Duration Swaps
Below are links to the following topics:
 Duration Swaps
 Interest Rate Risk for a Company
 Hedging the Asset and Liability Portfolios Individually
 Duration Gap Hedging
Duration Swap [viewable here in Excel]
A receivedfixed swap consists of a short position in a floatingrate instrument combined with a long position in a fixedrate coupon bond. Conversely, a payfixed swap consists of a short position in a coupon bond, coupled with a long position in a floatingrate 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 receivedfixed or a payfixed swap. Based on these reflections, we can state the following rules:
 Duration of a receivefixed swap = duration of the underlying coupon bond  duration of the underlying floatingrate bond > 0
 Duration of a payfixed swap = duration of the underlying floatingrate bond  duration of the underlying coupon bond <0
The duration of a floatingrate 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 halfyear. The calculations of the duration of the swap, then, really depends on finding the duration of the fixedrate coupon bond underlying the swap. As an example, consider an interest rate swap with an swapfixedrate (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 halfyear. Therefore, for this swap,
Duration of a receivedfixed swap = 5.651369  0.5 = 5.151369 years duration of a payfixed swap = 0.5  5.651369 = 5.151369 years
Table 22.3 Calculation of duration for the fixedrate side of an interest rate swap
Semiannual 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  Semiannual 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 tenyear average maturity. On the liability side, FSF relies largely on money market obligations and an FRN issue with a fiveyear 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 

Sixmonth money market (6% yield)  0.5  $75,000,000 
Marketable securities (6mo. 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 
Assets
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 sixmonth money market obligations are zerocoupon instruments that mature in six months, so their duration is one halfyear. The duration of the FRN is also one halfyear, 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 Semiannual period  Cash flow  Discount factor  Present Value  Weighted PV  Coupon Bond Semiannual 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:
D_{A} = $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
D_{L} = $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 sevenyear 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 halfyear. So the duration of the receivedfixed position in the swap is 5.151369, and the duration of the payfixed 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:
D_{X} x MV_{X} + D_{H} x MV*_{H} = 0
where D_{X} is the duration of the position to be hedged, D_{H} is the duration of the hedging instrument, MV_{X} 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 D_{X} x MV_{X} + D_{H} 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 payfixed swap with its duration of 25.151369. We apply D_{X} x MV_{X} + D_{H} x MV_{*H} = 0 to find the correct notional principal for the swap, MV_{*H}, and have the following:
D_{H}3.922013 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 payfixed swap. Therefore, we can hedge the asset side of the FSF balance sheet by entering a payfixed 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 receivefixed swap to complete the hedge. Again, applying D_{X} x MV_{X} + D_{H} 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 receivedfixed 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 payfixed swap with a notional principal of $118,365,451, and the liability portfolio required a receivefixed swap with a notional principal of $46,048,651. These two swaps are partially offsetting. Combined, the two swaps really equal a payfixed 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:
D_{G} = D_{A}  total liabilities/total assets x D_{L}
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:
D_{G} = 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 payfixed 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 payfixed 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 payfixed swap. This is the same result that we found by hedging the asset portfolio with a payfixed swap and the liability portfolio with a receivefixed 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} = D_{G} + D_{S} (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} = D_{G} + D_{S} (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 payfixed swap is required. (If one were sure that a payfixed swap would be required, the sign for DS could be shown as a negative, reflecting the duration of the payfixed 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 payfixed 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 = D_{G} + D_{S} (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 (payfixed swap is required)
So, a payfixed 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 riskneutral 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 = D_{G} + D_{S} (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 (Receivefixed swap is required)
To change the duration gap for the firm from 2.396201 to ten years, the firm would enter a receivefixed swap with a notional principal of $229,480,907. We have seen that a payfixed swap of modest proportions reduces the interest risk of FSF. A receivefixed 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 interestsensitive asset is as follows:
ΔP ~ D*(ΔAYTM/1 + AYTM) * P
where D is expressed in years, ATYM is an annual yieldtomaturity 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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