Week 7: Interest Rate Parity (IRP) & The Fisher Effect — Theory and Numerical Problems
Learning Objectives
By the end of this session, students will be able to:
Explain Covered Interest Rate Parity (CIRP) — the no-arbitrage condition that equates the forward premium/discount to the interest rate differential — and compute implied forward rates from given spot rates and interest rates, identifying arbitrage opportunities when CIRP is violated.
Distinguish between Covered and Uncovered Interest Rate Parity (UIRP), explain why UIRP is an expectation-based condition while CIRP is an arbitrage-based condition, and discuss why UIRP frequently fails empirically in the short run.
Apply the Fisher Effect — the decomposition of nominal interest rates into real rates and expected inflation — and the International Fisher Effect (IFE) to forecast exchange rate movements, solving numerical problems that link interest rates, expected inflation, and expected currency appreciation/depreciation.
Integrate the three parity conditions — PPP, IRP, and the Fisher Effect — into a unified framework for understanding the long-run relationships among inflation, interest rates, and exchange rates, and solve multi-step numerical problems that chain these relationships.
4-Hour Session Planner
This is a numerically intensive session. Faculty should allocate substantial time to worked examples on the board and to guided problem-solving. The key formulae must be displayed prominently throughout.
Opening Hook: "Why Would Anyone Lend in Yen at 0.5% When They Could Lend in Rupees at 7%?"
15 minStudents confront the apparent free lunch: borrow in JPY at 0.5%, convert to INR, invest at 7%. The missing piece — forward exchange rates and expected depreciation — introduces IRP and the Fisher Effect as the theoretical framework that explains why this is NOT a free lunch.
Section 1: Interest Rates and Exchange Rates — The Connection
20 minWhy interest rates matter for exchange rates: the carry trade, capital mobility, and arbitrage. The analytical distinction between the spot market, the forward market, and the money market — and how these three markets are linked by arbitrage.
Section 2: Covered Interest Rate Parity (CIRP)
35 minCIRP as an arbitrage condition: (F/S) = (1+iₕ)/(1+i_f). Derivation with the two-strategy comparison: domestic investment vs. covered foreign investment. Numerical examples computing implied forward rates. Identifying and exploiting CIRP arbitrage opportunities. The limits of CIRP: transaction costs, capital controls, and why CIRP may fail for currencies like the INR during periods of market stress.
Section 3: Uncovered Interest Rate Parity (UIRP)
25 minUIRP as an expectations condition: E(e₁)/e₀ = (1+iₕ)/(1+i_f). The distinction: CIRP uses the forward rate (known today); UIRP uses the expected future spot rate (unknown). UIRP and the carry trade puzzle — why high-interest-rate currencies tend to appreciate, not depreciate, in the short run (the "forward premium puzzle").
CQ Box 1: CIRP Arbitrage Identification
10 minStudents are given spot rates, forward rates, and interest rates for INR and USD. They must determine whether CIRP holds, compute the implied forward rate, and — if CIRP is violated — calculate the arbitrage profit from a USD 1 million position.
Section 4: Forward Premium and Discount
25 minComputing the forward premium/discount. Annualising the forward points. The relationship: (F−S)/S ≈ iₕ−i_f. Using the forward premium as the market's forecast of expected depreciation. When the forward rate is a biased predictor (the forward premium puzzle).
In-Lecture Quiz (4 Questions)
10 minQuiz covering CIRP, UIRP, forward premium/discount calculations, and the distinction between arbitrage-based and expectations-based parity conditions.
Section 5: The Fisher Effect — Domestic & International
30 minThe Fisher Equation: i = r + E(π). Real interest rates as the fundamental driver. The International Fisher Effect (IFE): expected exchange rate change equals the interest rate differential (≈ iₕ−i_f). Derivation and numerical examples. The IFE as the bridge between IRP and PPP: when all three parity conditions hold, the real interest rate is equalised across countries.
CQ Box 2: Integrating Parity Conditions
15 minStudents work through a multi-step problem: given spot rate, inflation expectations, and real interest rates, compute (a) nominal interest rates via Fisher, (b) forward rate via CIRP, (c) expected future spot rate via UIRP/IFE, and (d) expected PPP rate. The four answers are compared — when all parities hold, they are internally consistent.
Section 6: Numerical Problems on IRP & Fisher Effect
40 minGuided problem-solving: 10 numerical problems covering CIRP (implied forward, arbitrage), forward premium/discount, UIRP (expected rate), Fisher Effect (decomposing nominal rates), IFE (forecasting from interest differentials), and integrated parity problems. Problems solved on the board with student participation; final 3 problems for independent practice.
Key Concepts Glossary & Exit Ticket
15 minFaculty reviews 12 key terms. Students complete Exit Ticket with a multi-step parity problem. Preview of Week 8 (Central Banks & Currency Crises — final week of Unit 2).
"Right now, you can borrow Japanese yen at 0.5% interest. You can convert those yen to Indian rupees and invest in an Indian government bond yielding 7.0%. That's a spread of 6.5% — free money, right? Borrow JPY 100 million (about INR 5.5 crore), invest at 7%, and pocket the difference. If it's that easy, why isn't everyone doing it? Why haven't hedge funds already arbitraged this spread away?"
Making the Carry Trade Concrete
This icebreaker introduces the carry trade — borrowing in a low-interest currency and investing in a high-interest currency — which is the practical strategy that IRP explains. Key teaching moments:
- If a student says "exchange rate risk": "Exactly. You have INR assets and JPY liabilities. If the yen appreciates, your yen debt becomes more expensive in INR terms. Specifically: you borrowed 100 million yen. If the yen appreciates from 0.55 to 0.50 INR/JPY, your repayment costs 50 lakh more INR. The question is: does the forward market allow you to eliminate this risk? And if you don't hedge — if you bet that the yen won't appreciate — what does the market expect the yen to do, and why?"
- If a student says "everyone would do it and the opportunity would disappear": "Yes — that's the arbitrage mechanism. If everyone borrows yen and buys rupees, the yen interest rate rises (more demand for yen loans), the rupee interest rate falls (more supply of rupee deposits), the yen appreciates in the spot market (more demand for yen to repay loans), and the rupee depreciates in the forward market (more forward sales of rupees to hedge). These four price adjustments are exactly what CIRP describes."
1. Interest Rates and Exchange Rates — The Connection
Interest rates and exchange rates are linked through the most powerful force in financial markets: arbitrage. When capital is mobile across borders, investors compare the returns available in different currencies — and their collective actions, seeking the highest risk-adjusted return, create equilibrium relationships among spot exchange rates, forward exchange rates, and interest rates. These relationships — the parity conditions — are the subject of this session.
Three Markets, One Arbitrage Logic:
- The Spot FX Market: Where currencies are traded for immediate delivery. The spot rate (S) is the price of one currency in terms of another today.
- The Forward FX Market: Where currencies are traded for delivery at a specified future date, at a price (the forward rate, F) agreed today. The forward market allows investors to eliminate exchange rate risk on known future cash flows.
- The Money Market (Domestic and Foreign): Where deposits are placed and loans are raised in each currency. The domestic and foreign interest rates (iₕ and i_f) are the returns available on risk-free deposits in each currency.
The arbitrageur can move capital across these three markets. By borrowing in one currency, converting at the spot rate, investing in the other currency, and covering the future repatriation with a forward contract, the arbitrageur constructs a covered position — one with no exchange rate risk. The return on this covered position must equal the return on a pure domestic investment, or arbitrageurs would exploit the difference until prices adjusted. This is Covered Interest Rate Parity (CIRP).
2. Covered Interest Rate Parity (CIRP)
2.1 The No-Arbitrage Condition
Covered Interest Rate Parity (CIRP) is an arbitrage-based condition. It states that the return on a domestic-currency risk-free investment must equal the covered return on a foreign-currency risk-free investment — where "covered" means the foreign-currency proceeds are sold forward at a known rate, eliminating exchange rate risk. If CIRP is violated, a risk-free arbitrage profit (no capital at risk, no exchange rate exposure) exists. The arbitrage trades themselves force prices to adjust until CIRP holds.
Derivation — Two Strategies Compared:
Suppose an investor has 1 unit of domestic currency to invest for one year.
Strategy A — Domestic Investment: Invest 1 unit at the domestic interest rate iₕ. After one year, the investor has: 1 × (1 + iₕ) units of domestic currency.
Strategy B — Covered Foreign Investment (3 Steps):
- Convert to foreign currency at the spot rate (S): 1 unit of domestic currency buys 1/S units of foreign currency.
- Invest at the foreign interest rate (i_f): After one year: (1/S) × (1 + i_f) units of foreign currency.
- Cover the repatriation with a forward contract at rate F: Sell the foreign-currency proceeds forward at the rate F agreed today. After one year: (1/S) × (1 + i_f) × F units of domestic currency — with zero exchange rate risk, because F is fixed in advance.
The CIRP Equation: In equilibrium, the two strategies must yield the same return:
1 × (1 + iₕ) = (1/S) × (1 + i_f) × F
Rearranged: F/S = (1 + iₕ) / (1 + i_f)
Or, expressed as the forward premium/discount:
(F − S) / S = (iₕ − i_f) / (1 + i_f) ≈ iₕ − i_f (approximation for small i_f)
The Predictions of CIRP:
- If iₕ > i_f: F > S — the foreign currency is at a forward premium (it is more expensive in the forward market than in the spot market). The domestic currency is at a forward discount. This compensates the domestic investor for the lower foreign interest rate — the forward premium on the foreign currency offsets the interest loss.
- If iₕ < i_f: F < S — the foreign currency is at a forward discount. The domestic currency is at a forward premium.
- The currency with the higher interest rate always trades at a forward discount. The currency with the lower interest rate always trades at a forward premium.
2.2 Numerical Examples — CIRP
Example 1 — Computing the Implied Forward Rate: The INR/USD spot rate is INR 83/USD. The one-year Indian interest rate is 7.0%. The one-year US interest rate is 5.0%. What one-year forward rate is implied by CIRP?
Solution: F = S × (1 + iₕ) / (1 + i_f) = 83 × 1.07 / 1.05 = 83 × 1.01905 = INR 84.58/USD.
Interpretation: The dollar is at a forward premium of INR 1.58 (84.58 − 83 = 1.58), or 1.90% (1.58/83). This premium compensates a US investor for the lower US interest rate — they get fewer dollars by investing in the US at 5% than by investing in India at 7% and covering forward, so the forward rate adjusts to eliminate the advantage.
Example 2 — Identifying CIRP Arbitrage: The spot rate is INR 83/USD. The 12-month forward rate quoted by a bank is INR 84.20/USD. India 1-year rate = 7.0%. US 1-year rate = 5.0%. Does CIRP hold? If not, describe the arbitrage.
Step 1 — Compute CIRP-implied forward: FCIRP = 83 × 1.07/1.05 = INR 84.58/USD.
Step 2 — Compare: Actual forward (84.20) < CIRP-implied forward (84.58). The dollar is cheaper in the forward market than CIRP requires. The covered return on the US investment is too low relative to the Indian investment — or, equivalently, the covered return on the Indian investment is too high relative to the US investment.
Step 3 — Arbitrage Strategy (borrow USD, invest in INR, cover forward):
- Borrow USD 1 million at 5% for 1 year → will owe USD 1,050,000 at maturity.
- Convert USD to INR at spot: USD 1M × 83 = INR 83,000,000.
- Invest INR at 7% for 1 year → will have INR 83,000,000 × 1.07 = INR 88,810,000.
- Sell INR 88,810,000 forward at 84.20 → will receive USD 88,810,000 / 84.20 = USD 1,054,750.
- Repay USD loan: USD 1,054,750 − USD 1,050,000 = arbitrage profit of USD 4,750 per USD 1 million of capital — risk-free, zero net investment.
2.3 When Does CIRP Fail?
CIRP is the tightest of all parity conditions because it is enforced by pure arbitrage — no capital is at risk, and there is no exchange rate exposure (the position is covered). However, CIRP can fail when arbitrage is impeded:
- Transaction Costs: Bid-ask spreads in the spot market, forward market, and money markets create a "neutral band" within which CIRP deviations are not profitable after costs.
- Capital Controls: Restrictions on cross-border capital movements — India's capital account is not fully convertible; there are limits on how much Indian residents can invest abroad and how much foreign investors can invest in Indian debt markets — prevent arbitrageurs from executing CIRP trades at scale. The onshore INR forward market frequently deviates from CIRP because the arbitrage channel (foreign investors borrowing abroad and investing in India, or Indian residents borrowing in India and investing abroad) is constrained.
- Political / Sovereign Risk: The risk that the government will impose capital controls, freeze foreign-currency accounts, or default on domestic debt — a "sovereign risk premium" that CIRP (which assumes risk-free rates) does not capture. This was visible during the 2008 financial crisis and the 2020 COVID crisis, when even major-currency CIRP deviations appeared.
- Counterparty Risk: The risk that the counterparty to the forward contract defaults. This risk rose sharply during the 2008 crisis, causing CIRP deviations for even the most liquid currency pairs.
3. Uncovered Interest Rate Parity (UIRP)
3.1 The Expectations-Based Condition
Uncovered Interest Rate Parity (UIRP) replaces the known forward rate (F) in the CIRP equation with the expected future spot rate (E[S1]). The investor does not cover the foreign-currency proceeds with a forward contract — the position is "uncovered," leaving the investor exposed to exchange rate risk. UIRP states that, in equilibrium, the expected return on the domestic investment equals the expected (uncovered) return on the foreign investment.
E[S1] / S0 = (1 + iₕ) / (1 + i_f)
Approximation: (E[S1] − S0) / S0 ≈ iₕ − i_f
The expected percentage change in the exchange rate equals approximately the interest rate differential.
If iₕ > i_f, the domestic currency is expected to depreciate by approximately (iₕ − i_f).
Critical Distinction — CIRP vs. UIRP:
- CIRP uses the forward rate (F) — a price that is quoted in the market today and is certain. CIRP is an arbitrage condition — it must hold (within transaction costs) or risk-free profits exist.
- UIRP uses the expected future spot rate (E[S₁]) — a subjective forecast that no one knows with certainty. UIRP is an expectations (equilibrium) condition — it says what the market must expect on average, but deviations can persist because they do not create risk-free arbitrage opportunities.
- If both CIRP and UIRP hold, then F = E[S₁] — the forward rate is an unbiased predictor of the future spot rate.
3.2 The UIRP Puzzle (The Forward Premium Puzzle)
UIRP is one of the most thoroughly tested — and most thoroughly rejected — propositions in international finance. The empirical evidence consistently finds that:
- High-interest-rate currencies tend to appreciate, not depreciate, in the short run — the opposite of what UIRP predicts.
- The forward premium — which, under UIRP, should predict depreciation of the high-interest-rate currency — actually predicts appreciation over short to medium horizons (1–12 months).
This is the Forward Premium Puzzle (also called the UIRP Puzzle or the Fama Puzzle, after Eugene Fama's 1984 paper). It implies that the carry trade — borrowing in a low-interest currency and investing in a high-interest currency, without hedging the exchange rate risk — is profitable on average. Investors earn both the interest differential AND an exchange rate gain (as the high-interest currency appreciates). This is a puzzle because it contradicts the core prediction of UIRP.
Proposed Explanations: (a) The high-interest currency carries a risk premium — investors demand compensation for bearing the risk that the currency will crash (the "peso problem"); the interest differential is not an expected depreciation but a risk premium. (b) Slow adjustment of expectations — investors under-react to interest rate changes, so the initial exchange rate move is too small, and subsequent moves are in the "wrong" direction. (c) Central bank intervention — in managed-float currencies (like the INR), the central bank's intervention smooths depreciation, so the forward premium predicts depreciation over the long run but not the short run.
Numerical Example — UIRP: The INR/USD spot rate is 83. India's 1-year rate = 7%. US 1-year rate = 5%. Under UIRP, what is the expected spot rate in one year?
E[S₁] = 83 × 1.07/1.05 = INR 84.58/USD. The rupee is expected to depreciate by 1.58 (1.90%). But note: UIRP is an expectation, not an arbitrage condition. The actual spot rate in one year could be 80 (appreciation — UIRP failed) or 90 (depreciation more than predicted) or anywhere else. UIRP says what the market's expectation should be in equilibrium; it does not guarantee that the expectation is correct.
4. Forward Premium and Discount — Computation and Interpretation
4.1 Definitions and Formulae
The forward premium (or discount) is the percentage difference between the forward rate and the spot rate. It indicates whether a currency is more expensive (premium) or cheaper (discount) in the forward market than in the spot market.
Premium/Discount (%) = (F − S) / S × 100
Annualised Forward Premium/Discount:
Annualised (%) = [(F − S) / S] × (12 / n) × 100
where n = number of months to forward maturity
Under CIRP, the annualised forward premium/discount approximately equals the annual interest rate differential: (F − S)/S × 12/n × 100 ≈ iₕ − i_f
Numerical Example — Forward Premium: Spot: INR 83/USD. 3-month forward: INR 83.40/USD.
Forward premium on USD (the foreign currency) = (83.40 − 83) / 83 × 100 = 0.48% (for 3 months).
Annualised premium on USD = 0.48% × (12/3) = 1.93% per annum.
Under CIRP, this implies the Indian 3-month interest rate exceeds the US 3-month rate by approximately 1.93 percentage points (annualised).
The Forward Rate as a Forecast — CIRP + UIRP Together:
If both CIRP and UIRP hold simultaneously, then F = E[S₁] — the forward rate is the market's unbiased forecast of the future spot rate. Any deviation of the future spot rate from the forward rate is a random forecast error with mean zero. This is the Unbiased Forward Rate Hypothesis (UFRH). In practice, the UFRH is rejected for most currency pairs at short to medium horizons, for the same reasons UIRP is rejected (the forward premium puzzle). The forward rate is a biased predictor — it systematically overpredicts depreciation of the high-interest currency (or, equivalently, the high-interest currency systematically appreciates relative to the forward rate).
You observe the following market data:
• INR/USD spot rate (S): INR 83.00/USD
• 6-month INR/USD forward rate (F): INR 83.75/USD
• 6-month Indian interest rate (annualised): 7.20% p.a. → 3.60% for 6 months
• 6-month US interest rate (annualised): 5.00% p.a. → 2.50% for 6 months
(a) Does Covered Interest Rate Parity hold? Compare the actual forward rate to the CIRP-implied forward rate.
(b) If CIRP is violated, design an arbitrage strategy to exploit the mispricing. Assume you can borrow USD 5 million (or the INR equivalent). Specify: which currency do you borrow, which do you invest, at what rates, and what forward contract do you enter?
(c) Calculate the arbitrage profit in USD.
(d) What market adjustments would you expect to eliminate this arbitrage opportunity? Which rates would move, and in which direction?
For (a): CIRP-implied F = S × (1 + iₕ_period) / (1 + i_f_period). For 6 months, use the 6-month (not annualised) interest rates: 3.60% and 2.50%.
Guided Solution
(a) CIRP check: FCIRP = 83.00 × 1.0360 / 1.0250 = 83.00 × 1.01073 = INR 83.89/USD. Actual forward (83.75) < CIRP forward (83.89). The USD is undervalued (cheaper) in the forward market. CIRP is violated — an arbitrage opportunity exists.
(b) Arbitrage strategy: The USD forward is too cheap → buy USD forward (which means sell INR forward). To create a risk-free position, borrow USD, convert to INR at spot, invest INR, and sell INR forward to cover the repatriation.
Steps: (1) Borrow USD 5,000,000 at 2.50% for 6 months → will owe USD 5,125,000. (2) Convert to INR at spot: USD 5M × 83.00 = INR 415,000,000. (3) Invest INR at 3.60% for 6 months → INR 415M × 1.036 = INR 429,940,000. (4) Sell INR 429,940,000 forward at 83.75 → will receive USD 429,940,000 / 83.75 = USD 5,133,612.
(c) Profit: USD 5,133,612 − USD 5,125,000 = USD 8,612 risk-free profit per USD 5 million.
(d) Market adjustments: Arbitrageurs borrowing USD → US interest rate rises. Converting USD to INR at spot → INR appreciates (S falls). Investing INR → Indian interest rate falls. Selling INR forward → INR forward rate depreciates (F rises). All four adjustments push toward CIRP restoration: F rises, S falls, iₕ falls, i_f rises.
5. The Fisher Effect — Domestic and International
5.1 The Domestic Fisher Effect
Irving Fisher (1930) proposed that the nominal interest rate in any economy can be decomposed into two components: a real interest rate (r), which reflects the productivity of capital and society's rate of time preference; and expected inflation E(π), which compensates the lender for the erosion of purchasing power over the loan period.
(1 + i) = (1 + r) × (1 + E[π])
The Fisher Equation (Approximation):
i ≈ r + E[π]
The nominal interest rate approximately equals the real interest rate plus expected inflation.
Numerical Example — Domestic Fisher Effect: If the real interest rate in India is 2.5% and expected inflation is 4.5%, the nominal interest rate should be approximately: i = 2.5% + 4.5% = 7.0%. Using the exact formula: (1.025)(1.045) − 1 = 7.11%.
The Key Implication: Differences in nominal interest rates across countries primarily reflect differences in expected inflation, not differences in real returns. If real interest rates are approximately equal across countries (due to capital mobility), then a country with higher expected inflation must have higher nominal interest rates — the Fisher Effect.
5.2 The International Fisher Effect (IFE)
The International Fisher Effect (IFE) combines the Fisher Effect with the assumption that real interest rates are equalised across countries. When real rates are equal: iₕ − i_f = E[πₕ] − E[π_f] — the nominal interest differential equals the expected inflation differential. And from relative PPP, the expected inflation differential equals the expected exchange rate change: E[πₕ] − E[π_f] ≈ expected depreciation of the domestic currency. Therefore:
E[S₁]/S₀ = (1 + iₕ) / (1 + i_f)
Approximation: Expected %ΔS ≈ iₕ − i_f
Note: The IFE equation is identical to UIRP. The IFE is UIRP derived from the Fisher Effect + PPP, rather than from uncovered arbitrage. The IFE and UIRP are theoretically equivalent, but they highlight different causal mechanisms: the IFE emphasises inflation and real rates; UIRP emphasises expected returns and capital flows.
Numerical Example — IFE: India's nominal interest rate = 7.0%. US nominal interest rate = 5.0%. The IFE predicts the rupee will depreciate by approximately 2.0% against the USD over the relevant horizon. If the current spot rate is INR 83/USD, the IFE-expected future spot rate is approximately INR 84.66/USD (using the exact formula: 83 × 1.07/1.05 = INR 84.58/USD; using approximation: 83 × 1.02 = INR 84.66/USD).
5.3 The Unified Parity Framework
When all three parity conditions — PPP, IRP (CIRP), and the Fisher Effect — hold simultaneously, they form a unified framework linking four sets of variables:
| Parity Condition | Links These Variables | Key Prediction |
|---|---|---|
| Covered IRP (CIRP) | F, S, iₕ, i_f | F/S = (1+iₕ)/(1+i_f). Interest differential → Forward premium/discount. |
| Uncovered IRP (UIRP) / IFE | E[S₁], S, iₕ, i_f | E[S₁]/S = (1+iₕ)/(1+i_f). Interest differential → Expected depreciation. |
| Purchasing Power Parity (PPP) | E[S₁], S, E[πₕ], E[π_f] | E[S₁]/S = (1+E[πₕ])/(1+E[π_f]). Inflation differential → Expected depreciation. |
| Fisher Effect (each country) | i, r, E[π] | (1+i) = (1+r)(1+E[π]). Nominal rate = Real rate + Expected inflation. |
| Real Interest Rate Parity | rₕ, r_f | rₕ = r_f. Real rates equalised globally by capital mobility. (Holds if all three parities above hold simultaneously.) |
When all parities hold, the chain is closed: expected inflation differential → nominal interest differential (via Fisher) → forward premium/discount (via CIRP) → expected depreciation (via UIRP/IFE) → exactly matches the expected inflation differential (via PPP). The four variables — expected inflation, nominal interest rates, forward rates, and expected future spot rates — are internally consistent. In reality, this perfect consistency rarely holds, particularly in the short run. But the unified framework provides the analytical structure for understanding which parity is violated and why — knowledge that the financial manager can use to identify mispricing, anticipate corrections, and construct hedging strategies.
You are given the following data:
• INR/USD spot rate: INR 83.00/USD
• Expected inflation: India = 5.5%, US = 2.5%
• Real interest rate (assumed equal in both countries): 2.0%
Assuming all parity conditions hold simultaneously:
(a) Use the Fisher Effect to compute the nominal interest rate in India and the US (use approximation: i ≈ r + E[π]).
(b) Use CIRP to compute the 1-year forward rate (use exact formula: F = S × (1+iₕ)/(1+i_f)).
(c) Use PPP (exact formula) to compute the expected spot rate one year from now: E[S₁] = S × (1+E[πₕ])/(1+E[π_f]).
(d) Use UIRP/IFE (exact formula) to compute the expected spot rate: E[S₁] = S × (1+iₕ)/(1+i_f).
(e) Compare your answers to (b), (c), and (d). What do you observe, and why is this consistency expected when all parity conditions hold? If the actual market forward rate were 84.00 (instead of your CIRP-implied rate), which parity condition(s) would be violated, and what would that imply for an arbitrageur or a speculator?
When all parities hold, (b), (c), and (d) should give the same answer — the forward rate, the PPP-expected spot rate, and the UIRP-expected spot rate are all equal. This is the unified parity framework in action. For (e): if the actual forward rate differs from the CIRP rate, CIRP is violated → risk-free arbitrage exists (assuming no capital controls or transaction costs).
In-Lecture Formative Quiz
4 Questions • 10 MinutesSelect the best answer for each question, then click Check Answers.
1. Covered Interest Rate Parity (CIRP) differs from Uncovered Interest Rate Parity (UIRP) primarily because:
2. The INR/USD spot rate is 82. India's 1-year interest rate is 6.5%. The US 1-year rate is 4.5%. According to CIRP, what should the 1-year forward rate be (rounded to two decimals)?
3. According to the Fisher Effect (exact form), if the real interest rate is 2% and expected inflation is 4%, the nominal interest rate should be:
4. The International Fisher Effect (IFE) predicts that if India's nominal interest rate is 7% and the US nominal rate is 5%, then:
6. Numerical Problems on IRP & the Fisher Effect
Solve the following problems. For each, show the formula, substitute values, compute the result, and interpret. Problems 1–6 are guided; 7–10 are for independent practice.
CIRP (implied forward): F = S × (1 + iₕ_period) / (1 + i_f_period)
Forward Premium (annualised): [(F − S)/S] × (12/n) × 100
UIRP (expected future spot): E[S₁] = S × (1 + iₕ) / (1 + i_f)
Fisher (exact): (1 + i) = (1 + r)(1 + E[π])
IFE: Expected %ΔS ≈ iₕ − i_f (or exact form same as UIRP)
Guided Practice (Problems 1–6)
Problem 1 — CIRP Implied Forward: INR/USD spot = 83.00. India 1-year rate = 6.80%. US 1-year rate = 5.20%. (a) Compute the CIRP-implied 1-year forward rate (exact formula). (b) Interpret: which currency is at a forward premium? Why?
Solution: F = 83 × 1.068/1.052 = 83 × 1.01521 = INR 84.26/USD. The USD is at a forward premium (84.26 > 83.00) because the US interest rate is lower — CIRP compensates via the forward rate.
Problem 2 — Forward Premium Computation: Spot: INR 83/USD. 3-month forward: INR 83.55/USD. (a) Compute the 3-month forward premium on the USD. (b) Annualise it. (c) What annual interest rate differential does this imply under CIRP?
Solution: (a) 3-month premium = (83.55 − 83)/83 = 0.66%. (b) Annualised = 0.66% × 12/3 = 2.65% p.a. (c) Under CIRP, the annualised forward premium ≈ i_IN − i_US ≈ 2.65 percentage points. Indian rates exceed US rates by ~265 bps.
Problem 3 — CIRP Arbitrage: Spot: INR 82.50/USD. 6-month forward: INR 83.10/USD. India 6-month rate (annualised): 7.50% → 3.75% for 6 months. US 6-month rate (annualised): 5.50% → 2.75% for 6 months. Does CIRP hold? If not, compute the arbitrage profit from borrowing USD 2 million.
Solution: F_CIRP = 82.50 × 1.0375/1.0275 = INR 83.30/USD. Actual (83.10) < CIRP (83.30). USD forward is undervalued. Arbitrage: Borrow USD 2M at 2.75% → owe USD 2,055,000. Convert at spot to INR 165,000,000. Invest at 3.75% → INR 171,187,500. Sell INR forward at 83.10 → USD 2,059,600. Profit = 2,059,600 − 2,055,000 = USD 4,600.
Problem 4 — Fisher Effect: India's expected inflation = 5.0%. If the real interest rate is 2.0%, (a) compute the nominal interest rate using the exact Fisher equation. (b) Compute it using the approximation. (c) What is the cross-term (r × E[π])?
Solution: (a) i = (1.02)(1.05) − 1 = 1.071 − 1 = 7.10%. (b) Approx: i ≈ 2% + 5% = 7.00%. (c) Cross-term = 0.02 × 0.05 = 0.001 = 0.10 percentage points — the difference between exact (7.10%) and approximate (7.00%).
Problem 5 — IFE Forecasting: Current spot: INR 83/USD. India nominal rate = 7.0%. US nominal rate = 4.5%. Using the IFE (exact formula), compute the expected spot rate in one year.
Solution: E[S₁] = 83 × 1.07/1.045 = 83 × 1.02392 = INR 84.99/USD. The rupee is expected to depreciate by approximately 2.39%.
Problem 6 — Integrated Parities: Spot: INR 84/USD. India expected inflation = 5.5%. US expected inflation = 2.0%. Real rate (equal globally) = 1.8%. (a) Compute Indian and US nominal rates (Fisher). (b) Compute CIRP 1-year forward. (c) Compute PPP-expected spot rate in 1 year. (d) Verify that the three answers are consistent.
Solution: (a) i_IN = (1.018)(1.055) − 1 = 7.40%. i_US = (1.018)(1.02) − 1 = 3.84%. (b) F = 84 × 1.074/1.0384 = INR 86.88/USD. (c) E[S₁] via PPP = 84 × 1.055/1.02 = INR 86.88/USD. (d) They are identical — the integrated parity framework is internally consistent.
Independent Practice (Problems 7–10)
Problem 7: Spot: INR 85/USD. 1-year India rate = 6.5%, 1-year UK rate = 4.0%. The GBP/INR spot rate is INR 108/GBP. Compute the CIRP-implied 1-year GBP/INR forward rate.
Problem 8: A bank quotes a 1-month USD/INR forward at INR 83.25/USD. Spot is INR 83.00/USD. Compute the annualised forward premium on the USD. If the 1-month India T-bill rate is 6.80% p.a. and the 1-month US T-bill rate is 5.30% p.a., does CIRP approximately hold?
Problem 9: India's nominal interest rate for 5-year bonds is 7.2%. The US 5-year rate is 4.1%. The 5-year expected inflation rate is 4.5% in India and 2.2% in the US. (a) Compute the real interest rates in India and the US (exact formula). Are they approximately equal? (b) Using IFE, compute the expected INR/USD rate in 5 years if today's spot is 83.
Problem 10 (Challenge): An Indian MNC must pay USD 20 million in 6 months. The treasurer is considering two strategies: (A) Buy USD 20 million in the 6-month forward market at INR 84.20/USD. (B) Invest INR today in a USD-denominated deposit at 5.0% p.a. (2.5% for 6 months) sufficient to grow to USD 20 million. The spot rate is INR 83.50/USD. India's 6-month borrowing rate is 7.5% p.a. (3.75% for 6 months). Compute the INR cost under each strategy. Which is cheaper? Why? (Hint: Strategy B involves buying USD at the spot rate today and investing.)
7. Scenario Debate: Parity Conditions in Corporate Practice
Four persona cards. Each group applies IRP, Fisher Effect, and the unified parity framework to real-world financial decisions.
Sterling Motors must pay JPY 500 million in 3 months for imported robotic assembly equipment from a Japanese supplier. The current spot rate is INR 0.56/JPY. The 3-month forward rate quoted by Sterling's bank is INR 0.565/JPY. India's 3-month T-bill rate is 6.80% p.a. (1.70% for 3 months). Japan's 3-month T-bill rate is 0.20% p.a. (0.05% for 3 months). The CFO wants to know: (a) Should Sterling hedge using the forward contract? (b) Is the forward rate fair — does CIRP hold? If not, is there a cheaper way to lock in the JPY payment? (c) If Sterling does not hedge, what is the IFE-expected JPY/INR rate, and is the exposure worth bearing?
Compute the CIRP-implied 3-month forward rate. Compare it to the bank's quoted forward. Is the bank's quote fair, or is the bank extracting a margin beyond CIRP? Should Sterling hedge? If CIRP is violated, describe a money-market hedge — borrowing/investing in INR and JPY to create a synthetic forward — and compare its cost to the bank's forward.
GreenLeaf is bidding for a 25-year solar power purchase agreement (PPA) with the Rajasthan government. The PPA tariff will be fixed in INR for 25 years — no indexation to inflation or exchange rates. GreenLeaf is considering two financing options for the INR 1,200 crore project: (A) 100% INR debt at 8.5% fixed for 15 years from an Indian bank consortium; (B) 60% USD debt through an ECB (External Commercial Borrowing) at SOFR + 250 bps (~7.5% currently, floating) and 40% INR debt at 8.5%. The INR/USD spot rate is 83. India's 15-year expected inflation is 5%. US 15-year expected inflation is 2.5%.
Using the Fisher Effect (real vs. nominal rates) and IFE (expected long-term INR depreciation), evaluate the two financing options. What is the approximate real INR interest rate under Option A? Under Option B, what is the IFE-expected INR cost of the USD debt over 15 years? Which option exposes GreenLeaf to more risk — and which offers a lower expected cost? Is the "cheaper" USD debt truly cheaper when PPP-implied long-term depreciation is factored in?
NexGen Capital runs a systematic carry trade strategy: it borrows in low-interest-rate currencies (JPY at 0.25%, CHF at 1.5%) and invests in high-interest-rate currencies (INR at 7%, IDR at 6%) on an unhedged basis. The strategy has delivered annualised returns of 8–10% over the past 5 years — far above the interest differential alone — because the INR and IDR have appreciated against the JPY in several of those years. However, NexGen's risk committee is concerned: the strategy survived the 2013 Taper Tantrum and the 2020 COVID shock, but a prolonged yen appreciation (as Japan normalises interest rates from near-zero) could reverse years of profits in weeks. Suhas must present a risk assessment to the board.
Using UIRP and the Forward Premium Puzzle: (a) Explain why the carry trade has been profitable — why have high-interest-rate currencies systematically appreciated rather than depreciated as UIRP predicts? (b) Under what conditions would UIRP "reassert itself" — causing the INR and IDR to depreciate sharply against the JPY — and what macroeconomic triggers (interest rate policy changes, risk-off episodes, BOP shocks) could cause this? (c) Recommend a risk-management framework: should NexGen continue the unhedged carry trade, hedge partially, or exit entirely?
Meenakshi's team advises corporate clients on FX risk management. A client — a mid-sized Indian pharma exporter — has USD 50 million of forecast export receivables over the next 12 months. The client's board is split: half wants to hedge 100% using 12-month forwards at INR 84.50/USD (the current market quote); half wants to remain unhedged, arguing that the rupee consistently depreciates more than the forward rate implies, so hedging destroys value. The spot rate is INR 83/USD. India 1-year rate = 6.8%. US 1-year rate = 5.1%. India's expected inflation = 5.5%. US expected inflation = 2.5%.
As the adviser: (a) Compute the CIRP-implied 1-year forward rate. Is the bank's forward quote (84.50) consistent with CIRP? (b) Compute the PPP-implied expected spot rate (relative PPP). (c) Compare the market forward (84.50), the CIRP forward, and the PPP-expected rate. Which rate implies a greater rupee depreciation? What does this pattern suggest about the market's expectations vs. PPP fundamentals? (d) Advise the board: should the firm hedge? If yes, what percentage and why? If the board is divided, what analytical framework can resolve the disagreement?
Activity Structure
Setup (2 min): Four groups. 10 minutes analysis. 3-minute presentations. Synthesis (5 min).
Prompt cards:
- Sterling Motors (P1): "The money-market hedge is a synthetic forward — borrow INR, convert to JPY at spot, invest JPY, and use the JPY proceeds to pay the supplier. Its cost should equal the CIRP-implied forward rate. If the bank's forward quote deviates from CIRP, compare the two and choose the cheaper. This is CIRP in practical use."
- GreenLeaf (P2): "The IFE-expected depreciation of the rupee reduces the effective INR cost of USD debt over 25 years. But the IFE is an expectation, not a guarantee. If the rupee depreciates less than expected (or appreciates — as it has over some periods), the USD debt becomes more expensive in INR terms. The financing decision is a bet on whether PPP and IFE hold over the project's life."
- NexGen Capital (P3): "NexGen is the real-world embodiment of the UIRP puzzle. The carry trade's profitability is the puzzle — it should not work if UIRP holds, but it does. The question for the risk committee is: is this an enduring feature of FX markets (a risk premium that can be harvested systematically) or a temporary anomaly that will end painfully when the regime changes?"
- GlobalBank (P4): "Meenakshi's case is the corporate hedging dilemma in its purest form. The forward rate (84.50) implies depreciation of ~1.8%. PPP implies depreciation of ~3.0%. If the client believes PPP is the better long-run guide, the forward rate understates depreciation — and hedging would lock in a rate that is too strong. But the forward rate is the only rate the client can actually transact at today — PPP is a forecast, not a tradeable price. The analytical resolution: hedging is insurance, not speculation. Insure what you cannot afford to lose."
8. Fishbowl Debate: Is the Carry Trade a Legitimate Investment Strategy or Reckless Speculation?
Debate Proposition
"This House believes that the carry trade is a legitimate, risk-premium-based investment strategy that sophisticated institutions can deploy responsibly, and that its occasional blow-ups (2008, 2013) are manageable risks rather than evidence that the strategy is fundamentally flawed."
Position A: Carry Trade IS Legitimate
- The UIRP puzzle is real — the carry trade harvests a risk premium: Decades of empirical evidence show that high-interest currencies systematically deliver excess returns over low-interest currencies. This is not a free lunch — it is compensation for bearing "crash risk" (the risk that the high-yield currency depreciates sharply during global risk-off episodes). Investors who can bear this risk — pension funds, endowments, diversified portfolios — are compensated with a positive expected return.
- The strategy can be risk-managed: A disciplined carry trade programme uses position limits, stop-losses, diversification across 8–12 currency pairs, and dynamic hedging that increases during elevated volatility. The blow-ups are not inherent to the strategy — they are failures of risk management, not failures of the underlying logic.
- Carry trades provide liquidity to the FX market: When investors borrow in JPY/CHF and invest in INR/BRL/TRY, they provide liquidity to both sides of the market — they are natural buyers of high-yield currencies and sellers of low-yield currencies. This narrows spreads and deepens markets, benefiting all participants.
Position B: Carry Trade IS Reckless Speculation
- The strategy is picking up nickels in front of a steamroller: The carry trade generates small, steady gains for years and then suffers catastrophic losses — losses that can exceed decades of accumulated profits — during global risk-off episodes. The 2008 financial crisis (JPY carry trade unwinding — JPY appreciated 20%+ against high-yield currencies in weeks), the 2013 Taper Tantrum (INR depreciated 20%+), and the 2015 Swiss franc shock (CHF appreciated 30% against EUR in one day when the SNB removed the EUR/CHF floor) demonstrate that the strategy's risk is not normally distributed — it has severe negative skew and fat tails.
- UIRP fails but that doesn't mean the carry trade is a "premium" — it could be a bubble: The profitability of the carry trade could reflect a slow-moving capital-flow cycle rather than a genuine risk premium. As more investors pour into the carry trade, they push high-yield currencies higher (creating the very appreciation that makes the strategy profitable), attracting yet more investors — a self-reinforcing cycle that ends when the last marginal investor enters and the reversal begins.
- Carry trades amplify systemic risk: When a shock triggers a carry-trade unwind, the selling of high-yield currencies and buying of funding currencies is simultaneous, concentrated, and self-reinforcing — margin calls force more selling, which triggers more margin calls. This "carry crash" dynamic amplifies currency volatility and transmits stress across financial markets. Central banks in emerging economies (including the RBI) are forced to deploy reserves and raise rates to defend their currencies — imposing costs on the real economy.
Synthesis
"The carry trade debate is the practical manifestation of the UIRP puzzle. If UIRP held, the carry trade would have zero expected profit — the interest gain would be exactly offset by expected depreciation. That the carry trade has been profitable for decades tells us something important about FX markets — either there is a systematic risk premium, or UIRP fails in a predictable direction. For the corporate financial manager, the lesson is: do not base your hedging decisions on the assumption that UIRP holds in the short run. The forward rate is not necessarily the market's unbiased forecast. Understand the parity conditions, but respect their empirical limitations."
9. Key Concepts & Terminology — Week 7
Covered Interest Rate Parity (CIRP)
An arbitrage condition: the return on a domestic risk-free investment must equal the covered (hedged) return on a foreign risk-free investment. F/S = (1+iₕ)/(1+i_f). Enforced by arbitrage — if violated, risk-free profits exist (absent capital controls and transaction costs).
Uncovered Interest Rate Parity (UIRP)
An expectations condition: the expected return on a domestic investment equals the uncovered (unhedged) expected return on a foreign investment. E[S₁]/S₀ = (1+iₕ)/(1+i_f). Not enforced by arbitrage — empirically fails in the short run (the "forward premium puzzle").
Forward Premium/Discount
The percentage difference between the forward rate and the spot rate: (F−S)/S. A positive value means the foreign currency is at a forward premium (more expensive forward than spot). Under CIRP, the currency with the lower interest rate trades at a forward premium.
Forward Premium Puzzle (Fama Puzzle)
The empirical finding that the forward premium predicts appreciation of the high-interest-rate currency — the opposite of what UIRP predicts. High-interest-rate currencies tend to appreciate (not depreciate) in the short run, making the unhedged carry trade profitable. One of the most robust puzzles in international finance.
Fisher Effect (Domestic)
Irving Fisher's decomposition of the nominal interest rate into the real interest rate and expected inflation: (1+i) = (1+r)(1+E[π]). The nominal rate compensates the lender for both the real return to capital and the expected erosion of purchasing power.
International Fisher Effect (IFE)
The proposition that the expected exchange rate change between two currencies equals the nominal interest rate differential: E[S₁]/S₀ = (1+iₕ)/(1+i_f). Derived from combining the Fisher Effect with PPP. The currency with the higher nominal interest rate is expected to depreciate.
Real Interest Rate Parity
The condition that real interest rates are equalised across countries: rₕ = r_f. Holds when Fisher Effect, PPP, and CIRP all hold simultaneously. Implies that differences in nominal interest rates reflect only differences in expected inflation, not differences in real returns.
Carry Trade
An investment strategy of borrowing in a low-interest-rate (funding) currency and investing in a high-interest-rate (target) currency, typically unhedged. Profitable on average (the UIRP puzzle) but subject to crash risk — sharp reversals during global risk-off episodes when funding currencies appreciate rapidly.
Unbiased Forward Rate Hypothesis (UFRH)
The joint hypothesis that both CIRP and UIRP hold, implying F = E[S₁] — the forward rate is an unbiased predictor of the future spot rate. Rejected empirically: the forward rate is a biased predictor (the forward premium puzzle).
Money-Market Hedge
A synthetic forward contract created by borrowing in one currency, converting to another at the spot rate, and investing in that currency. The cost of a money-market hedge should equal the CIRP-implied forward rate. Used by firms when bank forward rates deviate from CIRP or when forward markets are thin or unavailable.
Covered Interest Arbitrage (CIA)
The arbitrage trade that enforces CIRP: borrowing in the currency with the lower covered return, converting at spot, investing in the currency with the higher covered return, and covering the future repatriation with a forward contract. Risk-free profit when CIRP is violated (net of transaction costs and capital controls). Studied in detail in Week 11.
Peso Problem
A proposed explanation for the UIRP puzzle: the expected depreciation of high-interest-rate currencies does occur, but only in rare, catastrophic events (currency crashes). During normal times, the depreciation does not materialise, making ex-post returns look anomalously high — but the premium is compensation for the crash that eventually (but infrequently) arrives. Named after the Mexican peso, which was consistently at a forward discount against the USD until it crashed in 1976.
Exit Ticket — Week 7
Complete each section. Estimated time: 7–10 minutes.
Describe the most important concept or insight from this session — CIRP, UIRP, the Fisher Effect, or the carry trade debate.
Identify one concept that remains unclear. If you struggle with the difference between CIRP and UIRP, or between the Fisher Effect and the International Fisher Effect, articulate what confuses you.
Spot = 84 INR/USD. r_real (global) = 1.5%. India expected inflation = 5.0%. US expected inflation = 2.0%. (a) Compute nominal rates via Fisher. (b) Compute CIRP 1-year forward. (c) Compute PPP-expected spot. (d) Compute IFE-expected spot. Verify consistency.
As a future finance professional, you will encounter forward rates, interest rate differentials, and exchange rate forecasts daily. In 3–4 sentences, explain: (a) What CIRP tells you that you can act on (an arbitrageable price relationship). (b) What UIRP/IFE tells you that you can use for planning but not bet on (an expected relationship). Use a specific example from this week's content.
10. Session References & Further Reading
Required Reading
- Eun, C., Resnick, B., & Chuluun, T. — International Financial Management, McGraw Hill. Chapter 6: "International Parity Relationships and Forecasting Foreign Exchange Rates."
- Apte, P. G., & Kapshe, S. — International Financial Management, McGraw Hill. Chapter 5: Sections on IRP and Fisher Effect.
Classic Works
- Fisher, I. (1930). The Theory of Interest. New York: Macmillan.
- Fama, E. F. (1984). "Forward and Spot Exchange Rates." Journal of Monetary Economics, 14(3), 319–338.
- Burnside, C., Eichenbaum, M., & Rebelo, S. (2011). "Carry Trade and Momentum in Currency Markets." Annual Review of Financial Economics, 3, 511–535.