The yield is priced, the coupon is rounded, and the reoffer price is whatever reconciles the two, which is why the coupon tells you nothing about the cost.
A new fixed-rate bond is priced as a yield, not a coupon. The reoffer yield is the mid-swap rate plus the reoffer spread agreed with investors; the coupon is that yield rounded down to the nearest eighth of a point; and the reoffer price is whatever makes the two consistent. For an illustrative EUR 500,000,000 seven-year bond at mid-swaps of 2.734 per cent plus 118 basis points, the yield is 3.914 per cent, the coupon 3.875 per cent and the reoffer price 99.765.
Worked in full in Capital Markets by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Investors bid in spread over mid-swaps, the order book tightens it, and the final spread is the number that gets announced. Everything after that is arithmetic done in a few minutes before the bond is launched, and it is worth being able to check.
| Input | Value |
|---|---|
| Size | EUR 500,000,000 |
| Tenor, annual coupons | 7 years |
| Seven-year mid-swap rate at pricing | 2.734% |
| Final reoffer spread | 118 bp |
| Coupon step | 0.125% |
| Underwriting fee | 0.25% |
Settlement is assumed on a coupon anniversary, so there is no accrued interest or broken first period to complicate the price.
Step 1: the reoffer yield. Add the spread to the swap rate fixed at pricing: 2.734 per cent plus 1.18 per cent is 3.914 per cent. This is the number investors bought.
Step 2: the coupon. Round the yield down to the nearest eighth: 3.914 per cent becomes 3.875 per cent. The coupon is a convenience for the bond's documentation, set just below the yield so the bond prices at or below par. Some issuers use finer steps, such as 0.05 per cent, which would give 3.90 per cent here; the logic is the same. The book's own case follows the eighth: the Ardennes Industries seven-year bond, priced at 3.10 per cent mid-swaps plus 122 basis points, yields 4.32 per cent, carries a 4.25 per cent coupon and prices at 99.585, the starting point of the new issue concession calculation.
Step 3: the reoffer price. Discount the coupons and the redemption at the reoffer yield.
Price = Σt=1..7 3.875 ÷ (1.03914)t + 100 ÷ (1.03914)7 = 99.765
In Excel: =-PV(3.914%,7,3.875,100), or with dates =PRICE(Settle,Maturity,3.875%,3.914%,100,1).
| Line | Value |
|---|---|
| Reoffer yield | 3.914% |
| Coupon | 3.875% |
| Reoffer price | 99.765 |
| Cash raised at the reoffer price | EUR 498,825,871 |
| Less underwriting fee at 0.25% | EUR 1,250,000 |
| Net proceeds (net price 99.515) | EUR 497,575,871 |
| Annual coupon payment | EUR 19,375,000 |
| Issuer's yield after the fee | 3.956% |
The price sits 0.235 points below par, EUR 1,174,129 on the size, because the coupon is 3.9 basis points below the yield for seven years. That discount is not a cost on top of the yield; it is the yield being delivered partly through the price. The fee is a cost on top, and at 0.25 per cent it adds 4.2 basis points to what the issuer pays. The other costs of the issue, and how to annualise them properly, are in how to calculate the all-in cost of a bond issue.
What a basis point is worth. At this yield one basis point moves the price by 0.0601 points, a modified duration of 6.03. On EUR 500,000,000 that is EUR 300,624 of proceeds per basis point of final spread. Every basis point given up in the last hour of the book is a number of that size.
The figures are fixed in a set order on pricing day: spread first, then the swap rate at the moment of pricing, then the coupon and price, all within minutes.
| Reoffer spread | Reoffer yield | Coupon | Reoffer price |
|---|---|---|---|
| 110 bp | 3.834% | 3.750% | 99.493 |
| 115 bp | 3.884% | 3.875% | 99.946 |
| 118 bp | 3.914% | 3.875% | 99.765 |
| 122 bp | 3.954% | 3.875% | 99.525 |
| 125 bp | 3.984% | 3.875% | 99.345 |
The coupon moves in steps and the price absorbs everything in between. Tightening from 118 to 115 basis points leaves the coupon at 3.875 per cent and lifts the price to 99.946; tightening to 110 drops the coupon a full eighth to 3.750 per cent and the price falls back to 99.493. A lower price is not a worse deal for the issuer: the 110 basis point bond is the cheapest funding in the table.
| Mid-swap | Reoffer yield | Coupon | Reoffer price |
|---|---|---|---|
| 2.500% | 3.680% | 3.625% | 99.666 |
| 2.734% | 3.914% | 3.875% | 99.765 |
| 3.000% | 4.180% | 4.125% | 99.672 |
Price the yield, round the coupon down to the eighth, and solve the price: here 3.914 per cent, 3.875 per cent and 99.765. Judge the deal by the yield and the fee, never by the coupon, and know that each basis point of spread is worth about EUR 300,624 on this bond. The free workbook for this book carries the pricing, the bond's full bill and the concession under four readings of the curve.
Convention in the euro market is a coupon in steps of an eighth of a point, set at or just below the reoffer yield so the bond is issued at or slightly below par. At a 3.914 per cent yield the coupon is 3.875 per cent and the price 99.765. Rounding up to 4.000 per cent would price the bond above par at 100.518, with the same 3.914 per cent yield.
No. The cost is the yield, not the coupon. A 3.875 per cent coupon at 99.765 and a 4.000 per cent coupon at 100.518 both yield 3.914 per cent. The lower coupon saves 625,000 a year of interest but raises less cash at issue; the two effects cancel at the yield. Fees then add about 4.2 basis points on this bond.
Roughly the price value of a basis point times the size. On a seven-year bond yielding 3.914 per cent with a 3.875 per cent coupon, one basis point moves the price by 0.0601 points, so on EUR 500,000,000 it is worth about EUR 300,624 of proceeds. That is the figure behind every negotiation over where the final spread lands.
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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