CMA Final · Strategic Financial Management

Forwards and Futures: formula sheet

Full chapter guide

Key formulas

Forward price, simple interest, no income: F = S × (1 + r × T); S = spot price, r = annual risk-free rate, T = time in years. Use when the question says simple interest or gives a period of months.
Forward price, annual compounding: F = S × (1 + r)^T; Use when the question says compounded annually.
Forward price, continuous compounding: F = S × e^(rT); Use when the question says continuously compounded.
Forward price with known income: F = (S − PV of income) × (1 + r)^T; Or F = S × (1 + r)^T − FV of income. Income may be dividends or coupons. Discount or compound each amount from its own date.
Forward price with dividend yield: F = S × e^((r − q)T); q = continuous dividend yield. With a discrete annual yield q, the exact form is F = S × [(1 + r) ÷ (1 + q)]^T. Otherwise use the form the question specifies. The approximation F ≈ S × (1 + r − q)^T is used only when the question specifies it.
Forward price with storage cost (commodity): F = (S + PV of storage costs) × (1 + r)^T; Storage and insurance are added. Convenience yield, if given, is subtracted.
Value of long forward: Value = (F_new − F_old) ÷ (1 + r)^(remaining T); F_new is the current forward price for the same maturity. F_old is the agreed price. Equivalent: Value = S − PV of income − [F_old ÷ (1 + r)^(remaining T)].
Implied repo rate: Implied r = [(F + FV of income) ÷ S − 1] × (12 ÷ months); For an asset with no income, FV of income is zero and this becomes (F ÷ S − 1) × (12 ÷ months). It is on a simple-interest basis. Compare it with your borrowing rate: if the implied repo rate is higher, buying spot with borrowed money and selling the forward (cash and carry) gives a profit.
Daily mark-to-market gain or loss (long): Daily MTM = (Today's settlement price − Previous settlement price) × Lot size × Number of lots; For the first day, use the entry price as the previous price. A positive result is credit to a long position.
Daily mark-to-market gain or loss (short): Daily MTM = (Previous settlement price − Today's settlement price) × Lot size × Number of lots; The sign is the reverse of the long position.
Contract value: Contract value = Futures price × Lot size; Initial margin is a percentage of this value, or an amount stated in the question.
Margin call rule: If margin balance < Maintenance margin, deposit = Initial margin − Current balance; Variation margin brings the account back to initial margin, unless the question says otherwise.
Total profit or loss on a position: Total P&L = Σ daily MTM = (Closing price − Entry price) × Quantity for a long; Use this to check your daily workings. For a short, reverse the sign.
Forward settlement (cash-settled): Payoff to buyer = Spot price at maturity − Forward price; Paid once, on the maturity date. Seller's payoff is the opposite sign.
Futures daily mark-to-market: Daily gain or loss (long) = (Today's settlement price − Previous settlement price) × Contract size; Credited or debited to the margin account every day. Short position has the opposite sign.
Total futures gain over the life: Σ daily MTM = (Final settlement price − Initial futures price) × Contract size (long); Ignoring interest on the daily cash flows, total gain matches the forward payoff if prices are equal.
Fair price, simple interest, no dividend: F = S × (1 + r × t); r is the annual rate, t is the time in years (e.g. 3 months = 0.25). Use when the question says simple interest or gives no compounding instruction.
Fair price, continuous compounding, no dividend: F = S × e^(rT); Use when the question says continuous compounding. The e^x value is normally given in the question.
Fair price with known dividends: F = S × (1 + r × t) − FV of dividends; Compound each dividend from its receipt date to the expiry date at r. A dividend received after expiry is ignored.
Fair price with dividend yield, continuous: F = S × e^((r − q)T); q is the continuous dividend yield. Typical for index futures.
Cost of carry: Cost of carry = Financing cost − Dividend income; Fair futures price = Spot + Cost of carry. Basis = Spot − Futures; it is zero at expiry.
Arbitrage profit: Profit = |Market futures price − Fair futures price| × lot size; Sell futures if market > fair. Buy futures if market < fair.
Basis: Basis = Spot price − Futures price; Some books define it the other way round. Follow the sign convention given in the question.
Minimum variance hedge ratio: h* = ρ × (σS ÷ σF); σS and σF are standard deviations of changes in spot and futures prices. ρ is their correlation.
Number of contracts (commodity or asset): N* = h* × Quantity of exposure ÷ Contract size; Round to the nearest whole contract.
Number of contracts (portfolio, full hedge): N = β × Portfolio value ÷ (Futures price × Lot size); Sell futures to hedge a long portfolio.
Number of contracts (change of beta): N = (β target − β current) × Portfolio value ÷ (Futures price × Lot size); Negative means sell. Positive means buy. Target beta 0 gives a full hedge.
Hedged outcome (short hedge): Effective price = Spot price at close + Futures gain (F0 − F1); With basis: effective price = F0 + closing basis.
Profit or loss on a long futures position: (Sell price − Buy price) × Lot size × Number of lots; Positive means profit. Brokerage and margin interest are ignored unless the question gives them.
Profit or loss on a short futures position: (Sell price − Buy price) × Lot size × Number of lots; You sell first, so sell price is the entry price and buy price is the exit price.
Fair futures price (discrete compounding): F₀ = S₀ × (1 + r)^T; T is in years. For a short period, S₀ × (1 + r × T) is often used. Follow what the question states.
Fair futures price (continuous compounding): F₀ = S₀ × e^(rT); Use when the question gives a continuously compounded rate or asks for it.
Fair futures price with known dividend: F₀ = (S₀ − PV of dividends) × (1 + r)^T, or S₀(1 + r)^T − FV of dividends at expiry; Both give the same result if the present value and future value are taken consistently.
Fair futures price with dividend yield: F₀ = S₀ × e^((r − q)T); q is the continuous dividend yield. Used mainly for index futures.
Cash and carry arbitrage profit: Quoted F − Fair F (when quoted F is higher); Per unit. Multiply by lot size and lots.
Reverse cash and carry arbitrage profit: Fair F − Quoted F (when quoted F is lower); Per unit. Needs the ability to short sell the asset or hold it already.
Currency futures fair price (discrete): F = S × (1 + r_d × t) ÷ (1 + r_f × t); S is the spot in domestic currency per unit of foreign currency. r_d is the domestic rate, r_f the foreign rate, t in years. Continuous form: F = S × e^((r_d − r_f) × t).
Interest rate future quote: Price = 100 − implied annual rate (%); Used for short-term rate futures. Rate up means price down.
Value of one tick (0.01) for a short-term rate future: Tick value = Contract size × 0.0001 × (contract months ÷ 12); A 3-month contract of ₹1,00,00,000 gives ₹250 per tick.
Number of currency futures contracts: N = Foreign currency exposure ÷ Lot size; Round to the nearest whole contract and state any unhedged amount.
Number of interest rate futures contracts: N = (Amount to hedge ÷ Contract size) × (Exposure months ÷ Contract months); Adjust the second ratio when your loan period differs from the contract period.
Duration-based hedge ratio (bond futures): N = (Modified duration of portfolio × Portfolio value) ÷ (Modified duration of futures × Futures contract value); Use when hedging a bond portfolio with bond futures.
Futures gain or loss: Long: (F_close − F_open) × quantity. Short: (F_open − F_close) × quantity; Net result = spot outcome + futures gain or − futures loss.
Basis: Basis = Spot price − Futures price; It falls to zero at expiry. If it does not, basis risk remains.

Quick revision

A forward is an over-the-counter, customised contract; a future is an exchange-traded, standardised contract.
Futures are marked to market daily; forwards are normally settled only at maturity.
Fair futures price = spot price + cost of carry, where cost of carry is financing cost less income from the asset.
With simple interest: F = S × (1 + r × t), where t is in years.
For a share paying a dividend before expiry, deduct the dividend (or its value) from the carry.
If market futures price is above fair price, sell the future and buy the asset; if below, buy the future and sell the asset.
Initial margin is paid at entry; maintenance margin triggers a margin call when the balance falls below it.
Number of contracts for a hedge = hedge ratio × value of exposure ÷ value of one futures contract.
To hedge a portfolio with an index future, scale by the portfolio's beta.
A long hedge protects against a price rise; a short hedge protects against a price fall.
Basis = spot price − futures price, and it narrows towards zero at expiry.
Always end a numerical answer with a clear decision and the net result.

Common mistakes

Using T in months in the formula. Fix: Always convert to years first. 6 months = 0.5.
Adding dividends instead of subtracting them. Fix: Income from the asset lowers the cost of carry. Subtract its value from the forward price.
Calculating profit only from entry price to final price and ignoring the daily balance. Fix: Build the day-by-day table. Margin calls depend on the path of prices, not only the end price.
Topping up only to the maintenance margin level. Fix: Unless the question says otherwise, the call restores the balance to the initial margin. Deposit = initial margin − current balance.
Saying futures have no counterparty risk at all. Fix: Write that counterparty risk is greatly reduced, not removed, by the clearing corporation and margins.
Saying forwards are settled daily. Fix: Remember: daily settlement belongs to futures. A forward settles once, on maturity.
Using the annual rate without converting the time period Fix: Always convert months to years first: t = months ÷ 12. Write t on the page before calculating.
Forgetting to subtract dividends Fix: Read the question for any dividend or yield. Subtract its future value from the carried-forward spot price.
Buying futures to hedge a share portfolio, or selling to hedge a future purchase. Fix: Ask what hurts you. If a price fall hurts, sell futures. If a price rise hurts, buy futures.
Using portfolio value ÷ index level without lot size, or ignoring the futures price. Fix: Always compute contract value = futures price × lot size first and divide by it.

Exam tips

Read the compounding basis first. Examiners often change between simple, annual and continuous across questions, and the answer differs by a few rupees.
Show the cash flows at each date in a short table or list. Marks are given for method even if the final figure differs.
In arbitrage questions, write both legs: the action today and the action at maturity. Then state the profit.
Learn the e^x values you may be given, such as e^0.05, and use them as supplied. Do not recompute from memory.
For MCQs, check direction before calculation. With positive rates and no income, the forward price is above spot. With high income, it can be below spot.
In MCQs, test each statement on the clearing house, standardisation and margins against the definitions. Watch words like 'full price' or 'only at expiry', which signal a wrong option.
Always show the daily table. Even if the final answer is wrong, you keep marks for method.
Read whether the question says the call restores initial margin or maintenance margin, and follow it.