FRM Part I · FRM Exam Part I

Foreign Exchange Markets: formula sheet

Full chapter guide

Key formulas

Quote convention: A/B = x means 1 unit of A (base) costs x units of B (quote); The base currency is always one unit. The quote currency is the price.
Inverse quote: B/A = 1 ÷ (A/B); Converts a direct quote to an indirect quote and back.
Inverting bid and ask: Bid(B/A) = 1 ÷ Ask(A/B); Ask(B/A) = 1 ÷ Bid(A/B); Bid and ask swap places when you invert. Do not invert each side into itself.
Bid-ask spread: Spread = Ask − Bid; Quoted in pips or as a percentage of the midpoint.
Percentage spread: Spread % = (Ask − Bid) ÷ Midpoint, where Midpoint = (Bid + Ask) ÷ 2; Allows comparison of liquidity across pairs.
Cross rate (both quoted against USD as base): A/B = (USD/B) ÷ (USD/A); Bid(A/B) = Bid(USD/B) ÷ Ask(USD/A); Ask(A/B) = Ask(USD/B) ÷ Bid(USD/A).
Cross rate (chain rule): A/C = (A/B) × (B/C); Bid × bid gives the bid and ask × ask gives the ask, but only when both quotes are already in chain form (A/B and B/C). Check that currency B cancels. For two USD-base quotes, do not multiply sides directly. Use Bid = Bid(USD/B) ÷ Ask(USD/A).
Percentage change in a currency: Change in base value = (S1 ÷ S0) − 1 for quote A/B; A rise in A/B means the base A appreciated and the quote B depreciated.
Forward rate from interest rates (CIP): F = S × (1 + r_quote × d/basis_quote) ÷ (1 + r_base × d/basis_base); S and F are quote currency per unit of base currency. d is the days to the forward date. Use simple interest with each currency's own day-count basis: 360 for USD and EUR, 365 for GBP. The formula reduces to d/360 for both currencies only when both currencies use 360. Use the day count the question states.
Forward points: Forward points (pips) = (F − S) × 10,000; Use × 100 when the quote currency is JPY. Positive points mean a forward premium for the base currency. Negative points mean a discount.
Forward rate from points: F = S + points ÷ 10,000; Use ÷ 100 for JPY quote pairs. Apply the bid points to the spot bid and the ask points to the spot ask.
Premium or discount rule: r_quote > r_base ⇒ F > S (base at premium); r_quote < r_base ⇒ F < S (base at discount); Compare rates for the same period as the forward. The higher-interest-rate currency trades at a forward discount.
Reading bid-ask points: If the first points figure < the second, add both to spot. If the first > the second, subtract both.; This is a market convention shorthand, not a derived formula. Check that the forward ask stays above the forward bid.
FX swap structure: Near leg at S, far leg at F, same notional, opposite directions; Net FX exposure is zero. The economic cost is the forward points.
CIP forward rate (discrete compounding): F = S × (1 + r_d × T) ÷ (1 + r_f × T); S and F are domestic per one foreign unit. r_d is the domestic rate, r_f the foreign rate, T in years using the money market day count.
CIP forward rate (continuous compounding): F = S × e^((r_d − r_f) × T); Use when the question gives continuously compounded rates.
Forward premium or discount: (F − S) ÷ S = (r_d − r_f) × T ÷ (1 + r_f × T) ≈ (r_d − r_f) × T; A positive value means the foreign currency is at a forward premium. The exact form is the first expression. The approximation drops the denominator (1 + r_f × T), so it is not exact. Use it to check direction and use the full formula for answers.
Uncovered interest parity: E(S_T) = S × (1 + r_d × T) ÷ (1 + r_f × T); Uses the expected future spot rate. It carries exchange rate risk, so it is not an arbitrage condition.
Cross-currency basis (idea): Basis = implied rate from FX swap − quoted money market rate; Zero under perfect CIP. A negative basis on a currency means its borrower pays extra to obtain the currency, usually USD, through FX swaps. Sign conventions vary, so read the question.
Covered interest parity (direct quote, simple rates): F ÷ S = (1 + r_d × T) ÷ (1 + r_f × T); S and F are domestic currency per one unit of foreign currency. r_d is the domestic rate, r_f the foreign rate. With continuous compounding, F = S × e^((r_d − r_f)T).
Uncovered interest parity (exact): E[S_T] ÷ S = (1 + r_d × T) ÷ (1 + r_f × T); Same quote convention as CIP. It uses the expected future spot rate, not the forward rate, and is not hedged.
Uncovered interest parity (approximation): Expected % change in S ≈ r_d − r_f; S is domestic per foreign. If r_d > r_f, the domestic currency is expected to depreciate, so S rises by about r_d − r_f and the foreign currency appreciates.
Absolute PPP: S = P_d ÷ P_f; P_d and P_f are price levels of the same basket in each country. S is domestic per foreign.
Relative PPP (approximation): Expected % change in S ≈ π_d − π_f; π is the inflation rate. The foreign currency appreciates if domestic inflation is higher.
Relative PPP (exact): S_T = S × (1 + π_d)^T ÷ (1 + π_f)^T; Use for multi-year horizons with annual inflation rates.
Real exchange rate: q = S × P_f ÷ P_d; Constant q means PPP holds. A rise in q is a real depreciation of the domestic currency.
Carry trade excess return: Return ≈ (r_high − r_low) − % depreciation of the high-rate currency; The trade profits when the high-rate currency depreciates by less than the rate gap.
Covered interest parity forward (continuous compounding): F = S × e^((r_d − r_f) × T); S and F are domestic per unit of foreign currency. r_d is the domestic rate and r_f the foreign rate. With annual compounding use F = S × (1 + r_d)^T ÷ (1 + r_f)^T.
Garman-Kohlhagen call: c = S × e^(−r_f × T) × N(d1) − K × e^(−r_d × T) × N(d2); Option on one unit of foreign currency, priced in domestic currency.
Garman-Kohlhagen put: p = K × e^(−r_d × T) × N(−d2) − S × e^(−r_f × T) × N(−d1); Same d1 and d2 as the call.
d1 and d2: d1 = [ln(S ÷ K) + (r_d − r_f + σ² ÷ 2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; Swapping r_d and r_f is the most common error.
Put-call parity for currency options: c − p = S × e^(−r_f × T) − K × e^(−r_d × T); European options with the same strike and maturity.
Option on a currency forward (Black model): c = e^(−r_d × T) × [F × N(d1) − K × N(d2)], with d1 = [ln(F ÷ K) + σ² × T ÷ 2] ÷ (σ × √T); Use when the option is written on a forward or futures rate F.
Value of a cross-currency swap: V = B_D − S × B_F (receive domestic, pay foreign); B_D and B_F are the present values of the domestic and foreign legs, including principal. Reverse the sign if you receive foreign and pay domestic. Value at inception is zero only if the swap is set at market (fair) rates. The worked example does not assume this.
Swap principals: Foreign principal = Domestic principal ÷ S0; Exchanged at the start and returned at maturity at the same rate S0.
Balance of payments identity: Current account + Capital account + Financial account + Errors and omissions = 0; Reserve changes sit inside the financial account. This sum-to-zero form uses the older convention: credits are positive, debits are negative, and financial inflows are positive. So a current account deficit implies a net financial account inflow. Under the IMF BPM6 convention the financial account is net acquisition of financial assets minus net incurrence of liabilities. The identity then reads: current account + capital account (+ errors and omissions) = financial account net lending (+) or net borrowing (−). A current account deficit therefore shows as a negative financial account balance. Check which convention a question uses.
Current account components: Current account = Trade balance (goods and services) + Net primary income + Net secondary income; Trade balance = Exports − Imports.
Saving-investment link: Current account balance = National saving − Domestic investment; A deficit means the country invests more than it saves.
Trilemma: Choose only two of: fixed exchange rate, free capital flows, independent monetary policy; Explains why pegs force policy to follow the anchor country.
Percentage change in a quoted rate: % change = (S1 − S0) ÷ S0 × 100%; This gives the percentage change in the quoted rate, which is the price of the base currency. For a direct quote (domestic per foreign), a rise means the foreign currency appreciated and the domestic currency depreciated. For an indirect quote (foreign per domestic), a rise means the domestic currency appreciated.

Quick revision

In a quote A/B, A is the base and B is the quote currency; the price is units of B per one A.
Inverting a quote flips base and quote: A/B = 1 ÷ (B/A).
Cross rate: multiply or divide so that the unwanted currency cancels.
Bid is where the dealer buys the base; ask is where the dealer sells it.
Forward points are the forward minus spot, adjusted for the market's scaling convention.
Covered interest parity: F = S × (1 + r quote × T) ÷ (1 + r base × T), with rates on a consistent basis and S in quote per base.
The currency with the higher interest rate trades at a forward discount under covered parity.
Covered parity is enforced by arbitrage; uncovered parity uses the expected future spot and carries risk.
Relative PPP links expected currency change to the inflation difference; it is a long-run tendency, not a short-term rule.
An FX swap pairs a spot deal with an opposite forward deal; a cross-currency swap exchanges interest, and usually principal, in two currencies.
A current account deficit must be matched by a net inflow on the capital and financial accounts.
Under a fixed peg, the central bank must defend the rate with reserves or interest rate policy.

Common mistakes

Mixing up the base and quote currency Fix: Always read A/B as 'one A costs x B'. The first currency is the one priced.
Using the bid when the client buys the base currency Fix: Quotes are from the dealer's view. The dealer buys at the bid and sells at the ask, so the client buying the base pays the ask.
Inverting the forward formula, putting the base rate in the numerator. Fix: Write the pair as base/quote first. The quote currency's rate goes in the numerator and the base currency's rate in the denominator.
Treating the forward rate as the expected future spot rate. Fix: The forward comes from the interest rate gap through no-arbitrage. It is not a forecast, and it can differ from the realised spot.
Inverting the quote direction and using the wrong rate in the numerator. Fix: Write 'domestic per foreign' before you start. The domestic rate goes in the numerator, the foreign rate in the denominator.
Using the interest rate difference with the approximation and treating it as exact. Fix: Use the full ratio formula for numeric answers. Use the approximation only to check direction.
Using the forward rate in place of the expected spot rate for UIP. Fix: CIP uses the forward rate F and is a no-arbitrage result. UIP uses the expected spot E[S_T] and is a hypothesis about expectations. Check which one the question names.
Flipping the quote convention, so the currency is predicted to move the wrong way. Fix: Write down what S means before calculating. If the quote is inverted, invert the ratio (swap domestic and foreign in the formula).
Using the foreign and domestic rates the wrong way round in the forward or d1 formula. Fix: Write S as domestic per foreign first. Then r_d is the rate of the currency in the numerator of S. Check the direction against the rule that the higher-rate currency trades at a forward discount.
Saying futures and forwards are the same. Fix: Remember: futures are standardised, exchange-traded, cleared and marked to market daily. Forwards are customised OTC contracts settled at maturity, with counterparty risk.

Exam tips

Read the question for who is buying and which currency is the base. Underline both before you touch the numbers.
Expect distractor options built from the wrong side of the spread or from not swapping bid and ask on inversion. Compute the correct side before looking at the options.
When inverting a two-way quote, confirm bid is still below ask. This catches most sign and side mistakes in seconds.
Keep four decimals through the calculation and round only at the end, since pip-level differences separate the answer options.
Know the vocabulary: interbank market, OTC, T+2 spot settlement, direct versus indirect quote, and spread as a liquidity measure.
Write the pair as base/quote before touching any number. Most formula errors come from mixing up the two currencies.
Check the sign of the forward points against the interest rate gap before choosing an answer. This removes two options quickly.
Read whether the question gives rates or points, and the pip size for the pair. JPY questions use 0.01.