Multiple Linear Regression 51
is F-distributed with 1 and n − k − 1 degrees of freedom under the null
hypothesis that the true model consists of k − 1 independent variables only.^9
Applications to Finance
We conclude this chapter with several applications of multiple linear regres-
sion analysis to various areas in finance.
Estimation of Empirical Duration
A commonly used measure of the interest-rate sensitivity of a financial asset’s
value is its duration. For example, if a financial asset has a duration of 5, this
means that the financial asset’s value or price will change by roughly 5%
for a 100 basis point change in interest rates. The direction of the change is
determined by the sign of the duration. Specifically, if the duration is posi-
tive, the price will decline when the relevant interest rate increases but will
increase if the relevant interest rate declines. If the duration is negative, the
price will increase if the relevant interest rate increases and fall if the rel-
evant interest rate decreases.
So suppose that a common stock selling at a price of $80 has a duration
of +5 and that the relevant interest rate that affects the value of the common
stock is currently 6%. This means that if that relevant interest rate increases
by 100 basis points (from 6% to 7%), the price of the financial asset will
decrease by 5%. Since the current price is $80, the price will decline by about
$4. On the other hand, if the relevant interest rate decreases from 6% to 5%
(a decline of 100 basis points), the price will increase by roughly 5% to $84.
Duration can be estimated by using a valuation model or empirically
by estimating from historical returns the sensitivity of the asset’s value to
changes in interest rates. When duration is measured in the latter way, it
is referred to as empirical duration. Since it is estimated using regression
analysis, it is sometimes referred to as regression-based duration.
The dependent variable in the regression model is the percentage change
in the value of the asset. We will not use individual assets in our illustration.
Rather we will use sectors of the financial market and refer to them as assets.
Effectively, these sectors can be viewed as portfolios that are comprised of
the components of the index representing the sector. The assets we will esti-
mate the duration for are (1) the electric utility sector of the S&P 500 index,
(2) the commercial bank sector of the S&P 500 index, and (3) Lehman U.S.
(^9) The chi-square and the F-distribution are covered in Appendix A and Appendix B,
respectively.