Final_1.pdf

advanced version of CAPM. Then, if we have access to the APT model, why
would we want to calculate the parameters of a simpler CAPM model? Is
that not going back full circle? That would indeed be true. However, if we
are looking to hedge our portfolio with the market portfolio, then the hedge
ratio that provides the best possible hedge is given by the beta of the port-
folio. Therefore, it does make sense to calculate beta. While the title may as
well read “Determination of Hedge Ratio,” having the word betain the title
helps make the association between beta and the hedge ratio explicit.
To see the relationship between beta and the hedge portfolio, consider
the linear combination of the two portfolios in the ratio 1:l. The return of
the linear combination is given by rp–lrm, where rpis the return on the port-
folio and rmis the return on the market. The value of lthat results in the least
return variance of the combined portfolio is indeed the best hedge ratio. To
evaluate the variance of the return, we will apply the algebraic identity

(3.21)

to the combined portfolio.

(3.22)

Applying expectations on both sides and using the formulas in the appendix
of Chapter 1, we have

(3.23)

To find the value for lthat minimizes the variance, we differentiate with
respect to land equate the differential to zero. It is easy to then see that

(3.24)

Equation 3.24 also happens to be the definition of beta, and we therefore
conclude that beta is the best hedge ratio. Equation 3.24 is composed of the
covariance and the variance terms, which we know how to evaluate in the
APT framework. Applying the substitutions, we have

λ= (3.25)

+

+

eVe h h

eVe h h

pm

T

pm

T

mm

T

mm

T

∆

∆

λ=

()

()

cov ,

var

rr

r

pm

m

var()rrpm−λλλ=var()rp+^2 var()rm− 2 cov()rrpm

()rr r r rrpm p m pm−λλλ=+ −

(^2222)
2
()ab a b ab+ =++
(^222)
2
48 BACKGROUND MATERIAL