Covered interest rate parity: forward rates from interest rates

Why must the forward match the two interest rates?

Two investors each hold one unit of currency B (the base currency of the A/B quote) and want the safest possible result over a period of length τ. Both have a risk-free choice.

1. Deposit at home. Earn the B interest rate $r_B$. End value: $1 + r_B\,\tau$ units of B.
2. Convert, deposit abroad, convert back. Swap the unit of B into A at the spot rate S (S units of A), deposit at $r_A$ to get $S(1 + r_A\,\tau)$ units of A, and sell those A forward at F, which gives back $S(1 + r_A\,\tau) \div F$ units of B. This route is called swap financing .

End value of one unit of B after the periodboth routes are risk free, so both must pay the same

• Route 1: deposit B at home Use it one step
  • Earn r_B end value 1 + r_B τ
• Route 2: swap financing Use it three steps, currency risk locked by the forward
  • Convert B to A at spot S units of A
  • Deposit A at r_A S (1 + r_A τ) units of A
  • Convert back at the forward F divide by F to get units of B
• Equal returns Memorize solving for F gives the parity forward

Both routes carry no risk: the interest rates are risk free, and the forward removes the currency risk. Equal risk means equal return. That is covered interest rate parity .

$$1 + r_B\,\tau = S \times (1 + r_A\,\tau) \times \frac{1}{F}$$

Solve for F and you have the forward rate.

$$F_{A/B} = S_{A/B} \times \frac{1 + r_A\,\tau}{1 + r_B\,\tau}$$

The day-count fraction τ converts an annual money-market rate into the rate for the period. Money-market rates use actual days over a 360-day year.

$$\tau = \frac{\text{days}}{360}$$

Exam tip Memory hook: the price currency’s rate goes on top, the base currency’s rate underneath. It mirrors the quote, with A over B.

How are forward points linked to the rates?

Use it

Subtract spot from the parity forward and the gap, the forward points, depends on the interest rate differential ($r_A - r_B$).

$$F - S = S \times \frac{(r_A - r_B)\,\tau}{1 + r_B\,\tau}$$

Three things follow. The points have the sign of $r_A - r_B$. They grow with the differential. And they grow with τ, only approximately in proportion, because τ also appears in the denominator.

Parity forward: F = S (1 + r_A τ) ÷ (1 + r_B τ)compare it with any quoted forward

• The higher-rate currency Memorize forward discount
• The lower-rate currency Memorize forward premium
• Dealer forward below parity Use it base currency B is cheap forward
  • Borrow B, convert to A, invest at r_A then buy B forward at the dealer rate
• Dealer forward above parity Use it base currency B is dear forward
  • Borrow A, convert to B, invest at r_B then sell B forward at the dealer rate
• Not a forecast Know the idea a pricing result of arbitrage, not a prediction of spot

What if a dealer quotes a forward away from parity?

Use it

Then a riskless arbitrage exists: compare the dealer’s forward with the parity value, buy the currency forward where it is cheap, and fund the other leg at the deposit rates.

Parity holds tightly in the major currencies because banks run this trade constantly. For the exam, treat any forward quoted away from parity as an arbitrage.

What does a forward not tell you?

Know the idea

A higher-rate currency at a forward discount is a pricing result. It is not a forecast that the currency will fall. If you do treat the forward as the expected future spot rate, you get an odd-looking result: the high-rate currency is expected to weaken.

$$\%\Delta S = \frac{r_A - r_B}{1 + r_B}$$

In practice, forwards are poor predictors of future spot rates. The exchange rate responds to far more than the interest rate gap, so read a forward as a price set by arbitrage and nothing more.

Go deeper Why swap financing costs about the domestic rate

Know the idea

Because route 2 is tied by arbitrage to route 1, the all-in cost of borrowing one currency, converting it and hedging the repayment with a forward is close to the plain borrowing rate in that currency. Any gap is an arbitrage that traders remove.

Go deeper The same carry argument in Derivatives

Know the idea

Forward pricing in the Derivatives topic uses the same logic: a forward price equals the spot price grown at the cost of carrying the asset. Here the carry is the interest rate gap between two currencies. Once you can do this lesson’s calculation, forward pricing is the same step with different inputs.