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Explain why the interest rate parity theorem follows from a non-arbitrage assumption in foreign exchange markets, and use it to compute forward exchange rates.

FRM® Part I · Financial Markets and Products · Foreign Exchange Markets · FRM1.FM.9.k

The idea

Interest rate parity follows from no arbitrage. An investor can earn the domestic rate at home. Or the investor can convert to foreign currency at spot, earn the foreign rate and sell the proceeds forward. The two covered returns must be equal. Here S is the direct quote, domestic currency per unit of foreign currency. The domestic and foreign interest rates for the period are iD and iF. With simple interest over the period, the forward rate is F = S × (1 + iD) ÷ (1 + iF). With continuously compounded rates rD and rF and a time T in years, the forward rate is F = S × e^((rD − rF) × T). With this quote, F is above S when iF is below iD, so the lower-rate currency is at a forward premium and the higher-rate currency at a discount, but quoted inversely F moves the other way while the lower-rate currency keeps the premium. Use the period fraction for contracts under a year. Inverting the ratio is the most frequent error. If the market forward differs from the parity forward, a covered arbitrage profit exists.

Formulas

Forward exchange rate under interest rate parity, simple rates

F = S × (1 + iD) ÷ (1 + iF)

S
the spot rate, domestic currency per unit of foreign currency
iD
the domestic interest rate for the period
iF
the foreign interest rate for the period

Rates are for the contract's period, so multiply an annual rate by the period fraction under a year. The lower-rate currency trades at a forward premium; putting the foreign rate on top is the classic error.

Forward exchange rate under interest rate parity, continuous rates

F = S × e^((rD − rF) × T)

S
the spot rate, domestic currency per unit of foreign currency
rD
the continuously compounded domestic rate
rF
the continuously compounded foreign rate
T
the time to delivery, in years

The foreign rate acts as the yield the currency pays. A market forward away from this value offers covered arbitrage.

A practice question

From the question bank, with the reason each wrong option is wrong.

EUR/USD spot is 1.0850 (USD per EUR). The one-year USD interest rate is 5% and the one-year EUR interest rate is 3%. The one-year forward EUR/USD rate implied by interest rate parity is closest to:

  1. 1.0643Why not: Inverts the interest rate ratio; the EUR has the lower rate so it should trade at a forward premium.
  2. 1.1393Why not: Applies only the USD interest rate and ignores the EUR rate.
  3. 1.1718Why not: Adds the two interest rates instead of forming the ratio of growth factors.
  4. 1.1061 - correct

F = S x (1 + i USD)/(1 + i EUR) = 1.0850 x 1.05/1.03 = 1.1061. The EUR, quoted here as USD per EUR, trades at a forward premium (F above S) because its interest rate is lower.

A harder one

USD/JPY spot is 148.00 (JPY per USD). The six-month interest rates (annualized, simple) are 4.5% in USD and 0.5% in JPY. The six-month forward USD/JPY rate implied by interest rate parity is closest to:

  1. 145.11 - correct
  2. 150.95Why not: Inverts the ratio; the JPY has the lower rate so the USD trades at a forward discount, a lower JPY per USD.
  3. 142.33Why not: Uses the full-year rates for a six-month contract instead of half of each.
  4. 153.89Why not: Inverts the ratio and also uses the full-year rates.

With JPY as the domestic currency: F = S x (1 + 0.005 x 0.5)/(1 + 0.045 x 0.5) = 148.00 x 1.0025/1.0225 = 145.11. The USD has the higher rate, so it trades at a forward discount.

The hardest tier

EUR/USD spot is 1.0850 (USD per EUR); one-year interest rates are 4% in USD and 3% in EUR. The market one-year forward is 1.1000. An arbitrageur borrows USD 1,085,000, converts to EUR at spot, invests for one year and sells the maturing EUR forward. What is the arbitrage profit in USD at maturity, closest to?

  1. USD 4,466Why not: Hedges only EUR 1,000,000, forgetting that the EUR investment grows to EUR 1,030,000 by maturity.
  2. USD -4,600Why not: Sign error; the market forward is above the parity forward, so selling EUR forward earns a profit.
  3. USD 4,600 - correct
  4. USD 48,000Why not: Forgets to repay the USD loan with interest; it deducts the USD 1,085,000 principal only.

Parity forward = 1.0850 x 1.04/1.03 = 1.0955. Sell EUR 1,030,000 forward at 1.1000 = USD 1,133,000; repay 1,085,000 x 1.04 = USD 1,128,400. Profit = USD 4,600, which equals (1.1000 - 1.0955) x 1,030,000.

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