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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

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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