# Duration Swaps

### Duration Swap [viewable here in Excel]

A received-fixed swap consists of a short position in a floating-rate instrument combined with a long position in a fixed-rate coupon bond. Conversely, a pay-fixed swap consists of a short position in a coupon bond, coupled with a long position in a floating-rate instrument. Therefore, an interest rate swap has a duration that equals the duration of the bond portfolio that is equivalent to the swap. The duration of a swap can be either positive or negative, depending on whether the swap is received-fixed or a pay-fixed swap. Based on these reflections, we can state the following rules:

- Duration of a receive-fixed swap = duration of the underlying coupon bond - duration of the underlying floating-rate bond > 0

- Duration of a pay-fixed swap = duration of the underlying floating-rate bond - duration of the underlying coupon bond <0

The duration of a floating-rate instrument equals the time between reset dates for the interest rate. Thus an FRN with semiannual payments would have a duration equal to six months, or one half-year. The calculations of the duration of the swap, then, really depends on finding the duration of the fixed-rate coupon bond underlying the swap. As an example, consider an interest rate swap with an swap-fixed-rate (SFR) of 7 percent, a tenor of seven years, and semiannual payments. Table 22.3 shows the calculation of the Macaulay duration for the coupon bond portion of the swap. As the table shows, the duration of the fixed side of the swap is 11.302738 semiannual periods, or 5.651369 years. For the floating side of the swap the duration is six months, or one half-year. Therefore, for this swap,

Duration of a received-fixed swap = 5.651369 - 0.5 = 5.151369 years duration of a pay-fixed swap = 0.5 - 5.651369 = -5.151369 years

Table 22.3 Calculation of duration for the fixed-rate side of an interest rate swap

 Semi-annual period Cash flows Discount Factor PV of cash flow Weighted PV of cash flow 1 35 1.0350 33.816 0.0338 2 35 1.0712 32.673 0.0653 3 35 1.1087 31.568 0.0947 4 35 1.1475 30.500 0.1220 5 35 1.1877 29.469 0.1473 6 35 1.2293 28.473 0.1708 7 35 1.2723 27.510 0.1926 8 35 1.3168 26.579 0.2126 9 35 1.3629 25.681 0.2311 10 35 1.4106 24.812 0.2481 11 35 1.4600 23.973 0.2637 12 35 1.5111 23.162 0.2779 13 35 1.5640 22.379 0.2909 14 1035 1.6187 639.40 8.9517 1000.00 11.30 Semi-annual Duration 5.65 Annual Duration

### Interest Rate Immunization with Swaps

We now turn to an application of swaps to manage the duration of ongoing business operations. Consider FSF, a financial services firm with the stylized balance sheet shown in table 22.4. All dollar values are market values. Most assets are held as amortizing loans with a ten-year average maturity. On the liability side, FSF relies largely on money market obligations and an FRN issue with a five-year maturity. FSF has also issued a coupon bond that has ten years remaining until maturity.

Table 22.4 Balance sheet - representing "market values"

 Assets Duration Balance Liabilities Duration Balance Cash 0.0 \$7,000,000 Six-month money market (6% yield) 0.5 \$75,000,000 Marketable securities (6-mo. Maturity: 7% yield) 0.5 18,000,000 FRN (5 yr maturity semiannual, 7.3% yield) 0.5 \$40,000,000 Amortizing loan, 10 yr avg. maturity, 8% 4.60 130,467,133 Coupon bond, 10 yr. maturity, 6.5% coupon, par \$25 mill, YTM = 7% 7.45 24,111,725 Total Liabilities \$139,111,725 Equity \$16,355,408 Assets \$155,467,133 Liabilities & Net Worth \$155,467,133

### The Interest Risk of a Company (FSF)

The interest rate risk of FSF can be analyzed by using the concept of duration. Most of the balance sheet items have a duration that is evident and requires no computation. Cash has a duration of zero years. The marketable securities and the six-month money market obligations are zero-coupon instruments that mature in six months, so their duration is one half-year. The duration of the FRN is also one half-year, because it has annual payments. This leaves the amortizing loan on the asset side and the coupon bond on the liability side with durations that require computation. Table 22.5 present detailed computations of the durations for the amortizing loan (asset item C) and the coupon bond (liability item F) from the balance sheet for FSF. The amortizing loan throw off \$9.6 million per semiannual period for a market value of \$130,467,133. The duration of these loans is 9.209125 semiannual periods, or 4.604562 years. The coupon bond has a market value of \$24,111,725 and a duration of 14.906737 semiannual periods, or 7.453369 years.

Table 22.5 Duration calculation for FSF balance sheet items

 Amortizing Semi-annual period Cash flow Discount factor Present Value Weighted PV Coupon Bond Semi-annual period Cash flow Discount factor Present Value Weighted PV 1 9,600,000 1.0400 9,230,769 0.07075 1 812,500 1.0350 785,024 0.0326 2 9,600,000 1.0816 8,875,740 0.13606 2 812,500 1.0712 758,477 0.0629 3 9,600,000 1.1249 8,534,365 0.19624 3 812,500 1.1087 732,828 0.0912 4 9,600,000 1.1699 8,206,120 0.25159 4 812,500 1.1475 708,047 0.1175 5 9,600,000 1.2167 7,890,500 0.30239 5 812,500 1.1877 684,103 0.1419 6 9,600,000 1.2653 7,587,019 0.34892 6 812,500 1.2293 660,969 0.1645 7 9,600,000 1.3159 7,295,211 0.39141 7 812,500 1.2723 638,618 0.1854 8 9,600,000 1.3686 7,014,626 0.43012 8 812,500 1.3168 617,022 0.2047 9 9,600,000 1.4233 6,744,833 0.46528 9 812,500 1.3629 596,156 0.2225 10 9,600,000 1.4802 6,485,416 0.49709 10 812,500 1.4106 575,997 0.2389 11 9,600,000 1.5395 6,235,977 0.52577 11 812,500 1.4600 556,518 0.2539 12 9,600,000 1.6010 5,996,132 0.55151 12 812,500 1.5111 537,699 0.2676 13 9,600,000 1.6651 5,765,511 0.57449 13 812,500 1.5640 519,516 0.2801 14 9,600,000 1.7317 5,543,761 0.59488 14 812,500 1.6187 501,948 0.2914 15 9,600,000 1.8009 5,330,539 0.61286 15 812,500 1.6753 484,974 0.3017 16 9,600,000 1.8730 5,125,518 0.62857 16 812,500 1.7340 468,574 0.3109 17 9,600,000 1.9479 4,928,383 0.64217 17 812,500 1.7947 452,728 0.3192 18 9,600,000 2.0258 4,738,830 0.65380 18 812,500 1.8575 437,418 0.3265 19 9,600,000 2.1068 4,556,567 0.66358 19 812,500 1.9225 422,626 0.333 20 9,600,000 2.1911 4,381,315 0.67164 20 25,812,500 1.9898 12,972,482 10.76 Duration - semiannual 130,467,133 9.21 Duration - semiannual 24,111,725 14.91 Duration - years 4.60 Duration - years

Table 22.6 summarizes the market value and durations of the balance sheet items. Based on the information, we can compute the duration of the assets and liabilities. The duration are weighted averages of the duration of the individual items, weighted by the fraction of assets or liabilities. Letting DA represent the duration of the assets and DL the duration of the liabilities, we have:

DA = \$7,000,000/\$155,457,133 x 0.0 + \$18,000,000/\$155,457,133 x 0.50 + \$130,467,133/\$155,467,133 x 4.604562 = 3.922013

DL = \$75,000,000/\$139,111,725 x0.50 + \$40,000,000/\$139,111,725 x0.50 + \$24,111,725/\$139,111,725 x0.50 = 1.705202

### Hedging the Asset and Liability Portfolios Individually [viewable here in Excel]

Armed with this duration analysis of its balance sheet, FSF can use the swaps market to protect itself against unanticipated changes in interest rates. We first consider how the asset and liability portfolios can be protected against changing interest rates individually. In the next section, we show how to synthesize the entire analysis into an integrated solution. For both the asset and liability portfolio, we use the swap of the preceding section as a hedging instrument. That swap had a seven-year tenor, semiannual payments, and an SFR of 7 percent. As discussed in the preceding section, and computed in Table 22.3, the duration of the fixed side of the swap is 5.651369 years. With semiannual payments, the duration of the floating side is one half-year. So the duration of the received-fixed position in the swap is 5.151369, and the duration of the pay-fixed position in the swap is just the negative, or -5.151369 years.

In general, the solution for hedging an existing asset or portfolio, X, with hedging vehicle, H, using the duration approach is given by:

DX x MVX + DH x MV*H = 0

where DX is the duration of the position to be hedged, DH is the duration of the hedging instrument, MVX is the market value of the position to be hedged, and MV*H is the market value (or notional principal) of the hedging vehicle. The market value of a swap, in the sense of Equation 22.2 equals the notional principal. To completely immunize a position from changes in value due to changes in interest rates, the desired duration should be zero. Given the existing position and the choice of a hedging instrument, the problem is to find the amount of the hedging instrument, MV*H, that satisfies DX x MVX + DH x MV*H = 0.

Therefore, FSF can protect the market value of the asset and liability sides of the balance sheet by combining each with the interest rate swap of the preceding section. For the asset portfolio, the duration is 3.922013 years, so we need a pay-fixed swap with its duration of 25.151369. We apply DX x MVX + DH x MV*H = 0 to find the correct notional principal for the swap, MV*H, and have the following:

DH3.922013 x \$155,467,133 - 5.151369 x MV*H = 0

=(3.922013*155467133)/5.151369 = \$118,365,451

MV*H for the asset = \$118,364,451 in a pay-fixed swap. Therefore, we can hedge the asset side of the FSF balance sheet by entering a pay-fixed swap with a notional principal of \$118,365,451.

Similarly, we can use the same swap to hedge the liability side of the balance sheet. The total liabilities are \$139,111,725. Because these are liabilities, FSF has a short position in these instruments. Therefore, considering the liabilities on their own, FSF will need a receive-fixed swap to complete the hedge. Again, applying DX x MVX + DH x MV*H = 0 to the liabilities, we have the following:

1.705202 x (-\$130,111,725) + 5.151369 x MV*H = 0

=(1.705202*-139111725)/5.151369 = \$-46,048,651

MV*H for the liabilities is \$46,048,651 in a received-fixed swap.

### Duration Gap Hedging [viewable here in Excel]

In the above section, we showed how to eliminate the interest rate risk in the asset and liability portfolio separately. As we saw, the assets required a pay-fixed swap with a notional principal of \$118,365,451, and the liability portfolio required a receive-fixed swap with a notional principal of \$46,048,651. These two swaps are partially offsetting. Combined, the two swaps really equal a pay-fixed swap with a notional principal of \$118,365,451 - \$46,048,651 = \$72,316,800. This section shows how to reach this same solution by using an integrated approach to the entire risk position of FSF.

As we have noted for FSF, the value of the assets is \$155,467,133 with a duration of 3.922013 years, while the value of the liabilities is \$139,111,725 with a duration of 1.705202 years. The difference in durations is 2.2869999 years, but the assets exceed the liabilities by the net worth of the firm. We need an integrated measure of the duration difference, or duration gap, between the assets and liabilities that reflects the difference in market value between the assets and liabilities. This measure is called the duration gap, DG , which is defined as follows:

DG = DA - total liabilities/total assets x DL

The ratio of total liabilities to total assets acts as a scale factor to reflect the difference in market value between the assets and liabilities For FSF, the duration gap is as follows:

DG = 3.922013 - \$139,111,725/\$155,465,133 x 1.705202 = 2.396201

The duration gap is greater than the difference in durations because the market value of the assets exceeds the market value of the liabilities. Because the duration gap embraces both the assets and liabilities and reflects the difference in market value between the two, it summarizes the entire risk position of the firm. Because the duration gap of FSF is 2.396201, the entire firm has an interest rate risk that behave like a long position in a bond with a duration of 2.396201 years.

To hedge the entire value of the firm, FSF could use a pay-fixed swap to set the duration gap of the entire firm, including the swap, so that it equals zero. Using the duration gap that we just computed and our sample pay-fixed swap, we have:

2.396201 x \$155,467,133 - 5.1513699 x MV*H = 0

=(2.396201*155467133)/5.151369 = \$72,316,796

MV*H for hedging the entire firm is \$72,316,800 in a pay-fixed swap. This is the same result that we found by hedging the asset portfolio with a pay-fixed swap and the liability portfolio with a receive-fixed swap and noting the offsetting positions that were created.

### Setting Interest Rate Sensitivity

So far, we have seen how to immunize FSF against changing interest rates which amounted to setting the duration gap to zero. Let us now assume that FSF wants to reduce, but not eliminate, the interest rate risk inherent in the firm's operations. FSF management decides to make the firm behave like a bond with a duration of one year, instead of behaving like a bond with a duration equal to the firm's duration gap of 2.396201.

In general, we can use swaps to set the duration gap of the firm to any desired level as follows:

D*G = DG + DS (MV*H / total assets)

where D*G is the desired duration gap, DS is the duration of the swap, and MV*H is the required market value (notional principal) for the swap. To set the duration gap of the firm's interest sensitive assets to one year, the required solution is as follows:

D*G = DG + DS (MV*H / total assets)

= 2.396201 + 5.151369 (MV*H / \$155,467,133) MV*H =((1 - 2.396201)*155467133)/5.151369

=((1 - 2.396201)*155467133)/5.151369 = -\$42,137,025

The negative sign on MV*H indicates that a pay-fixed swap is required. (If one were sure that a pay-fixed swap would be required, the sign for DS could be shown as a negative, reflecting the duration of the pay-fixed position.)

This result makes intuitive sense, given what we have already seen. The duration gap of the firm was 2.396201 initially, and this position combined with a pay-fixed swap having a notional principal of \$72,316,000 moved the duration gap of the firm to zero. Changing the duration gap from 2.267588 to 1.0 alters it by 58.2673 percent (1.396201/2.396201). Not surprisingly, the necessary swap position is 58.2672 percent as large (\$42,137,025/\$72,316,800) as the swap necessary to move the duration gap to zero.

With a duration gap greater than zero, the firm's value is exposed to the danger of rising interest rates. If the firm expects rates to rise and wishes to SPECULATE on that eventuality, it might wish to set its duration gap to less than zero. For instance, a modest speculative position could be achieved with a duration gap of -0.5 years. The swap position to achieve this exposure would be follows:

D*G = -0.5 = DG + DS (MV*H / total assets)

= 2.396201 + 5.151369 (MV*H / \$155,467,133) MV*H =((-0.5 - 2.396201)*155467133)/5.151369

=((-0.5 - 2.396201)*155467133)/5.151369 = -\$87,406,681 (pay-fixed swap is required)

So, a pay-fixed swap with a notional principal of \$87,406,681 is required to change the duration gap to 20.5 years. This procedure moves the duration gap of the firm from its original position value of 2.396201, indicating an exposure to rising interest rates, beyond the risk-neutral duration gap of zero, to a negative duration gap of 20.5. Now, if rates rise, the firm will benefit. However, it is exposed to losses if rates fall.

Finally, we consider how swaps could be used to take an extreme speculative position for FSF. A duration of ten years world be an extremely large duration for any bond in the market, indicating a large sensitivity to rising interest rates. FSF could use a swap to create this position with startling ease. Applying the same formula, the solution would be as follows:

D*G = 10 = DG + DS (MV*H / total assets)

= 2.396201 + 5.151369 (MV*H / \$155,467,133)

MV*H =((10 - 2.396201)*155467133)/5.151369

=((10 - 2.396201)*155467133)/5.151369 = \$229,480,907 (Receive-fixed swap is required)

To change the duration gap for the firm from 2.396201 to ten years, the firm would enter a receive-fixed swap with a notional principal of \$229,480,907. We have seen that a pay-fixed swap of modest proportions reduces the interest risk of FSF. A receive-fixed swap increases the duration gap for FSF.

These examples emphasis the power and flexibility of swaps for changing the interest rate exposure of a firm. The risk of extreme positions should not be neglected either. A formula for the approximate price change of an interest-sensitive asset is as follows:

ΔP ~ -D*(ΔAYTM/1 + AYTM) * P

where D is expressed in years, ATYM is an annual yield-to-maturity on the asset, and P is the current price of the asset. If FSF were to set its duration gap to 10 years, its net worth would be extremely vulnerable to interest rate risk. Taking the average rate on all of FSF's positions at 7 percent for convenience, we see that an interest rate increase of only ten basis points would cost FSF almost \$152,854, which is about 1 percent of its net worth of \$16,355,408.

ΔP ~ -D*(ΔAYTM/1 + AYTM) * P

= -10 x +0.0010/1.07 x \$16,355,408 = -\$152,854

= -10 *(0.001/1.07)*16355408 = \$-152,854.28

A major rise in rates, say an increase of five percentage points, would cost FSF \$7,642,714, or
about 46.7% of its net worth, if FSF's duration gap were set to ten years.

ΔP ~ -D*(ΔAYTM/1 + AYTM) * P

= -10 x +0.05/1.07 x \$16,355,408 = -\$152,854

= -10 *(0.05/1.07)*16355408 = \$-7,642,714.02

### Swaps, like other powerful tools, require caution in their use.

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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