One new seven-year euro bond measured against swaps and against government bonds, with the interpolation, the swap spread that links them and why the choice of reference moves the number.
The I-spread is a bond's yield minus the swap rate interpolated to its maturity; the G-spread is the same yield minus the interpolated government bond yield. An illustrative seven-year euro bond yielding 3.914 per cent, with mid-swaps at 2.734 per cent and an interpolated Bund at 2.512 per cent, is 118.0 bp over swaps and 140.2 bp over Bunds. The 22.2 bp between them is the swap spread, and it belongs to the rates market, not to the issuer's credit.
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Both numbers describe the same price. The euro syndicate desk prices corporate and financial issuers over mid-swaps, so the I-spread is what appears on the term sheet. Rates investors, central bank purchase screens and almost every make-whole clause work over government bonds, so the G-spread is what they see. Confusing the two, or quoting one as if it were the other, is a 20 to 30 basis point error on an ordinary euro curve.
| Input | Value |
|---|---|
| Reoffer yield of the new bond | 3.914% |
| Seven-year mid-swap rate | 2.734% |
| Five-year Bund yield | 2.380% |
| Ten-year Bund yield | 2.710% |
| Nearest benchmark Bund: residual maturity, yield | 6.4 years, 2.470% |
| Bond size; DV01 of the issue | EUR 500,000,000; EUR 300,624 per bp |
Step 1: the I-spread. The swap curve quotes a rate at every annual maturity, so for a bond of exactly seven years no interpolation is needed. For a bond of 7.3 years you would interpolate between the seven and eight-year swaps in the same way as step 2.
I-spread = Yield − Mid-swap = (3.914% − 2.734%) × 100 = 118.0 bp
Step 2: interpolate the government curve. Government bonds exist only at the maturities issued, so a seven-year point has to be built from the five and ten-year Bunds. Linear in maturity, the seven-year point sits 0.4 of the way between them.
Weight = (7 − 5) ÷ (10 − 5) = 0.4
Bund7y = 2.380% + 0.4 × (2.710% − 2.380%) = 2.380% + 0.132% = 2.512%
Excel: =FORECAST(7,{2.380,2.710},{5,10})
Step 3: the G-spread and the identity that checks it.
G-spread = 3.914% − 2.512% = 140.2 bp
Swap spread = Mid-swap − Bund = 2.734% − 2.512% = 22.2 bp
G-spread = I-spread + swap spread = 118.0 + 22.2 = 140.2 bp
If your two figures do not reconcile through the swap spread at the same maturity, one of the interpolations is wrong.
| Reference | Reference yield | Spread, bp |
|---|---|---|
| Five-year Bund | 2.380% | 153.4 |
| Benchmark Bund, 6.4 years | 2.470% | 144.4 |
| Interpolated seven-year Bund, linear (G-spread) | 2.512% | 140.2 |
| Interpolated seven-year Bund, logarithmic | 2.540% | 137.4 |
| Ten-year Bund | 2.710% | 120.4 |
| Seven-year mid-swap (I-spread) | 2.734% | 118.0 |
The spread over the nearest benchmark is 4.2 bp wider than the G-spread only because that Bund matures 0.6 of a year earlier on an upward-sloping curve. Logarithmic interpolation, with a weight of 0.4854 rather than 0.4, takes another 2.8 bp off. Every one of these figures is quoted in practice; only the reference and the method make them comparable. At EUR 300,624 a basis point, the 22.2 bp swap spread alone is worth EUR 6,673,853 of price on this EUR 500m bond, which is why a spread quoted without its reference is not a number.
Two other measures appear on dealer runs. The asset swap spread is the floating margin an investor earns by buying the bond and swapping its fixed coupons, and it sits close to the I-spread for a bond priced near par. The Z-spread is the constant spread over the whole zero-coupon swap curve that discounts every cash flow to the price; on a flat or gently sloping curve it differs from the I-spread by a few basis points at most. Neither changes the core point: every spread is a yield minus a reference, and the reference must be stated.
The issuer's credit can stand still while the G-spread moves, because the swap spread is set by bank funding, collateral and government supply. Holding the yield and the I-spread constant:
| Swap spread, bp | Bund 7y | I-spread | G-spread |
|---|---|---|---|
| 0.0 | 2.734% | 118.0 | 118.0 |
| 10.0 | 2.634% | 118.0 | 128.0 |
| 22.2 | 2.512% | 118.0 | 140.2 |
| 35.0 | 2.384% | 118.0 | 153.0 |
Comparing a bond's G-spread today with its G-spread a year ago mixes a change in credit with a change in the swap spread. For the issuer's own credit trend, use the I-spread; for comparison with sovereign or supranational paper, use the G-spread.
Make-whole clauses are written over the government curve, typically Bund plus 15 to 50 bp. Here Bund plus 20 discounts at 2.712 per cent, 120.2 bp inside the bond's own G-spread of 140.2 bp. A treasurer who sets "plus 20" against the 118.0 bp I-spread on the term sheet, rather than the 140.2 bp G-spread the clause is actually measured against, understates how expensive the call is: exercised on day one, the 3.875 per cent bond issued at 99.765 would be redeemed at 107.325, EUR 36,625,602 over par. The full calculation is in how a make-whole call price is calculated.
How the choice of interpolation on the issuer's own curve moves the fair value and the concession, with the euros it is worth, is rebuilt in the free workbook for this case.
Because issuers and many investors swap their fixed coupons to floating, so the swap curve is the funding reference for both sides. On an illustrative seven-year bond at 3.914 per cent, the syndicate quotes 118.0 bp over mid-swaps; the same bond is 140.2 bp over the interpolated Bund, a figure rates investors and make-whole clauses use instead.
The G-spread uses a government yield interpolated to the bond's exact maturity; the benchmark spread uses one specific government bond, usually with a different maturity. Against a 6.4-year Bund yielding 2.470 per cent the illustrative bond is 144.4 bp wide, against 140.2 bp on the interpolated seven-year point.
Because the call discounts at a spread far below the bond's own G-spread. With the interpolated Bund at 2.512 per cent, Bund plus 20 is 2.712 per cent, 120.2 bp under the bond's 140.2 bp. Redeemed at once, a 3.875 per cent seven-year bond would be bought back at 107.325, EUR 36,625,602 over par on EUR 500,000,000.
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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