Hedging Portfolio Risk

As an example, recall from Chapter 7 that we estimated β = 2 against the S&P 500 for Citigroup. Suppose an investor wants to hedge downside risk on an investment in this stock. Suppose there is no futures contract available for Citigroup, but there is a liquid futures market for the S&P 500. This investor could short S&P futures contracts to hedge tail risk on a 2:1 basis. So, if she has a portfolio P = $100,000 in Citigroup stock and one-month S&P 500 futures are at F = $1,000, then she needs to short 200 futures contracts on the S&P 500. That is, the required short futures position N is equal to , where P is the value of the Citigroup portfolio and F is the futures price on the S&P 500.

It is worthwhile noting that short hedges like the one just described effectively change the portfolio's beta. In this case, the portfolio had a beta of 2, which was reduced to zero with the hedge. Thus, the hedge effectively eliminated the portfolio's exposure to broad market movements. In fact, investors can use short and long futures exposures to achieve any desired beta and, with it, the associated risk and return. In the previous example, shorting 100 futures contracts on the S&P 500, for example, would result in hedging out half the downside risk, effectively pulling beta back to 1 on the portfolio. On the other hand, a long futures position equal to 100 contracts will increase the portfolio beta to 3. In general, if β∗ is the desired beta on the portfolio and β > β∗, then the investor needs to adopt a short position equal to:

If, on the other hand, β < β∗, then the investor adopts a long futures position equal to:

Forward with Cost of Carry

Commodities generally cannot be stored at zero cost. Oil for example, was stored in offshore tankers during the runup in oil futures in the summer of 2008. Grain must be siloed in a conditioned environment to prevent spoilage, and livestock require a carrying cost to cover feed, water, grazing, and vet costs. These costs of carry must therefore be reflected in the futures price. Let c(k) equal the per unit per time carrying costs. Then the price of the forward contract F will be the sum of S/d(0,M) plus carry costs compounded over the carry period reflecting the opportunity cost of those expenses, that is, Σc(k)/d(k,M) over k = 0,..., M – 1 (M periods). Therefore, the spot price must be S = d(0,M) F – Σc(k)/d(k,M)∗d(0,M) = d(0,M)F – Σc(k)d(0,k). This says that the spot price is the sum of the discounted value at time M minus the discounted carrying costs.