No existing bond ever sits at the new maturity, so the concession is a measurement made with a chosen ruler, and four common rulers disagree by five basis points.
The new issue concession is the reoffer spread minus the fair value of a new bond at the same maturity, read off the issuer's own secondary curve. Ardennes Industries, the issuer in Capital Markets, priced a seven-year euro bond at 122 basis points over mid-swaps; interpolating linearly between its 2030 and 2036 bonds gives a fair value of 117.81, so the concession is 4.19 basis points, worth EUR 1,241,953 in present value on EUR 500 million. A different, equally defensible interpolation gives minus 0.78.
Worked in full in Capital Markets by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The concession is the price an issuer pays to clear a new deal in size: investors are asked to buy something new, in one go, and they want a little more spread than the existing bonds offer. It is reported in every syndicate's pricing summary. It is also the one line of the cost of a bond that depends entirely on a modelling choice, because no existing bond ever sits at exactly the new maturity.
| Bond | Maturity, years | Spread over mid-swaps, bp |
|---|---|---|
| 2.500% March 2028 | 1.58 | 62 |
| 3.250% September 2030 | 4.12 | 88 |
| 1.875% January 2036 | 9.24 | 141 |
| New bond, 4.25% coupon | 7 | 122 (reoffer) |
The seven-year mid-swap rate is 3.10 per cent, so the reoffer yield is 4.32 per cent and the bond prices at 99.585. The size is EUR 500,000,000.
Fair value = S1 + (T − M1) ÷ (M2 − M1) × (S2 − S1), using the two bonds that bracket the target maturity T
Concession = reoffer spread − fair value
EUR cost = concession × size × 0.0001 × annuity factor at the reoffer yield
In Excel: =B3+(7-A3)/(A4-A3)*(B4-B3) for the fair value and =Size*0.0001*(1-(1+y)^-7)/y for the value of one basis point.
Step 1, bracket the maturity. Seven years lies between the 2030 bond at 4.12 years and the 2036 bond at 9.24. The new bond is 2.88 years past the shorter one, out of a gap of 5.12 years: a weight of 0.5625.
Step 2, interpolate. The spreads differ by 53 basis points, so fair value is 88 + 0.5625 × 53 = 117.8125.
Step 3, the concession. 122 − 117.8125 = 4.1875 basis points.
Step 4, put it in euros. The concession is extra coupon paid every year for seven years: 4.1875 basis points on EUR 500 million is 209,375 a year, 1,465,625 undiscounted. Discounted at the reoffer yield of 4.32 per cent, the annuity factor is 5.9317 and one basis point is worth EUR 296,586. The concession is 4.1875 × 296,586 = EUR 1,241,953; discounting takes 223,672 off the simple sum.
Linear interpolation in maturity is one choice among several. Credit curves are usually concave, steep at the short end and flattening out, so practitioners also interpolate in the square root or the logarithm of maturity, or fit a line through all the bonds.
| Convention | Fair value, bp | Concession, bp | Present value, EUR |
|---|---|---|---|
| Linear in maturity | 117.81 | 4.19 | 1,241,953 |
| Least squares, three points | 117.84 | 4.16 | 1,232,589 |
| Linear in square root of maturity | 120.32 | 1.68 | 496,881 |
| Linear in log of maturity | 122.78 | −0.78 | −231,927 |
The four fair values span 4.97 basis points, worth EUR 1,473,879 at this size: 84 per cent of an illustrative 35 basis point underwriting commission of EUR 1,750,000. On the logarithmic reading the bond priced inside its own curve and the issuer paid no concession at all. Nothing about the market changed between the rows; only the line drawn between two bonds.
The least-squares fit is not an interpolation. Its slope is 10.3188 basis points a year and it passes through none of the three bonds. Here it lands almost on the linear answer; on a curve with one off-the-run bond it can land a long way from it.
Move the 2036 bond from 141 to 132 basis points, nine basis points tighter, with nothing else changed.
| Convention | Fair value, bp | Concession, bp |
|---|---|---|
| Linear in maturity | 112.75 | 9.25 |
| Least squares, three points | 112.30 | 9.70 |
| Linear in square root of maturity | 114.84 | 7.16 |
| Linear in log of maturity | 116.88 | 5.12 |
The logarithmic concession turns from minus 0.78 to plus 5.12. A single stale or illiquid quotation at the long end moves the concession by more than its whole reported level, which is why the quotations used, and their time stamps, belong in the pricing record. Size matters too: at EUR 1 billion one basis point is worth 593,171 and the span between conventions 2,947,759; at EUR 250 million they are 148,293 and 736,940.
The common mistake is to report the concession as an observation. It is a measurement made with a chosen ruler, and the syndicate that reports it also chose the ruler. On an upward-sloping curve, linear interpolation in maturity gives the lowest fair value of the three interpolations, and so the largest concession; the logarithm gives the smallest. An issuer shown a single reading cannot tell which end of that range it is looking at. Ask for all four readings, and convert each into euros at the annuity factor rather than quoting basis points.
Concession = reoffer spread − interpolated fair value, and its cost = concession × size × 0.0001 × the annuity factor. Compute it under more than one convention before the mandate is signed. The workbook in the free companion files for Capital Markets returns all four readings from any three quotations and a target maturity, and how much of a placement's cost is uninvoiced applies the same discipline to an equity raise.
It varies with market conditions and is always a matter of measurement. On the Ardennes seven-year bond, the illustrative case in Capital Markets, the concession is 4.19 basis points on linear interpolation and minus 0.78 on logarithmic interpolation of the same three quotations, a span worth EUR 1,473,879 on EUR 500 million.
Multiply the concession in basis points by the size, by 0.0001 and by the annuity factor at the reoffer yield over the bond's life. At 4.32 per cent over seven years the factor is 5.9317, so one basis point on EUR 500 million is worth EUR 296,586 and 4.19 basis points EUR 1,241,953.
None is uniquely right, so compute several. On an upward-sloping curve, linear in maturity gives the lowest interpolated fair value and therefore the largest concession: 117.81 against 120.32 in square root of maturity and 122.78 in logarithm on the Ardennes curve. State the method and the quotations with the figure.
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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