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FRM Exam Part I · Using Futures for Hedging

Stock Index Futures and Beta Hedging for FRM Part I

Updated 11 October 2026 · Fact-checked

Beta hedging uses index futures to offset the market risk of an equity portfolio. The number of contracts to short for a full hedge is N = β × V ÷ F, where V is portfolio value and F is one contract's value. For target beta, use N = (β* − β) × V ÷ F; negative means short.

Understand Stock Index Futures and Beta Hedging

An equity portfolio carries two kinds of risk. Systematic risk moves with the market. Unsystematic risk is specific to individual stocks. Diversification removes most of the second kind. Index futures let you deal with the first kind.

Beta measures how much the portfolio tends to move when the market moves. A portfolio with beta 1.2 tends to move 1.2% for each 1% move in the index. If you short index futures, you gain when the index falls. That gain offsets the portfolio loss caused by market moves.

You do not need to sell the stocks. This is why managers use futures: they are cheap to trade, liquid, and quick to reverse. A fund can cut market exposure for a few weeks and keep its stock selection.

The hedge removes market risk only. Stock-specific risk stays. If the portfolio is not well diversified, or its beta is poorly estimated, the hedge is imperfect. Under CAPM, the hedged portfolio is expected to earn approximately the risk-free rate (plus any alpha). This is an expectation, not a guarantee, because basis risk and specific risk remain.

You can also use futures to raise or lower beta to a target. To lower beta, short futures. To raise it, go long. The same idea works for any target beta, including zero (a full hedge).

Key formulas to remember

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.

How to solve Stock Index Futures and Beta Hedging questions

Use this method for any question on hedging or changing equity exposure with index futures.

  1. 1Identify the portfolio value V and current beta β from the question.
  2. 2Work out the contract value F = futures index level × multiplier. Use the futures level if given.
  3. 3Decide the target beta β*. For a full hedge, β* = 0.
  4. 4Compute N = (β* − β) × V ÷ F. A negative answer means short; a positive answer means long.
  5. 5Round to the nearest whole contract, since you cannot trade fractions.
  6. 6If asked for the outcome, estimate the portfolio change using beta and the index move, then add the futures gain or loss.
  7. 7Check the sign: a short hedge should gain when the market falls.

Quickest way: Contract count in three lines

When to use it: Use when the question gives V, β, the index level and the multiplier, and asks only for the number of contracts.

  1. Compute F = index level × multiplier.
  2. Compute (β* − β) × V and divide by F.
  3. Read the sign: negative means short. Round to a whole number and move on.

Common mistakes in Stock Index Futures and Beta Hedging

  • Using β × V ÷ F when the target beta is not zero.

    Students memorise the full-hedge formula only.

    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.

    The index level looks like the contract price.

    Fix: Multiply the index level by the multiplier first. Write F down as a separate step.

  • Going long when you should be short.

    Sign errors when lowering beta.

    Fix: To reduce beta, short futures. To raise beta, go long. Check with the sign of β* − β.

  • Claiming a beta hedge removes all risk.

    The hedge sounds complete.

    Fix: It removes market risk only. Stock-specific risk and basis risk remain, and beta estimates can be wrong.

  • Dividing by the portfolio's index-equivalent instead of the futures contract value.

    Mixing up V and F.

    Fix: V is the portfolio in currency. F is the value of one contract. N is the ratio of exposures.

  • Leaving the answer as a fraction of a contract.

    Treating the formula result as final.

    Fix: Round to the nearest whole contract, as MCQ options are usually whole numbers.

Worked examples

Example 1

A portfolio worth $10,000,000 has a beta of 1.2. The S&P 500 futures price is 4,000 and the multiplier is $50. How many contracts must you trade to fully hedge market risk?

Show the solution
  1. F = 4,000 × 50 = $200,000 per contract.
  2. Target beta β* = 0, current beta β = 1.2.
  3. N = (0 − 1.2) × 10,000,000 ÷ 200,000.
  4. N = −12,000,000 ÷ 200,000 = −60.
  5. Negative sign means short.

Answer: Short 60 index futures contracts.

Example 2

A $20,000,000 portfolio has a beta of 0.8. The index futures price is 2,500 with a multiplier of $250. The manager wants to raise beta to 1.4. What position is needed?

Show the solution
  1. F = 2,500 × 250 = $625,000 per contract.
  2. β* − β = 1.4 − 0.8 = 0.6.
  3. N = 0.6 × 20,000,000 ÷ 625,000.
  4. N = 12,000,000 ÷ 625,000 = 19.2.
  5. Round to the nearest whole contract: 19.
  6. Positive sign means long.

Answer: Go long about 19 contracts.

Exam tips

  • Write F as its own line. The multiplier is the most common lost mark.
  • Read the target beta carefully. 'Hedge fully' means 0, but many questions ask for a non-zero target.
  • Check the sign of N before choosing an option. Wrong-direction options are usually among the choices.
  • Remember what the hedge leaves behind: specific risk, basis risk and beta estimation error.
  • If the question gives CAPM inputs, expect the hedged portfolio's expected return to be approximately the risk-free rate. It is an expectation, not a guarantee, because basis risk and specific risk remain.

Practice questions from Using Futures for Hedging

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 you calculate the number of index futures contracts to hedge a portfolio?

Use N = β × V ÷ F, where F is index futures level times the multiplier. This is the number of contracts to short for a full hedge. It is the magnitude of (0 − β) × V ÷ F. Round to the nearest whole number.

How do you change portfolio beta using futures?

Use N = (β* − β) × V ÷ F. If β* is lower than β, the result is negative and you short. If β* is higher, you go long.

Why hedge a portfolio with stock index futures instead of selling the stocks?

Futures are cheaper and faster to trade, and they keep the stock positions intact. A manager can remove market risk temporarily and keep the stock-picking view. Selling stocks can trigger costs and tax.

Does a beta hedge remove all risk?

No. It removes market risk based on the estimated beta. Stock-specific risk, basis risk and beta estimation error remain.