The DV01 of a swap two ways, by the annuity shortcut and by bumping and repricing a bootstrapped curve, with the tenor table and the hedge sizing that follows from it.
The DV01 of an interest rate swap is the change in its value when rates move by one basis point. For a swap at the current market rate it equals notional times the annuity, the sum of the discount factors over the payment dates, times 0.0001. On an illustrative 100,000,000 five-year par swap at 4.75 per cent, that is 43,693 per basis point, and bumping and repricing the whole curve gives the same figure. On an older swap paying 3.00 per cent the true DV01 is 41,567, 4.87 per cent below the shortcut.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A company pays fixed and receives floating on 100,000,000, annual payments, single curve for projection and discounting. The par swap rates are illustrative, and the discount factors are bootstrapped from them in the usual way: each par swap is worth zero, so each rate fixes one more factor.
| Year | Par rate | Discount factor |
|---|---|---|
| 1 | 4.40% | 0.957854 |
| 2 | 4.55% | 0.914794 |
| 3 | 4.65% | 0.872357 |
| 4 | 4.70% | 0.831886 |
| 5 | 4.75% | 0.792456 |
| 7 | 4.85% | 0.716751 |
| 10 | 4.95% | 0.614666 |
The years in between are bootstrapped too, from 4.80, 4.88 and 4.91 per cent at six, eight and nine years; they matter for the seven- and ten-year rows further down.
A swap at today's par rate is worth zero. Move the par rate by one basis point and the swap is worth one basis point a year, on the notional, for each remaining year, discounted: notional times annuity times 0.0001.
Annuity = DF1 + … + DF5 = 0.957854 + 0.914794 + 0.872357 + 0.831886 + 0.792456 = 4.369348
DV01 ≈ 100,000,000 × 4.369348 × 0.0001 = 43,693
Excel: =SUM(C2:C6)*100000000*0.0001
The pay-fixed side gains about 43,693 when rates rise one basis point. That is 437 per million of notional, and it is almost exactly the DV01 of a five-year bond with a 4.75 per cent coupon trading at par: a fixed payer is short a fixed-rate bond and long a floating-rate note, and the note, which resets to par, has almost no rate sensitivity.
The robust method does not rely on the swap being at par. Shift every par rate up one basis point, bootstrap again, revalue; repeat with a downward shift; DV01 is half the difference.
DV01 = (V+1bp − V−1bp) / 2 = (43,681.33 − (−43,705.63)) / 2 = 43,693
V = N × (1 − DF5) − N × K × Σ DFi, recomputed on each bumped curve
For a par swap the two methods agree to the unit. The difference shows when the swap is off-market.
Suppose the same company also holds a five-year swap struck years ago at 3.00 per cent. It is worth 7,646,359 to the fixed payer, which is (4.75% − 3.00%) × 4.369348 × 100,000,000. Its DV01 by bump and reprice is 41,567, not 43,693. The shortcut is 2,126 too high: 4.87 per cent of the shortcut, or 5.11 per cent of the true figure.
The reason is in the formula: the swap's value is the rate gap times the annuity. When rates rise, the gap widens but the annuity shrinks, because every discount factor falls. On a par swap the gap is zero and the second effect vanishes; on a swap 175 basis points in the money it takes away almost 5 per cent of the sensitivity. A hedge book with old swaps sized on the annuity shortcut is overhedged.
| Tenor | Par rate | Annuity | DV01 | Per million |
|---|---|---|---|---|
| 2 years | 4.55% | 1.8726 | 18,726 | 187 |
| 3 years | 4.65% | 2.7450 | 27,450 | 275 |
| 5 years | 4.75% | 4.3693 | 43,693 | 437 |
| 7 years | 4.85% | 5.8402 | 58,402 | 584 |
| 10 years | 4.95% | 7.7845 | 77,845 | 778 |
DV01 grows with tenor but less than in proportion, because later years are discounted more. The table is how a hedge is sized. A company with a 60,000,000 seven-year floating loan has a DV01 of 35,041 to hedge. If only five-year swaps are available, 35,041 divided by 437 per million gives 80,197,439 of notional, 1.337 times the loan. That hedge matches a parallel move. If five-year rates rise and seven-year rates do not, it over-hedges; DV01 by bucket, not in total, is what shows that risk.
DV01 is also a linear measure. A move of 100 basis points up changes the par swap's value by 4,250,472, and a move of 100 down by −4,493,563, against 4,369,348 predicted by DV01 times 100. For the fixed payer convexity works against the position: negligible at one basis point, it costs 118,877 on the way up and 124,214 on the way down at a hundred.
How a swap's value, rather than its sensitivity, moves after rates have changed is worked in how to value an interest rate swap, on the same first four years of this curve.
Both of the book's curves are bootstrapped from their ten par rates in the free workbook for this case.
Usage varies. Many desks use PV01 for the value of one basis point on the fixed rate, notional times the annuity times 0.0001, and DV01 for the value change when the whole curve moves one basis point. For a par swap they coincide: 43,693 on 100,000,000 for five years here. For an off-market swap they differ: 41,567 against 43,693 on a swap paying 3.00 per cent.
Sum the discount factors over the swap's payment dates to get the annuity, then multiply by notional and 0.0001: =SUM(DF1:DF5)*100000000*0.0001 returns 43,693 here. For an exact figure, bootstrap the curve twice with every par rate moved up and down one basis point, revalue the swap on each, and take half the difference.
Divide the DV01 to be hedged by the DV01 of one unit of the hedging swap. A 60,000,000 seven-year exposure has a DV01 of 35,041; a five-year swap carries 437 per million, so 80,197,439 of five-year notional matches it. The match holds for parallel moves only, not for a steepening of the curve between five and seven years.
This article is one calculation from Derivatives. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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