FRM Part I · FRM Exam Part I

Futures Markets: formula sheet

Full chapter guide

Key formulas

Futures contract value at initiation: Value of a new futures contract = 0; No cash changes hands for the contract. Margin is a deposit, not a price.
Long futures payoff at maturity: Long payoff = S(T) − F(0); S(T) is the spot price at maturity and F(0) is the price agreed at the start. Ignores daily settlement timing.
Short futures payoff at maturity: Short payoff = F(0) − S(T); The short gains when the price falls. Long and short payoffs always sum to zero.
Total gain on a position: Gain = (F(close) − F(entry)) × contract size × number of contracts (long); reverse sign for short; Daily settlement splits this gain across days but the total is the same, ignoring interest.
Margin call test: Margin call if account balance < maintenance margin; deposit is needed to restore the initial margin level; Standard exchange practice: top up to initial margin, not just to maintenance.
Open interest: Open interest = number of long positions outstanding = number of short positions outstanding; Count each contract once, not both sides. It rises when a new long and new short both open.
Daily variation margin (long): Variation margin = (Settlement price today − Settlement price yesterday) × Contract size × Number of contracts; Positive means cash credited to a long. A short receives the negative of this amount.
Daily variation margin (short): Variation margin = (Settlement price yesterday − Settlement price today) × Contract size × Number of contracts; Short gains when prices fall.
Margin account balance: New balance = Old balance + Variation margin + Deposits − Withdrawals; Check against maintenance margin each day.
Margin call amount: Call = Initial margin level − Current balance (if balance < maintenance margin); You restore to initial margin, not to maintenance. Scale both levels by the number of contracts.
Price at which a margin call occurs (long): Price drop that triggers call = (Balance − Maintenance margin) ÷ Contract size, per contract; For a short, the trigger is a price rise of the same size.
Total gain over life: Sum of daily settlements = (Final price − Original futures price) × Contract size (long); Ignores interest earned on the daily flows.
Novation: Buyer ↔ Seller becomes Buyer ↔ CCP and CCP ↔ Seller; The CCP is counterparty to both sides. It has no net market position when all members perform.
Variation margin: Variation margin = (F_t − F_t-1) × contract size × number of contracts (for a long); Paid daily. The long receives it when the futures price rises and pays when it falls. Signs reverse for a short.
Margin account balance: New balance = old balance + gains − losses − withdrawals + deposits; A margin call is issued when the balance falls below the maintenance margin. The deposit must restore it to the initial margin level (in the standard exchange convention).
Default waterfall (typical order): Defaulter's margin → defaulter's default fund share → CCP capital → survivors' default fund → other tools; Order is a typical design, not a universal rule. Use the CCP's stated order if given.
Multilateral vs bilateral netting: Net exposure to CCP = |Σ of a member's positions in the same product| versus bilateral netting within each pair; CCP netting is multilateral. Bilateral netting applies only between two parties under a netting agreement.
Convergence at expiry: Futures price at expiry = Spot price at expiry; Holds at delivery for the deliverable asset. Before expiry, the gap is the basis: spot − futures.
Basis: Basis = Spot price − Futures price; Basis tends toward zero as expiry approaches. Sign conventions vary, so read the question.
Cash settlement flow: Long gain = (Final settlement price − Previous settlement price) × contract size; Final day is marked to the reference price. Short gets the opposite.
Cheapest to deliver (T-bond): Cost to deliver = Quoted bond price − (Settlement futures price × Conversion factor); Choose the bond with the lowest value. The short receives the futures settlement price × conversion factor plus accrued interest, and pays the quoted price plus accrued interest, so accrued interest cancels in the comparison.
Cash received by short at delivery: Cash = (Settlement futures price × Conversion factor) + Accrued interest; Applies to Treasury bond futures delivery.
Daily price limit band: Limit range = Previous settlement ± Daily limit; Orders cannot trade outside the band on that day.
Basis: Basis = Spot price − Futures price; Use the sign convention stated in the question. Some texts define it the other way round.
Effective price with a hedge: Effective price = F1 + b2 (b2 = S2 − F2); F1 is the futures price at start. b2 is the basis when the hedge is closed. Same form for long and short hedges.
Minimum variance hedge ratio: h* = ρ × σS ÷ σF; ρ is the correlation between changes in spot and futures prices. σS and σF are standard deviations of those changes over the hedge period.
Number of contracts: N* = h* × QA ÷ QF; QA is the size of the exposure in units. QF is the size of one futures contract in the same units.
Hedge effectiveness: Effectiveness = ρ²; The share of variance of the spot change removed by the optimal hedge.
Hedge ratio as regression slope: h* = Cov(ΔS, ΔF) ÷ Var(ΔF); Equivalent to ρ × σS ÷ σF. Slope of ΔS regressed on ΔF.
Variance of hedged position (per unit): σ²hedged = σS² × (1 − ρ²); Residual variance when the optimal ratio is used.
Value of one index futures contract: A = futures price × multiplier; Use the futures price, not the spot index, if the question gives both. Check the multiplier given in the question.
Contracts to hedge fully (beta to zero): N = β × P ÷ A; Sell N contracts when you hold the portfolio long.
Contracts to change beta: N = (β* − β) × P ÷ A; β* is target beta. Positive N means buy, negative N means sell.
Futures hedge of a portfolio with beta: Hedge ratio h = β, in units of portfolio value per unit of futures value; The minimum variance hedge ratio equals the regression slope of portfolio returns on futures returns, which is close to beta.
Expected return of hedged portfolio (CAPM view): E(R) ≈ Rf + α; When beta is hedged to zero, market return drops out. The result is about the risk-free rate plus any alpha.

Quick revision

Futures are standardised and exchange-traded; forwards are customised and over the counter.
Marking to market settles gains and losses daily; the long gains when the futures price rises.
A margin call arises when the balance falls below maintenance margin; you must restore to initial margin, not just to maintenance.
The clearinghouse stands between buyer and seller, and margin plus default funds protect it.
Most futures positions are closed out before delivery; cash-settled contracts involve no physical delivery of the asset.
Basis = spot − futures (the convention in Hull; check the definition given in the question); basis risk is uncertainty in the basis when you close the hedge.
Minimum-variance hedge ratio h* = ρ × σS ÷ σF.
Beta hedge: N = (β* − β) × P ÷ A; negative means short futures.
Treasury futures: the short chooses the cheapest-to-deliver bond, using conversion factors.
Cost of carry for an asset with no income: F = S × e^(rT) under continuous compounding.
Rolling a hedge exposes you to the basis or spread between contract months.
A perfect hedge removes price risk only if asset, amount and timing match; otherwise basis risk remains.

Common mistakes

Saying futures have no credit risk at all. Fix: Say credit risk is greatly reduced by the clearinghouse, margining and daily settlement, not eliminated.
Treating margin as a payment for the contract or as a down payment on the price. Fix: Margin is performance collateral. A new futures contract has zero value and no price is paid at initiation.
Restoring the account only to the maintenance margin. Fix: The call brings the balance back to initial margin. Call = initial margin − current balance.
Using the wrong sign for a short position. Fix: For a short, a price rise is a loss. Reverse the sign before updating the balance.
Saying a CCP eliminates counterparty risk. Fix: Say it reduces and mutualises counterparty risk. Risk moves to the CCP and its members, and a CCP can fail if losses exceed its resources.
Putting survivors' default fund contributions before the defaulter's own resources in the waterfall. Fix: The defaulter pays first: its margin, then its default fund share. Survivors' contributions come after the CCP's own capital in the typical order.
Saying the long chooses which bond to deliver. Fix: The short holds the delivery options. They lower the futures price.
Picking the bond with the lowest quoted price as cheapest to deliver. Fix: Compute quoted price − (futures price × conversion factor) for each bond. Choose the smallest.
Using the wrong side of the market, for example going long futures to protect a holding. Fix: Ask what hurts you. If a price fall hurts, you need a gain when price falls, so go short futures.
Swapping σS and σF in the hedge ratio. Fix: Remember spot on top: h* = ρ × σS ÷ σF. It is the regression of spot changes on futures changes.

Exam tips

Know the forward vs futures table cold: venue, standardization, settlement, credit risk, flexibility. Many questions are just this comparison.
Read each option for absolute words like always, never or eliminates. Futures reduce credit risk but do not remove it.
In margin problems, the call restores the initial margin, not just maintenance margin, unless the question says otherwise.
Classify participants by motive: hedger reduces risk, speculator takes risk, arbitrageur locks in riskless profit.
Use a financial calculator or simple arithmetic for price change × units, and check the sign for a short before choosing.
Always compute totals for the whole position first. Many wrong options come from using per-contract margin.
Memorize that a call restores the account to initial margin. Options often include the gap to maintenance as a trap.
Check whether the balance is below maintenance, not merely below initial margin.