We know from the numerical example above that when the swap fixed rate falls, the fixed-rate payer loses market value and the fixed-rate receiver gains. Therefore, the swap has negative duration to the payer (i.e., the long position or the “buyer” of the swap) and positive duration to the receiver

(the short position or the “seller”). We see in Chapter 10 that adding a pay- fixed swap to a fixed-income investment portfolio reduces average portfolio duration while adding a receive-fixed swap increases average duration. How much of an increase or decrease depends on the duration of the swap and the amount of notional principal.

The duration of a plain vanilla interest rate swap is derived by recognizing that the net settlement cash flows on the derivative are the same (assuming no default) as on a pair of bonds, one a fixed-rate bond and the other a floater. The swap to Party A in Figure 8.1 is as if it has purchased a 2-year floater paying 3-month LIBOR flat and has issued a 2-year, 3.40% quarterly payment, fixed- rate bond. Both bonds are priced at par value because the initial value of the swap is zero. In the same manner, the swap to Party B provides the same net cash flows as if it buys a 3.40% fixed-rate bond financed by issuing the LIBOR flat floater.

Equation 6.13 from Chapter 6 provides a closed-form formula for the Macaulay duration (MacDur) of a standard fixed-rate bond. It is repeated here as equation 8.2.

(8.2)

The current date is t days into the T-day period. The yield per period prevailing on date t is y; the fixed coupon rate per period is c; the number of periods to maturity as of the beginning of the period is N.

Equation 7.10 in Chapter 7 shows the general formula for the Macaulay duration of a floater that might be trading at a premium or discount on a payment date. Following market practice, I assume that the floating-rate note component of the swap always is priced at par value on a payment date, so that PVANN = 0. That dramatically simplifies the equation for floater duration.

(8.3)

As shown in equations 8.4 and 8.5, the Macaulay duration of an interest rate swap (MacDurSWAP) subtracts one formula from the other because one bond is an implicit asset and the other a liability.

(8.4)

(8.5)

It's interesting that the t/T term drops out of the two expressions. Fixed-rate bonds and floaters have a “saw-tooth” pattern for the duration statistic during the period. The duration of each declines smoothly (assuming no change in market interest rates) and then jumps up on the payment date. Because a swap can be interpreted as a “long-short” combination of two bonds, the “saw-teeth” are smoothed out. That's not to say that the duration is constant – it still is inversely related to the yield (i.e., the fixed rate on the mark-to-market swap). Remember that y is the yield on the implicit fixed- rate bond that prevails on date t and likely will change during the period. The other terms in the equation, c and N, are constants.

Equations 8.4 and 8.5 are formulas for rate duration, not for credit duration. That is, they can be used to estimate the change in the market value of the swap arising from a change in benchmark interest rates, in particular, the forward curve for the money market reference rate. Counterparty credit risk is not an issue. This justifies the simplifying assumption for the duration of the floater. The pair-of-bonds interpretation of the swap is fine for assessing the impact of market rates but is inappropriate for default risk. A swap is an exchange of interest cash flows, not of principal. For example, if Party A defaults, the loss to Party B is limited to the fixed-rate cash flows no longer received less the floating-rate cash flows that no longer need to be paid.

Let's use these duration statistics to see how well they estimate the actual change in the value of the swap once three months go by and the LIBOR forward curve has shifted and twisted from Table 8.1 to Table 8.2. We know that leads to a loss of $410,233 to Party A, the fixed-rate payer, and an equivalent gain to Party B, the fixed-rate receiver. Let the contractual fixed rate on the swap c = 0.0085 (= 0.0340/4), the number of periods to maturity as of the beginning of the period N = 8, and the fixed rate on the mark-to-market swap y = 0.0075 (= 0.0300/4).

Pay-Fixed Swap:

Receive-Fixed Swap:

It's no surprise that the pay-fixed swap has negative duration and the receive-fixed swap positive duration. Annualized, the Macaulay durations are -1.69225 and +1.69225 after dividing by four periods in the year. The annual modified durations are -1.67965 and +1.67965, the Macaulay durations divided by 1.0075.

For a notional principal of $60 million and a 40-basis-point decrease in the swap rate, duration estimates the change in market value (ΔMV) to be a gain of $403,116 to Party B, the fixed-rate receiver, and a loss to Party A, the fixed-rate payer, for the same amount.

Another version of this calculation is to use the basis-point-value (BPV) for the swap, which is its modified duration times the notional principal, times one basis point (0.0001). For the fixed-rate payer, the BPV is -$10,077.90 (= -1.67965 * $60,000,000 * 0.0001) whereas for the fixed-rate receiver it is +$10,077.90. Then, for a 40 basis point change in the swap fixed rate, the estimated change in value to the payer is -$10,077.90 * 40 = -$403,116. The BPV, as a measure of money duration, relates directly to a change in value in currency units. Modified duration relates to a percentage change in value – that can be awkward for a newly initiated swap that has a value of zero. Therefore, some derivatives analysts prefer to work with the BPV (or the DV01 or PV01, which are very similar statistics) rather that modified duration. In any case, the modified duration and the BPV of the swap contain the same information and produce the same estimated change in value.

When the swap fixed rate goes down from 3.40% to 3.00%, the estimated change in value of $403,116 is not a bad approximation for the actual change, which we determined above to be $410,233. The reason for the difference between the estimated and actual results concerns the change in the swap rate, 40 basis points in this example. (Also, the convexity of the swap has been neglected. It, too, can be inferred from the convexities of the implied fixed-rate and floating-rate bonds.) The new swap fixed rate of 3.00% could have resulted from many twists and shifts to the LIBOR forward curve. Each one of those twists and shifts would produce a different implied spot curve and a different present value of the annuity. However, the only input into our estimation is the 40 basis point change in the swap fixed rate. Although we do not have to assume a parallel shift to the forward curve to use duration, we do have to keep in mind that we are estimating outcomes with error.

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