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FRM Exam Part I · Futures Markets

Stock Index Futures and Beta Hedging for FRM Part I

Updated 11 October 2026 · Fact-checked

A stock index future lets you add or remove market exposure without trading the stocks. To change a portfolio's beta from β to β*, trade N = (β* − β) × P ÷ A contracts, where P is portfolio value and A is the futures price times the multiplier. A negative N means sell.

Understand Stock Index Futures and Beta Hedging

A stock index future is a futures contract on an equity index such as the S&P 500. It is cash settled. One contract is worth the futures price multiplied by the contract multiplier. For example, if the index future is 4,500 and the multiplier is $50, one contract covers 4,500 × 50 = $225,000 of exposure.

A portfolio's beta measures how much it moves for a 1% move in the market index. A beta of 1.2 means a 1% market fall is expected to cost the portfolio about 1.2%. You can reduce this market risk by selling index futures. If the market falls, the futures gain and offset the loss on the stocks.

Beta hedging works on the market (systematic) part of risk only. Stock-specific risk stays. So a hedge to beta zero removes expected market moves but leaves the portfolio with idiosyncratic risk and basis risk. Expected return of the hedged portfolio is then about the risk-free rate plus alpha, not zero.

You do not have to hedge fully. To move beta from 1.2 to 0.8, you sell fewer contracts. To raise beta, you buy futures. The same formula covers all cases: the number of contracts is the change in beta wanted, times portfolio value, divided by the value of one contract.

Key formulas to remember

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.

How to solve Stock Index Futures and Beta Hedging questions

Use this order for any index futures or beta hedging question.

  1. 1Write down portfolio value P, current beta β and target beta β*.
  2. 2Identify the futures price and the multiplier. Compute A = price × multiplier.
  3. 3Decide the target: full hedge means β* = 0. Partial hedge or beta increase means use the stated β*.
  4. 4Compute N = (β* − β) × P ÷ A.
  5. 5Read the sign: negative means sell contracts, positive means buy. Round to the nearest whole contract only if the question asks.
  6. 6If asked for a result, apply the market move: portfolio gain = β × market move × P, futures gain = −N_sold × change in futures price × multiplier.
  7. 7Check that your answer makes sense: a long portfolio hedged down should need short futures.

Quickest way: One-line contract count

When to use it: Any multiple-choice question asking how many contracts to buy or sell to reach a target beta.

  1. Compute the beta change: β* − β.
  2. Multiply by P and divide by futures price × multiplier.
  3. Attach the sign: negative equals sell.
  4. Eliminate options with the wrong sign first, then check the size against a rough estimate such as P ÷ A.

Common mistakes in Stock Index Futures and Beta Hedging

  • Forgetting the multiplier and dividing by the index level only.

    The index level looks like the contract price.

    Fix: Always compute A = futures price × multiplier before dividing.

  • Using full beta when the question asks to reduce beta to a target.

    Students memorise N = β × P ÷ A as the only formula.

    Fix: Use (β* − β) every time. Full hedge is the special case β* = 0.

  • Getting the sign wrong: buying futures to hedge a long portfolio.

    The word hedge is not linked to direction.

    Fix: Long stocks need short futures to cut beta. Buy only to raise beta.

  • Claiming a beta-zero hedge removes all risk.

    Systematic and specific risk are blurred.

    Fix: State that stock-specific risk and basis risk remain.

  • Using the spot index instead of the futures price.

    Both are quoted and look similar.

    Fix: Use what the question gives for the futures contract. If only spot is given, follow the question's instruction.

Worked examples

Example 1

A fund holds $20,000,000 of equities with beta 1.3. The index future is 4,000 with a multiplier of $250. How many contracts must be traded to reduce beta to zero?

Show the solution
  1. A = 4,000 × 250 = $1,000,000 per contract.
  2. N = (0 − 1.3) × 20,000,000 ÷ 1,000,000.
  3. N = −1.3 × 20 = −26.
  4. Negative sign means sell.

Answer: Sell 26 contracts.

Example 2

A $12,000,000 portfolio has beta 0.9. The manager wants beta 1.5 using index futures priced at 3,000 with multiplier $100. What trade is needed?

Show the solution
  1. A = 3,000 × 100 = $300,000 per contract.
  2. Beta change = 1.5 − 0.9 = 0.6.
  3. N = 0.6 × 12,000,000 ÷ 300,000.
  4. N = 7,200,000 ÷ 300,000 = 24.
  5. Positive sign means buy.

Answer: Buy 24 contracts.

Exam tips

  • Read whether the question asks for beta to zero or to a target. This changes the formula input.
  • Check the multiplier in the question stem. Different contracts use different multipliers.
  • Use the sign to remove wrong options before doing arithmetic.
  • If the question asks about hedged return, think CAPM: market return drops out, risk-free rate plus alpha remains.
  • Watch for basis risk and imperfect beta estimates as reasons a hedge is not perfect.

Practice questions from Futures Markets

Stock Index Futures and Beta Hedging in other exams

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

Stock Index Futures and Beta Hedging: frequently asked questions

How do I change portfolio beta using index futures?

Compute N = (target beta − current beta) × portfolio value ÷ (futures price × multiplier). Sell if N is negative and buy if positive. This works for raising, lowering or fully removing market exposure.

What is the contract multiplier?

It is the dollar amount per index point. Multiply it by the futures price to get the value of one contract. It is set by the exchange and given in exam questions.

Does a beta hedge remove all risk?

No. It removes the expected market-driven part only. Stock-specific risk, basis risk and beta estimation error remain.

Should I buy or sell futures to hedge a long equity portfolio?

Sell. A short futures position gains when the market falls, offsetting losses on the stocks. You buy futures only to increase beta.