If Nigerian prices rise 25% a year and American prices only 3%, the naira cannot keep buying the same number of dollars. If Nigerian deposits pay 20% and American ones 4%, the market cannot offer both the high rate and a steady exchange rate for free. Four linked theories explain how inflation, interest rates and exchange rates move together, and they give the formulas used to forecast spot rates and set forward rates.
This topic covers the four-way equivalence, purchasing power parity (PPP), interest rate parity (IRP), the Fisher effect and the international Fisher effect, expectations theory and covered interest arbitrage. The previous topic covered exchange rates and exposure. The next covers forward contracts, money market hedges and netting. The May 2014 Pathfinder paper included a short-answer question on interest rate parity.
In 30 seconds
Four relationships link spot rates, forward rates, inflation and interest rates. PPP: future spot = spot × (1 + inflation of the currency being priced) ÷ (1 + inflation of the other); the currency with higher inflation depreciates. IRP: forward = spot × (1 + interest rate of the priced currency) ÷ (1 + interest rate of the other); the currency with the higher interest rate trades at a forward discount. Fisher: (1 + nominal) = (1 + real)(1 + inflation).
International Fisher: real returns are equal across countries, so interest rate gaps match inflation gaps. Expectations theory: the forward rate is the expected future spot rate. Covered interest arbitrage earns a riskless profit when the actual forward rate departs from the IRP rate: borrow, convert, invest and cover forward. For periods under a year, use the part-year rate (annual rate × months ÷ 12).
Key terms in plain English
| Term | What it means |
|---|---|
| Purchasing power parity | Exchange rates adjust so that the same goods cost the same in each country |
| Interest rate parity | Forward rates adjust so that hedged returns are equal in each currency |
| Fisher effect | The nominal interest rate equals the real rate plus expected inflation (compounded) |
| Expectations theory | The forward rate is the market’s expected future spot rate |
| Covered interest arbitrage | A riskless profit from interest and forward rate differences |
| Forward premium / discount | The forward rate is stronger / weaker than spot |
Syllabus check
This topic is in syllabus area E (treasury and advanced risk management). It covers:
- describing and applying purchasing power parity and interest rate parity;
- adjusting a spot rate to arrive at a forward rate.
CAMA 2020 and currency forecasts
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Directors should base forecasts on reasoned methods for the benefit of the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should be able to explain the assumptions behind exchange rate forecasts |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Four-way equivalence
Four concepts together explain exchange rate changes and let us predict spot rates: purchasing power parity, interest rate parity, the Fisher effect (and with PPP, the international Fisher effect) and expectations theory. They hold when the foreign exchange market is in equilibrium. In a perfect capital market the model would readjust instantly to any change.
Purchasing power parity
PPP says the exchange rate adjusts to inflation differences, so the same amount of goods can be bought with equivalent money in any country. ICAN’s example: a basket costs ₦25,000 in Nigeria and £100 in the UK, implying ₦250/£1. Inflation is 8% in Nigeria and 5% in the UK. At year-end the basket costs ₦27,000 and £105, so the rate must be 27,000 ÷ 105 = ₦257.14/£1. Higher inflation means a weaker currency.
Formula: St = S0 × (1 + inumerator currency) ÷ (1 + iother currency). The inflation rate on top belongs to the currency whose units appear in the quote (the variable currency). Direct quote (₦ per £): St = 250 × 1.08 ÷ 1.05. Indirect from the UK view (£1 = $): the dollar inflation is on top.
ICAN’s examples: ₦150/$ with 3% inflation in Nigeria and 5% in the US gives 150 × 1.03 ÷ 1.05 = 147.14 after one year, then 144.34 and 141.59 (ICAN rounds the first to 147.13). £1 = $2 with 3% UK and 2% US inflation gives 2 × 1.02 ÷ 1.03 = 1.9806, then 1.9614. In reality other factors, especially in the short term, move rates, so PPP is a guide to long-term direction.
Interest rate parity
IRP says exchange rates adjust to remove interest rate differences. ICAN’s example: ₦25,000 invested in Nigeria at 8% becomes ₦27,000; £100 invested in the UK at 5% becomes £105; so the rate must be 27,000 ÷ 105 = ₦257.14/£1. The currency with the higher interest rate depreciates against the one with the lower rate.
Formula: F = S × (1 + ipriced currency) ÷ (1 + iother), the same shape as PPP with interest rates. It gives a forward rate; if the market is in equilibrium it equals the expected future spot rate, though in practice forward rates are poor predictors. ICAN’s examples: R7.40/$ with a 6% rand rate and 2% dollar rate gives 7.4 × 1.06 ÷ 1.02 = 7.6902 in one year, then 7.9918 and 8.3052; £1 = €1.2115 with 5% sterling and 2.5% euro rates gives 1.2115 × 1.025 ÷ 1.05 = 1.1827, then 1.1601 and 1.1273. For periods under a year, use the part-year rates.
The Fisher effects and expectations theory
Fisher effect: (1 + m) = (1 + r)(1 + i). With a nominal rate of 4% and inflation of 2.5%, 1 + r = 1.04 ÷ 1.025, so the real rate is 1.46%. International Fisher effect: investors everywhere expect the same real return, so differences in nominal interest rates reflect differences in inflation. ICAN’s example: Country X has a nominal rate of 6% and inflation of 5%, a real return of 1.06 ÷ 1.05 − 1 = 0.95%; Country Y has 8.02% and 7%, so 1.0802 ÷ 1.07 − 1 = 0.95%.
Expectations theory: all relevant information is in market rates, so the forward rate reflects the expected future spot rate. With $1 = 4.00 francs, PPP gives 4.00 × 1.07 ÷ 1.05 = 4.0762; IRP gives 4.00 × 1.0802 ÷ 1.06 = 4.0762; the one-year forward rate is the same.
Covered interest arbitrage
When the actual forward rate differs from the IRP rate, an investor can borrow in one currency, convert, invest and cover forward, for a riskless profit. IRP: (1 + Rdomestic) = (1 + Rforeign) × F ÷ S. If the two sides differ, arbitrage exists.
ICAN’s example. UK and US interest rates are 6% and 4%. Spot is $1.6000/£ and the 3-month forward rate is $1.5865/£. A dealer may buy or sell $10 million.
- Period rates.
US 4% × 3÷12 = 1%
UK 6% × 3÷12 = 1.5%.
IRP forward = 1.6000 × 1.01 ÷ 1.015 = $1.5921.
The actual forward ($1.5865) is lower, so forward pounds are undervalued. - Borrow £6,250,000 (= $10m ÷ 1.6000) at 1.5%; owe £6,343,750 in three months.
Buy £6,343,750 forward at 1.5865. - Convert to $10m and invest at 1% to get $10,100,000.
The forward pounds cost 6,343,750 × 1.5865 = $10,064,360.
Profit = about $35,640 (formula: $10m × [1.01 − 1.015 × 1.5865 ÷ 1.6000]).
As arbitrageurs buy forward pounds and sell spot pounds, the forward rate rises and the spot rate falls until parity returns. Limits: transaction costs, liquidity at quoted rates, exchange controls and interest rate changes.
The rules in more depth
Why inflation drives long-term rates
If a currency loses purchasing power faster than another, it should buy fewer units of the other over time.
Why interest rates drive forward rates
Otherwise investors could earn more in one currency, hedged, than the other, risk free.
Why the numerator rule works
The currency quoted as the number (the one you buy with) loses value when its inflation or interest rate is higher, so its rates go on top.
Why the theories fail in the short term
Capital flows, news and expectations move rates faster than prices.
Effect on analysis
Identify which currency is priced, put its rate on top, use part-year rates for short periods and state the result as a forecast, not a certainty.
Worked examples
Example 1: PPP. Spot is ₦1,500/$. Nigerian inflation is 25% and US inflation is 3%.
- S1 = 1,500 × 1.25 ÷ 1.03.
- S1 = ₦1,820.39/$.
- The naira is expected to weaken.
Example 2: IRP forward. Spot is ₦1,500/$. One-year rates are 20% in Nigeria and 4% in the US.
- F = 1,500 × 1.20 ÷ 1.04.
- F = ₦1,730.77/$.
- The dollar is at a forward premium.
Example 3: a part-year forward. The same rates, for six months.
- Half-year rates: 10% and 2%.
- F = 1,500 × 1.10 ÷ 1.02.
- F = ₦1,617.65/$.
Example 4: the Fisher equation. The nominal rate is 20% and expected inflation is 15%.
- 1 + r = 1.20 ÷ 1.15.
- r = 1.0435 − 1.
- Real rate = 4.35%.
How to score full marks
- Name the theory and the formula.
- Put the priced currency’s rate in the numerator.
- Convert annual rates for part-years.
- Show each year of a multi-year forecast.
- For arbitrage, compare actual and IRP forward rates before the steps.
- State the limits of the theories.
Common mistakes in the exam
- Putting the wrong currency’s rate on top.
- Using annual rates for a three-month forward.
- Applying PPP with interest rates or IRP with inflation.
- Forgetting to compound over several years.
- Treating forward rates as accurate predictions.
- Ignoring transaction costs in arbitrage.
Quick revision
- PPP: inflation gap; IRP: interest rate gap; both give the same shape of formula.
- Higher inflation or interest rate: weaker currency, forward discount.
- Fisher.
(1 + m) = (1 + r)(1 + i). - Expectations: forward = expected spot.
- Covered interest arbitrage: borrow, convert, invest, cover forward.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. PPP says that a currency with higher inflation will:
- Depreciate
- Appreciate
- Stay fixed
- Pay more interest
Show the answer
A. To restore purchasing power parity.
Question 2. IRP says the currency with the higher interest rate trades at a forward:
- Discount
- Premium
- Par always
- Floor
Show the answer
A. It is expected to weaken.
Question 3. Spot ₦200/$, Nigeria 10%, US 5%. The one-year forward rate is:
- ₦209.52
- ₦190.48
- ₦210.00
- ₦195.00
Show the answer
A. 200 × 1.10 ÷ 1.05.
Question 4. The Fisher equation is:
- (1 + m) = (1 + r)(1 + i)
- m = r − i
- m = r ÷ i
- r = m + i
Show the answer
A. Real and inflation compound.
Question 5. Under the international Fisher effect, nominal rate gaps reflect:
- Inflation gaps
- Tax gaps
- Size gaps
- Trade gaps
Show the answer
A. Real returns are equal.
Question 6. Expectations theory says the forward rate is:
- The expected future spot rate
- Always wrong
- The bank’s profit
- The spot rate
Show the answer
A. In equilibrium.
Question 7. For a three-month forward, a 12% annual interest rate becomes:
- 3%
- 12%
- 36%
- 1%
Show the answer
A. 12 × 3 ÷ 12.
Question 8. Covered interest arbitrage exists when:
- The actual forward differs from the IRP forward
- Rates are equal
- There are no banks
- Spot equals forward
Show the answer
A. Otherwise parity holds.
Question 9. Arbitrage profits are limited by:
- Transaction costs and exchange controls
- Nothing
- Tax only
- The spot rate
Show the answer
A. Among other limits.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) State the four relationships in the four-way equivalence. (4 marks) (b) Explain why PPP is only a guide to long-term movements. (3 marks) (c) State the formula for the Fisher effect. (3 marks)
Question 12 (20 marks). (a) Spot is ₦1,500/$. Nigerian inflation is 18% and US inflation 3%. Forecast the rate at the end of years 1 and 2 using PPP. (5 marks) (b) With one-year rates of 20% and 4%, calculate the one-year IRP forward rate. (4 marks) (c) US and UK interest rates are 4% and 6%. Spot is $1.6000/£ and the 3-month forward $1.5865/£. A dealer may use $10m. Calculate the IRP forward and the arbitrage profit. (8 marks) (d) Explain the limits of covered interest arbitrage. (3 marks)
Question 13 (10 marks). (a) Calculate the real rate if the nominal rate is 20% and inflation 15%. (3 marks) (b) Explain expectations theory and one reason forward rates predict poorly. (3 marks) (c) Calculate the six-month forward rate with spot ₦1,500/$ and annual rates of 20% and 4%. (4 marks)
Model answers
Show the model answer to Question 11
(a) Relationships (4 marks). Purchasing power parity; interest rate parity; the Fisher effect (with PPP, the international Fisher effect); expectations theory. (1 each)
(b) PPP (3 marks). Other factors apart from inflation, such as capital flows and news, affect rates. (1.5) These matter especially in the short term, so PPP is a guide to long-term direction. (1.5)
(c) Fisher (3 marks). (1 + m) = (1 + r)(1 + i), where m is the nominal rate, r the real rate and i inflation. (3)
Show the model answer to Question 12
(a) PPP (5 marks).
- Year 1: 1,500 × 1.18 ÷ 1.03 = ₦1,718.45. (2.5)
- Year 2: 1,718.45 × 1.18 ÷ 1.03 = ₦1,968.71. (2.5)
(b) IRP (4 marks). F = 1,500 × 1.20 ÷ 1.04 = ₦1,730.77. (4)
(c) Arbitrage (8 marks).
- IRP forward.
1.6000 × 1.01 ÷ 1.015 = $1.5921. (2) - Actual forward 1.5865 is lower, so buy forward pounds and borrow pounds now. (1)
- Borrow £6,250,000; owe £6,343,750. (1.5)
- Convert to $10m, invest at 1% for $10,100,000. (1.5)
- Cost of forward pounds.
6,343,750 × 1.5865 = $10,064,360. (1) - Profit = about $35,640. (1)
(d) Limits (3 marks). Transaction costs; limited liquidity at the quoted rates; exchange controls; interest rate changes. Any three. (1 each)
Show the model answer to Question 13
(a) Real rate (3 marks). 1 + r = 1.20 ÷ 1.15 = 1.0435, so r = 4.35%. (3)
(b) Expectations (3 marks). The forward rate reflects the market’s expectation of the future spot rate. (1.5) In practice news, capital flows and risk premiums move the actual rate away from it. (1.5)
(c) Forward (4 marks).
- Half-year rates 10% and 2%. (2)
- F = 1,500 × 1.10 ÷ 1.02 = ₦1,617.65. (2)
Pathfinder on PPP, IRP, Fisher effects, expectations theory and covered interest arbitrage
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Short-answer question 5 tested interest rate parity. Practise the forward rate formula and the worked examples on IRP and part-year rates.
Revise first: Interest rate parity
Case questions ask whether a quoted forward rate offers an arbitrage profit. See the section on covered interest arbitrage.
Revise first: Covered interest arbitrage
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: May 2014 short-answer 5.
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