One basis point turned into euros on a seven-year new issue, three ways, and why the years-times-notional shortcut overstates it by a sixth.
DV01 is the change in a bond's price for a one basis point move in its yield: reprice the bond one basis point higher and one lower, halve the difference, and scale it to the size. On an illustrative seven-year EUR 500 million issue with a 3.875 per cent coupon priced at 99.765, the DV01 is 0.0601 per 100 of nominal, or EUR 300,624 per basis point. That is the number that turns every basis point of spread argued over at pricing into euros.
Worked in full in Capital Markets by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Investors use DV01 to size hedges and risk limits. Issuers need it for a different reason: the negotiation on a new bond is conducted in basis points, and a basis point on its own means nothing until it is multiplied by the right factor. The usual shortcut, one basis point times the years times the notional, overstates it by about a sixth on this bond.
| Input | Value |
|---|---|
| Nominal issued | EUR 500,000,000 |
| Maturity | 7 years |
| Coupon, annual | 3.875% |
| Reoffer yield | 3.914% |
| Reoffer price | 99.765 |
The coupon sits slightly below the reoffer yield, which is why the price is a little under par: the coupon is set by rounding the yield down to the nearest eighth and the price absorbs the difference, as worked in how a new bond's coupon and reoffer price are set. Settlement falls on a coupon date, so there is no accrued interest to strip out.
Step 1. Price the bond at its yield
P = Σ 3.875 / 1.03914t + 100 / 1.039147 = 99.7652. In Excel: =PRICE(settle, maturity, 3.875%, 3.914%, 100, 1).
Step 2. Bump the yield both ways
At 3.924 per cent the price is 99.7051; at 3.904 per cent it is 99.8253.
Step 3. Take half the difference
DV01 per 100 = (99.8253 − 99.7051) / 2 = 0.06012. A central difference cancels most of the curvature that a one-sided bump would leave in.
Step 4. Scale to the issue
0.06012 / 100 × 500,000,000 = EUR 300,624, or EUR 601 per million.
The same figure comes out of modified duration. Macaulay duration on this bond is 6.2625 years; dividing by 1.03914 gives a modified duration of 6.0266. DV01 is modified duration times price times one basis point: 6.0266 × 99.7652 / 10,000 = 0.0601 per 100, and EUR 300,624 on the issue. In Excel, =MDURATION(settle, maturity, 3.875%, 3.914%, 1)*B5/10000 with the price in B5. How duration itself is built, and how convexity corrects it on larger moves, is in how to calculate the modified duration of a bond.
From the issuer's side the question is slightly different: what does it cost to pay one more basis point of yield for seven years? The answer is 0.01 per 100 a year, discounted at the yield, which is the annuity factor of 6.0212 times a basis point times the nominal: EUR 301,059. It differs from the investor's DV01 by EUR 435 because one discounts a running cost and the other measures a price; at pricing they are the same number for every practical purpose.
| Method | Per basis point |
|---|---|
| DV01, central bump | 300,624 |
| Annuity factor × 1 bp × nominal | 301,059 |
| Macaulay duration instead of modified | 312,391 |
| Shortcut: 1 bp × 7 years × nominal | 350,000 |
Applied to a new issue concession of 4 basis points, the DV01 says the issuer gave away EUR 1,202,497 against fair value, about EUR 1.20 million that appears on no invoice. The method for measuring the concession itself is set out in how to calculate a new issue concession.
DV01 is a first-order measure. On a 25 basis point rise the bond loses EUR 7,446,519 and on a 25 basis point fall it gains EUR 7,585,676, against EUR 7,515,606 for 25 times the DV01. The gap is convexity, and on moves of a few basis points it can be ignored.
| Maturity, years | Price | Modified duration | DV01 per 100 | DV01, EUR | Shortcut, EUR |
|---|---|---|---|---|---|
| 3 | 99.892 | 2.78 | 0.0278 | 138,878 | 150,000 |
| 5 | 99.826 | 4.47 | 0.0446 | 222,885 | 250,000 |
| 7 | 99.765 | 6.03 | 0.0601 | 300,624 | 350,000 |
| 10 | 99.682 | 8.16 | 0.0813 | 406,494 | 500,000 |
| 15 | 99.564 | 11.20 | 0.1115 | 557,748 | 750,000 |
DV01 grows with maturity but more slowly than the years, because distant coupons are discounted harder. At fifteen years the shortcut overstates a basis point by EUR 192,252, a third of the true figure. That matters when an issuer compares tenors: a basis point of concession on a fifteen-year deal costs roughly four times what it costs on a three-year deal, not five.
Multiplying the basis point by the years and the nominal. On this bond the shortcut gives EUR 350,000 against EUR 300,624, an overstatement of EUR 49,376, or 16.4 per cent, because it treats a euro of yield paid in year seven as worth a euro today. The opposite error is subtler: using Macaulay duration where modified belongs inflates DV01 by EUR 11,766 here, the factor of 1.03914 that converts one into the other. Neither error looks large on a single basis point. Both compound once they are multiplied by a concession, a fee comparison or a hedge ratio. For the full cost of the bond with fees as well as spread, see how to calculate the all-in cost of a bond issue.
Bump the yield one basis point each way, halve the price difference and scale it: on a seven-year EUR 500 million bond at 99.765, one basis point is worth EUR 300,624. Put that figure on the first page of the pricing note, before the spread discussion starts, and every basis point conceded or saved becomes a sum of money rather than a rounding. The free workbook for this case carries the interpolation, the concession and the full bill of a new bond with every figure live.
Modified duration is the percentage price change for a 100 basis point yield move; DV01 is the money change for one basis point. DV01 equals modified duration times price times 0.0001. On the seven-year bond, a modified duration of 6.0266 at a price of 99.7652 gives 0.0601 per 100, EUR 300,624 on EUR 500 million.
Close but not identical. DV01 moves the bond's yield; PV01, as issuers often use it, is the present value of one basis point paid every year. On the seven-year bond PV01 is the annuity factor of 6.0212 times a basis point times the nominal, EUR 301,059, against a DV01 of EUR 300,624. At pricing the two are interchangeable.
Multiply the concession by the DV01. On a seven-year EUR 500 million bond with a DV01 of EUR 300,624, 4 basis points is EUR 1,202,497, about EUR 1.20 million transferred to investors. On a fifteen-year bond of the same size the DV01 is EUR 557,748, so the same concession costs 4.02 times what it would on a three-year deal.
This article is one calculation from Capital Markets. 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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