Two legs, two curves, one spot rate: the mark-to-market of a fixed-for-fixed GBP/USD swap with notional exchange, and why the final exchange carries most of the value.
A cross-currency swap is valued leg by leg: discount each leg's remaining cash flows, final notional included, on its own currency's curve, convert the foreign leg at today's spot, and subtract. A UK company receiving 4.80 per cent on USD 50,000,000 and paying 4.30 per cent on GBP 40,000,000, with three years left and spot at 1.32, holds a swap worth minus 1,825,907 sterling. More than all of it, minus 2,079,279, comes from the final exchange of notionals.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Two years ago the company issued a five-year dollar bond paying 4.80 per cent and swapped it into sterling, because its revenues are in pounds. The swap mirrors the bond: the company receives dollar coupons and, at maturity, the dollar principal it needs to repay the bondholders, and pays sterling coupons and a sterling principal fixed at the inception spot of 1.25. Since then sterling has risen and both curves have fallen.
| Input | Value |
|---|---|
| USD notional received at maturity | 50,000,000 |
| USD fixed coupon received, annual | 4.80% |
| Inception spot, dollars per pound | 1.25 |
| GBP notional paid at maturity | 40,000,000 |
| GBP fixed coupon paid, annual | 4.30% |
| Years remaining | 3 |
| Spot today | 1.32 |
| USD discount rate, flat | 4.20% |
| GBP discount rate, flat | 4.00% |
Flat annual rates keep the arithmetic visible; a production valuation uses bootstrapped discount factors for each payment date, as in how to value an interest rate swap after rates move. The cross-currency basis is assumed to be inside the discount rates. All figures are illustrative.
PVUSD = Σ (USD coupon + final USD notional) × DFUSD(t)
PVGBP = Σ (GBP coupon + final GBP notional) × DFGBP(t)
Value to the company in GBP = PVUSD / spot − PVGBP
In Excel, with cash flows in B2:B4 and C2:C4, years in A2:A4, rates in F1 and F2, spot in F3:
=SUMPRODUCT(B2:B4,(1+F1)^-A2:A4)/F3-SUMPRODUCT(C2:C4,(1+F2)^-A2:A4)
Step 1, the cash flows. Dollar coupons are 50,000,000 × 4.80% = 2,400,000 a year. The sterling notional is 50,000,000 / 1.25 = 40,000,000, so sterling coupons are 1,720,000. Each leg pays its notional back in year three.
| Year | USD cash flow | DF at 4.20% | PV, USD | GBP cash flow | DF at 4.00% | PV, GBP |
|---|---|---|---|---|---|---|
| 1 | 2,400,000 | 0.9597 | 2,303,263 | 1,720,000 | 0.9615 | 1,653,846 |
| 2 | 2,400,000 | 0.9210 | 2,210,425 | 1,720,000 | 0.9246 | 1,590,237 |
| 3 | 52,400,000 | 0.8839 | 46,315,689 | 41,720,000 | 0.8890 | 37,088,928 |
| Total | 50,829,377 | 40,333,011 |
Step 2, convert at spot. The dollar leg is worth 50,829,377 / 1.32 = 38,507,104 pounds. Spot is the right rate here because both legs are already present values; the forward points are inside the two discount curves.
Step 3, net. 38,507,104 less 40,333,011 = minus 1,825,907, or 4.56 per cent of the sterling notional. The company would have to pay that amount to close the swap today, and under a credit support annex it is roughly the collateral it has posted.
| Component | USD leg in GBP | GBP leg | Net |
|---|---|---|---|
| Final exchange of notionals | 33,480,576 | 35,559,854 | -2,079,279 |
| Coupons | 5,026,528 | 4,773,157 | 253,372 |
| Swap | 38,507,104 | 40,333,011 | -1,825,907 |
The principal exchange is 113.9 per cent of the value. It was struck at 1.25 and spot is now 1.32, so the dollars the company will receive buy fewer pounds than it will pay. This split is an accounting of legs, not of market moves: the coupons were already positive at inception, because the dollar coupon is higher than the sterling one, and the higher dollar rate discounted the dollar principal more heavily to offset it. The split by market move is cleaner. Holding spot at 1.25, the fall in both curves alone would make the swap worth plus 330,491, because dollar rates have fallen further below the 4.80 per cent received than sterling rates have below the 4.30 per cent paid; the currency move takes away 2,156,398.
None of this is a loss to the company as a whole. Its dollar bond is cheaper to repay in sterling by roughly the same amount, which is exactly what the swap was for. The cash consequence is real, though: a swap that moves 293,947 for each cent of spot can call several million of collateral in a quarter.
| Spot | Value, GBP | Share of GBP notional |
|---|---|---|
| 1.16 | 3,485,418 | 8.71% |
| 1.20 | 2,024,803 | 5.06% |
| 1.25 | 330,491 | 0.83% |
| 1.32 | -1,825,907 | -4.56% |
| 1.36 | -2,958,469 | -7.40% |
The swap is worth zero at 1.2602, the ratio of the two leg values, not at the inception rate. The basis is the smaller lever: discounting sterling at 3.85 per cent instead of 4.00 takes the value to minus 1,993,838, and at 4.15 per cent to minus 1,658,929, about 167,000 for 15 basis points. That is why two banks valuing the same swap with different basis curves rarely agree to the pound.
Valuing a cross-currency swap like a single-currency swap, coupons only, with the notionals assumed to cancel. In an interest rate swap they do; here they are in different currencies and were fixed at a rate that no longer prevails. The coupons-only figure is plus 253,372, wrong in sign and short by 2,079,279. The same error shows up in hedge accounting files that test only the interest component, and in collateral forecasts that underestimate what a currency move will call.
A second error is to convert the dollar leg at the forward rate for each date and then discount in sterling while also discounting in dollars. Each cash flow is discounted once, on its own curve, and only the total is converted at spot.
Derivatives follows a hedge from the price on the confirmation to the cash it consumes; the swap, the rolling programme and the collateral ledger are rebuilt in the free workbooks for the book.
Because it is fixed at the inception spot and is the largest cash flow, so it carries most of the currency move. In this case the final exchange of USD 50,000,000 for GBP 40,000,000 is worth minus 2,079,279 at spot 1.32, more than the swap's total value of minus 1,825,907, while the coupons are worth plus 253,372.
Roughly the present value of the foreign leg divided by spot squared for each unit of spot. Here a one-cent move in GBP/USD changes the value by about 293,947 sterling. The swap is worth zero at a spot of 1.2602, not at the 1.25 inception rate, because dollar and sterling rates have also moved.
In the discount rates. Market practice discounts one leg on a curve adjusted for the basis quoted for that currency pair. In this illustration, moving the sterling discount rate by 15 basis points either way changes the value by about 167,000, so the basis is a second-order but real input.
This article is one calculation from Derivatives. 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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