Interest rate parity
The principle that a currency pair's forward or futures price differs from spot by the two currencies' interest rate differential, so borrowing in one and lending in the other earns no riskless profit.
In plain language
If rupee deposits pay more than dollar deposits, why not borrow dollars, convert them to rupees, and pocket the difference? Because you have to convert the rupees back to dollars to repay the loan — and the rate at which you can lock that in already has the interest difference built into it.
That is interest rate parity. In the workbook's words, it says the spot price and futures price of a currency pair incorporate any interest rate differentials between the two currencies, assuming there are no transaction costs or taxes.
Its most useful consequence is a rule about premium and discount:
The currency with the higher interest rate trades at a forward discount; the currency with the lower interest rate trades at a forward premium.
India's rates are typically higher than US rates, so USDINR futures usually trade above spot: the dollar is at a premium, the rupee at a discount.
How it works
The no-arbitrage argument (Chapter 3, section 3.6). Borrow USD 1 at the US rate, convert at spot S into rupees, deposit at the Indian rate for a year, and lock in the conversion back to dollars at the one-year futures price F. For there to be no riskless profit:
S × (1 + R_INR) = F × (1 + R_USD)
For traders. Everything else constant, if the USD interest rate is expected to rise while the INR rate stays put, the differential narrows and USDINR futures prices fall — so a trader would go short USDINR futures.
For the exchange. The theoretical price of a currency future — used as the base price on the first day and as the daily settlement price for illiquid contracts — is computed in continuous form: F = S × e^((r − rf) × t), with r the quote currency rate and rf the base currency rate.
Two workbook inconsistencies to be aware of. The section opens with Indian rates at 7% and US rates at 3%, then works the example with US rates at 5%; and the trading illustration speaks of the USD rate rising "from 3.0% to 4.0%" before computing a rise from 5% to 5.25%. The method is the same throughout — use whatever rates a question gives.
The formula
Exact: F = S × (1 + R_QC × Period) / (1 + R_BC × Period)
Approximate: F ≈ S + S × (R_QC − R_BC) × Period
Continuous (exchange theoretical price): F = S × e^((r − rf) × t)
QC = quotation currency, BC = base currency, Period in years
A worked example
The workbook's figures. USDINR spot 83; India 7%, USA 5%.
1. One year.
F = 83 × 1.07 / 1.05 = 84.5810 (the workbook truncates to 84.5809)
Check the arbitrage is gone: USD 1 borrowed → Rs 83 deposited → Rs 88.81 after a year → at 84.5810 buys USD 1.05, exactly the loan plus interest.
2. Six months.
Approximate: 83 + 1% × 83 = 83.83
Exact: 83 × (1 + 0.07 × 0.5) / (1 + 0.05 × 0.5) = 83.8098
The workbook prints 83.8097 (truncated rather than rounded).
3. USD rate rises to 5.25%.
F = 83 × 1.035 / 1.02625 = 83.7077 (workbook: 83.7076)
The forward falls — so a trader expecting this goes short.
4. Low-rate currency at a premium. EURJPY spot 123.40; JPY rate 0.25%, EUR rate 1%.
F = 123.40 × 1.0025 / 1.01 = 122.4837
EUR, the higher-rate currency, is at a discount to JPY.
In rupees. On one USDINR lot of USD 1,000, the six-month forward points of 0.8098 are worth Rs 809.80.
Why NISM asks about it
Chapter 3, section 3.6 (Interest Rate Parity and Pricing of Currency Futures), derives the relationship, gives the exact and approximate formulas, the 6-month illustration, the effect of a US rate rise, the EURJPY premium/discount example and the exchange's theoretical price formula. Chapter 5's calendar spread discussion relies on it. Expect a forward-price calculation and a premium/discount question.
Common exam traps
- Higher interest rate → forward discount. The rupee, with higher rates, is at a discount to USD; USD is at a premium.
- In the exact formula the quotation currency rate is on top, the base currency rate below.
- Scale annual rates by the period. 7% for six months is 3.5%.
- A rise in the base currency rate lowers the forward (USD rate up → USDINR forward down → go short).
- The exchange's theoretical price is continuous: F = S × e^((r − rf) × t).
- The workbook's printed results are truncated to four decimals (83.8097, 83.7076). Pick the option closest to your answer.
- Parity assumes no transaction costs, taxes or capital restrictions.
Check yourself
1.Which assumption does the workbook make in deriving the arbitrage-free futures price through interest rate parity?
- a)The spot rate will stay unchanged for the year
- b)Interest rates are equal in both countries
- c)The futures contract is physically settled
- d)Money can be transferred easily from one country to another without restriction of amount and without taxes
Show the answer
Answer: (d) Money can be transferred easily from one country to another without restriction of amount and without taxes
Assumptions listed: "1. Spot exchange rate of USDINR is 83 (S); 2. One year future rate for USDINR is 'F'; 3. Risk free interest rate for one year in USA is 5% (RUSD); 4. Risk free interest rate for one year in India is 7% (RINR); 5. Money can be transferred easily from one country into another without any restriction of amount, without any taxes etc."
The spot rate need not stay unchanged — the point is to lock the future conversion with F: "you would lock the conversion rate of INR into USD using the one-year future prices of USDINR."
The rates are deliberately different (5% and 7%) — the differential is what parity prices.
Physical settlement is not an assumption.
The conclusion: "interest rate parity says that the spot price and futures price of a currency pair incorporate any interest rate differentials between the two currencies assuming there are no transaction costs or taxes."
2.How does the futures price relate to the spot price of a currency pair?
- a)It can be higher or lower than spot, but converges with the spot price on the expiry date
- b)It is always higher than spot until expiry
- c)It must equal spot at all times, or arbitrage arises
- d)It is always lower than spot, since futures are cash settled
Show the answer
Answer: (a) It can be higher or lower than spot, but converges with the spot price on the expiry date
"Futures price: The current price of the specified futures contract. The future price can be higher or lower than the spot price. However, on the expiry date, the futures price converges with the spot price."
It is not always higher. Whether futures trade above or below spot depends on the interest rates: "The future value of a currency with a high interest rate is at a discount to the value of the currency with a low interest rate." The EURJPY example gives a forward (122.4837) below spot (123.4).
It need not equal spot before expiry. Interest rate parity says the two prices "incorporate any interest rate differentials between the two currencies" — a gap is the arbitrage-free outcome, not an arbitrage.
Cash settlement has nothing to do with the direction of the gap.
Convergence is also why the final settlement price comes from the spot market: "The Final settlement price / rate is mainly derived from the underlying/spot market."
3.USDINR spot is 83, the 6-month Indian rate is 7% per annum and the US rate is 5% per annum. Using the exact formula, the 6-month futures price is approximately:
- a)82.1903
- b)83.8097
- c)83.0000
- d)84.5809
Show the answer
Answer: (b) 83.8097
6-month futures price = 83 x (1+0.070.5) / (1+0.050.5) = 83.8097.
The formula: F = S × (1 + RQC × Period) / (1 + RBC × Period), where RBC = interest rate on base currency and RQC = interest rate on quoting currency.
In USDINR, USD is the base and INR the quote, so the rupee rate goes on top.
The quick estimate confirms it: Spot + 6-month interest difference = 83 + 1% of 83 = 83+0.83 = 83.83. Close to the exact figure, which is what F = S + (S × (RQC – RBC) × Period) is for.
Option D, 84.5809, is the one-year answer from the chapter's earlier derivation: F/83 = (1+0.07)/(1+0.05), F = 84.5809. Here the period is half a year.
Option A inverts the ratio, putting the base currency rate on top — the single most common error in this calculation.
What the number means: it means that INR trades at a discount to USD and USD quotes at a premium to INR.
The intuition: to buy 1 USD you had to pay INR 83 now but you have to pay 83.8097 after 6 months i.e., you have to pay more INR to buy the same 1 USD. Hence the future value of INR is at a discount to USD.
And the general rule: the future value of a currency with a high interest rate is at a discount to the value of the currency with a low interest rate.
This is interest rate parity — the spot price and futures price of a currency pair incorporate any interest rate differentials between the two currencies assuming there are no transaction costs or taxes.
Where this is taught
Free preparation for NISM Series IRelated terms
- Base currencyThe first currency in a pair, fixed at one unit and priced in the second (quotation) currency. In USDINR the base is USD, so the quote is the rupee price of one dollar.
- Calendar spread tradingTaking opposite positions in two maturities of the same currency futures (or options) contract to trade the change in the spread between them, rather than the direction of the currency.
- Currency futuresA standardised exchange-traded contract to buy or sell one currency against another on a future date at a price fixed today; in India, cash-settled in rupees, USD 1,000 a lot for USDINR.
- Fair valueThe theoretical futures price — spot plus the cost of carrying the commodity to expiry — at which a buyer is indifferent between buying today and buying forward.
- ArbitrageSimultaneously buying and selling an asset, or a replica of it, in different markets to profit from a price discrepancy — a trade that aims to lock in a profit rather than bet on direction.
- Quotation currencyThe second currency in a pair — the one that does the pricing. Its amount varies as the market moves, and a contract's value is always expressed in it.
- Carry tradeBorrowing in a currency with low interest rates and investing the money in assets of a currency with high interest rates, to earn the interest rate differential.