36 The Basics of financial economeTrics
Step 2.Beta-adjusted equivalent market index units = 1.05 × 0.745 × $121,581
= $95,106Step 3. The multiple for the S&P 500 contract is 250. Therefore,Number of contractstobesold==$,95 106
250
3880
During the period of the hedge, the DJIA actually lost $11,720,000.
This meant a loss of 11.72% on the portfolio consisting of the compo-
nent stocks of the DJIA. Since 380 S&P 500 futures contracts were sold
and the gain per contract was 88.3 points, the gain from the futures posi-
tion was $8,388,500 ($88.3 × 380 × 250). This means that the hedged
position resulted in a loss of $3,331,500, or equivalently, a return of
–3.31%.
We already analyzed why this was not a perfect hedge. In the previous
illustration, we explained how changes in the basis affected the outcome.
Let’s look at how the relationship between the DJIA and the S&P 500 Index
affected the outcome. As stated in the previous illustration, the S&P 500
over this same period declined in value by 10.99%. With the beta of the
portfolio relative to the S&P 500 Index (1.05), the expected decline in the
value of the portfolio based on the movement in the S&P 500 was 11.54%
(1.05 × 10.99%). Had this actually occurred, the DJIA portfolio would have
lost only $10,990,000 rather than $11,720,000, and the net loss from the
hedge would have been $2,601,500, or –2.6%. Thus, there is a difference
of a $730,000 loss due to the DJIA performing differently than predicted
by beta.
Linear Regression of a Nonlinear Relationship
Sometimes, the original variables do not allow for the concept of a linear
relationship. However, the assumed functional relationship is such that a
transformation h(y) of the dependent variable y might lead to a linear func-
tion between x and the transform, h. This is demonstrated by hypotheti-
cal data in Figure 2.5, where the y-values appear to be the result of some
exponential function of the x-values. The original data pairs in Table 2.3 are
indicated by the symbols in Figure 2.5.
We assume that the functional relationship is of the form
ye=αβx (2.14)