Hedging using index futures

Index futures are widely used to hedge a well diversified portfolio. Recall that the CAPM states that for a well diversified portfolio --where $\rho _{\Delta P/P,\Delta M/M}=1$,

$$r_{P}-r = \beta \left( r_{M}-r\right)$$

$$\sigma _{P} = \beta \sigma _{M}$$

where $r_{P}$ is the return of a portfolio and $r_{M}$ of the market. $%% \sigma _{P}$ and $\sigma _{M}$ denote the respective standard deviation of the returns. Using futures contracts on the market index, one can change the beta of this well diversified portfolio. In general, to change the beta from $\beta$ to $\beta ^{\star }$ it required to

$$\begin{tabular}{lll} short & $\left( \beta -\beta ^{\star }\right... ...ght) \frac{P}{A}$ & contracts when $%% \beta <\beta ^{\star }$%% \end{tabular}$$

where $P$ denotes the value of the portfolio and $A$ is the value of assets in one futures contract.

It is clear that by shorting $\beta \frac{P}{A}$ contracts, the beta of the portfolio becomes zero. Then the volatility of the portfolio is zero and of course the value of the portfolio will grow at the risk free rate.

Example 8 (Perfect hedge)   A company wants to perfectly hedge a well diversified portfolio worth £1.2m for two months using FTSE100 futures with four months to maturity. The beta of the portfolio is 1.5 and the level of the FTSE100 index is 6000 points. The FTSE100 contract is valued as £10 per point^3.2 This means that the value of the assets underlying one futures contract is

$$6,000\times \pounds 10=\pounds 60,000$$.

In order to perfectly hedge, the company should short

$$1.5\times \frac{\pounds 1.2m}{\pounds 60k}=30$$ contracts.

Example 9 (cont. Perfect hedge)   Suppose that over the course of the next two months the interest rate is 6%p.a., or 1% over the two month period. Suppose that the market collapses in these two months --perhaps what the company's fears were!!-- and offers a return of -9%. The CAPM will therefore dictate that the return of the portfolio is

$$r_{P} = r+\beta \left( r_{M}-r\right)$$

$$= 1\%+1.5\times \left( -9\%-1\%\right) =-14\%$$

If the dividend yield is 3%p.a. or 0.5% per two months, it is implied that the ftse100 index has declined by 9.5% over these two months, down to 5430 points. The initial and final futures prices are respectively

$$6,000e^{\left( 6\%-3\%\right) \times 4/12} = 6,060.30$$

$$5,430e^{\left( 6\%-3\%\right) \times 2/12} = 5,457.20$$

The total income of the company over these two months due to the shorting of the futures is therefore

$$30\times \left( 6,060.30-5,457.20\right) pts\times 10\frac{\pounds }{pt}%% =\pounds 180,930$$.

The loss of the portfolio value is

$$14\%\times \pounds 1.2m=\pounds 168,000$$,

and the total profit of the hedged position is

$$\pounds 180,930-\pounds 168,000 = \pounds 12,930$$

$$\approx 1\% \pounds 1.2m$$

The differences in the above example occur because we have ignored the distinction between continuously and discretely compounding returns, and we did not take into account the daily settlements --tailing the hedge. Nevertheless, the return of the hedge is roughly equal to the risk free rate of return.