T.

Historical Default Rates

Definitions

MT – The number of issuers rated R that default in year T

NT – The number of issuers rated R that have no default by the

beginning in year T.

dT – The marginal default rate during year T – the proportion

of issuers, relative to the number at the beginning of year T.

ST – The survival rate - The number of issuers rated R that

will not have default by T.

PT – The probability of defaulting in year 2.

CT – The cumulative default rate at the end of year T

Historical Default Rates

The marginal default rate during year T:

The survival rate:

The probability of defaulting in year 1:

In order to default in year 2, the firm must have survived the

first year and default in the second

T

TT N

Md

)d1(ST

1ttT

11 dp

212 dSp

Cumulative Default Rates

Thus, the cumulative default rate at end of year 2:

In order to default in year 3, the firm must have survived the

first and the second years and default in year 3.

323 dSp

211212 dSdpCC

32211323 dSdSdpCC

Default Process

Default

Default

Default

No default

No default

d1

1-d1

d2

d3

1-d3

1-d2

322113

2112

11

dSdSdC

dSdC

dC

Historical Default Rates

Numerical Example

Consider a BBB rated firm that has default rates of d1=4%,

d2=6% and d3=8%

What are the survival rates at the end of years 1,2 and 3?

What is the probability of defaulting in years 1,2 and 3?

What is the cumulative default rates at the end of years 1,2 and

3?

Historical Default Rates

Numerical Example

%4dpC 111

%9604.01d1S 11

%76.506.096.0dSp 212

%24.90)06.01(96.0)d1(S)d1)(d1(S 21212

%2.708.09024.0dSp 323

%76.9%76.5%4pCC 212

%96.16%2.7%76.9pCC 323

Recovery Rates

Credit rating agencies measure recovery rates using the

historical observations of the value of the debt right after default.

The historical observations reveal that the RR depend on:

The state of the economy

The seniority of debtor – the proceeds from liquidation

should be divided according to the absolute priority rule

Recovery Rates

Credit rating agencies measure recovery rates using the

historical observations of the value of the debt right after default.

The historical observations reveal that the RR depend on:

The state of the economy

The seniority of debtor – the proceeds from liquidation

should be divided according to the absolute priority rule

Recovery Rates

Priority rule

Secured creditors – up to the extent of secured collateral

Priority creditors – post-bankruptcy creditors and taxes.

General creditors – unsecured creditors before bankruptcy

Shareholders

Recovery Rates

S&P’s Historical RR for Corporate Debt

Seniority RankingWeighted Average

Senior secured49.32

Senior unsecured47.09

Subordinated32.46

Junior subordinated 35.51

Total40.23