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Describe how stock index futures can be used to alter the beta of a stock portfolio.

FRM® Part I · Financial Markets and Products · Using Futures for Hedging · FRM1.FM.8.g

The idea

Stock index futures let a manager change the beta of an equity portfolio without trading the shares. To move beta from its current level to a target level, the number of contracts is N = (target beta − current beta) × P ÷ A. Here P is the portfolio value and A is the value of the stocks underlying one contract (index level × multiplier). A positive result means buy futures to raise beta, and a negative result means sell futures to lower it. Setting the target beta to zero gives a full hedge, N = −current beta × P ÷ A. Common errors are using the target beta alone instead of the difference, ignoring the sign, and averaging betas without weighting by value. The beta of a mixed portfolio is the value-weighted average of the betas.

Formulas

Index futures needed to move a portfolio's beta

N = (target beta − current beta) × P ÷ A

target beta
the beta wanted
current beta
the portfolio's beta now, value-weighted across holdings
P
the portfolio value
A
the value of one contract, index level × multiplier

Positive means buy futures, negative means sell. A target of zero is a full hedge; using the target beta alone rather than the difference is the usual error.

A practice question

From the question bank, with the reason each wrong option is wrong.

A portfolio manager wants to raise the beta of an equity portfolio from 1.0 to 1.4 without trading the shares. The appropriate action is to:

  1. sell stock index futuresWhy not: Short index futures reduce the portfolio's market exposure and beta, so this lowers beta.
  2. buy Treasury bond futuresWhy not: Bond futures carry interest-rate risk and do not add equity-market beta.
  3. do nothing, since beta can only change by trading the stocksWhy not: Index futures are used precisely to change beta without trading the underlying shares.
  4. buy stock index futures - correct

Long index futures add equity-market exposure and raise beta; short futures reduce it.

A harder one

A manager holds USD 75 million of equities with a beta of 1.2 and wants to reduce the beta to 0.8 using index futures that each represent USD 1,250,000. The correct trade is to:

  1. sell 24 contracts - correct
  2. sell 48 contractsWhy not: Uses the target beta alone instead of the difference between current and target beta.
  3. sell 72 contractsWhy not: Computes the full hedge, which would bring beta to zero rather than to 0.8.
  4. buy 24 contractsWhy not: Right size but wrong direction; reducing beta requires selling futures.

N = (target - current) x P / A = (0.8 - 1.2) x 75,000,000 / 1,250,000 = -24, i.e. sell 24.

The hardest tier

A manager holds USD 60 million of equities with a beta of 1.1. The manager sells 25 index futures contracts, each representing USD 1.2 million of stock. The resulting portfolio beta is closest to:

  1. 0.60 - correct
  2. 1.60Why not: Adds the futures effect instead of subtracting it; selling futures lowers beta.
  3. 0.50Why not: Reports the beta reduction from the futures and omits the portfolio's own beta of 1.1.
  4. 0.55Why not: Scales beta by (1 - futures value share), but the futures change beta additively, not proportionally.

Futures notional = 25 x 1.2 = 30 million = 0.5 of the portfolio. New beta = 1.1 - 0.5 = 0.60. Check: N = (0.6 - 1.1) x 60 / 1.2 = -25.

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