Simple Linear Regression 35
was selling at 822.5. The basis was equal to 3.38 index units (the cash index
of 825.88 minus the futures price of 822.5). At the same time, it was calcu-
lated that, based on the cost of carry, the theoretical basis was 1.45 index
units. That is, the theoretical futures price at the time the hedge was placed
should have been 824.42. Thus, according to the pricing model, the futures
contract was mispriced by 1.92 index units.
When the hedge was removed at the close of February 27, 2009, the
cash index stood at 735.09, and the futures contract at 734.2. Thus, the
basis changed from 3.38 index units at the time the hedge was initiated
to 0.89 index units (735.09 – 734.2) when the hedge was lifted. The basis
had changed by 2.49 index units (3.38 – 0.89) alone, or $622.5 per con-
tract (2.49 times the multiple of $250). This means that the basis alone
cost $225,538 for the 362 contracts ($622.5 × 362). The index dropped
90.79 index units, for a gain of $22,698 per contract, or $8,223,532. Thus,
the futures position cost $225,538 due to the change in the basis risk, and
$8,223,532 due to the change in the index. Combined, this comes out to be
the $7,997,994 gain in the futures position.
Illustration 2 We examined basis risk in the first illustration. Because we
were hedging a portfolio that was constructed to replicate the S&P 500
index using the S&P 500 futures contract, there was no cross-hedging risk.
However, most portfolios are not matched to the S&P 500. Consequently,
cross-hedging risk results because the estimated beta for the price behav-
ior of the portfolio may not behave as predicted by BPI. To illustrate this
situation, suppose that an asset manager owned all the stocks in the Dow
Jones Industrial Average (DJIA) on January 30, 2009. The market value of
the portfolio held was $100 million. Also assume that the portfolio man-
ager wanted to hedge the position against a decline in stock prices from
January 30, 2009, to February 27, 2009, using the March 2009 S&P 500
futures contract. Since the S&P 500 futures September contract is used here
to hedge a portfolio of DJIA to February 27, 2009, this is a cross hedge.
Information about the S&P 500 cash index and futures contract when
the hedge was placed on January 30, 2009, and when it was removed on
February 27, 2009, was given in the previous illustration. The beta of the
index relative to the futures contract (BIF) was 0.745. The DJIA in a regres-
sion analysis was found to have a beta relative to the S&P 500 of 1.05
(with an R-squared of 93%). We follow the three steps enumerated above
to obtain the number of contracts to sell:
Step 1.Equivalentmarketindexunits=