Covered interest parity worked on one-year GBP/USD, proved by replication, and why the forward points a treasury report calls a cost are not one.
An FX forward rate is spot × (1 + interest rate of the quote currency) ÷ (1 + interest rate of the base currency), over the forward's term. With GBP/USD spot at 1.2700, a one-year dollar rate of 4.5 per cent and a sterling rate of 4.0 per cent, the one-year forward is 1.2761, or 61.1 points above spot. The points are the interest differential and nothing else, which is why they are not a cost of hedging.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The formula is covered interest parity, and it is not a forecast. It is the only rate at which a bank cannot be beaten by a customer who borrows in one currency, converts at spot and deposits in the other. Understanding that one sentence settles most arguments about what a currency hedge "cost".
| Input | Value |
|---|---|
| Spot GBP/USD | 1.2700 |
| One-year USD rate (quote currency) | 4.5% |
| One-year GBP rate (base currency) | 4.0% |
| Dollars to be received in one year by a UK exporter | 10,000,000 |
| Dealer spread on the forward | 5 pips |
F = S × (1 + rquote) ÷ (1 + rbase)
=Spot*(1+r_USD)/(1+r_GBP)Which rate goes on top is fixed by the quote, not by which currency you are selling. The quote currency, the one the price is expressed in, goes in the numerator. Because the dollar pays more interest here, a pound buys more dollars forward than spot: sterling trades at a forward premium, the dollar at a forward discount.
The exporter will receive USD 10,000,000 in a year and wants a fixed sterling amount. Two ways to get it:
| Route | Steps | Sterling in one year |
|---|---|---|
| Forward | Sell 10,000,000 dollars at 1.2761 | 7,836,341 |
| Money market | Borrow 9,569,378 dollars at 4.5%, sell at 1.2700 spot for 7,534,943 pounds, deposit at 4.0%; repay the loan with the 10,000,000 received | 7,836,341 |
The two routes land on the same pound. If the forward were any higher, a bank customer could buy dollars forward and sell them through the money market, borrowing dollars, converting at spot and depositing sterling, for a riskless profit; any lower, and the trade runs the other way. The forward rate is pinned to the two interest rates by arbitrage.
Converted at today's spot, the 10,000,000 dollars would be worth 7,874,016 pounds. The forward delivers 7,836,341: 37,675 less, 0.48 per cent. Treasury reports often label that difference the cost of hedging. It is not. The company cannot have today's spot rate for dollars it receives next year without borrowing them now, and borrowing them costs exactly the same 0.5 points of interest differential. The 37,675 is the price of time, paid by anyone, hedged or not.
The real cost of the forward is the bank's margin. A spread of 5 pips takes the rate to 1.2766 and the proceeds to 7,833,272 pounds: 3,069, less than a tenth of the figure usually reported.
For less than a year, money-market rates are simple rates prorated on each currency's own day count: ACT/360 for the dollar, ACT/365 for sterling.
F = S × (1 + rUSD × d ÷ 360) ÷ (1 + rGBP × d ÷ 365)
Three months, 91 days: 1.2700 × 1.011375 ÷ 1.009973 = 1.2718, 17.6 points.
| Tenor | Days | Money-market forward | Points | Read as annual effective | Points |
|---|---|---|---|---|---|
| 1 month | 30 | 1.2706 | 5.9 | 1.2705 | 5.0 |
| 3 months | 91 | 1.2718 | 17.6 | 1.2715 | 15.2 |
| 6 months | 182 | 1.2735 | 34.9 | 1.2730 | 30.4 |
| 12 months | 365 | 1.2769 | 68.7 | 1.2761 | 61.1 |
Beyond a year, rates compound: F = S × ((1 + rUSD) ÷ (1 + rGBP))n. Five years gives 1.3008, 308.2 points; prorating simply instead would give 1.2965, 43.7 points too few. And the sign follows the differential: with dollar rates at 3.5 per cent and sterling at 4.5, the one-year forward is 1.2578, 121.5 points below spot.
Inverting the rates. Putting the base-currency rate on top gives 1.2639, points of minus 60.8 instead of plus 61.1, a 121.8-pip error that flips the premium into a discount. On the exporter's 10,000,000 dollars it overstates the sterling proceeds by 75,531. Check the direction with the rule: the higher-yielding currency is always worth less forward.
Ignoring the day count. If the 4.5 per cent dollar rate is a money-market quote, a year of it is 4.5625 per cent once multiplied by 365 ÷ 360. Treating it as annual effective prices the one-year forward 7.6 pips too low, a difference of 4,684 pounds on the deal.
Calling the points a cost. It leads to bad decisions in both directions: leaving exposures unhedged because the forward "costs" 0.48 per cent, or praising a hedge in a currency whose points happen to run in the company's favour. Measure the hedge on its spread and on its credit and collateral terms, which are where the money actually goes.
The free workbook on the Derivatives companion page reproduces the book's treasury report "cost of hedging" on a GBP/USD exposure, then dismantles it with this identity, and prices the three real cost lines over five years. For what happens when a hedge shows a loss, see did my currency hedge fail if it lost money.
Forward points are the forward rate minus spot, quoted in pips of 0.0001. They come entirely from the interest rate differential between the two currencies. With GBP/USD at 1.2700 and dollar rates 0.5 points above sterling, one-year points are plus 61.1: the currency with the higher interest rate trades at a forward discount, so the pound buys more dollars forward than spot.
No. Selling USD 10,000,000 forward at 1.2761 rather than spot at 1.2700 gives 37,675 GBP less, but borrowing the dollars and converting today gives exactly the same sterling a year later. The difference is interest, which the company earns or pays either way. The real cost of the forward is the dealer's spread: 5 pips here cost 3,069 GBP.
Money-market rates are simple rates on the currency's own day count: ACT/360 for the dollar, ACT/365 for sterling. Treating a 4.5 per cent ACT/360 dollar rate as an annual effective rate understates the one-year forward by 7.6 pips, 1.2761 instead of 1.2769, which on USD 10,000,000 is 4,684 GBP.
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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