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K. Valuation of Credit Default Swaps

K. Valuation of Credit Default Swaps

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TABLE K.1



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RISK MANAGEMENT AND FINANCIAL INSTITUTIONS

Unconditional Default Probabilities and Survival Probabilities



Time (years)



Default Probability



Survival Probability



0.0200

0.0196

0.0192

0.0188

0.0184



0.9800

0.9604

0.9412

0.9224

0.9039



1

2

3

4

5



TABLE K.2



8:58



Calculation of the Present Value of Expected Payments (payment = s per



annum)

Time

(years)

1

2

3

4

5



Probability

of Survival



Expected

Payment



Discount

Factor



PV of Expected

Payment



0.9800

0.9604

0.9412

0.9224

0.9039



0.9800s

0.9604s

0.9412s

0.9224s

0.9039s



0.9512

0.9048

0.8607

0.8187

0.7788



0.9322s

0.8690s

0.8101s

0.7552s

0.7040s



Total



TABLE K.3

Time

(years)

0.5

1.5

2.5

3.5

4.5



4.0704s



Calculation of the Present Value of Expected Payoff (notional principal = $1)

Probability

of Default



Recovery

Rate



Expected

Payoff ($)



Discount

Factor



PV of Expected

Payoff ($)



0.0200

0.0196

0.0192

0.0188

0.0184



0.4

0.4

0.4

0.4

0.4



0.0120

0.0118

0.0115

0.0113

0.0111



0.9753

0.9277

0.8825

0.8395

0.7985



0.0117

0.0109

0.0102

0.0095

0.0088



Total



0.0511



TABLE K.4



Calculation of the Present Value of Accrual Payment



Time

(years)



Probability

of Default



Expected Accrual

Payment



Discount

Factor



PV of Expected

Accrual Payment



0.0200

0.0196

0.0192

0.0188

0.0184



0.0100s

0.0098s

0.0096s

0.0094s

0.0092s



0.9753

0.9277

0.8825

0.8395

0.7985



0.0097s

0.0091s

0.0085s

0.0079s

0.0074s



0.5

1.5

2.5

3.5

4.5

Total



0.0426s



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Appendix K: Valuation of Credit Default Swaps



553



From Tables K.2 and K.4 the present value of the expected payments is

4.0704s + 0.0426s = 4.1130s

From Table K.3, the present value of the expected payoff is 0.0511. Equating the

two, the CDS spread for a new CDS is given by

4.1130s = 0.0511

or s = 0.0124. The mid-market spread should be 0.0124 times the principal or 124

basis points per year. (This is roughly what we would expect from the relationship

in equation 16.3: the recovery rate is 40% and the hazard rate is about 2%.)

This example is designed to illustrate the calculation methodology. In practice,

we are likely to find that calculations are more extensive than those in Table K.2 to

K.4 because (a) payments are often made more frequently than once a year and (b)

we want to assume that defaults can happen more frequently than once a year.



Marking to Market a CDS

At the time it is negotiated, a CDS like most other swaps is worth close to zero. At later

times it may have a positive or negative value. Suppose, for example, the credit default

swap in our example had been negotiated some time ago for a spread of 150 basis

points; the present value of the payments by the buyer would be 4.1130 × 0.0150 =

0.0617 and the present value of the payoff would be 0.0511. The value of swap to the

seller would therefore be 0.0617 − 0.0511 or 0.0106 times the principal. Similarly,

the mark-to market value of the swap to the buyer of protection would be −0.0106

times the principal.

The software DerivaGem that accompanies this book and can be downloaded from the author’s website includes a worksheet that carries out the above

calculations.



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APPENDIX



L



Synthetic CDOs and

Their Valuation



ynthetic collateralized debt obligations (CDOs) consist of tranches where one

party (Party A) agrees to make payments to another party (Party B) that are equal

to those losses on a specified portfolio of debt instruments that are in a certain range.

In return, Party B agrees to make payments to Party A that are a certain proportion

of the amount of principal that is being insured.

Suppose that the range of losses for a particular tranche is from ␣L to ␣H . The

variables ␣L and ␣H are known as the attachment point and detachment point,

respectively. If ␣L is 8% and ␣H is 18%, Party A pays to Party B the losses on the

portfolio, as they are incurred, in the range 8% to 18% of the total principal of the

portfolio. The first 8% of losses on the portfolio does not therefore affect the tranche.

The tranche is responsible for the next 10% of losses and its notional principal

(initially 18 − 8 = 10% of the portfolio principal) reduces as these losses are incurred.

The tranche is wiped out when losses exceed 18%. The payments that are made by

Party B to Party A are made periodically at a specified rate applied to the remaining

notional tranche principal. This specified rate is known as the tranche spread.

The usual assumption is that all the debt instruments in the portfolio have the

same probability distribution for the time to default. Define Q(t) as the probability

of a debt instrument defaulting by time t. The one-factor Gaussian copula model of

time to default presented in Section 11.5 has become the standard market model for

valuing a tranche of a collateralized debt obligation (CDO). From equation (11.12)



S



Q(t|F ) = N



N−1 [Q(t)] −

1−␳







␳F



(L.1)



where Q(t|F ) is the probability of the ith entity defaulting by time t conditional on

the value of the factor, F . In the calculation of Q(t) it is usually assumed that the

hazard rate for a company is constant. When a CDS spread or other credit spread

is available, it can be used to determine the hazard rate using calculations similar to

those in Appendix K in conjunction with a search procedure.

Suppose that the hazard rate is ␭. Then

Q(t) = 1 − e−␭t



(L.2)



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RISK MANAGEMENT AND FINANCIAL INSTITUTIONS



From the properties of the binomial distribution, the probability of exactly k defaults

by time t, conditional on F is

P(k, T|F ) =



n!

Q(t|F )k[1 − Q(t|F )]n−k

(n − k)!k!



(L.3)



Define

nL =



␣Ln

1− R



and



nH =



␣H n

1− R



where R is the recovery rate (assumed constant). Also, define m(x) as the smallest

integer greater than x. The tranche suffers no losses when the number of defaults, k,

is less than m(nL). It is wiped out when k is greater than or equal to m(n H ). Otherwise

the tranche principal at time t is a proportion

␣H − k(1 − R)/n

␣H − ␣L

of the initial tranche principal. These results can be used in conjunction with equations (L.1), (L.2), and (L.3) to calculate the expected tranche principal at all times

conditional on F . We can then integrate over F to find the (unconditional) expected

tranche principal. This integration is usually accomplished with a procedure known

as Gaussian quadrature. (The author’s website provides the tools for integrating over

a normal distribution using Gaussian quadrature.)

It is usually assumed that defaults happen at the midpoint of the intervals

between payments. Similarly to Appendix K, we are interested in the following

quantities

1. The present value of the expected spread payments received by Party A.

2. The present value of the expected payments for tranche losses made by Party A.

3. The present value of accrual payments received by Party A.

The spread payments received by Party A at a particular time are linearly dependent

on the tranche principal at that time. The tranche loss payments made by Party A

(assumed to be at the midpoint of an interval) is the change in the principal during

the interval. The accrual payment received by Party A is proportional to the tranche

loss payments. For any assumption about spreads, all three quantities of interest can

therefore be calculated from the expected tranche principal. The breakeven spread

can therefore be calculated analogously to the way it is calculated for CDSs in

Appendix K.

Derivatives dealers calculate the implied copula correlation, ␳ , from the spreads

quoted in the market for tranches of CDOs and tend to quote these rather than

the spreads themselves. This is similar to the practice in options markets of quoting Black–Scholes–Merton implied volatilities rather than dollar prices. There is a



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