Foreign Exchange Markets and Currency Forwards › Interest Rate Parity

Interest rate parity (IRP) is the fundamental relationship connecting spot exchange rates, forward exchange rates, and interest rates across countries. It explains why currencies with higher interest rates tend to trade at forward discounts. Understanding this relationship is essential because it reveals why forward rates are not forecasts of future spot rates but rather mathematical consequences of interest rate differentials.

A simple thought experiment explains interest rate parity. Consider a situation where you have $1 to invest for a period $T$ (in years). You have two strategies available, each starting with the same capital and ending at the same future date. For markets to be in equilibrium, both strategies must produce identical returns, otherwise arbitrage would be possible.

Strategy A: Invest domestically

• Invest in a US dollar deposit earning $r_d$ (domestic rate)
• After period $T$: $1 \times (1 + r_d \times T)$ dollars

This is the straightforward approach. You keep your money in dollars and earn the US interest rate.

Strategy B: Invest abroad

• Convert dollars to euros at spot rate $S$ (USD per EUR, so you get $1/S$ euros)
• Invest in a euro deposit earning $r_f$ (foreign rate)
• Lock in a forward contract to convert back to dollars at forward rate $F$
• After period $T$: $(1/S) \times (1 + r_f \times T) \times F$ dollars

This alternative path requires more steps but involves no exchange rate risk because the forward contract locks in the conversion rate for the return journey.

For no arbitrage, both strategies must yield the same return. If they didn't, investors would pile into the superior strategy, driving rates until equality was restored. By equating the returns and solving for $F$, we derive the fundamental pricing relationship:

$$\begin{aligned} 1 + r_d \times T &= \frac{1}{S} \times (1 + r_f \times T) \times F && \text{(equate returns)} \\ F &= S \times \frac{1 + r_d \times T}{1 + r_f \times T} && \text{(solve for F)} \end{aligned}$$

• $F$: forward exchange rate
• $S$: spot exchange rate (domestic per foreign)
• $r_d$: domestic interest rate
• $r_f$: foreign interest rate
• $T$: time to maturity in years

This is the covered interest rate parity (CIP) condition. The word "covered" refers to the fact that exchange rate risk is eliminated (covered) by the forward contract.

The relationship between spot and forward rates is expressed as:

$$F = S \times \frac{1 + r_d \times T}{1 + r_f \times T}$$

where:

• $F$: forward exchange rate
• $S$: spot exchange rate
• $r_d$: domestic interest rate
• $r_f$: foreign interest rate
• $T$: time to maturity in years

This formula has important implications for understanding currency markets. When $r_d > r_f$, the forward rate $F > S$, meaning the foreign currency trades at a forward premium (it will be worth more in terms of domestic currency in the forward market than spot). Intuitively, this makes sense: if you can earn more interest in the domestic currency, the market compensates by making the foreign currency relatively more valuable in forward transactions.

Conversely, when $r_d < r_f$, the forward rate $F < S$, meaning the foreign currency trades at a forward discount. The higher foreign interest rate is offset by expected currency depreciation embedded in the forward rate.

The forward premium or discount can be annualized to express it as a comparable rate:

$$\text{Forward Premium (annualized)} = \frac{F - S}{S} \times \frac{360}{D}$$

where:

• $F$: forward exchange rate
• $S$: spot exchange rate
• $D$: number of days to forward settlement
• $\frac{F - S}{S}$: the percentage return (yield) locked in by the forward contract
• $\frac{360}{D}$: scaling factor to convert the period return to an annualized rate

The annualization allows comparison across different tenor forwards and against interest rate differentials.

In practice, forward rates are often quoted as "forward points," the difference between the forward rate and spot rate, multiplied by 10,000 (for most pairs) or 100 (for JPY pairs). This convention exists because forward points change more frequently than spot rates and allow dealers to quote forward adjustments without constantly updating full prices.

When CIP is violated, arbitrage opportunities arise. The arbitrage mechanism ensures CIP holds in well-functioning markets. Constructing this arbitrage shows how trading pressure restores the interest rate parity relationship.

Suppose the forward rate is "too high" relative to CIP (i.e., the euro is overpriced in the forward market). You can exploit this mispricing through the following sequence of transactions:

1. Borrow dollars at $r_{USD}$
2. Convert to euros at spot rate $S$
3. Invest euros at $r_{EUR}$
4. Sell euros forward at the (overpriced) rate $F$
5. At maturity: receive euros, convert to dollars, repay dollar loan
6. Pocket the risk-free profit

This arbitrage is "covered" because the forward contract eliminates exchange rate risk. You know exactly how many dollars you will receive when you convert your euro proceeds at maturity.

The positive profit in Case 2 confirms that when the market forward rate is higher than the theoretical CIP rate, a risk-free arbitrage is possible by borrowing domestic currency and investing in the foreign currency. The arbitrage activity itself pushes rates back toward equilibrium: borrowing pressure raises domestic rates, selling euros forward pushes the forward rate down, until no profit remains.

While covered interest parity uses forward contracts to eliminate exchange rate risk, uncovered interest parity (UIP) makes a prediction about future spot rates based on economic theory rather than arbitrage:

$$E[S_T] = S \times \frac{1 + r_d \times T}{1 + r_f \times T}$$

where:

• $E[S_T]$: expected future spot rate
• $S$: current spot exchange rate
• $r_d$: domestic interest rate
• $r_f$: foreign interest rate
• $T$: time to maturity in years

UIP states that the expected future spot rate equals the current forward rate. In other words, high-yielding currencies are expected to depreciate against low-yielding currencies, exactly offsetting the interest rate advantage. The intuition is that if a currency offered both higher interest rates and appreciation, capital would flood in until one or both advantages disappeared.

Unlike CIP (which is a no-arbitrage condition and holds tightly), UIP is an equilibrium hypothesis that doesn't hold well empirically. The "forward premium puzzle" refers to the finding that high-yield currencies tend to appreciate rather than depreciate, contrary to UIP. This anomaly underlies the popular "carry trade" strategy, where investors borrow low-yielding currencies to invest in high-yielding ones, profiting from both the interest differential and (often) favorable currency movements.