32 The Basics of financial economeTrics
The foregoing points about hedging will be made clearer in the next
illustrations.
hedge ratio To implement a hedging strategy, it is necessary to determine
not only which stock index futures contract to use, but also how many of
the contracts to take a position in (i.e., how many to sell in a short hedge
and buy in a long hedge). The number of contracts depends on the relative
return volatility of the portfolio to be hedged and the return volatility of the
futures contract. The hedge ratio is the ratio of volatility of the portfolio to
be hedged and the return volatility of the futures contract.
It is tempting to use the portfolio’s beta as a hedge ratio because it is an
indicator of the sensitivity of a portfolio’s return to the stock index return. It
appears, then, to be an ideal way to adjust for the sensitivity of the return of
the portfolio to be hedged. However, applying beta relative to a stock index
as a sensitivity adjustment to a stock index futures contract assumes that
the index and the futures contract have the same volatility. If futures were
always to sell at their theoretical price, this would be a reasonable assump-
tion. However, mispricing is an extra element of volatility in a stock index
futures contract. Since the futures contract is more volatile than the underly-
ing index, using a portfolio beta as a sensitivity adjustment would result in
a portfolio being overhedged.
The most accurate sensitivity adjustment would be the beta of a portfo-
lio relative to the futures contract. It can be shown that the beta of a portfo-
lio relative to a futures contract is equivalent to the product of the portfolio
relative to the underlying index and the beta of the index relative to the
futures contract.^9 The beta in each case is estimated using regression analy-
sis in which the data are historical returns for the portfolio to be hedged, the
stock index, and the stock index futures contract.
The regression to be estimated is
rP = aP + BPIrI + ePwhere rP=the return on the portfolio to be hedged
rI=the return on the stock index
BPI=the beta of the portfolio relative to the stock index
aP=the intercept of the relationship
eP=the error term
(^9) See Edgar E. Peters, “Hedged Equity Portfolios: Components of Risk and Return,”
Advances in Futures and Options Research 1, part B (1987): 75–92.