Strategic Financial Management · Forwards and Futures

Hedging with Futures and Optimal Hedge Ratio

Updated 11 October 2026 · Fact-checked

Hedging with futures means taking a futures position opposite to your spot exposure, so that gains on one offset losses on the other. For a portfolio, number of contracts = (target beta − current beta) × portfolio value ÷ (futures price × lot size). A negative answer means sell contracts; a positive one means buy.

Understand Hedging with Futures and Hedge Ratio

A hedge is a futures position taken to cut the risk of an existing or planned spot position. You do not try to profit from the futures. You try to fix the price or value you will realise.

A short hedge means selling futures. You use it when you hold an asset, or will sell one later, and fear a price fall. A fund manager holding shares sells index futures. A long hedge means buying futures. You use it when you must buy later, or have a short position, and fear a price rise.

Basis = spot price − futures price. At expiry the basis is close to zero. Before expiry it can change. Basis risk is the risk that the basis changes between the day you set the hedge and the day you close it. A hedge is therefore rarely perfect. It replaces price risk with the smaller basis risk. Basis risk is also larger when the asset hedged differs from the futures underlying (cross hedge) or when the hedge ends before expiry.

The hedge ratio is the size of the futures position relative to the exposure. The minimum variance hedge ratio is h* = ρ × (σS ÷ σF), where ρ is the correlation between changes in spot and futures prices, σS is the standard deviation of spot price changes and σF that of futures price changes. It is also the slope of a regression of spot changes on futures changes.

For a diversified equity portfolio hedged with index futures, the hedge ratio is the portfolio beta. To remove market risk fully, target beta is zero. To change beta to some other level, you hold only part of the full hedge. Beta is measured against the index on which the futures are written.

Key rules to remember

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.

How to solve Hedging with Futures and Hedge Ratio questions

Use this order for any question on hedging with futures.

1Identify the exposure: do you hold assets (fear a fall) or must you buy later (fear a rise)? This fixes short or long hedge.
2Note the underlying, the futures price, the lot size and the hedge period. Check that the futures expiry is on or after the hedge end.
3Choose the ratio: 1 for a simple hedge, h* = ρσS/σF if given, or beta for a portfolio. For a partial change, use target beta minus current beta.
4Compute the contract value = futures price × lot size, then the number of contracts. Round sensibly and state buy or sell.
5Find the spot result at the close date: gain or loss on the portfolio or the purchase cost.
6Find the futures result: (sell price − buy price) × lot size × contracts. Sign it correctly for a short or long position.
7Add the two results and state the net outcome or effective price. Compare with the unhedged result if asked.
8Comment on any residual basis risk or imperfect beta, and give a clear conclusion.

Quickest way: Beta contract count in four lines

When to use it: Use for portfolio hedging with index futures when value, beta, futures price and lot size are given.

Compute the contract value: futures price × lot size.
Compute the beta gap: target beta − current beta. Target is 0 for a full hedge.
N = beta gap × portfolio value ÷ contract value. The sign tells you buy or sell.
For the outcome, take the index fall or rise in percent, apply beta to the portfolio, and add the futures gain on N contracts.

Common mistakes in Hedging with Futures and Hedge Ratio

Buying futures to hedge a share portfolio, or selling to hedge a future purchase.
Students match the futures action to the spot action instead of opposing it.
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.
Contract value is forgotten, so the count is wrong by a large factor.
Fix: Always compute contract value = futures price × lot size first and divide by it.
Using beta × value ÷ contract value when the target beta is not zero.
The full-hedge formula is memorised without the target-beta term.
Fix: Use (target − current) beta. If the target is 0, it reduces to the full-hedge formula.
Assuming a futures hedge removes all risk.
Basis is treated as constant, and beta as exact.
Fix: State that basis risk and beta error remain. Compute the actual outcome using the closing futures price given.
Confusing the hedge ratio h* with the correlation or swapping σS and σF.
The formula is learnt as a pattern without meaning.
Fix: Remember h* = ρ × spot SD ÷ futures SD. It is a regression slope of spot on futures.
Taking the futures gain with the wrong sign for a short position.
Profit is computed as closing minus opening for every position.
Fix: Short: opening price − closing price. Long: closing price − opening price.

Worked examples

Example 1

Ramesh Traders holds a diversified equity portfolio worth ₹5,00,00,000 with beta 1.2. Nifty futures trade at 25,000 and the lot size is 50. He wants to remove market risk fully for one month. (a) How many contracts should he trade? (b) If the Nifty falls 4% and the futures also fall 4% to 24,000, find the net result. Assume the portfolio moves by beta times the index move.

Show the solution

Contract value = 25,000 × 50 = ₹12,50,000.
Target beta 0, current beta 1.2, so beta gap = −1.2.
N = −1.2 × 5,00,00,000 ÷ 12,50,000 = −6,00,00,000 ÷ 12,50,000 = −48. So sell 48 contracts.
Portfolio fall = 1.2 × 4% = 4.8% of ₹5,00,00,000 = ₹24,00,000 loss.
Futures gain = (25,000 − 24,000) × 50 × 48 = 1,000 × 2,400 = ₹24,00,000.
Net = −24,00,000 + 24,00,000 = ₹0.

Answer: Sell 48 Nifty futures contracts. Under the stated assumptions the loss of ₹24,00,000 on the portfolio is exactly offset by a futures gain of ₹24,00,000, so the net result is nil.

Example 2

A fund holds a portfolio worth ₹10,00,00,000 with beta 0.8. The manager expects the market to fall and wants to cut beta to 0.2 using Sensex futures at 80,000, lot size 10. How many contracts should she trade? Then find the number needed for a full hedge.

Show the solution

Contract value = 80,000 × 10 = ₹8,00,000.
Reduce beta: gap = 0.2 − 0.8 = −0.6.
N = −0.6 × 10,00,00,000 ÷ 8,00,000 = −6,00,00,000 ÷ 8,00,000 = −75. Sell 75 contracts.
Full hedge: gap = 0 − 0.8 = −0.8.
N = −0.8 × 10,00,00,000 ÷ 8,00,000 = −8,00,00,000 ÷ 8,00,000 = −100. Sell 100 contracts.

Answer: Sell 75 contracts to bring beta down to 0.2. A full hedge needs 100 contracts sold.

Exam tips

Write contract value first. Most marks in the beta numerical sit in that one line and the beta gap.
State the action in words: sell or buy, and the number of contracts. A bare negative sign can lose marks.
If the question gives actual closing prices, compute the real hedged result and mention the residual basis risk. Do not claim a perfect hedge.
For the minimum variance ratio, show the formula h* = ρσS/σF, substitute, then convert to contracts using exposure ÷ contract size.
Check whether the question gives portfolio beta against the same index as the futures. If not, mention the cross-hedge risk in your comment.

Practice questions from Forwards and Futures

Hedging with Futures and Hedge Ratio in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Hedging with Futures and Hedge Ratio: frequently asked questions

How do I calculate the number of futures contracts to hedge a portfolio?

Compute the contract value as futures price × lot size. Then N = (target beta − current beta) × portfolio value ÷ contract value. A negative result means sell that many contracts.

What is basis risk in futures hedging?

Basis is spot price minus futures price. Basis risk is the chance that it changes between opening and closing the hedge, so gains and losses do not fully offset. It is zero only if you hold the hedge to expiry on the same underlying.

What is the optimal hedge ratio formula for minimum variance?

h* = ρ × (σS ÷ σF), where ρ is the correlation of spot and futures price changes and σS, σF are their standard deviations. It minimises the variance of the hedged position. Multiply by the exposure and divide by contract size to get contracts.

When do I use a long hedge instead of a short hedge?

Use a long hedge when you will buy the asset later or hold a short position, and a price rise would hurt. Use a short hedge when you hold the asset or will sell it later and a fall would hurt.