The Derivatives-on-Line Pages

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Futures prices and future spot prices

One very important question that one can ask is: Is the futures price an unbiased estimator of the future spot price? The answer is no in general. Remember the relation between risk and return as stated by the

CAPM: there are two types of risk in the economy, namely the systematic and the nonsystematic risk. The nonsystematic risk can be eliminated by holding a well diversified portfolio, which is perfectly correlated with the market. The systematic risk cannot be eliminated, since it is the risk of the portfolio that is inherited from the market as a whole and it cannot be diversified away. The CAPM formula dictates that

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

We have already seen that a futures contract, when seen as a riskless investment will grow in value with the risk free rate of return.

Example 10 (Futures risk)   Suppose that an investor takes a long futures position. She puts the present value of the futures position into a risk free investment, to meet the requirements when the contract matures, in order to buy the asset on the delivery date. The cash flows of the speculator are

$\displaystyle \begin{tabular}{ll} $-F\left( t,\tau \right) e^{-r\left( \tau -t\... ... at time $t$, and \\ $S\left( \tau \right) $ & at time $\tau $. \end{tabular}$

The present value of this investment is

$\displaystyle -F\left( t,\tau \right) e^{-r\left( \tau -t\right) }+\mathbf{E}_{t}\left[ S\left( \tau \right) \right] e^{-r_{I}\left( \tau -t\right) }$,

where $ \mathbf{E}_{t}$ is the conditional expectations operator, and $ r_{I}$ is the discount rate appropriate for the investment --meaning the expected return required from investors in order to compensate for the risks that are beard. The fact that the present value of all investment opportunities is equal to zero will give

$\displaystyle F\left( t,\tau \right) =\mathbf{E}_{t}\left[ S\left( \tau \right) \right] e^{\left( r-r_{I}\right) \left( \tau -t\right) }$.

It is straightforward to observe that the relationship of the futures with the expected spot price will depend on the relationship between the two returns, which in turn depends on the correlation of the investment with the market due to the CAPM.

Example 11 (cont. Futures risk)   Consider the case that $ S_{T}$ is positively correlated with the market --as is the usual case. Then, from the definition of $ \beta _{I}=%% \frac{cov\left( r_{I},r_{M}\right) }{var\left( r_{M}\right) }$ it is implied that $ \beta _{I}$ is positive. The CAPM dictates that an investment with positive $ \beta $ will have required return higher than the risk free rate. This in turn will give $ F\left( t,\tau \right) <\mathbf{E}_{t}\left[ S\left( \tau \right) \right] $. The inverse will also hold: If $ S_{T}$ is negatively correlated with the market, then $ F\left( t,\tau \right) >%% \mathbf{E}_{t}\left[ S\left( \tau \right) \right] $. What is the case where the futures price is an unbiased estimator of the future spot price? This happens only when the investment is not correlated with the market, or equivalently when the investment does not exhibit systematic risk. If fact in this case the above feature is more of an accident: it is not the case that the futures price became an unbiased estimator, it is more that the asset price happens to grow at the risk free rate of return.

next up previous contents
Next: Appendix: Hedging with BLOOMBERG Up: Futures and Forwards Previous: Cost of carry   Contents
Kyriakos 2003-03-17