Contracts such as swaps that can become either assets or liabilities are more complicated to analyze than those considered in the previous section. One popular approach for swaps that has been used by regulatory authorities is to compare the average expected exposure on a swap with the average expected exposure on a loan that has the same principal as the swap. If the average expected exposure on the swap is, say, 5 percent of that on the loan, this is an indication that the financial institution should require the swap to contribute about 5 percent as much as towards profits as a loan with the same counterparty.
The average expected exposure on a swap during its life can be calculated using Monte Carlo simulation. Consider first an interest rate swap where the financial institution is receiving fixed and paying floating. The exposure at a future time is equal to max(5fiXed - ^floating, 0)
where fifixed is the price of the fixed bond underlying the swap, and fifloating is the price of the floating rate bond underlying the swap. Since ^floating is relatively constant, this is similar to the payoff from a call option on the fixed-rate bond. Consider next a currency swap when the financial institution is receiving the domestic interest and principal while paying the foreign interest and principal. The exposure at a future time is equal to max(BD - BfS, 0)
Bf max
Bf max
where Bp is the price of the foreign bond underlying the swap and BD is the price of the domestic bond underlying the swap. If interest rates are assumed constant, Bd and Bf are relatively constant and this is similar to the payoff from a put option on the currency.
Figure 18.2 compares the expected exposure on a matched pair of interest rate swaps with the expected exposure on a matched pair of currency swaps. The expected exposure on a matched pair of interest rate swaps starts at zero, increases, and then decreases to zero. By contrast, the expected exposure on a matched pair of currency swaps increases steadily with the passage of time.
Figure 18.2 Expected Exposure on Matched Pairs of Interest Rate Swaps and Currency Swaps
There are theoretical problems in assessing the impact of credit risk on a contract by looking at the average expected exposure during the life of the contract. For a start, it is difficult to know whether we should evaluate exposures in a risk-neutral world or the real world. A risk-neutral world is easier, but the real world is arguably more relevant. It is also difficult to know whether we should take a straight average of the exposures or weight them in some way. Arguably more
Figure 18.2 Expected Exposure on Matched Pairs of Interest Rate Swaps and Currency Swaps
Time
Time weight should be given to later exposures because some time must elapse for the fortunes of the company to decline sufficiently for default to occur.4
A more precise approach to the problem of quantifying the impact of default risk can be developed by extending the results in the previous section. Hull and White show that when the independence assumption is made5
where u(r) = £[e-F(T-0 max(/*, 0)], yT and y* are the yields on vulnerable and riskless bonds maturing at time r, r is the average interest rate between t and r, and f* is the no-default value of the security at time r. The term v(r) is the present value of a claim that pays off the exposure at time r.
The function v(t) can be valued by using Monte Carlo simulation or one of the other derivative security pricing procedures that have been discussed in this book. The value of f* — f can then be calculated from Equation (18.2) using numerical integration.
Consider a fixed-for-fixed foreign currency swap in which interest in sterling at the sterling risk-free rate is exchanged for interest in dollars at the dollar risk-free rate. Principals are also exchanged at the end of the life of the swap. Suppose that the swap details are as follows
Life of swap: 5 years Frequency of payments: Annual Sterling Principal: £100 million Dollar Principal: $100 million Initial exchange rate: 1.0000
Dollar riskless rate: 5% per annum (assumed constant)
4Figure 18.2 shows that the exposure on a matched pair of currency swaps increases with time whereas the exposure on an interest rate swap first increases and then decreases. Moving to a scheme where more weight is given to later exposures therefore increases the credit risk of a currency swap relative to that of an interest rate swap.
5See J. Hull and A. White, "The Impact of Default Risk on the Prices of Options and Other Derivative Securities," Working Paper, University of Toronto, 1991. We are here assuming that the counterparty may default, but that the company from whose viewpoint the derivative security is being valued will not default.
Sterling riskless rate: 10% per annum (assumed constant)
Volatility of exchange rate: 15%
We suppose that 1-, 2-, 3-, 4-, and 5-year zero coupon bonds issued by the counterparty would have yields that are 25, 50, 75, 85, and 95 basis points above the corresponding riskless rate.
When the company is receiving domestic and paying fixed, Equation (18.2) shows that the cost of defaults is $0.73 million; when it is receiving foreign and paying domestic the cost of defaults is $0.13 million.6 In total, the impact of credit risk on the value of a matched pair of swaps is 0.73+0.13=$0.86 million. The financial institution should therefore require a spread that has a present value equal to at least $0.86 million. Using a discount rate of 5 percent per annum, an annuity of $0.198 million for 5 years has a present value of $0.86 million. Since the principal is $100 million, this indicates that a total spread of at least 19.8 basis points is required on the matched pair of swaps.
The impact of default risk on interest rate swaps is generally considerably less than that on currency swaps. Using the similar data to that for the currency swap, the required total spread for a matched pair of interest rate swaps is only 2 to 3 basis points.
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