Week 11: Cross Rates, Chain Rule & Arbitrage — Two-Point, Three-Point & Covered Interest Arbitrage
Learning Objectives
By the end of this session, students will be able to:
Compute cross rates between two non-USD currencies from their respective USD exchange rates, apply the chain rule (rule of three) for multi-currency conversions, and determine the most efficient conversion path for a given currency transaction.
Identify and exploit two-point (locational) arbitrage opportunities — where the same currency pair is quoted at different rates in different locations — and three-point (triangular) arbitrage opportunities — where cross rates are inconsistent across three currencies, calculating the arbitrage profit.
Compute and execute Covered Interest Arbitrage (CIA) — identifying when CIRP is violated, determining which currency to borrow and which to invest, and calculating the risk-free arbitrage profit from a given principal amount.
Distinguish between fundamental, technical, and market-based exchange rate forecasting approaches and explain why all forecasting methods have severe practical limitations — connecting this to the corporate hedging philosophy from Week 10.
4-Hour Session Planner
This is the most numerically intensive session of the course — the culmination of Unit 3's computational spine. Faculty should reserve at least 60 minutes for board-based arbitrage problem-solving.
Opening Hook: "The INR 10 Crore Question — Can You Make Money From a Misquoted Cross Rate?"
15 minA bank in Singapore quotes USD/INR = 83.00. A bank in London quotes EUR/USD = 1.0850. A bank in Mumbai quotes EUR/INR = 90.00. Can you profit? Students compute the implied EUR/INR cross rate (83.00 × 1.0850 = 90.06) and discover the Mumbai quote is 6 paise below — an arbitrage opportunity. This visceral demonstration of cross-rate arbitrage hooks students into the session's core logic: prices must be consistent, or arbitrageurs will enforce consistency.
Section 1: Cross Rates — Concept & Computation
30 minThe concept of a cross rate: deriving the exchange rate between two non-USD currencies from their respective USD rates. Two methods: the direct computation method and the chain rule. Computing cross rates with bid-ask spreads: the rules for multiplying and dividing two-way quotes to obtain the correct cross-rate bid and ask.
Section 2: The Chain Rule (Rule of Three)
25 minThe chain rule for multi-currency conversions through an arbitrary number of intermediate currencies. Structuring the chain: starting currency on the left, each intermediate rate expressed so that each currency appears in both numerator and denominator, cancelling to the target currency. Worked examples with 3, 4, and 5 currencies.
Section 3: Two-Point (Locational) Arbitrage
20 minThe simplest form of arbitrage: the same currency pair is quoted at different rates in two different locations. Buy where the base currency is cheap, sell where it is expensive. The no-arbitrage condition: the bid in one location must not exceed the ask in another (net of transaction costs). Worked examples with and without transaction costs.
Section 4: Three-Point (Triangular) Arbitrage
35 minTriangular arbitrage: exploiting inconsistent cross rates among three currencies. The logic: start with Currency A, convert to Currency B, convert to Currency C, convert back to Currency A — if you end with more A than you started, an arbitrage exists. Identifying the profitable direction. Computing the arbitrage profit. The cross-rate consistency condition: S(A/C) = S(A/B) × S(B/C).
In-Lecture Quiz (4 Questions)
10 minQuiz covering cross-rate computation, two-point arbitrage identification, triangular arbitrage logic, and the no-arbitrage condition.
Section 5: Covered Interest Arbitrage (CIA)
25 minCIA as the practical application of CIRP (Week 7). Identifying CIA opportunities: when the actual forward rate deviates from the CIRP-implied rate. The CIA strategy: borrow in the currency with the lower covered return, convert, invest, and cover forward. Computing the arbitrage profit. Why CIA opportunities are rare and fleeting in major currencies — and more persistent in INR due to capital controls.
CQ Box: Multi-Step Arbitrage
15 minStudents are given spot rates, forward rates, and interest rates for four currencies. They must: (a) identify triangular arbitrage opportunities, (b) identify CIA opportunities, (c) determine which opportunity offers the highest profit, and (d) explain why the CIA opportunity persists (capital controls).
Section 6: Comprehensive Arbitrage Problem Set
50 min10 problems: cross rates (3), two-point arbitrage (2), triangular arbitrage (3), covered interest arbitrage (2). Problems of increasing complexity solved on the board with student participation.
Key Concepts, Exit Ticket & Unit 3 Consolidation
15 minFaculty reviews 12 key terms. Exit Ticket with arbitrage computation. Unit 3 synthesis: FX Market → Spot/Forward → Cross Rates/Arbitrage. Preview of Unit 4 (International Investments).
"You are an FX trader at a hedge fund. Your Bloomberg terminal shows: (a) a bank in Singapore quoting USD/INR = 83.00, (b) a bank in London quoting EUR/USD = 1.0850, (c) a bank in Mumbai quoting EUR/INR = 90.00. In the 3 seconds it took you to read this, you've spotted an inconsistency. Can you make a risk-free profit from INR 10 crore? How much — and what trade do you do?"
Key Teaching Moments
(1) Speed matters: In real FX markets, this arbitrage would be identified and exploited by algorithms in microseconds. The profit would narrow to zero almost instantly. The fact that the opportunity exists on the board for students to find is a pedagogical device — in reality, cross rates are continuously arbitraged by the market-makers themselves. (2) Direction matters: What if the Mumbai quote were 90.10? Then EUR/INR in Mumbai would be more expensive than the cross rate (90.055). The arbitrage would run in the opposite direction: sell EUR in Mumbai, buy EUR via the cross route. Identifying the correct direction is the core skill. (3) Transaction costs: If the spread on each leg were 2 paise, the 5.5-paise gross deviation might narrow to below transaction costs — no arbitrage profit after spreads. Real-world arbitrage is about identifying deviations that exceed transaction costs.
1. Cross Rates — Concept & Computation
1.1 What Is a Cross Rate?
A cross rate is an exchange rate between two currencies, neither of which is the US dollar, derived from their respective exchange rates against the USD. In the global FX market, most non-USD currency pairs are not traded directly — they are traded as "two legs through the dollar." The USD is the vehicle currency. Therefore, the rate between, say, the Indian rupee and the Japanese yen is typically derived from USD/INR and USD/JPY rather than quoted directly as JPY/INR.
However, not all non-USD pairs are traded only via the dollar. Major crosses — EUR/JPY, EUR/GBP, GBP/JPY — have active direct markets. For these pairs, the cross rate computed from USD rates serves as a consistency check: if the direct EUR/JPY quote deviates from the EUR/USD × USD/JPY cross rate, triangular arbitrage (Section 4) will force it back into line.
1.2 Computing Cross Rates — Two Methods
Method 1 — Direct Computation: The simplest method. For two currencies A and B, given S(X/A) and S(X/B) — both expressed in terms of a common currency X (typically USD) — the cross rate S(A/B) is:
Where S(X/A) = units of X per unit of A, and S(X/B) = units of X per unit of B.
Or equivalently: S(A/B) = S(A/USD) × S(USD/B) — converting A → USD → B.
Worked Example 1 — JPY/INR: USD/INR = 83.00. USD/JPY = 150.00. Compute the JPY/INR cross rate (INR per JPY).
INR per JPY = (INR per USD) / (JPY per USD) = 83.00 / 150.00 = INR 0.5533/JPY. One Japanese yen buys 0.5533 Indian rupees. Alternatively: JPY per INR = 150.00 / 83.00 = JPY 1.8072/INR — one rupee buys 1.81 yen.
Worked Example 2 — EUR/GBP: EUR/USD = 1.0850 (European terms). GBP/USD = 1.2650 (European terms). Compute EUR/GBP.
EUR per GBP = (EUR per USD) / (GBP per USD) = Wait — both are quoted in European terms (USD per unit). Actually: EUR/USD = 1.0850 means EUR 1 = USD 1.0850. In American terms: USD/EUR = 1 / 1.0850 = 0.9217. Similarly: USD/GBP = 1 / 1.2650 = 0.7905.
EUR/GBP = (USD/GBP) / (USD/EUR) = 0.7905 / 0.9217 = 0.8577. One GBP buys 0.8577 EUR. Or equivalently: EUR/GBP = (EUR/USD) / (GBP/USD) = 1.0850 / 1.2650 = 0.8577. When both are in European terms (both have USD as the quote currency), simply divide.
1.3 Cross Rates with Bid-Ask Spreads
When computing cross rates from two-way (bid-ask) quotes, the financial manager must determine the correct bid and ask for the cross rate. The general principle: the cross-rate bid is the rate at which the dealer buys the base currency — which means the dealer follows the path that gives the lowest cost. The cross-rate ask is the rate at which the dealer sells the base currency — the path that gives the highest revenue.
Case 1 — Both in American terms (USD/XXX and USD/YYY):
S(XXX/YYY)_bid = (USD/YYY_bid) / (USD/XXX_ask)
S(XXX/YYY)_ask = (USD/YYY_ask) / (USD/XXX_bid)
Case 2 — Both in European terms (XXX/USD and YYY/USD):
S(XXX/YYY)_bid = (XXX/USD_bid) / (YYY/USD_ask)
S(XXX/YYY)_ask = (XXX/USD_ask) / (YYY/USD_bid)
Memory aid: The bid is always the lower number; the ask is always the higher. Use the combination that gives the lowest cross bid and the highest cross ask.
Worked Example 3 — Cross Rate with Spreads: USD/INR = 82.95/83.05. USD/JPY = 149.80/149.90. Compute JPY/INR (bid and ask).
JPY/INR = INR per JPY. Both quotes are in American terms. JPY is the base currency in the cross.
JPY/INR_bid = USD/INR_bid / USD/JPY_ask = 82.95 / 149.90 = 0.5534. (The dealer buys JPY — pays fewer INR — at this rate.)
JPY/INR_ask = USD/INR_ask / USD/JPY_bid = 83.05 / 149.80 = 0.5544. (The dealer sells JPY — charges more INR — at this rate.)
Quote: JPY/INR = 0.5534 / 0.5544. Cross-rate spread = 0.0010 (10 pips).
2. The Chain Rule (Rule of Three)
2.1 What Is the Chain Rule?
The Chain Rule — also called the Rule of Three — is a systematic method for computing an exchange rate between two currencies through an arbitrary number of intermediate currencies. It is particularly useful when: (a) there is no direct market between the two currencies, (b) the conversion involves 3 or more intermediate steps, or (c) the financial manager needs to determine the cheapest conversion path among multiple alternatives.
2.2 Structuring the Chain
The chain is structured as an equation where each currency appears as both a numerator and a denominator — cancelling through the chain until only the target currency pair remains.
Start with: X units of Currency A = ? units of Currency Z
Chain: (A → B) × (B → C) × (C → D) × ... × (Y → Z)
Each link is expressed so that the intermediate currency cancels: if link 1 is B/A (B per A) and link 2 is C/B (C per B), then (B/A) × (C/B) = C/A — B cancels.
Result: ? = X × (exchange rates through the chain, with all intermediate currencies cancelling).
Worked Example 4 — 3-Currency Chain: An Indian firm needs to pay THB 5 million to a Thai supplier. The firm has INR. No direct INR/THB market exists. Given: USD/INR = 83.00, USD/THB = 35.50. How many INR are needed?
Chain: INR → USD → THB. First, INR to USD: 1 INR = 1/83.00 USD. Then USD to THB: 1 USD = 35.50 THB. So 1 INR = 35.50/83.00 = 0.4277 THB. THB 5M = 5,000,000 / 0.4277 = INR 11,690,000 (approximately). Alternatively: INR needed = THB 5M × (83.00 / 35.50) = INR 11,690,141.
Worked Example 5 — 4-Currency Chain with Bid-Ask: An Indian importer must pay BRL 2 million. No direct INR/BRL market. Available quotes: USD/INR = 82.95/83.05, USD/BRL = 4.9500/4.9700. Compute the INR cost.
Chain: INR → USD → BRL. The importer sells INR to buy USD (dealer sells USD → transact at USD/INR ask = 83.05), then sells USD to buy BRL (dealer sells BRL → transact at USD/BRL ask = 4.9700 — wait, in American terms, ask is the higher number, meaning more BRL per USD, which means the dealer gives more BRL... actually for USD/BRL, the ask is the rate at which the dealer sells BRL. The customer buying BRL transacts at the ask = 4.9700).
INR per USD = 83.05. BRL per USD = 4.9700. INR per BRL = 83.05 / 4.9700 = 16.71. Total INR = 2,000,000 × 16.71 = INR 33,420,000 (approx). Precise: 2,000,000 × (83.05 / 4.9700) = INR 33,416,499.
3. Two-Point (Locational) Arbitrage
3.1 The Concept
Two-point arbitrage — also called locational or spatial arbitrage — exploits the fact that the same currency pair is quoted at different rates in two different locations (e.g., a bank in Mumbai and a bank in Singapore quoting different USD/INR rates). The arbitrageur simultaneously buys the base currency where it is cheaper and sells it where it is more expensive, locking in a risk-free profit.
Ask price in Location X < Bid price in Location Y
Buy at the lower ask in Location X; sell at the higher bid in Location Y. The profit per unit = Bid_Y − Ask_X.
With transaction costs: Ask_X + transaction costs < Bid_Y − transaction costs.
Worked Example 6 — Two-Point Arbitrage: Bank A in Mumbai quotes USD/INR = 82.95/83.05. Bank B in Singapore quotes USD/INR = 83.08/83.18. Is there an arbitrage opportunity?
Analysis: Can I buy USD cheaply at one bank and sell it expensively at the other? Buy USD at the lowest ask: Bank A ask = 83.05. Sell USD at the highest bid: Bank B bid = 83.08. Since 83.05 < 83.08, an arbitrage exists! Buy USD 1M from Bank A at 83.05 (cost: INR 8,30,50,000). Simultaneously sell USD 1M to Bank B at 83.08 (receive: INR 8,30,80,000). Profit = INR 30,000 — risk-free, instantaneous, zero capital at risk (assuming the trades are simultaneous).
Important nuance: Two-point arbitrage in modern FX markets is virtually extinct for major currency pairs — electronic trading and high-speed connections mean prices are synchronised across locations within microseconds. For INR, however, locational arbitrage can exist between the onshore (Mumbai) and offshore (NDF in Singapore/London) markets due to capital account restrictions that prevent free arbitrage between the two.
4. Three-Point (Triangular) Arbitrage
4.1 The Concept
Triangular arbitrage exploits inconsistent cross rates among three currencies. If the direct exchange rate between two currencies (say, EUR/INR) differs from the cross rate implied by their respective USD rates (EUR/USD × USD/INR), an arbitrage opportunity exists. The arbitrageur executes three trades — converting Currency A → B → C → A — and ends with more of Currency A than they started with.
S(A/C) = S(A/B) × S(B/C)
If the equality does not hold — if the market quotes S(A/C) at a rate different from the cross rate — arbitrage exists. The profit comes from taking the cheaper path (direct vs. cross) for one direction and the more expensive path for the reverse.
4.2 Identifying the Profitable Direction
The key question: should the arbitrageur go clockwise (A → B → C → A) or counter-clockwise (A → C → B → A)? The rule: compare the implied cross rate to the quoted direct rate.
Worked Example 7 — Triangular Arbitrage with a Single Exchange Rate: Given: USD/INR = 83.00, EUR/USD = 1.0850, EUR/INR = 90.05 (quoted directly). Is there arbitrage?
Step 1 — Compute the implied cross rate: EUR/INR (implied via USD) = EUR/USD × USD/INR = 1.0850 × 83.00 = 90.055.
Step 2 — Compare: Implied = 90.055. Direct quote = 90.05. The direct EUR/INR is slightly lower than implied — EUR is cheaper via the direct route.
Step 3 — Arbitrage (start with INR 10 Cr): (a) Buy EUR directly at 90.05: INR 10,00,00,000 / 90.05 = EUR 1,110,494. (b) Convert EUR to USD at 1.0850: EUR 1,110,494 × 1.0850 = USD 1,204,886. (c) Convert USD to INR at 83.00: USD 1,204,886 × 83.00 = INR 10,00,05,538. Profit = INR 5,538.
Worked Example 8 — Triangular Arbitrage with Bid-Ask Spreads: Quotes: USD/INR = 82.96/83.04. EUR/USD = 1.0848/1.0852. EUR/INR = 90.00/90.10.
Step 1 — Compute the EUR/INR cross bid and ask: EUR/INR_bid (implied) = USD/INR_bid × EUR/USD_bid = 82.96 × 1.0848 = 89.99. EUR/INR_ask (implied) = USD/INR_ask × EUR/USD_ask = 83.04 × 1.0852 = 90.11.
Implied cross: EUR/INR = 89.99 / 90.11. Market direct: 90.00 / 90.10.
Step 2 — Compare: Market bid (90.00) > implied bid (89.99). Market ask (90.10) < implied ask (90.11). The market's direct quote is inside the implied cross — no arbitrage (the spreads overlap).
Teaching point: Most apparent triangular arbitrage opportunities disappear when bid-ask spreads are properly accounted for. The spread on the cross rate (implied bid to implied ask) is typically wider than the direct market spread — because computing a cross from two pairs compounds their spreads. Arbitrage exists only when the direct quote is entirely outside the implied range.
4.3 The Mechanics of Triangular Arbitrage in the Real World
In the modern FX market, triangular arbitrage is executed by algorithms within microseconds. Dealer banks' pricing engines continuously compute cross rates and adjust quotes to eliminate inconsistencies. A human trader cannot compete with algorithmic arbitrageurs — the opportunities exist for milliseconds and are captured by the fastest computers with the lowest-latency connections. The study of triangular arbitrage is therefore primarily about understanding the no-arbitrage logic that ensures price consistency — the same logic that underlies CIRP, PPP, and all other parity conditions studied in this course.
5. Covered Interest Arbitrage (CIA)
5.1 CIA — The Practical Application of CIRP
Covered Interest Arbitrage (CIA) is the arbitrage trade that enforces Covered Interest Rate Parity (CIRP, Week 7). When the actual forward rate deviates from the CIRP-implied forward rate, the arbitrageur borrows in one currency, converts at the spot rate, invests in the other currency, and covers the future repatriation with a forward contract — locking in a risk-free profit with zero net investment.
Identifying a CIA Opportunity:
- Compute the CIRP-implied forward rate: F_CIRP = S × (1 + i_h × t) / (1 + i_f × t) for period t.
- Compare to the actual market forward rate (F_market).
- If F_market > F_CIRP: the foreign currency is overvalued in the forward market. The covered return on the foreign investment is too high. Borrow domestic, invest foreign, cover forward.
- If F_market < F_CIRP: the foreign currency is undervalued in the forward market. The covered return on the domestic investment is too high. Borrow foreign, invest domestic, cover forward.
Worked Example 9 — CIA: Spot USD/INR = 83.00. 12-month forward (market) = 84.20. India 1-year rate = 6.80%. US 1-year rate = 5.00%.
Step 1: F_CIRP = 83.00 × 1.068 / 1.05 = 83.00 × 1.01714 = 84.42. Market forward (84.20) < CIRP forward (84.42). The USD is undervalued in the forward market — the covered return on the USD investment is too low, meaning the covered return on the INR investment is too high.
Step 2 — Arbitrage (borrow USD, invest INR, cover forward): (a) Borrow USD 1M at 5% for 1 year → will owe USD 1,050,000. (b) Convert USD 1M to INR at spot 83.00 → INR 83,000,000. (c) Invest INR at 6.80% → INR 83,000,000 × 1.068 = INR 88,644,000. (d) Sell INR 88,644,000 forward at 84.20 → will receive USD 88,644,000 / 84.20 = USD 1,052,779. (e) Repay USD loan: USD 1,052,779 − USD 1,050,000 = arbitrage profit of USD 2,779.
5.2 Why CIA Opportunities Persist for INR
For major currencies (EUR, JPY, GBP), CIA opportunities are eliminated within seconds by algorithmic arbitrageurs. For INR, CIA deviations can persist because: (a) capital account restrictions prevent foreign investors from freely borrowing INR and investing abroad (the INR leg of the arbitrage is constrained); (b) the onshore INR forward market is subject to RBI intervention and regulatory limits; (c) the onshore/offshore segmentation (NDF market) creates persistent pricing gaps. These constraints mean that the CIRP equation F = S × (1+i_INR)/(1+i_USD) does not hold exactly for INR — there is a "convenience yield" or "capital-control premium" embedded in INR forward points.
You observe the following market quotes simultaneously:
Spot Rates: USD/INR = 82.90/83.10. EUR/USD = 1.0845/1.0855. EUR/INR = 90.10/90.30.
6-Month Forward (USD/INR): 83.70/83.90.
6-Month Interest Rates (annualised): INR = 7.00% p.a. (3.50% for 6 months). USD = 5.20% p.a. (2.60% for 6 months).
(a) Triangular arbitrage: Compute the EUR/INR cross rate from the USD quotes. Is there a triangular arbitrage opportunity? If yes, starting with INR 10 crore, compute the profit. If no, explain why the bid-ask spreads eliminate the opportunity.
(b) Covered Interest Arbitrage: Compute the CIRP-implied 6M forward rate (use the midpoints: S = 83.00). Compare with the market forward midpoint (83.80). Is there a CIA opportunity? If yes, starting with USD 1M, compute the profit. If no, explain why.
(c) Policy question: CIA opportunities in INR are common because onshore INR forward rates frequently deviate from CIRP. Why doesn't arbitrage eliminate these deviations? What specific Indian capital account restrictions prevent the CIA trade from being executed at scale?
For (a): EUR/INR cross bid = USD/INR_bid × EUR/USD_bid. Cross ask = USD/INR_ask × EUR/USD_ask. Compare to the direct EUR/INR quote. For (c): think about which direction the CIA trade requires — does it require borrowing INR and investing abroad (restricted), or borrowing abroad and investing in INR (permitted with limits)?
Guided Solution
(a) Triangular: EUR/INR cross bid = 82.90 × 1.0845 = 89.91. Cross ask = 83.10 × 1.0855 = 90.20. Implied cross: 89.91/90.20. Market direct: 90.10/90.30. The ranges overlap (89.91–90.20 and 90.10–90.30 share 90.10–90.20). No risk-free arbitrage after spreads — the market is efficient within the bid-ask bounds.
(b) CIA: F_CIRP = 83.00 × 1.035 / 1.026 = 83.00 × 1.00877 = 83.73 (midpoint). Market forward midpoint = (83.70 + 83.90)/2 = 83.80. Market (83.80) > CIRP (83.73). The USD is overvalued in the forward market. Strategy: borrow INR (3.50%), convert to USD at spot (83.00), invest USD (2.60%), sell USD forward (83.70 bid — the arbitrageur sells USD at the bid). Compute: INR 8,30,00,000 → USD 1,000,000 → invest → USD 1,026,000 → sell forward at 83.70 → INR 8,58,76,200. Repay INR loan: 8,30,00,000 × 1.035 = INR 8,59,05,000. Result: INR 8,58,76,200 − INR 8,59,05,000 = −INR 28,800 — a loss! After accounting for the bid-ask spread, the CIA opportunity disappears.
(c) INR CIA constraints: The typical CIA trade to exploit INR forward mispricing requires either: (i) borrowing INR and investing abroad (restricted — Indian residents cannot freely borrow INR for overseas investment beyond LRS limits), or (ii) borrowing USD abroad and investing in INR (permitted but subject to FPI debt limits, withholding taxes, and RBI approval for large amounts). The capital account asymmetry means that one direction of the arbitrage is constrained, allowing deviations to persist. This is the "onshore-offshore" segmentation premium embedded in INR forward points.
In-Lecture Formative Quiz
4 Questions • 10 MinutesSelect the best answer for each question, then click Check Answers.
1. USD/INR = 83.00. USD/JPY = 150.00. What is the JPY/INR cross rate (INR per JPY)?
2. Bank A quotes USD/INR = 83.00/83.06. Bank B quotes USD/INR = 83.08/83.14. Is two-point arbitrage possible?
3. In triangular arbitrage, if EUR/INR (direct) = 90.10 and EUR/USD × USD/INR (cross) = 90.25, which of the following is correct?
4. In Covered Interest Arbitrage, if F_market < F_CIRP (the actual forward rate is below the CIRP-implied rate), the correct arbitrage strategy is:
6. Comprehensive Arbitrage Problem Set
Solve the following problems. Show all steps, compute the arbitrage profit, and identify the direction. Problems 1–6 guided; 7–10 independent.
Cross rate: S(A/B) = S(X/B) / S(X/A) for common currency X.
No-arbitrage: S(A/C) = S(A/B) × S(B/C).
CIA: F_CIRP = S × (1 + i_h × t) / (1 + i_f × t).
Guided Practice (1–6)
P1 — Simple Cross Rate: USD/INR = 83.50. USD/SGD = 1.3400. Compute SGD/INR (INR per SGD). Ans: 83.50/1.34 = INR 62.31/SGD.
P2 — Cross Rate with Bid-Ask: USD/INR = 83.10/83.20. EUR/USD = 1.0820/1.0830. Compute EUR/INR bid-ask. Ans: Bid = 83.10 × 1.0820 = 89.91. Ask = 83.20 × 1.0830 = 90.11. EUR/INR = 89.91/90.11.
P3 — Chain Rule (4 currencies): An Indian firm pays CHF 500,000. USD/INR = 83.00, USD/CHF = 0.8850. Compute INR cost. Ans: INR/CHF = 83.00/0.8850 = 93.79. INR = 500,000 × 93.79 = INR 4,68,95,000.
P4 — Two-Point Arbitrage: Bank X: USD/INR = 82.80/82.88. Bank Y: USD/INR = 82.90/82.98. Arbitrage? Starting with INR 5 Cr, compute profit. Ans: Buy USD at Bank X (ask 82.88), sell to Bank Y (bid 82.90). 82.88 < 82.90 — arbitrage! INR 5 Cr → USD 603,282 → convert back at 82.90 → INR 5,00,12,072. Profit = INR 12,072.
P5 — Triangular Arbitrage: USD/INR = 83.00, EUR/USD = 1.0860, EUR/INR = 90.00. Starting with INR 10 Cr, compute profit. Ans: Implied EUR/INR = 83 × 1.086 = 90.14. Direct (90.00) < implied (90.14). Buy EUR direct (cheap): INR 10Cr / 90 = EUR 1,111,111. EUR → USD → INR: 1,111,111 × 1.086 × 83 = INR 10,01,55,556. Profit ≈ INR 1,55,556.
P6 — CIA: Spot = 83.00. 6M forward = 83.70. INR 6M rate = 7.0% p.a. (3.5% per 6M). USD 6M rate = 5.0% p.a. (2.5% per 6M). Starting with USD 2M, is there CIA? Ans: F_CIRP = 83 × 1.035/1.025 = 83.81. Market (83.70) < CIRP (83.81). USD forward undervalued. Borrow USD 2M at 2.5% → owe USD 2,050,000. Convert to INR: 2M × 83 = INR 166,000,000. Invest INR at 3.5% → INR 171,810,000. Buy USD forward at 83.70: 171,810,000/83.70 = USD 2,052,688. Profit = 2,052,688 − 2,050,000 = USD 2,688.
Independent Practice (7–10)
P7: GBP/USD = 1.2650/1.2660. USD/INR = 83.00/83.10. Compute GBP/INR bid-ask.
P8: Three banks quote: Bank A (Mumbai) EUR/INR = 89.50/89.60. Bank B (London) EUR/USD = 1.0820/1.0830. Bank C (Singapore) USD/INR = 82.95/83.05. Is triangular arbitrage possible? If yes, starting with EUR 1M, compute the profit via the optimal route.
P9: Spot USD/INR = 83.20. 3M forward = 83.60. INR 3M rate = 6.80% p.a. (1.70% per 3M). USD 3M rate = 5.30% p.a. (1.325% per 3M). Does CIA exist? If yes, starting with INR 5 Cr, compute profit.
P10 (Challenge): The RBI imposes a 5% withholding tax on INR interest paid to foreign investors. Spot = 83.00. 12M forward = 84.50. INR rate = 7.0% (before tax; 6.65% after tax for foreign investor). USD rate = 5.0%. (a) Compute F_CIRP before and after tax. (b) Does the tax create or eliminate a CIA opportunity? (c) What does this tell you about the role of tax policy in creating or eliminating cross-border arbitrage?
7. Exchange Rate Forecasting — A Brief Overview
Unit 3 closes with a brief examination of exchange rate forecasting — the practical activity that all the theory, computation, and arbitrage logic is ultimately directed toward. The financial manager needs an exchange rate forecast to make hedging decisions, to price cross-border contracts, and to evaluate international investments. Yet forecasting is notoriously unreliable. The key is not to abandon forecasting but to understand its severe limitations.
| Approach | Method | Strengths | Weaknesses |
|---|---|---|---|
| Fundamental Analysis | Uses economic fundamentals (PPP, IRP, BOP, monetary models) to determine the "equilibrium" exchange rate and forecast its movement toward equilibrium. | Provides a logical, theory-grounded anchor for long-horizon forecasts. Useful for identifying substantial over/undervaluation. | The Meese-Rogoff (1983) finding: fundamental models are worse than a random walk at short-to-medium horizons (1–18 months). Equilibrium may take years to be restored. Model parameters are unstable. |
| Technical Analysis | Uses past price patterns, trends, support/resistance levels, and technical indicators (moving averages, RSI, MACD) to forecast future movements. | Widely used by short-term traders. Can identify momentum and trend-following opportunities. Does not require economic data. | No theoretical foundation — purely statistical pattern recognition. Vulnerable to false signals, regime changes, and data-snooping bias (finding patterns in noise). |
| Market-Based Forecasting | Uses market prices — forward rates and interest rate differentials — as forecasts. The forward rate as the expected future spot rate (UFRH). | Readily available, costless. Embeds the collective wisdom of all market participants. No model risk. | The forward premium puzzle: forward rates are biased predictors — they systematically mispredict the direction of change. Using the forward rate as a forecast would have led to systematic losses in the carry trade. |
8. Scenario Debate: Arbitrage and Cross Rates in Practice
Rajiv runs an algorithmic triangular arbitrage strategy across USD/INR, EUR/USD, and EUR/INR on three electronic platforms (EBS, Reuters, Bloomberg). His algorithms scan for cross-rate mispricing and execute only when the arbitrage profit exceeds transaction costs (bid-ask spreads + brokerage + settlement). The algorithms execute trades in under 50 microseconds. In a typical day, the algo finds 200+ apparent mispricings, but only 5–10 exceed the transaction-cost threshold and are executable. The average profit per executed trade is USD 200. The algo has been profitable every month for 3 years.
(a) If triangular arbitrage is so straightforward, why are the profits so small — and why do opportunities persist at all? (b) What are the risks that could cause Rajiv's algo to suffer a large loss — counterparty failure, "flash crash" liquidity evaporation, or regulatory intervention? (c) How does the onshore/offshore INR segmentation create additional triangular arbitrage opportunities between the deliverable INR market and the NDF market?
IndoGlobal pays and receives in 15+ currencies across its global freight operations. The firm maintains bank accounts in INR, USD, EUR, GBP, SGD, AED, and JPY. Every day, the treasury must decide how to route payments: should a THB payment to a Thai trucking company be made by converting INR→USD→THB, or by drawing from the firm's SGD account in Singapore and converting SGD→THB directly? The chain rule can identify the cheapest path — but with 15 currencies, the number of possible paths is enormous. Priya is evaluating FX treasury management systems (TMS) that can automatically compute optimal conversion paths.
(a) For a THB 2M payment, compare three conversion paths: (i) INR → USD → THB, (ii) SGD → THB (direct), (iii) INR → SGD → THB. Use realistic rates. Which is cheapest and why? (b) How do the firm's existing currency balances affect the "optimal" path — should Priya use existing SGD balances even if the SGD→THB rate is slightly worse, to avoid converting INR? (c) How does the corporate treasurer's problem differ from a pure arbitrageur's problem — what non-price factors (relationship banking, credit lines, regulatory compliance) affect the routing decision?
National Bank of India's FX trading desk has been cited by the RBI for "excessive speculative activity" in the INR forward market. The RBI's investigation found that the desk was running a systematic CIA strategy: borrowing USD abroad through the bank's London branch at 5.0%, converting to INR at the onshore spot rate (83.00), investing in INR government bonds at 7.0%, and selling INR forward (locking in the forward rate of 84.50) — capturing a spread of approximately 0.5% above the CIRP-implied return. The RBI has directed the bank to demonstrate that all its forward trades are linked to "genuine underlying exposures" — client hedging, not proprietary arbitrage.
(a) Is the bank's CIA strategy illegal under FEMA, or merely "excessive" from a regulatory perspective? What specific FEMA provisions govern proprietary trading by banks in the FX market? (b) Why does the RBI restrict banks from engaging in CIA — what policy objective does this serve? (c) If the bank cannot run CIA directly, can it achieve the same economic result by offering clients "structured hedging products" that embed the same arbitrage logic?
Neha manages a USD 2 billion global macro fund. The fund runs a multi-strategy FX book: (a) a systematic triangular arbitrage strategy across G10 currencies (EUR, JPY, GBP, CHF, AUD, NZD, CAD, USD), (b) a discretionary carry trade strategy across EM currencies (INR, BRL, MXN, ZAR, TRY), and (c) a relative-value strategy that trades deviations from CIRP in EM forward markets. The triangular arb in G10 generates small but consistent profits (~2% annualised, Sharpe ratio 3.5). The carry trade generates higher returns (~6%) but with severe drawdowns during risk-off events (Sharpe ratio 0.8). The EM CIRP deviation strategy is the highest-return book (~8% annualised) but requires maintaining relationships with onshore banks in each EM country and navigating capital controls.
(a) Why doesn't the fund's G10 triangular arb get competed away — aren't these markets perfectly efficient? (The answer: the fund's advantage is speed — co-located servers, microwave links, FPGA hardware — not mispricing identification.) (b) For the INR CIRP deviation strategy, what specific barriers prevent other hedge funds from competing away the profits? (c) If the RBI were to fully liberalise the INR Capital Account, what would happen to Neha's INR strategies? Would profits increase (more opportunities) or decrease (more competition)?
Activity Structure
Four groups, 10 min, 3-min presentations. Synthesis: "Arbitrage is the force that makes markets efficient — but it is not costless or riskless. Speed, scale, regulatory access, and capital constraints determine who can arbitrage and how much profit remains. For the corporate financial manager, understanding arbitrage is not about becoming an arbitrageur — it is about understanding how prices are determined, why they are (mostly) consistent, and where the hidden costs and risks lie."
9. Unit 3 Synthesis — The Foreign Exchange Market
Week 10 (Spot & Forward Rates): The pricing mechanics — how spot and forward rates are quoted, how the forward premium/discount is computed and annualised, and how the forward rate serves as both an arbitrage price (CIRP) and the primary corporate hedging instrument.
Week 11 (Cross Rates & Arbitrage): The consistency enforcement — how cross rates are derived, how two-point, triangular, and covered interest arbitrage eliminate price inconsistencies, and why understanding the no-arbitrage logic is essential even when the opportunities are captured by algorithms.
Bridge to Unit 4 (International Investments): Unit 4 applies everything you have learned — about exchange rate determination (Unit 2) and FX market operations (Unit 3) — to the big decisions of international financial management: raising capital across borders (ADRs, GDRs, Masala Bonds, ECBs), constructing internationally diversified portfolios, financing foreign subsidiaries, and navigating geopolitical risks. The parity conditions, quotation conventions, and arbitrage logic you have mastered are the analytical toolkit for every decision in Unit 4.
10. Key Concepts & Terminology — Week 11
Cross Rate
An exchange rate between two currencies, neither of which is the USD, derived from their respective USD rates. S(A/B) = S(USD/B) / S(USD/A). Most non-USD pairs are traded as two "legs" through the dollar; only major crosses (EUR/JPY, EUR/GBP) have deep direct markets.
Chain Rule (Rule of Three)
A systematic method for computing an exchange rate through an arbitrary number of intermediate currencies. Each link is structured so that intermediate currencies cancel: (A→B) × (B→C) × (C→D) = (A→D). Used to find the cheapest conversion path among multiple alternatives.
Two-Point (Locational) Arbitrage
Exploiting different quotes for the same currency pair in two different locations. Buy the base currency where the ask is lowest; simultaneously sell where the bid is highest. Profit = Bid_high − Ask_low per unit. Virtually extinct in major currencies due to electronic trading; persists between onshore and offshore INR markets.
Triangular (Three-Point) Arbitrage
Exploiting inconsistent cross rates among three currencies. Converting A → B → C → A, ending with more of A than started. The no-arbitrage condition: S(A/C) = S(A/B) × S(B/C). Enforces cross-rate consistency; executed by algorithms in microseconds in modern markets.
Covered Interest Arbitrage (CIA)
The arbitrage trade enforcing CIRP: borrowing in one currency, converting at spot, investing in another, and covering the repatriation with a forward contract. Risk-free profit when the actual forward rate deviates from F_CIRP = S × (1+i_h)/(1+i_f). Constrained in INR by capital account restrictions.
No-Arbitrage Condition
The principle that identical assets or cash flows must trade at identical prices — otherwise risk-free profits are possible. Underlies CIRP (F must equal the interest-differential-implied rate), triangular arbitrage (cross rates must be consistent), and ultimately all parity conditions.
Implied Cross Rate
The cross rate computed from two USD-based exchange rates. For EUR/INR: implied = EUR/USD × USD/INR (if EUR/USD is in European terms and USD/INR in American terms). The implied rate is the benchmark against which the direct quote is compared for triangular arbitrage.
Fundamental Analysis (FX)
Exchange rate forecasting using economic fundamentals — PPP, IRP, BOP, monetary models, terms of trade. Identifies equilibrium exchange rates and predicts movement toward equilibrium. Best for long horizons (3–10 years); poor for short horizons (the Meese-Rogoff puzzle).
Technical Analysis (FX)
Forecasting using past price patterns, trends, and technical indicators. No economic theory — purely statistical. Widely used by short-term traders. Vulnerable to false signals and regime changes. No consistent evidence of predictive power beyond transaction costs.
Market-Based Forecasting
Using market prices — forward rates, interest rate differentials — as exchange rate forecasts. The forward rate as the expected future spot rate (UFRH). Simple and costless, but empirically biased — the forward premium puzzle means forward rates systematically mispredict.
Meese-Rogoff Puzzle
The finding (Meese & Rogoff, 1983) that a random walk (no-change forecast) outperforms structural exchange rate models at horizons up to 12–18 months. Survived decades of academic assault. The single most important empirical finding for corporate FX hedging policy: if the best models cannot forecast, the corporate treasurer certainly cannot.
Onshore-Offshore Arbitrage (INR)
Arbitrage between the deliverable onshore INR market (Mumbai) and the Non-Deliverable Forward (NDF) offshore INR market (Singapore, London). The two markets can price the same INR differently because capital controls prevent free arbitrage between them. The RBI intervenes to close the gap but a persistent basis exists.
Exit Ticket — Week 11 (End of Unit 3)
Complete each section. Estimated time: 7–10 minutes.
Most important concept from this session — cross rates, chain rule, a specific arbitrage type, or the forecasting overview.
What remains unclear — triangular arbitrage direction, CIA strategy selection, or the chain rule?
USD/INR = 83.00, EUR/USD = 1.0820, EUR/INR (direct) = 89.70. (a) Compute the implied EUR/INR cross rate. (b) Is there triangular arbitrage? (c) If yes, starting with INR 5 Cr, compute the profit and show the three trades.
Unit 3 (W9: FX Market, W10: Spot/Forward, W11: Cross Rates/Arbitrage) has equipped you with the operational skills to work in the FX market. In 3–4 sentences, explain how these three weeks' content collectively prepare you for a career in corporate treasury, FX trading, or international finance.
11. Session References
- Eun, C., Resnick, B., & Chuluun, T. — IFM, McGraw Hill. Chapter 5: Cross rates and arbitrage. Chapter 6: Forecasting.
- Apte, P. G., & Kapshe, S. — IFM, McGraw Hill. Chapter 7: Arbitrage in FX markets.