8.1 Credit Risk Lecture n.8. 8.2 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.

8.1

Credit Risk

Lecture n.8

8.2

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”

• Another provider of ratings is Fitch

8.3

Some basic questions

• What is rating? It is a system to provide an easy to understand signal about the probability of default of a certain financial instrument and the loss caused by this default

• What is default? – tricky question, it depends on the contract

• debt restructuring, failure to pay, bankruptcy

• What is loss given default?

8.4

Average Cumulative Default Rates (%) (S&P Credit Week, April 15, 1996, Table 23.2, page 627)

Yrs 1 2 3 4 5 7 10

AAA 0.00 0.00 0.07 0.15 0.24 0.66 1.40

AA 0.00 0.02 0.12 0.25 0.43 0.89 1.29

A 0.06 0.16 0.27 0.44 0.67 1.12 2.17

BBB 0.18 0.44 0.72 1.27 1.78 2.99 4.34

BB 1.06 3.48 6.12 8.68 10.97 14.46 17.73

B 5.20 11.00 15.95 19.40 21.88 25.14 29.02

CCC 19.79 26.92 31.63 35.97 40.15 42.64 45.10

8.5

from default statistics to bond pricing• we need to know, what is the default definition:

• changes according to the contract

• ISDA effort has not yet produced a standard

• changes according to the nature of the issuer (sovereign vs private, corporate vs bank, etc)

• changes according to the legislations

• we need to know what is the loss caused by a default

• changes according to financial instruments

• changes according to covenants

•changes according to legislations

• changes according to sectors

• changes according to economic cycles

8.6

Example:let’s start from these 2 ZCB curves

Maturity(years)

Risk-freeyield

Corporatebond yield

1 5% 5.25%

2 5% 5.50%

3 5% 5.70%

4 5% 5.85%

5 5% 5.95%

8.7

implied losses from defaults• One-year Treasury bond (principal=$1) sells for

• One-year corporate bond (principal=$1) sells for

or at a 0.2497% discount• This indicates that the holder of a corporate bond

expects to lose 0.2497% from defaults in the first year

e 0 05 1 0951229. .

e 0 0525 1 0948854. .

8.8

Notationh(T1,T2): Expected proportional loss between

times T1 and T2 as seen at time zero

y (T ): yield on a zero-coupon corporate bond maturing at time T

y *(T ) yield on a zero-coupon Treasury bond maturing at time T

P (T ): price of a zero-coupon corporate bond paying $1 at time T

P*(T ): price of a zero-coupon Treasury bond paying $1 at time T

8.9

Estimating Default Statistics from Bond Prices

h T

P T P T

P T

P T e P T e

h T e

h T T h T h T

h T T e e

y T T y T T

y T y T T

y T y T T y T y T T

( , )( ) ( )

( )

( ) ( )

( , )

( , ) ( , ) ( , )

( , )

*

*

( ) * ( )

[ ( ) ( )]

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

*

*

* *

0

0 1

0 01 2 2 1

1 21 1 1 2 2 2

Because and

Because

8.10

implied forward losses

• Similarly the holder of a A-rated bond expects to lose

or 0.9950% in the first two years

• Between years one and two the expected loss is 0.7453%

e e

e

0 0550 2 0 05 2

0 05 2 0 009950. .

. .

8.11 another example

Suppose that the spreads over Treasuries for

the yields on 5 - and 10 - year BBB - rated zero

coupon bonds are 130 and 170 basis points.

In this case:

so that

The expected loss on a BBB bond between years

5 and 10 is 9.34% of the no - default value

h e

h e

h

( , ) .

( , ) .

( , ) . . . .

.

.

0 5 1 0 0629

010 1 01563

510 01563 0 0629 0 0934

0 013 5

0 017 10

8.12

Bond Prices vs Historical Default Experience

• The estimates of the probability of default calculated from bond prices are much higher than those from historical data

• Consider for example A-rated bonds

• These typically yield at least 50 bps more than Treasuries

8.13

Bond Prices vs Historical Default Experience

This means that we expect to lose at least

or 2.47% of the bond's value over a 5 - year period.

Assume a low recovery rate of 30%.

The probability of default is then

This is over five times greater than the 0.67%

historical probability

1 0 0247

2 47 0 7 353%

0 005 5

e . .

. . .

8.14

Possible Reasons for These Results

• The liquidity of corporate bonds is less than that of Treasury bonds

• Bonds traders may be factoring into their pricing depression scenarios much worse than anything seen in the last 20 years

8.15

A Key Theoretical Reason

• The default probabilities estimated from bond prices are risk-neutral default probabilities

• The default probabilities estimated from historical data are real-world default probabilities

• In the real world we earn an extra 40 bps per year

8.16

Real World vs Risk Neutral World

• When we infer default probabilities or expected losses from security prices they are “risk-neutral”. When we infer them from historical data they are “real-world”

8.17

Quantifying the Cost of Default

The cost of default has to be estimated not only for traditional assets such as bonds, but also for Derivatives:

• Those that are always assets to one party and liabilities to the other (e.g., options)

• Those that can become assets or liabilities (e.g., swaps, forward contracts)

8.18

Notation

f

f

*:

:

value of derivative assuming defaults

are impossible

actual value of derivative

8.19

Independence Assumption

• The independence assumption states that the variables affecting the price of a derivative are independent of the variables determining defaults

• This assumption (although not perfect) makes pricing for default risk possible

8.20

Contracts that are Assets

• Consider a contract that promises a payoff at time T

• The contract is assumed to rank equally with an unsecured bond in the event of a default

8.21

Contracts that are Assets continued

f f

fh T

f f e y T y T T

*

*

* [ ( ) ( )]

( , )

*

0

8.22

A Simple Interpretation

• Use the “risky” discount rate rather than the risk-free discount rate when discounting cash flows in a risk-neutral world

• Note that this does not mean we simply increase the interest rate in option pricing formulas

8.23Credit Exposure for ContractsThat Can be Assets or Liabilities

Exposure

Contract value

8.24

Netting

• Most derivative contracts state that, if a company defaults on one contract with a counterparty, it must default on all contracts with that counterparty

• This means that the total exposure is an option on a portfolio rather than a portfolio of options

8.25

Incremental Effect of a New Contract

• The incremental effect of a new contract on credit risk is found by calculating expected credit losses with and without the contract

• The incremental effect can be negative

8.26

Reducing Exposure to Credit Risk

• Set credit limits

• Ask counterparty to post collateral

• Design contract to reduce credit risk (eg margins)

• Include a downgrade trigger in contract

• use credit derivatives

8.27

Credit Derivatives

Examples:

• Credit default swap

• Total return swap

• Credit spread option

• CDO

• ABS

8.28

Credit Default Swap

• Company A has the right to sell a reference bond for its face value to company B in the event there is a default on the bond

• In return, A makes periodic payments to B

• The reference bond is issued by a third party, C

8.29

Total Return Swap

• The total return from one asset or group of assets is swapped for the total return from another asset or group of assets

• If the deal is fair, the assets have the same market value at the beginning of the life of a total return swap

8.30

Uses of Total Return Swap

• Consider a bank in Texas lending primarily to oil companies and a bank in Michigan lending primarily to auto manufacturers and their suppliers

• A total return swap allows them to achieve credit risk diversification without buying or selling assets

8.31

Credit Spread Option

• This is an option on the spread between the yields earned on two assets.

• The option provides a payoff whenever the spread exceeds some level

8.32

Attraction of Credit Derivatives

• Allows credit risks to be exchanged without the underlying assets being exchanged

• Allows credit risks to be managed

8.33

Does Credit risk mean only default?

• Credit risk is not a one-dimensional problem of default

• It is also a problem of credit deterioration, i.e. the worsening of the rating (and then increased def.prob.)

• An example is constituted by mutual funds investing in Investment Grade bond. Their problem is downgrading to sub-investment grades

8.34One-Year Transition MatrixYear End Rating

InitRat

AAA AA A BBB BB B CCC Def

AAA 90.81 8.33 0.68 0.06 0.12 0.00 0.00 0.00

AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0.00

A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

BBB 0.02 0.33 5.95 86.93 5.30 1.17 0.12 0.18

BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06

B 0.00 0.11 0.24 0.43 6.48 83.46 4.07 5.20

CCC 0.22 0.00 0.22 1.30 2.38 11.24 64.86 19.79

Def 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100

8.35

Default probability obtained via option pricing: 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

8.36

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

8.37

Volatilities

E V

V

E

E

V 0

0

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