Covered Interest Parity

You have a million euros to park for a year and no appetite for currency risk. There are two ways to do it. You can leave the money in euros and collect the euro deposit rate. Or you can sell the euros for dollars today, collect the higher dollar deposit rate, and — crucially — agree today, in a forward contract, on the rate at which you will turn those dollars back into euros in a year.

The second route touches a foreign currency but takes no currency risk, because every single price is fixed before you start. That is what "covered" means: the exchange-rate leg is covered by a forward. And here is the point. Both routes begin with a million euros, both end with a certain number of euros, and both are free of risk. They must end with the same number, or someone would run the cheap route and the expensive route in opposite directions and collect the difference for nothing.

Think of two flights to the same city, both with the ticket price, the taxi fare and the exchange booth agreed in advance. If one route is reliably cheaper, everybody takes it until the prices adjust.

The square

Write $S$ for the spot rate, $F$ for the forward rate, $r_b$ for the interest rate on the base currency and $r_q$ for the rate on the quote currency, over a period $T$ measured in years. Matching the two routes gives

$$F = S \times \frac{1 + r_q T}{1 + r_b T}$$

In plain English: start from today's exchange rate, then scale it up by however much faster the quote currency's cash grows than the base currency's. If the quote currency pays the higher rate, $F$ sits above $S$; if it pays less, $F$ sits below.

Worked example: pricing the forward

EURUSD spot is 1.0800. One-year deposit rates are 5.00 percent in dollars and 3.00 percent in euros. The base currency is the euro, so $r_b = 0.03$ and $r_q = 0.05$.

$$F = 1.0800 \times \frac{1.05}{1.03} = 1.0800 \times 1.019417 = 1.10097$$

Check it the long way, which is the only way to really believe it.

• Stay in euros. EUR 1,000,000 at 3 percent becomes EUR 1,030,000.
• Go via dollars. EUR 1,000,000 sold at 1.0800 gives USD 1,080,000. At 5 percent that becomes USD 1,134,000. Divide by the forward: $1{,}134{,}000 / 1.10097 = 1{,}030{,}000$.

Identical, to the euro. Notice also the shortcut: the forward is about 1.94 percent above spot, close to the 2 percent interest-rate gap. For short tenors and small rates, forward premium ≈ rate differential is accurate enough for mental arithmetic.

Worked example: arbitraging a bad forward

Now suppose a bank quotes the one-year forward at 1.1150 rather than 1.10097. Euros are too expensive forward, so you sell them forward and buy them spot. With USD 10,000,000 of borrowing:

1. Borrow USD 10,000,000 for a year at 5 percent. You will owe USD 10,500,000.
2. Buy euros spot at 1.0800: $10{,}000{,}000 / 1.0800 = 9{,}259{,}259$ euros.
3. Deposit the euros at 3 percent: they grow to EUR 9,537,037.
4. Sell that exact amount forward at 1.1150, agreed today: $9{,}537{,}037 \times 1.1150 = 10{,}633{,}796$ dollars.
5. Repay the loan. USD 10,633,796 minus USD 10,500,000 leaves USD 133,796 of profit.

Every price was locked on day one, so there is no market risk anywhere in that chain. Trades like this are why the forward does not stay mispriced for long.

What this means in practice

Because parity holds so tightly, the forward market is really a funding market in disguise. A treasurer with euros who needs dollars can borrow euros and use an FX Swaps package instead of borrowing dollars directly; the implied dollar rate is whatever the forward points say. When the two differ, the gap is the cross-currency basis, and since 2008 it has been persistently non-zero. Parity assumes you can borrow and lend freely at the quoted rates, but bank balance sheet is scarce and regulated, so the arbitrage in the second example is not always free to put on. The basis is the price of that constraint, and it widens at quarter-end and in stress.

Key terms

• Covered — the currency leg is locked with a forward, so no exchange-rate risk remains.
• Spot / forward rate — the rate for immediate settlement, and the rate agreed today for a future date.
• Forward premium / discount — the forward above or below spot, driven by the interest-rate gap.
• Cross-currency basis — the residual mispricing when parity fails, a measure of dollar funding stress.