Currency Management: An Introduction: formula sheet

Key formulas

Quote convention

P/B = units of price currency per 1 unit of base currency

Base currency is the one being bought or sold; if P/B rises, the base appreciates.

Inverse quote

B/P = 1 ÷ (P/B); bid(B/P) = 1 ÷ offer(P/B); offer(B/P) = 1 ÷ bid(P/B)

Inverting swaps bid and offer.

Bid-ask spread

Spread = offer − bid; percent spread = (offer − bid) ÷ offer

Percent spread is usually taken on the offer in this curriculum; the sign-free spread in pips is also common.

Cross rate (common currency opposite sides)

A/C = (A/B) × (B/C)

The common currency B must cancel; invert one quote if needed.

Cross rate bid-offer

bid(A/C) = bid(A/B) × bid(B/C); offer(A/C) = offer(A/B) × offer(B/C)

If an inverse is needed, invert first and swap bid and offer.

Forward rate from points

F = S + (points ÷ scale factor)

Scale factor is usually 10,000, and 100 for yen pairs. Points can be negative.

Forward premium/discount

Premium (discount) = F − S (in P/B terms); in % = (F − S) ÷ S

Positive means the base is at a forward premium.

Covered interest rate parity

F = S × (1 + iP × Act/360) ÷ (1 + iB × Act/360)

iP is the price currency rate and iB the base currency rate. Adjust day count to the currency's convention.

Domestic currency return

RDC = (1 + RFC)(1 + RFX) − 1 ≈ RFC + RFX

RFX is the change in the domestic-currency value of one unit of foreign currency (foreign currency as base).

Portfolio currency exposure

Exposure to a currency = Σ (asset value in that currency ÷ portfolio value)

Measures weight of the portfolio sensitive to that currency.

Domestic-currency return of a foreign asset (approximate)

R_DC ≈ R_FC + R_FX

R_FC is the asset's return in foreign currency, R_FX is the foreign currency's change against the domestic currency. The exact form is (1 + R_FC)(1 + R_FX) − 1.

Minimum-variance hedge ratio

h* = Cov(R_DC, R_FX) ÷ Var(R_FX)

The slope from regressing R_DC on R_FX. Result is the share of the foreign-asset value to hedge (the slope gives the exposure to hedge).

Variance of domestic-currency return

Var(R_DC) = Var(R_FC) + Var(R_FX) + 2 × Cov(R_FC, R_FX)

Uses the approximate return relationship. Shows why a positive covariance between asset and currency raises risk.

Unhedged exposure

Unhedged share = 1 − hedge ratio

Use this to size the currency position left open after hedging.

Spectrum order

Passive → Rule-based → Discretionary (active) → Highly active

Potential alpha, tracking error, cost and complexity generally tend to rise with greater activeness. This is not guaranteed. Currency overlay is a separate dimension (who runs currency), not a further step.

Passive hedging

Fixed hedge ratio, rebalanced mechanically, no currency views

Aim is risk reduction or benchmark match, not alpha.

Rule-based hedging

Hedge ratio or position set by a predefined rule or signal

Manager has little or no discretion; it is systematic and repeatable.

Discretionary management

Manager judgment sets positions within IPS limits

Aim is to add value relative to a benchmark; depends on skill.

Highly active management

Currency treated as an asset class or profit centre

Positions need not offset portfolio currency exposure; highest potential alpha, cost and tracking error.

Currency overlay

Specialist manages currency exposure separately from the asset decisions

An implementation structure that can run any approach, including simple hedging.

Covered interest rate parity (forward rate)

F = S × (1 + i_price) ÷ (1 + i_base), with S and F quoted as price currency per 1 unit of base currency

The base currency is the foreign currency you are hedging. i_price is the interest rate of the price (home) currency. Use the rate for the forward period, not an annual rate unless the term is one year.

Forward premium or discount

(F − S) ÷ S ≈ i_price − i_base

Quote is price per base. If the foreign currency is the base and its rate is below the home rate, it trades at a forward premium.

Hedged return (approximation)

R_hedged ≈ R_foreign asset (local) + (i_home − i_foreign)

Valid when the hedge ratio is 100% and the hedge is sized to the asset value. It ignores the small cross term.

Domestic-currency return

R_DC = (1 + R_FC) × (1 + R_FX) − 1

R_FX is the % change in the home-currency value of the foreign currency.

Forward contract value before expiry (long base)

V = PV of (F_t − F_0) × notional, discounted at the price-currency rate over the remaining term

Use the current forward for the remaining term. Short position value is the negative.

Put payoff for a foreign-currency holder

Payoff = max(K − S_T, 0) per unit of foreign currency; net = payoff − premium

Floor on the home value of the asset is about K minus the premium, ignoring financing of the premium.

Quick revision

• Foreign asset return in domestic terms combines the local return and the currency return, so compute both.
• Check the quote direction before every calculation: price currency per one unit of base currency.
• Forward rate comes from covered interest rate parity: the forward premium or discount reflects the interest rate difference.
• With the quote as price currency per base currency, the base currency with the higher interest rate trades at a forward discount (F < S), and the base currency with the lower interest rate trades at a forward premium (F > S).
• A forward hedge locks in the forward rate, so it removes the currency gain or loss on the hedged amount. The hedge ratio sets how much is hedged. Residual exposure remains on any unhedged part and on the asset's changing value.
• Options protect against one direction and keep the upside, but you pay a premium.
• Hedge ratio choice depends on the client's objectives, constraints, risk tolerance and cost.
• The strategy spectrum runs from passive hedging through rule-based and discretionary to active currency management.
• Active currency management seeks added return and takes extra risk, so it needs a clear mandate and a justification.
• Hedging cost is driven by interest rate differences, and rolling a hedge adds cash flow and liquidity needs.
• In essays, answer the command word exactly, show the calculation, and give only the number of answers asked for.
• Link every recommendation to the client's stated needs, not to a generic rule.

Common mistakes

• Reading the quote backwards, treating the price currency as the base. Fix: Always say it aloud: 'one base costs X price.' Label base and price currencies first.
• Using the bid when you should use the offer, or vice versa. Fix: The bid and offer are the dealer's prices for the base. You buy the base at the offer, sell it at the bid.
• Saying the minimum-variance hedge ratio is always 100%. Fix: A 100% hedge removes the currency exposure but is not always the lowest-variance choice. For equities, correlation with the currency can make a lower ratio better.
• Regressing the wrong variables, such as the foreign-currency return on the exchange rate. Fix: Regress the domestic-currency return of the asset on the currency return. Use Cov(R_DC, R_FX) ÷ Var(R_FX).
• Calling a 50% hedge ratio kept constant a rule-based or active strategy. Fix: A fixed ratio with no views is passive, whatever the ratio. Rule-based needs the ratio to change by a rule.
• Saying rule-based strategies use manager judgment. Fix: Rule-based means mechanical and repeatable. Discretion belongs to discretionary management.
• Putting the interest rates the wrong way round in the forward formula. Fix: Write the quote as price per base first. The price-currency rate goes on top and the base-currency rate goes on the bottom.
• Thinking a forward discount or premium predicts where spot will go. Fix: Remember it is set by no-arbitrage from interest rates. It is the carry on the hedge, not a prediction.

Exam tips

• Write base and price currency next to every quote in the vignette before calculating. It takes seconds and prevents most errors.
• For calculation prompts in essay sets, show the formula and the inputs. A correct number alone earns credit, but a clear line helps if the number is off.
• Watch the command word. 'Calculate' needs a number, 'Explain' needs a reason tied to the client, and 'Identify' needs only a label.
• Check the reasonableness of every cross rate: the offer must exceed the bid, and the result should sit between the plausible values.
• Link exposure to the client. State whether the currency risk helps or hurts the client's objectives, not just what the number is.
• Start every recommendation with the client's risk tolerance, horizon and IPS rules. Then give the technical point.
• If asked to calculate, show Cov ÷ Var and the final number. A correct number alone earns full credit on a calculation, but showing the work helps if you slip.
• Follow the command word. 'Determine' wants a number. 'Justify' wants a reason tied to the facts. Give only the number of answers requested.