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Question 5.18: Show that the beta factor βV of a portfolio consisting of n ...

Show that the beta factor β_V of a portfolio consisting of n securities with weights w_1, . . . , w_n is given by β_V = w_1β_1+· · ·+w_nβ_n, where β_1, . . . , β_n are the beta factors of the securities.

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The expected return on the portfolio can be expressed as K_V = w_1K_1 +· · ·+w_nK_n in terms of the expected returns on the individual securities. Because covariance is linear in each of its arguments,

\beta _V=\frac{Cov(K_V ,K_M)}{\sigma ^{2}_{M} }=w_1\frac{Cov(K_1,K_M)}{\sigma ^{2}_{M}} +…+w_n\frac{Cov(K_n,K_M)}{\sigma ^{2}_{M}}
=w_1β_1 + · · · + w_nβ_n.

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