Interest Rate Parity and Pricing of Currency Futures
This sub‑topic covers Interest Rate Parity (IRP) and how it is used to price exchange‑traded currency futures. Understanding IRP helps you calculate the theoretical futures price and evaluate whether a quoted price is mis‑priced. The concept links foreign exchange rates with domestic and foreign interest rates, a frequent exam question in NISM Series I. Mastery of IRP also clarifies the relationship between forwards and futures, which is essential for the certification.
Learning Objectives
- 1Define Interest Rate Parity and its assumptions.
- 2Derive the IRP formula using simple compounding.
- 3Apply IRP to compute the theoretical price of a currency future.
- 4Distinguish between forward contracts and exchange‑traded currency futures.
Interest Rate Parity (IRP) – Concept
Interest Rate Parity states that the forward (or futures) exchange rate must reflect the difference between the domestic and foreign risk‑free interest rates for the same maturity. If the parity does not hold, arbitrageurs could earn a risk‑free profit by borrowing in the cheaper‑rate currency, converting at the spot rate, investing in the higher‑rate currency, and closing the position in the forward market.
The parity condition assumes perfect capital mobility, no transaction costs, and that both currencies are freely convertible. In the Indian context, the domestic rate is the RBI’s benchmark or the yield on a risk‑free INR instrument, while the foreign rate is typically the US Treasury yield for USD.
For the NISM exam, questions often present spot rate, domestic and foreign rates, and the time to maturity, asking you to compute the forward or futures price. Remember that the rates must be expressed on the same time basis (annualised) and that the period T is expressed in years.
- Spot rate (S) – current market price of 1 foreign currency unit in INR.
- Domestic rate (r₁) – annual risk‑free rate in India.
- Foreign rate (r₂) – annual risk‑free rate in the foreign currency’s country.
Students often mix up the numerator and denominator when applying IRP, placing the domestic rate in the denominator. The correct formula has the domestic rate in the numerator because you are effectively converting INR into the foreign currency.
Mathematical Expression of IRP
Where:
F= Forward (or futures) exchange rate in INR per unit of foreign currencyS= Spot exchange rate in INR per unit of foreign currencyr_{\text{dom}}= Annual domestic risk‑free interest rate (in decimal)r_{\text{for}}= Annual foreign risk‑free interest rate (in decimal)T= Time to maturity in years (e.g., 0.5 for six months)Worked Example
Given S = 75 INR/USD, r_{\text{dom}} = 6% (0.06), r_{\text{for}} = 2% (0.02), T = 0.5 years: Step 1: Compute numerator = 1 + 0.06 \times 0.5 = 1.03 Step 2: Compute denominator = 1 + 0.02 \times 0.5 = 1.01 Step 3: F = 75 \times (1.03 / 1.01) = 75 \times 1.01980198 = 76.485149 Verification: 75 \times \frac{1 + 0.06 \times 0.5}{1 + 0.02 \times 0.5} = 76.49 (rounded to two decimals).
The IRP formula uses simple interest because the period T is usually short (days to months) and the rates are quoted on an annual basis. If the exam explicitly mentions continuous compounding, you must switch to the exponential form, but most NISM questions use the simple‑interest version.
All rates must be expressed in the same units. For example, a 6‑month contract requires converting the annual rates to a half‑year factor by multiplying by 0.5. Forgetting this conversion leads to a systematic over‑ or under‑estimation of F.
IRP is the theoretical benchmark; actual quoted futures prices may deviate due to market frictions, margin requirements, or supply‑demand imbalances. The exam may ask you to identify the direction of the deviation (e.g., futures price higher than theoretical implies market expects INR depreciation).
Worked Example of IRP
Scenario
An Indian distributor sees a spot rate of 74.50 INR per USD. The RBI 3‑month money market rate is 5.5% p.a., and the US 3‑month Treasury yield is 1.8% p.a. The contract expires in 3 months (0.25 years). Compute the theoretical futures price.
Solution
Step 1: Convert annual rates to the 3‑month period: r_{\text{dom}} = 0.055 \times 0.25 = 0.01375, r_{\text{for}} = 0.018 \times 0.25 = 0.0045. Step 2: Numerator = 1 + 0.01375 = 1.01375. Step 3: Denominator = 1 + 0.0045 = 1.0045. Step 4: F = 74.50 \times (1.01375 / 1.0045) = 74.50 \times 1.0091978 = 75.19 (rounded to two decimals). Verification: 74.50 \times \frac{1 + 0.055 \times 0.25}{1 + 0.018 \times 0.25} = 75.19.
Conclusion
The theoretical futures price is 75.19 INR/USD. If the exchange lists the futures at 75.60, the market expects a slight INR depreciation or includes a liquidity premium.
Pricing Currency Futures Using IRP
Exchange‑traded currency futures are priced using the same parity relationship, but they are settled daily through the clearing house. This daily marking‑to‑market means the futures price incorporates the cost of carry, which is the interest‑rate differential between the two currencies.
When the exam asks for the futures price, you can use the continuous‑compounding version for precision, especially when the time horizon is expressed in days. The formula is F = S \times e^{(r_{\text{dom}} - r_{\text{for}})T} where e is the exponential function.
Remember that SEBI requires all listed currency futures to be quoted in INR per foreign unit, and the contract size is standardized (e.g., 12,500 USD). The parity formula gives the price per unit; multiply by the contract size only if the question asks for the total contract value.
Where:
F= Theoretical futures price in INR per foreign currency unitS= Spot exchange rate in INR per foreign currency unitr_{\text{dom}}= Domestic risk‑free rate (decimal, annual)r_{\text{for}}= Foreign risk‑free rate (decimal, annual)T= Time to maturity in yearse= Base of natural logarithm, approximately 2.71828Worked Example
Given S = 75 INR/USD, r_{\text{dom}} = 0.06, r_{\text{for}} = 0.02, T = 0.5 years: Step 1: Compute exponent = (0.06 - 0.02) \times 0.5 = 0.02. Step 2: e^{0.02} \approx 1.020201. Step 3: F = 75 \times 1.020201 = 76.515075. Verification: 75 \times e^{(0.06-0.02)\times0.5} = 76.52 (rounded to two decimals).
The continuous‑compounding formula is especially useful when the exam provides rates as continuously compounded yields, which the RBI sometimes publishes for money‑market instruments.
Note the direction of the exponent: if the domestic rate exceeds the foreign rate, the exponent is positive and the futures price will be higher than the spot rate, reflecting the higher cost of holding INR.
In practice, the quoted futures price on the NSE or BSE will be rounded to two decimal places. The exam may ask you to choose the closest quoted price from multiple‑choice options.
Forward vs Futures – Quick Comparison
Key differences between Forward Contracts and Exchange‑Traded Currency Futures
| Feature | Forward Contract | Currency Futures |
|---|---|---|
| Settlement | Single settlement at maturity (cash or physical) | Daily mark‑to‑market settlement via clearing house |
| Counterparty Risk | Bilateral credit risk; mitigated by collateral agreements | Clearing house guarantees performance; minimal credit risk |
| Liquidity | Generally lower; depends on dealer network | High liquidity on NSE/BSE; transparent order book |
| Pricing Model | IRP with simple interest; may include credit spread | IRP with continuous compounding; includes margin cost |
| Regulation | Subject to contract law; less SEBI oversight | Regulated by SEBI; listed on recognized stock exchanges |
Effect of Interest Rate Changes on Futures Price
Theoretical Futures Price for Varying Indian Interest Rates (USD Rate = 2% p.a., Spot = 75 INR/USD, T = 0.5 yr)
Always convert annual rates to the contract's exact time horizon before plugging them into the formula. Using the raw annual percentage directly will give a wrong answer.
Scenario
A 6‑month INR/EUR futures contract was originally priced when the RBI 6‑month rate was 5% p.a. and the Euro‑zone rate was 1.5% p.a. Spot = 88.00 INR/EUR. Mid‑way through the contract, RBI raises its 6‑month rate to 7% p.a. while the Euro rate stays unchanged. Compute the new theoretical futures price for the remaining 3 months.
Solution
Step 1: Remaining time T = 0.25 years. Step 2: Updated domestic rate r_{\text{dom}} = 0.07, foreign rate r_{\text{for}} = 0.015. Step 3: Use simple‑interest IRP: F = 88.00 \times \frac{1 + 0.07 \times 0.25}{1 + 0.015 \times 0.25} = 88.00 \times \frac{1.0175}{1.00375} = 88.00 \times 1.01368 = 89.20 (rounded). Verification: 88 \times (1 + 0.07*0.25)/(1 + 0.015*0.25) = 89.20. Step 4: Compare with original price (using 5% domestic): Original F = 88 \times (1+0.05*0.5)/(1+0.015*0.5) = 88 \times 1.025/1.0075 = 88 \times 1.0173 = 89.52. The new price is slightly lower, reflecting the higher domestic cost of carry over the remaining period.
Conclusion
A rise in the domestic interest rate increases the cost of carry, pushing the futures price higher for the same remaining maturity. The exam often tests this directional relationship.
⭐Exam Takeaways
- Interest Rate Parity links spot, forward/futures price and the interest‑rate differential between two currencies.
- Use the simple‑interest IRP formula F = S × (1 + r_dom × T) / (1 + r_for × T) unless the question specifies continuous compounding.
- For currency futures, the continuous‑compounding version F = S × e^{(r_dom - r_for)T} is the standard NISM expression.
- Always express rates on the same time basis as the contract’s maturity; convert annual rates to the exact fraction of a year.
- Forward contracts settle once at maturity, while futures settle daily, enjoy higher liquidity, and are SEBI‑regulated.
Practice Questions
9 questions on Interest Rate Parity and Pricing of Currency Futures
What does Interest Rate Parity (IRP) state about the relationship between the forward exchange rate and interest rates?
Which of the following is the correct simple‑interest IRP formula for the forward (or futures) exchange rate?
According to the assumptions underlying IRP, which of the following is presumed to be absent?
Spot = 78 INR/USD, domestic rate = 5% p.a., foreign rate = 1.5% p.a., and T = 0.5 years. Using the simple‑interest IRP formula, what is the theoretical futures price (rounded to two decimals)?
Which statement correctly describes the settlement mechanism of exchange‑traded currency futures?
Spot = 80 INR/USD, domestic rate = 5% p.a., foreign rate = 1% p.a., T = 0.25 years. Using the continuous‑compounding IRP formula, what is the theoretical futures price (rounded to two decimals)?
An INR/EUR 6‑month futures contract was priced when the RBI rate was 4% p.a. and the Euro‑zone rate was 1% p.a., with Spot = 90 INR/EUR. After 3 months, the RBI rate rises to 6% p.a. while the Euro rate stays at 1% p.a. What is the new theoretical futures price for the remaining 3 months (rounded to two decimals)?
If the domestic risk‑free rate increases while the foreign risk‑free rate remains unchanged, the theoretical futures price will:
When a quoted futures price is higher than the theoretical price derived from IRP, what does this most likely indicate?