Simple Linear Regression 25
Two Applications in Finance
In this section, we provide two applications of simple linear regression anal-
ysis to finance.
estimating the Characteristic Line of a Mutual Fund
We discuss now a model for security returns. This model suggests that secu-
rity returns are decomposable into three parts. The first part is the return of
a risk-free asset. The second is a security-specific component. And finally, the
third is the return of the market in excess of the risk-free asset (i.e., excess
return) which is then weighted by the individual security’s covariance with
the market relative to the market’s variance. Formally, this is
RRSf=+αβSS+−,MM()RRf (2.11)
where RS = the individual security’s return
Rf = the risk-free return
αS = the security-specific term
βSM, =cov(,RRSM)/var()RM = the so-called beta factor
The beta factor measures the sensitivity of the security’s return to the
market. Subtracting the risk-free interest rate Rf from both sides of equation
(2.11) we obtain the expression for excess returns:
RRSf−=αβSS+−,MM()RRfor equivalently
rrSS=+αβSM, M (2.12)
which is called the characteristic line where rS = RS – Rf and rM = RM – Rf
denote the respective excess returns of the security and the market.
This form provides for a version similar to equation (2.3). The model
given by equation (2.12) implies that at each time t, the observed excess
return of some security rS,t is the result of the functional relationship
rrSt,,=+αβSSMM,,tS+ε t (2.13)
So, equation (2.13) states that the actual excess return of some security S is com-
posed of its specific return and the relationship with the market excess return,