We show how a generic or “plain-vanilla” interest-rate swap can be mapped by decomposing it into a portfolio of a fixed-rate note and a floating-rate note based on LIBOR, using an example of a hypothetical pay-fixed, receive-floating interest-rate swap with a notional principal of $100 million and payments every six months based on six-month LIBOR. The swap fixed rate is 8% per year, and six payments remain, with payment dates occurring and 2.75 years in the future. This situation might arise if the swap originally had a tenor of three years and was entered into three months in the past. The timing of reset dates and payments fol- lows the usual convention; for example, the payment at is based on LIBOR quoted six months previously, at 0.25. In general, the pay- ment at time t is based on six-month LIBOR quoted at time 0.5, denoted .
With these assumptions, the cash flows of the fixed and floating legs of
the swap at time t are million and mil-
lion, respectively, where is six-month LIBOR quoted at time for a loan from to t and 0.08 or 8% is the fixed rate. The coefficient 0.5 is the length of the payment period and reflects a decision to ignore the details of day-count conventions. Similarly, we ignore certain other details of the timing, for example, the fact that LIBOR quoted at time t typically covers a deposit period beginning two banking days after time t. Table 5.2 shows these swap cash flows for the six payment dates as functions of the floating rates observed on the reset dates.
t = 0.25, 0.75, . . . ,
t = 0.75 t =
t– rt–0.5
$100 0.5 0.08[ ( )]
– $100 0.5[ rt–0.5]
rt–0.5 t–0.5
t–0.5
Table 5.3 decomposes the swap cash flows shown in the second col- umn into a set of fixed cash flows (third column) and the cash flows that would ensue from buying a floating-rate note at time 0.25 (fourth column). In particular, in the fourth column the cash flow of -$100 at time 0.25 is the amount paid to purchase the floating-rate note, the cash
flows of the form , , etc., are the interest
payments, and the cash flow of at time 2.75 repre- sents the final interest payment and the return of the principal. Because time 0.25 is a reset date, an investor who purchases the note on this date will receive the prevailing market six-month rate over each six-month period during the remaining life of the note, implying that the note should trade at par on this date. As a result, the payment of $100 to purchase the note is exactly equal to the present value of the remaining cash flows, and the set of cash flows in the fourth column has present value zero as of time 0.25. This then implies that the set of cash flows in the fourth column has present value zero as of time 0.
But if the swap is equivalent to the sum of the cash flows in the third and fourth columns and the present value of the cash flows in the fourth column is zero, then the present value of the swap cash flows must equal the present value of the cash flows in the third column. Thus, for purposes of val- uation, we can substitute the fixed cash flows in the third column for those of the swap. If the market factors consist of zero-coupon bonds, these fixed cash flows can then be mapped onto the zero-coupon bonds using the approach discussed in this chapter.
TABLE 5.2 Cash flows of a hypothetical three-year interest rate swap entered into three months prior to the current date
Time Swap Cash Flows
($million) 0.25
0.75 1.25 1.75 2.25 2.75
100 0.5 0.075[ ( )–0.5 0.08( )] 100 0.5r[ 0.25–0.5 0.08( )]
100 0.5r[ 0.75–0.5 0.8( )] 100 0.5r[ 1.25–0.5 0.8( )] 100 0.5r[ 1.75–0.5 0.8( )] 100 0.5r[ 2.25–0.5 0.8( )]
100 0.5r[ 0.25] 100 0.5r[ 0.75] 100 0.5r[ 2.25]+100
89 TABLE 5.3Decomposition of the swap into fixed cash flows and the purchase of a floating rate note TimeSwap Cash FlowsFixed Cash FlowsCash Flows of Floating Note Bought at Time 0 0.25 0.75 1.25 1.75 2.25 2.75
1000.50.075()0.50.08()–[]1001000.50.8()0.50.075()–[]+100– 1000.5r0.250.50.8()–[]1000.50.8()[]–1000.5r0.25[] 1000.5r0.750.50.8()–[]1000.50.8()[]–1000.5r0.75[] 1000.5r1.250.50.8()–[]1000.50.8()[]–1000.5r1.25[] 1000.5r1.750.50.8()–[]1000.50.8()[]–1000.5r1.75[] 1000.5r2.250.50.8()–[]1000.50.08()[]–100–1000.5r2.25[]100+
6
91
Monte Carlo Simulation
The Monte Carlo simulation approach has a number of similarities to his- torical simulation. Most importantly, the method is able to capture the risk of portfolios that include options and other instruments whose values are nonlinear functions of the underlying market factors. Like historical simu- lation, Monte Carlo simulation accomplishes this by repeatedly revaluing the portfolio, using hypothetical new values of the underlying market fac- tors that determine the portfolio value. Because the exact portfolio value is computed for every realization of the market factors considered, the method captures any nonlinearities in the value of the portfolio.
The main difference between the two approaches is that the Monte Carlo method does not conduct the simulation using the observed changes in the market factors over the last N periods to generate N hypothetical portfolio profits or losses. Instead, one chooses a statistical distribution that is believed to adequately capture or approximate the possible changes in the market fac- tors. Then, a pseudo-random number generator is used to generate thousands, or perhaps tens of thousands, of hypothetical changes in the market factors.
These are then used to construct the distribution of possible portfolio profit or loss. Finally, the value-at-risk is determined from this distribution.
The approach is illustrated using the example portfolio analyzed using the delta-normal method in Chapter 3. It consists of $110 million invested in a well-diversified portfolio of large capitalization U.S. equities, where it is assumed that the returns on this portfolio are perfectly correlated with changes in the S&P 500 index. The portfolio manager has reduced his exposure to the U.S. market by shorting 200 of the S&P 500 index futures contracts and obtained exposure to the U.K. market by establishing a long position of 500 FT-SE 100 index futures contracts. In addition, the portfo- lio manager has written 800 of the September S&P 500 index call options with a strike of 1100 and has written 600 of the September FT-SE 100 index call options with a strike price of 5875. Combining the written options positions with the portfolio of U.S. equities, the net value of the portfolio is approximately $101,485,220.
To analyze this portfolio, we perform the same steps we carried out in the historical simulation method, except that collecting data on the past realizations of changes in the basic market factors will be replaced by selecting a statistical distribution from which to draw pseudo-random changes. The steps consist of:
(i) identifying the market factors; (ii) selecting a statistical distribution from which to draw pseudo-random hypothetical changes in the value of the market factors; (iii) applying these hypothetical pseudo-random changes in the market factors to the current portfolio; and (iv) identifying the value-at-risk.
Once again, we use a probability of 5% and a holding period of one month.