Cross rate
Also written Cross currency rate · Cross exchange rate
The price of a currency pair that is not directly quoted, derived by combining two pairs that share a common currency — EURINR from EURUSD and USDINR, for example.
In plain language
Some currency pairs trade actively and have their own quotes. Many do not. For those, the price is worked out from two other pairs that share a common currency — usually the US dollar.
The workbook's definition: the price of the currency pair for which direct quotes are not available is called the "cross rate".
The logic is a conversion path. To buy euros with rupees when there is no direct euro-rupee market, you sell rupees for dollars, then sell those dollars for euros. The rupee cost of one euro at the end of that chain is the cross rate.
Crossing can mean multiplying the two underlying prices or dividing one by the other. Which one depends on whether the common currency is the base in one pair and the quote in the other, or the base in both. The workbook deliberately avoids memorised shortcuts such as the chain rule and derives each cross rate by that commercial logic instead.
How it works
Three steps (Chapter 1, section 1.4):
- Write the conversion path. Which currency do you sell, which do you buy, and in what order?
- Pick the right side of each quote. When you buy the base currency of a pair you pay the offer; when you sell it you receive the bid.
- Multiply or divide so the result is the price of one unit (or 100 units, for JPY) of the new base in the new quotation currency.
| Target pair | Built from | Common currency is… | Operation |
|---|---|---|---|
| EURINR | EURUSD, USDINR | Quote in EURUSD, base in USDINR | Multiply |
| GBPINR | GBPUSD, USDINR | Quote in GBPUSD, base in USDINR | Multiply |
| JPYINR | USDJPY, USDINR | Base in both | Divide |
The same logic can run backwards: from a cross rate and one underlying rate, you can recover the other underlying rate — EURUSD from EURINR and USDINR, for instance.
Cross rates are also the benchmark for triangular arbitrage: when a quoted rate strays from the implied cross rate, a three-leg arbitrage exists.
The formula
EURINR offer = EURUSD offer × USDINR offer
EURINR bid = EURUSD bid × USDINR bid
JPYINR (per 100 JPY) offer = USDINR offer / USDJPY bid × 100
JPYINR (per 100 JPY) bid = USDINR bid / USDJPY offer × 100
A worked example
The workbook's quotes: EURUSD 1.1125 / 1.1150, GBPUSD 1.3300 / 1.3325, USDJPY 115.08 / 115.09, USDINR 75.64 / 75.65.
EURINR. To buy 1 EUR: buy USD at 75.65 (offer), then buy EUR at 1.1150 USD (offer).
Offer = 1.1150 × 75.65 = 84.3498
Bid = 1.1125 × 75.64 = 84.1495
EURINR = 84.1495 / 84.3498
GBPINR.
Offer = 1.3325 × 75.65 = 100.8036
Bid = 1.3300 × 75.64 = 100.6012
GBPINR = 100.6012 / 100.8036
JPYINR (per 100 JPY). To buy JPY: buy USD at 75.65, then sell each USD for JPY at the USDJPY bid of 115.08.
Offer = 75.65 / 115.08 = 0.6574 per JPY → 65.74 per 100 JPY
Bid = 75.64 / 115.09 = 0.6572 per JPY → 65.72 per 100 JPY
JPYINR = 65.72 / 65.74
In rupees. An importer paying EUR 10,000 at the EURINR offer pays 10,000 × 84.3498 = Rs 8,43,498.
Why NISM asks about it
Chapter 1, section 1.4 (Exchange Rate Arithmetic — Cross Rate), works EURINR, GBPINR and JPYINR exactly as above. Chapter 5, section 5.5, uses the implied cross rate to test for triangular arbitrage. Chapter 3 and Chapter 7 rely on crossing to value cross-currency futures in rupees using FBIL rates. Expect a calculation with bid and offer sides.
Common exam traps
- Use the offer on both legs for the offer, the bid on both legs for the bid when multiplying. Mixing sides gives a rate inside the true spread.
- When USD is the base in both pairs (JPYINR), divide, and the sides cross over: the JPYINR offer uses the USDJPY bid.
- JPYINR is quoted per 100 JPY. 0.6574 per yen is 65.74 per 100.
- A cross rate's spread is wider than either underlying spread, because both spreads are paid.
- The workbook derives crosses by conversion logic, not by the chain rule — show the path and the sides take care of themselves.
Check yourself
1.EURUSD is 1.1125 / 1.1150 and USDINR is 75.64 / 75.65. What is the EURINR cross rate?
- a)84.3498 / 84.1495
- b)84.1495 / 84.3498
- c)67.98 / 68.00
- d)1.1125 / 1.1150
Show the answer
Answer: (b) 84.1495 / 84.3498
Therefore, the cross rate for EURINR would be 84.1495 / 84.3498.
The buy side: to buy 1 unit of USD, the applicable price is 75.65 INR (offer side)... The price for buying 1 unit of EUR is 1.1150 USD (offer side). Therefore, price of buying 1.1150 units of USD would be 1.1150 x 75.65 INR i.e. 84.3498 INR.
The sell side: the price comes to 84.1495 (1.1125 x 75.64).
So offer × offer gives the buying price and bid × bid gives the selling price.
Option A has the two the right way round arithmetically but presented in the wrong order. The bid price (lower price) is quoted first followed by offer price (higher price) — a two-way quote always reads low/high.
Option C divides instead of multiplying. That is the JPYINR method, which applies only because in case of JPYINR, USD is base currency for both the currency pairs. For EURINR, USD is the quotation currency in EURUSD and the base currency in USDINR, so the two multiply cleanly.
The conversion path is what tells you which sides to use: you need to sell INR to buy USD; and further sell the USD received to buy EUR. It is important to identify this FX conversion path of selling one currency and buying another to calculate the cross rate.
GBPINR works the same way: assume GBPUSD price as 1.3300 / 1.3325 and USDINR as 75.64 / 75.65, the price for GBPINR works out to be 100.6012 / 100.8036.
And the method reverses: using EURINR and USDINR rate, EURUSD rate could be calculated.
2.For a trader in India, USDINR = 83.20 is what kind of quote?
- a)An indirect quote — the domestic currency is the base
- b)A direct quote — the foreign currency is the base and the local currency the quote
- c)A cross rate, since USD is not the domestic currency
- d)An indirect quote, since USD is the universal base currency
Show the answer
Answer: (b) A direct quote — the foreign currency is the base and the local currency the quote
"In a direct quote, the foreign currency is the base currency, while the local currency is the quote currency. An indirect quote is just the opposite: the domestic currency is the base currency, and the foreign currency is the quote currency."
In USDINR, USD (foreign) is the base and INR (local) the quote — a direct quote for an Indian trader.
An indirect quote would be INRUSD — the price of one rupee in dollars.
A cross rate is the price of a pair "for which direct quotes are not available" and is derived by crossing two quotes — USDINR is quoted directly.
USD being the universal interbank base does not make the quote indirect; direction depends on which currency is domestic. "Most pairs using the U.S. dollar are direct quotes."
3.In the exchange-traded currency derivatives context, a "cross currency" derivative is one:
- a)On any currency pair other than USD-INR, such as EUR-INR
- b)Whose price is derived by crossing two quotes, such as JPY-INR from USD-JPY and USD-INR
- c)On a currency pair not involving the Indian rupee, such as EUR-USD
- d)That is settled in a currency other than the Indian rupee
Show the answer
Answer: (c) On a currency pair not involving the Indian rupee, such as EUR-USD
"In the exchange-traded currency derivatives context, cross currency derivatives means derivatives on currency pair not involving the Indian rupee." The listed ones are EUR-USD, GBP-USD and USD-JPY.
EUR-INR is not a cross currency contract in this sense — it involves the rupee, and it is a rupee pair like USD-INR, GBP-INR and JPY-INR.
Crossing two quotes describes how a cross rate is calculated in Chapter 1, not the exchange-traded definition. JPY-INR is derived that way in the OTC market, but on the exchange it is an INR pair.
Settlement currency is the trap for the last option. Even cross currency contracts are "Cash Settled in Indian Rupees" — so no cross currency contract is settled in another currency.
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.
- FBIL reference rateThe daily USD/INR (and EUR, GBP, JPY vs INR) benchmark rate published by Financial Benchmarks India Pvt Ltd around 1.30 pm; it is the final settlement price of INR currency futures and options.
- 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.
- Triangular arbitrageExploiting a misalignment among three exchange rates by converting currency A to B, B to C and C back to A, ending with more of A than you started with.
- Vehicle currencyA common third currency — most often the US dollar — through which two other currencies are traded, instead of trading them directly against each other.