Options, Futures, and Other Derivatives

Transcript Options, Futures, and Other Derivatives

Credit Risk

Chapter 20

Options, Futures, and Other Derivatives

20.1

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”

Options, Futures, and Other Derivatives

20.2

Historical Data

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

Options, Futures, and Other Derivatives

20.3

Cumulative Ave Default Rates (%)

(1970-2003, Moody’s, Table 20.1, page 482)

Aaa Aa A Baa Ba B Caa 1 2 3 0.00 0.00 0.00 4 0.04 5 7 10 0.12 0.29 0.62 0.02 0.03 0.06 0.02 0.09 0.23 0.15 0.38 0.24 0.43 0.68 0.54 0.91 1.59 0.20 0.57 1.03 1.62 2.16 3.24 5.10 1.26 3.48 6.00 8.59 11.17 15.44 21.01 6.21 13.76 20.65 26.66 31.99 40.79 50.02 23.65 37.20 48.02 55.56 60.83 69.36 77.91

Options, Futures, and Other Derivatives

20.4

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 0.20% of defaulting by the end of the first year, 0.57% by the end of the second year, and so on

Options, Futures, and Other Derivatives

20.5

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

Options, Futures, and Other Derivatives

20.6

Default Intensities vs Unconditional Default Probabilities

(page 482-483)

 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 rate company in the third year?

Options, Futures, and Other Derivatives

20.7

Probability of default

(

)  1 

  (

)

(20.1) Q (t) - probability of default by time t  (

) - Average default intensity between time 0 and time t

Options, Futures, and Other Derivatives

20.8

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

Options, Futures, and Other Derivatives

20.9

Recovery Rates

(Moody’s: 1982 to 2003, Table 20.2, page 483)

Class Senior Secured Senior Unsecured Senior Subordinated Subordinated Junior Subordinated Mean(%) 51.6 36.1 32.5 31.1 24.5

Options, Futures, and Other Derivatives

20.10

Estimating Default Probabilities

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

Options, Futures, and Other Derivatives

20.11

Using Bond Prices

(Equation 20.2, page 484)

Average default intensity over life of bond is approximately

 1 

s R

(20.2) where

is the default intensity per year,

is the spread of the bond’s yield over the risk free rate and

is the recovery rate

Options, Futures, and Other Derivatives

20.12

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

per year and that defaults always happen half way through a year (immediately before a coupon payment).

Options, Futures, and Other Derivatives

20.13

Calculations

(Table 20.3, page 485)

Time (yrs) 0.5

Def Prob

Recovery Amount 40 Risk-free Value 106.73

1.5

2.5

3.5

4.5

Total

Q Q Q Q

40 40 40 40 105.97

105.17

104.34

103.46

LGD 66.73

65.97

65.17

64.34

63.46

Discount Factor 0.9753

PV of Exp Loss 65.08

0.9277

0.8825

0.8395

0.7985

61.20

57.52

54.01

50.67

288.48

Q Options, Futures, and Other Derivatives

20.14

Calculations

continued

 We set 288.48

= 8.75 to get

= 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

Options, Futures, and Other Derivatives

20.15

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

Options, Futures, and Other Derivatives

20.16

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

Options, Futures, and Other Derivatives

20.17

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

Options, Futures, and Other Derivatives

20.18

Real World vs Risk Neutral Default Probabilities, 7 year averages

(Table 20.4, page 487)

Aaa Aa A Baa Ba B Caa Rating Real-world default probability per yr (bps) 4 6 13 47 240 749 1690 Risk-neutral default probability per yr (bps) 67 78 128 238 507 902 2130 Ratio 16.8 13.0 9.8 5.1 2.1 1.2 1.3 Difference 63 72 115 191 267 153 440

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20.19

Risk Premiums Earned By Bond Traders

(Table 20.5, page 488)

Rating Aaa Aa A Baa Ba B Caa Bond Yield Spread over Treasuries (bps) 83 90 120 186 347 585 1321 Spread of risk-free rate used by market over Treasuries (bps) 43 43 43 43 43 43 43 Spread to compensate for default rate in the real world (bps) 2 4 8 28 144 449 1014 Extra Risk Premium (bps) 38 43 69 115 160 93 264

Options, Futures, and Other Derivatives

20.20

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

Options, Futures, and Other Derivatives

20.21

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

Options, Futures, and Other Derivatives

20.22

Merton’s Model

(page 489-491)

 Merton’s model regards the equity as an option on the assets of the firm  In a simple situation the equity value is max(

V T

, 0) where

V T

is the value of the firm and the debt repayment required

Options, Futures, and Other Derivatives

20.23

Equity vs. Assets

An option pricing model enables the value of the firm’s equity today,

0 , to be related to the value of its assets today,

0 , and the volatility of its assets, s

V E

0  where ( 1 ) 



rT d

1  ln (

)  (

 s

2 s

V T

2 ) 2 )

;

2 

1  s

V T Options, Futures, and Other Derivatives

20.24

Volatilities

E E

0  



V V

0  1 ) s

V V

0 This equation together with the option pricing relationship enables

0 determined from

0 and and s

to be

Options, Futures, and Other Derivatives

20.25

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

0 =12.40 and s

=21.23%

Options, Futures, and Other Derivatives

20.26

Example continued

 The probability of default is

2 ) or 12.7%  The market value of the debt is 9.40

 The present value of the promised payment is 9.51

 The expected loss is about 1.2%  The recovery rate is 91%

Options, Futures, and Other Derivatives

20.27

The Implementation of Merton’s Model (e.g. Moody’s KMV)

 Choose time horizon   Calculate cumulative obligations to time horizon. This is termed by KMV the “default point”. We denote it by

Use Merton’s model to calculate a theoretical probability of default  Use historical data or bond data to develop a one-to-one mapping of theoretical probability into either real-world or risk-neutral probability of default.

Options, Futures, and Other Derivatives

20.28

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

Options, Futures, and Other Derivatives

20.29

General Result

  Assume that default probability is independent of the value of the derivative Consider times

1 ,

2 ,…

t n

and default probability is

q i

at time

t i

. The value of the contract at time

t i

f i

and the recovery rate is

  The loss from defaults at time

t i

q i

(1-

)

[max(

f i

,0)]. Defining

u i

q i

(1-

) and

v i

as the value of a derivative that provides a payoff of max(

f i

,0) at time

t i

, the cost of defaults is

i n

  1

u i v i Options, Futures, and Other Derivatives

20.30

Credit Risk Mitigation

 Netting  Collateralization  Downgrade triggers

Options, Futures, and Other Derivatives

20.31

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.

Options, Futures, and Other Derivatives

20.32

Measurement

 There is no generally accepted measure of default correlation  Default correlation is a more complex phenomenon than the correlation between two random variables

Options, Futures, and Other Derivatives

20.33

Gaussian Copula Model

(page 496-499)

   Define a one-to-one correspondence between the time to default,

t i

, of company

and a variable

x i

Q i

(

t i

) =

(

x i

) or

x i

-1 [

(

t i

)] where

is the cumulative normal distribution function. This is a “percentile to percentile” transformation. The

percentile point of the

Q i p

percentile point of the

i normal distribution distribution is transformed to the distribution.

x i

has a standard We assume that the

x i

are multivariate normal. The default correlation measure, r

between companies

and

is the correlation between

x i

and

x j Options, Futures, and Other Derivatives

20.34

Example of Use of Gaussian Copula

(Example 20.3, page 498)

Suppose that we wish to simulate the defaults for

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

Options, Futures, and Other Derivatives

20.35

Use of Gaussian Copula

continued

 We sample from a multivariate normal distribution to get the

x i

 Critical values of

N x i

are -1 (0.01) = -2.33,

-1 (0.03) = -1.88,

-1 (0.06) = -1.55,

-1 (0.10) = -1.28,

-1 (0.15) = -1.04

Options, Futures, and Other Derivatives

20.36

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

Options, Futures, and Other Derivatives

20.37

A One-Factor Model for the Correlation Structure

(Equation 20.7, page 498)

x i



a i M

 1 

a i

Z i

  The correlation between

x i

and

x j

a i a j

The

th company defaults by time

when or

x i

-1 [

i (

)]

Z i



 1 [

Q i

(

) 

a i M

] 1 

a i

2  The probability of this is

Q i

(

T M

) 

 

 1 

Q i

(

1  )  

a i M a i

2  

Options, Futures, and Other Derivatives

20.38

Binomial Correlation Measure

(page 499)

   One common default correlation measure, between companies

and

is the correlation between A variable that equals 1 if company

between time 0 and time

and zero otherwise defaults  A variable that equals 1 if company

between time 0 and time

and zero otherwise defaults The value of this measure depends on

. Usually it increases at

increases.

Options, Futures, and Other Derivatives

20.39

Binomial Correlation

continued

Denote

Q i

(

) as the probability that company

will default between time zero and time

, and

P ij

(

) as the probability that both

and

will default. The default correlation measure is 

(

) 

P ij

(

) 

Q i

(

)

Q j

(

) [

Q i

(

) 

Q i

(

) 2 ][

Q j

(

) 

Q j

(

) 2 ]

Options, Futures, and Other Derivatives

20.40

Survival Time Correlation

  

Define

t i

and

Q i

(

t i

as the time to default for company ) as the probability distribution for

t i

The default correlation between companies

and

can be defined as the correlation between

t i

and

t j

But this does not uniquely define the joint probability distribution of default times

Options, Futures, and Other Derivatives

20.41

Binomial vs Gaussian Copula Measures

(Equation 20.10, page 499)

The measures can be calculated from each other

P ij

(

) 

[

x i

x j

; r

] so that 

(

) 

[

x i

x j

; r

] 

Q i

(

)

Q j

(

) [

Q i

(

) 

Q i

(

) 2 ][

Q j

(

) 

Q j

(

) 2 ] where

is the cumulative bivariate probabilit y distributi on function normal

Options, Futures, and Other Derivatives

20.42

Comparison

(Example 20.4, page 499)

 The correlation number depends on the correlation metric used   Suppose

value of r

= 1,

Q i

(

) =

Q j

(

) value of 

(

) equal to 0.024.

= 0.01, a equal to 0.2 corresponds to a In general 

(

) < r

and 

(

) is an increasing function of

T Options, Futures, and Other Derivatives

20.43

Credit VaR

(page 499-502)

 Can be defined analogously to Market Risk VaR  A

-year credit VaR with an

% confidence is the loss level that we are

% confident will not be exceeded over

years

Options, Futures, and Other Derivatives

20.44

Calculation from a Factor-Based Gaussian Copula Model

(equation 20.11, page 500)

  Consider a large portfolio of loans, each of which has a probability of

(

) of defaulting by time

. Suppose that all pairwise copula correlations are r so that all

a i

’s are r We are −

-1 (

)

% certain that

is less than

-1 (1 −

) =  It follows that the VaR is

(

) 

N N

 1 

(

)   1  r r

 1 (

)

Options, Futures, and Other Derivatives

20.45

CreditMetrics

(page 500-502)

 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

Options, Futures, and Other Derivatives

20.46

Options, Futures, and Other Derivatives

Transcript Options, Futures, and Other Derivatives

Credit Risk

Chapter 20

Credit Ratings

Historical Data

Cumulative Ave Default Rates (%)

Interpretation

Do Default Probabilities Increase with Time?

Default Intensities vs Unconditional Default Probabilities

Probability of default

Recovery Rate

Recovery Rates

Estimating Default Probabilities

Using Bond Prices

More Exact Calculation

Calculations

Calculations

The Risk-Free Rate

Real World vs Risk-Neutral Default Probabilities

A Comparison

Real World vs Risk Neutral Default Probabilities, 7 year averages

Risk Premiums Earned By Bond Traders

Possible Reasons for These Results

Which World Should We Use?

Merton’s Model

Equity vs. Assets

Volatilities

Example

Example continued

The Implementation of Merton’s Model (e.g. Moody’s KMV)

Credit Risk in Derivatives Transactions

General Result

Credit Risk Mitigation

Default Correlation

Measurement

Gaussian Copula Model

Example of Use of Gaussian Copula

Use of Gaussian Copula

Use of Gaussian Copula

A One-Factor Model for the Correlation Structure

Binomial Correlation Measure

Binomial Correlation

Survival Time Correlation

Binomial vs Gaussian Copula Measures

Comparison

Credit VaR

Calculation from a Factor-Based Gaussian Copula Model

CreditMetrics

Directory