A dollar coupon 86 basis points above the euro yield turns out to be 32 basis points cheaper: the cross-currency comparison, step by step.
Bring each bond back to its own swap curve, then convert the dollar spread into euros with the cross-currency basis. On Ardennes Industries' seven-year figures, a dollar coupon of 5.18 per cent looks 86 basis points dearer than a 4.32 per cent euro reoffer, but it is 108 bp over dollar mid-swaps, 90 bp over euro mid-swaps after an 18 bp basis, and therefore 32 bp cheaper than the euro bond at mid-swaps plus 122. That is worth about 5.69 million in present value on the USD 300 million indicated.
Worked in full in Capital Markets by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Ardennes Industries is the fictional issuer that runs through Capital Markets. At the same time as its seven-year euro bond, its banks gave a dollar indication. All market levels are illustrative and taken at 30 June 2026, the date the book uses for its swap rates.
| Input | Value |
|---|---|
| Euro mid-swap, seven years | 3.10% |
| Euro reoffer spread over mid-swaps | 122 bp |
| Euro reoffer yield | 4.32% |
| Dollar coupon indicated | 5.18% |
| Dollar mid-swap, seven years | 4.10% |
| EUR/USD cross-currency basis, seven years, on the euro leg | −18 bp |
| Dollar amount indicated | USD 300,000,000 |
The first instinct is to put the two yields side by side: 5.18 per cent against 4.32 per cent, so the dollar bond costs 86 bp more. That comparison mixes two things. Part of the dollar coupon is the issuer's credit, and part is simply that dollar rates are higher than euro rates. An issuer that wants euros will not keep the dollar fixed rate: it will swap the proceeds and the coupons back into euros, and the rate gap goes with them.
Euro spread = 4.32% − 3.10% = 122 bp over euro mid-swaps
Dollar spread = 5.18% − 4.10% = 108 bp over dollar mid-swaps
Excel: =(5.18%-4.10%)*10000
On credit alone the dollar market already prices Ardennes 14 bp tighter. That is before the swap does anything.
In a cross-currency swap the issuer receives dollars to pay its dollar coupons and pays euros in return. The basis is the adjustment on the euro leg. Here it is negative, so the euro leg costs 18 bp less than a flat swap would.
Euro-equivalent spread = dollar spread + basis = 108 + (−18) = 90 bp over euro mid-swaps
Euro-equivalent yield = 3.10% + 0.90% = 4.00%
Advantage of the dollar bond = 122 − 90 = 32 bp a year
The additive conversion is the standard first cut and the one the book uses. A swap desk would refine it for the difference between the two curves' annuity factors and for the dollar deal's own fees and documentation, which are not the same as the euro deal's. Those refinements change the decimals, not the sign.
The naive comparison said dollars cost 86 bp more; the swapped comparison says they cost 32 bp less. The reversal is 118 bp, and it splits cleanly.
| Source | bp | Share |
|---|---|---|
| Gap between the dollar and euro swap curves, 4.10% − 3.10% | 100 | 84.7% |
| Cross-currency basis | 18 | 15.3% |
| Reversal | 118 | 100.0% |
The point in the book is the order of those two lines. The basis does not reverse the comparison; at a basis of zero the dollar bond is already 14 bp cheaper. It widens a gap that was reversed the moment each coupon was set against its own curve.
A spread advantage is paid every year, so it is valued like a coupon: multiply by the amount and by the annuity factor over the tenor. At the euro reoffer yield of 4.32 per cent over seven years the factor is 5.9317.
Annual saving = 0.32% × 300,000,000 = 960,000 a year
Present value = 960,000 × 5.9317 = 5,694,445
Per 100 million: 320,000 a year, 1,898,148 in present value
This is the book's first cut: the saving is measured on the dollar notional and discounted at the euro yield. Strictly, the swapped coupons are paid on the euro leg, whose notional is the dollar amount divided by the EUR/USD spot rate at issue, so a treasury paper should convert at spot before quoting the saving in euros. That moves the size of the figure, not its sign.
Note the scale: 300 million is what the dollar market indicated, not the whole of what the issuer needed. A comparison per basis point per 100 million, 59,317 here, travels better between deals of different sizes. For how the same annuity factor turns a new issue concession into euros, see how to calculate a new issue concession.
| Basis, bp | Euro-equivalent spread | All-in euro yield | Advantage, bp | PV on 300 million |
|---|---|---|---|---|
| −40 | 68 | 3.78% | +54 | 9,609,377 |
| −30 | 78 | 3.88% | +44 | 7,829,863 |
| −18 | 90 | 4.00% | +32 | 5,694,445 |
| −10 | 98 | 4.08% | +24 | 4,270,834 |
| 0 | 108 | 4.18% | +14 | 2,491,320 |
| +10 | 118 | 4.28% | +4 | 711,806 |
The basis would have to move to +14 bp before the two markets cost the same. The dollar spread has more room: at an 18 bp negative basis the dollar bond stays cheaper up to 140 bp over dollar swaps, a coupon of 5.50 per cent. At that break-even the naive comparison would show the dollar bond 118 bp dearer while the two cost exactly the same.
| Dollar spread | Dollar coupon | Naive gap | Swapped advantage |
|---|---|---|---|
| 108 | 5.18% | +86 | +32 |
| 120 | 5.30% | +98 | +20 |
| 130 | 5.40% | +108 | +10 |
| 140 | 5.50% | +118 | 0 |
| 150 | 5.60% | +128 | −10 |
Comparing coupons across currencies. Every board paper that says "dollars are more expensive" because the dollar coupon is higher has compared two interest rate markets, not two credit spreads. The second mistake is the opposite: treating the basis as the whole story and attributing the reversal to it, when here it is 15.3 per cent of the reversal. The third is forgetting the sign convention: state on which leg the basis is quoted every time it is written down, or the 36 bp between −18 and +18 will turn up as an error in someone's model.
The comparison, with every convention that sits inside it, is set out in the register in the free workbooks for this case.
It is the adjustment on one leg of a cross-currency swap that converts funding in one currency into another. Quoted on the euro leg at -18 bp, it lowers the euro-equivalent cost of dollar funding by 18 bp. On Ardennes' seven-year bond it turns 108 bp over dollar swaps into 90 bp over euro swaps, but it accounts for only 15.3 per cent of the reversal against the naive comparison.
Because most of the coupon gap reflects the difference between dollar and euro interest rates, which the swap removes. A 5.18 per cent dollar coupon against a 4.32 per cent euro yield is 86 bp higher, but the dollar swap curve sits 100 bp above the euro curve. Measured as spreads, dollars cost 108 bp against 122 before the basis and 90 after it.
When the dollar spread plus the basis equals the euro spread. With a euro spread of 122 bp and a basis of -18 bp, the break-even is 140 bp over dollar swaps, a coupon of 5.50 per cent at a 4.10 per cent swap rate. At that level the naive coupon comparison would show the dollar bond 118 bp dearer.
The comparison sits in the register of every convention, the workbook for chapter 16 of 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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