FRM Part I · FRM Exam Part I

Using Futures for Hedging: formula sheet

Full chapter guide

Key formulas

Basis: Basis = Spot price of asset to be hedged − Futures price of contract used; This is the usual convention. Check the sign convention in the question.
Effective price, short hedge: Price realised = S2 + F1 − F2 = F1 + b2; F1 is the initial futures price, F2 and S2 are the futures and spot prices at close-out, b2 = S2 − F2 is the final basis.
Effective price, long hedge: Price paid = S2 + F1 − F2 = F1 + b2; Same algebra as the short hedge, but now it is the cost you pay. Lower is better.
Futures gain or loss: Short: (F1 − F2) × contract size × contracts. Long: (F2 − F1) × contract size × contracts; Rising futures prices help longs and hurt shorts.
Perfect hedge: Basis at close-out known with certainty, so price is locked at F1 + b2; Basis risk is the uncertainty in b2.
Basis: Basis = S − F; Spot of the asset being hedged minus futures price of the contract used. Some books use the reverse sign, so follow the question's definition.
Effective price received (short hedge): Effective price = S2 + (F1 − F2) = F1 + b2; F1 is the initial futures price, F2 the futures price at close-out, S2 the spot at close-out, b2 = S2 − F2.
Effective price paid (long hedge): Effective price = S2 − (F2 − F1) = F1 + b2; Same expression as the short hedge. Only b2 is uncertain.
Basis direction rule: Short hedger: gains if basis strengthens. Long hedger: gains if basis weakens.; Strengthening means S − F increases; weakening means it decreases.
Convergence: Basis → 0 at delivery (same asset); Does not hold if the hedged asset differs from the futures underlying asset.
Minimum variance hedge ratio: h* = ρ × σS ÷ σF = Cov(ΔS, ΔF) ÷ Var(ΔF); σS and σF are standard deviations of changes in spot and futures prices over the hedge period. h* is the slope of the regression of ΔS on ΔF.
Optimal number of contracts: N* = h* × QA ÷ QF; QA is the size of the position being hedged (units of the asset). QF is the size of one futures contract in the same units.
Hedge effectiveness: Effectiveness = ρ² = R² = 1 − Var(hedged) ÷ Var(unhedged); It is the proportion of variance eliminated.
Variance of hedged position (per unit): Var = σS² + h² σF² − 2 h ρ σS σF; At h = h*, this reduces to σS² × (1 − ρ²).
Minimum hedged standard deviation: σhedged = σS × √(1 − ρ²); Per unit of spot exposure. Multiply by the quantity hedged for the total.
Number of contracts using values (tailing form): N* = h* × VA ÷ VF; VA is the value of the position and VF is the value of one futures contract (futures price × contract size). Use it when the question gives values rather than units.
Value of one index futures contract: F = Futures index level × Contract multiplier; Example: S&P 500 E-mini has a multiplier of $50. Use the futures price, not the spot level, unless the question says otherwise.
Contracts for a full beta hedge: N = β × V ÷ F; This is the number of contracts to short, a magnitude. It equals the absolute value of (0 − β) × V ÷ F, so it follows the same sign convention as the general formula below. V is the portfolio value.
Contracts to change beta to a target: N = (β* − β) × V ÷ F; β* is target beta, β is current beta. Positive N means go long. Negative N means go short. Setting β* = 0 gives the full hedge.
Expected portfolio return (CAPM): E(Rp) = Rf + β × (E(Rm) − Rf); Used to find the expected return of the portfolio before hedging, or after hedging at the new beta. With β = 0 it gives Rf, but that is an expectation, not a guarantee, because basis risk and specific risk remain.
Approximate hedged outcome: Hedged value change ≈ Portfolio gain/loss + Futures gain/loss; Futures gain on a short = (F0 − F1) × multiplier × N.

Quick revision

A short hedge protects an asset you own or will sell; a long hedge protects a future purchase.
Basis = spot price − futures price; a hedger is exposed to changes in basis. A strengthening basis benefits a short hedger; a weakening basis benefits a long hedger.
Basis risk arises when the asset hedged differs from the futures asset or the dates do not match. It is why hedges are imperfect. Hedging with a different but related asset can also make h* differ from 1.
Minimum variance hedge ratio: h* = ρ × σS ÷ σF.
h* differs from 1 when ρ < 1 or σS ≠ σF.
Number of contracts: N* = h* × QA ÷ QF.
Hedge effectiveness equals ρ², the share of spot variance removed.
h* is the slope from regressing spot price changes on futures price changes.
If ρ = 1 and σS = σF, then h* = 1.
Beta hedge contracts: N = (β* − β) × P ÷ A; a negative result means sell futures.
To take beta to zero, sell β × P ÷ A contracts.
Tailing adjusts the contract count by the ratio of the spot value of the exposure to the value of one futures contract: N* = h* × VA ÷ VF, where VA = QA × S and VF = QF × F. This equals h* × QA ÷ QF × (S ÷ F) and reflects daily settlement. The count falls when the futures price exceeds the spot price.
Rolling a hedge forward adds basis risk at each roll date.

Common mistakes

Choosing the wrong direction, for example going long to hedge an inventory you own. Fix: Hedge the exposure you want to cancel. If you lose when the price falls, sell futures. If you lose when the price rises, buy futures.
Assuming a futures hedge locks in the price exactly. Fix: State that the locked price is F1 + b2. It is exact only when b2 is known, such as when the contract expires on the transaction date.
Using the wrong sign for the basis Fix: Use S − F unless the question states otherwise, and write the definition before calculating.
Saying a short hedger benefits from a weakening basis Fix: Short hedger: effective price = F1 + b2, so a larger b2 helps. Strengthening helps the short hedger.
Using h* = ρ × σF ÷ σS (ratio inverted). Fix: Remember it is the slope of ΔS on ΔF, so spot volatility is on top. If futures are more volatile than spot, you need fewer than ρ per unit.
Reporting effectiveness as ρ instead of ρ². Fix: Effectiveness is variance reduction, so it is ρ². ρ = 0.8 gives 64%.
Using β × V ÷ F when the target beta is not zero. Fix: Use (β* − β) × V ÷ F for every case. For a full hedge it gives −β × V ÷ F, and β × V ÷ F is just the magnitude, the number of contracts to short.
Forgetting the contract multiplier. Fix: Multiply the index level by the multiplier first. Write F down as a separate step.

Exam tips

First decide who is exposed: a seller of the asset needs a short hedge and a buyer needs a long hedge. Many questions test only this.
When a question gives the final basis, use effective price = F1 + b2 and skip the long calculation.
For the why-hedge discussion, be ready with both sides: stable cash flows and lower distress costs for, and shareholder diversification, costs and competitive effects against.
Read carefully whether the answer asks for the futures gain, the effective price or the total amount, and whether the hedge is in contracts or in units.
Watch for statements that a hedge eliminates all risk. They are usually the wrong option, because basis risk remains.
Write the basis definition first. Questions may define it as S − F, and the strengthen/weaken answer depends on it.
Use the shortcut F1 + b2 to cross-check your longer calculation.
Remember short hedger = strengthening is good; long hedger = weakening is good. Questions often test this directly.