Anon

34 The Basics of financial economeTrics

Illustration 1 Consider a portfolio manager on January 30, 2009, who is
managing a $100 million portfolio that is identical to the S&P 500. The
manager wants to hedge against a possible market decline. More specifically,
the manager wants to hedge the portfolio until February 27, 2009. To hedge
against an adverse market move during the period January 30, 2009, to
February 27, 2009, the portfolio manager decides to enter into a short hedge
by selling the S&P 500 futures contracts that settled in March 2009. On
January 30, 2009, the March 2009 futures contract was selling for 822.5.
Since the portfolio to be hedged is identical to the S&P 500, the beta
of the portfolio relative to the index (BPI) is, of course, 1. The beta relative
to the futures contract (BIF) was estimated to be 0.745. Therefore, the num-
ber of contracts needed to hedge the $100 million portfolio is computed as
follows:
Step 1.

Equivalentmarketindexunits=

$,100 000 000,

82 22 5

121 581

.

=$,

Step 2.

Beta-adjusted equivalent market index units = 1 × 0.745 × $121,581

= $90,578

Step 3. The multiple for the S&P 500 contract is 250. Therefore,

Numberofcontractstobesold==

$,

$

90578

250

3362

This means that the futures position was equal to $74,500,000 (362 ×
$250 × 822.5). On February 27, 2009, the hedge was removed. The portfo-
lio that mirrored the S&P 500 had lost $10,993,122. At the time the hedge
was lifted, the March 2009 S&P 500 contract was selling at 734.2. Since the
contract was sold on January 30, 2009, for 822.5 and bought back on Febru-
ary 27, 2009, for 734.2, there was a gain of 88.3 index units per contract. For
the 362 contracts, the gain was $7,997,994 (88.3 × $250 × 362). This results
in a smaller loss of $2,995,129 ($7,997,994 gain on the futures position and
$10,993,122 loss on the portfolio). The total transaction costs for the futures
position would have been less than $8,000. Remember, had the asset man-
ager not hedged the position, the loss would have been $10,993,122.
Let’s analyze this hedge to see not only why it was successful, but also
why it was not a perfect hedge. As explained earlier, in hedging, basis risk is
substituted for price risk. Consider the basis risk in this hedge. At the time
the hedge was placed, the cash index was at 825.88, and the futures contract