Purchasing Power Parity, Interest Rate Parity and Four-Way Equivalence

Why should a currency lose value? The simplest answers are that its goods have become dearer than abroad, or that the interest it pays is higher than elsewhere.

If the naira buys fewer goods at home because prices have risen faster than in the United States, it should buy fewer dollars; if naira deposits pay 20% while dollar deposits pay 4.5%, there must be a reason that investors do not all move their money into naira. The reason is that the naira is expected to fall by about that difference.

These two ideas, purchasing power parity and interest rate parity, with the Fisher effect that links interest rates to inflation and the expectations theory that links the forward rate to the future spot rate, make up the four-way equivalence: four relationships that should all hold at once if markets are in equilibrium, and that together give a method of forecasting exchange rates.

This topic explains each, the formulae for direct and indirect quotes (and the rule that makes them easy to remember), how to forecast a rate for several years, how the relationships tie together, why they do not hold in the short run, and what this means for Nigeria, with worked examples. The next topic covers covered interest arbitrage and forward rates.

Interest rate parity was part of May Diet, 2026, Question 4 (20 marks), where part (a) (5 marks) asked about it and the later parts tested it with arbitrage. Read causes of exchange rate movements first.

The four-way equivalence

Four relationships should hold together when the foreign exchange market is in equilibrium:

1. Purchasing power parity: the expected change in the spot rate is explained by the difference in inflation rates.
2. Interest rate parity: the forward rate is explained by the difference in interest rates.
3. The Fisher effect (and, between countries, the international Fisher effect): the difference in interest rates is explained by the difference in inflation rates.
4. Expectations theory: the forward rate is the market’s expected future spot rate.

In a perfect capital market, if one variable changed (say a country’s interest rate rose), the others would readjust at once. The model shows what would happen if a variable moved away from the equilibrium.

Purchasing power parity

Purchasing power parity (PPP) says that an exchange rate will adjust so that the same amount of goods can be bought with a given amount of money in any country. A country with higher inflation sees its prices rise relative to others, so its currency must fall to restore parity. Using the relative form (the change in the rate follows the difference in inflation):

Direct quote: S_t = S₀ × (1 + I_d) ÷ (1 + I_f) Indirect quote: S_t = S₀ × (1 + I_f) ÷ (1 + I_d)

where S₀ is the current spot rate, S_t the expected spot rate after the period, I_d the inflation in the domestic country and I_f in the foreign country. Memory rule: the numerator is the inflation of the currency whose units the quote counts.

In a direct quote such as ₦ per $1 the units counted are naira, the domestic currency; in an indirect quote such as $ per ₦1 they are dollars, the foreign currency.

Example (Nigerian view): the spot rate is ₦1,345/$1; expected inflation is 14% in Nigeria and 3% in the United States (illustrative figures). After one year S₁ = 1,345 × 1.14 ÷ 1.03 = ₦1,488.64. After two years, 1,345 × (1.14 ÷ 1.03)² = ₦1,647.62, and after three, ₦1,823.58. The dollar rises 10.68% a year in naira, which means the naira loses 1 − 1.03 ÷ 1.14 = 9.65% of its value in dollars. Using the indirect quote gives the same answer: 1 ÷ 1,345 = $0.000743 per ₦1, and 0.000743 × 1.03 ÷ 1.14 = $0.000672 per ₦1, which is 1 ÷ 0.000672 = ₦1,488.6 per $1.

Example with an indirect quote: the rate is $1.30 per £1, UK inflation is 3.5% and US inflation 2%. Here the domestic currency is the pound, so I_d = 3.5% and I_f = 2%: S₁ = 1.30 × 1.02 ÷ 1.035 = $1.2812 per £1. The pound is expected to fall 1.45% against the dollar because UK inflation is higher.

Interest rate parity

Interest rate parity (IRP) says that exchange rates adjust to remove the advantage of a higher interest rate: the currency with the higher interest rate trades at a forward discount by about the interest differential.

An investor with ₦1,345 can invest at the naira rate for a year, or convert to one dollar and invest at the dollar rate; if both give the same result in naira after a year, then, with a naira rate of 20% and a dollar rate of 4.5%, the forward rate is such that ₦1,345 × 1.20 = $1 × 1.045 × F, so:

Direct quote: F = S₀ × (1 + i_d) ÷ (1 + i_f) Indirect quote: F = S₀ × (1 + i_f) ÷ (1 + i_d)

F = 1,345 × 1.20 ÷ 1.045 = ₦1,544.50. The dollar is at a forward premium of 14.83% and the naira at a forward discount. Over two years, F = 1,345 × (1.20 ÷ 1.045)² = ₦1,773.59. The formula is the same as the PPP formula with interest rates in place of inflation rates. The forward rate does not forecast; it is a rate fixed by interest rates and a price at which a bank will deal.

If the market is in equilibrium, however, the forward rate equals the expected spot rate (see expectations theory below). If the forward rate does not line up with the interest rates, a risk-free profit is available (covered interest arbitrage, the next topic).

The Fisher effect and the international Fisher effect

The Fisher effect links nominal and real interest rates: 1 + nominal rate = (1 + real rate) × (1 + inflation rate). A nominal rate of 22% with inflation of 17% gives a real rate of 1.22 ÷ 1.17 − 1 = 4.27%. The international Fisher effect says that investors everywhere expect the same real return, so differences in nominal rates between countries reflect differences in expected inflation, and, through PPP, the expected change in the exchange rate.

All four together

Suppose the real rate is 3% in both countries, inflation is 12% in Nigeria and 2% in the United States. By Fisher, the nominal rates are 1.12 × 1.03 − 1 = 15.36% in naira and 1.02 × 1.03 − 1 = 5.06% in dollars. PPP gives S₁ = 1,345 × 1.12 ÷ 1.02 = ₦1,476.86. IRP gives F = 1,345 × 1.1536 ÷ 1.0506 = ₦1,476.86, the same. By expectations theory the forward rate is also the expected spot rate. This is equilibrium.

If the real rates differ (for example, a naira real rate of 5.26% from a 20% nominal rate and 14% inflation, against a dollar real rate of 1.46% from 4.5% and 3%), the PPP forecast (₦1,488.64) and the IRP forward rate (₦1,544.50) differ, which signals that capital would flow to take advantage of the higher real return in naira until the difference disappears, or that the inputs are expectations that may not come true.

Expectations theory and the limits of all four

Expectations theory says that all relevant information is reflected in the market rates, so the forward rate is the market’s expectation of the spot rate at that date. In practice the forward rate is a poor predictor of future spot rates. The relationships hold better in the long term than in the short term, because:

• exchange rates respond in the short run to capital flows, news, speculation and government intervention, not only to inflation;
• goods are not all traded and are not identical, and there are transport costs, tariffs and quotas;
• capital markets are not perfect, with controls, taxes and transaction costs;
• inflation and interest rates are forecasts, themselves uncertain; and
• in Nigeria, the exchange rate has been affected by policy, the supply of foreign currency, oil prices and the reserves of the Central Bank, as well as by inflation.

In an examination the forecasts are a guide, with the assumptions stated, to the likely direction and size of a long-term move.

The ideas in more depth

Why higher inflation weakens a currency

If prices in Nigeria rise 14% and prices in the United States 3%, a Nigerian good that cost ₦1,345 (one dollar’s worth) now costs ₦1,534, while the American good still costs about $1.03. To keep Nigerian goods competitive and prevent a flow of imports and a loss of exports, the naira must fall by about the difference.

The adjustment is not automatic or immediate, but over years the countries with the higher inflation have tended to see their currencies depreciate.

Why the forward rate is not a forecast

The forward rate is fixed by the interest rates in the two currencies, because a bank can always borrow one, lend the other, and hedge, and arbitrage prevents any other rate. It would equal the expected spot rate only if investors were indifferent to risk and all the relationships held.

Where they do not, the forward rate and the future spot rate may differ in either direction. A company cannot assume that a forward contract will give the best rate in hindsight.

Sources and references

• The theories are those of Cassel (purchasing power parity, 1918), Keynes (interest rate parity, 1923) and Fisher (1930). The inflation and interest figures in the examples are illustrative; the Central Bank of Nigeria’s Monetary Policy Rate was 23% from September 2026.

These points are stated as at 5 November 2026.

Worked examples

1. Formula. 1,345 × 1.14 ÷ 1.03.
2. Rate. ₦1,488.64.

1. Formula. 1,345 × 1.20 ÷ 1.045.
2. Rate. ₦1,544.50.

1. Formula. 1.22 ÷ 1.17 − 1.
2. Real rate. 4.27%.

How to answer in the exam

• Identify the quote and the domestic and foreign currencies before using a formula.
• Put the right rate on top: the inflation or interest rate of the currency whose units the quote counts.
• Compound for several years or apply the one-year factor repeatedly.
• Comment on the assumptions and on why the forecast may not come true.

Common mistakes in the exam

• Inverting the ratio, so that the high-inflation currency appears to strengthen.
• Using the formula for a direct quote with an indirect quote.
• Using inflation rates in the IRP formula, or interest rates in the PPP formula.
• Adding rates instead of dividing the compounded factors.
• Treating the forward rate as a certain forecast.
• Forgetting to compound over more than one year.

Quick revision

• PPP.
  S_t = S₀ × (1 + I_d) ÷ (1 + I_f) for a direct quote.
• IRP.
  F = S₀ × (1 + i_d) ÷ (1 + i_f); higher-interest currency at a forward discount.
• Fisher.
  1 + m = (1 + r)(1 + i); international Fisher: equal real rates.
• Expectations: forward rate is the expected spot rate; all weak in the short term.

Practice questions

These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.

Recall check: multiple choice

Section B style: written questions

A further written question

Model answers

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