108156.pdf

5. Portfolio Management 121

α+βKMis called theresidual random variable. The condition defining the
line of best fit is that

E(ε^2 )=E(K^2 V)− 2 βE(KVKM)+β^2 E(KM^2 )+α^2 − 2 αE(KV)+2αβE(KM)

as a function ofβandαshould attain its minimum atβ=βV andα=αV.
In other words, the line of best fit should lead to predictions that are as close
as possible to the true values ofKV. A necessary condition for a minimum is
that the partial derivatives with respect toβandαshould be zero atβ=βV
andα=αV. This leads to the system of linear equations

αVE(KM)+βVE(KM^2 )=E(KVKM),

αV+βVE(KM)=E(KV),

which can be solved to find the gradientβVand interceptαVof the line of best
fit,

βV=

Cov(KV,KM)

σ^2 M

,αV=μV−βVμM.

Here we employ the usual notationμV =E(KV),μM=E(KM)andσM^2 =
Var(KM).

Exercise 5.17

Suppose that the returnsKVon a given portfolio andKMon the market

portfolio take the following values in different market scenarios:

Scenario Probability ReturnKV ReturnKM

ω 1 0. 1 −5% 10%

ω 2 0. 3 0% 14%

ω 3 0. 4 2% 12%

ω 3 0. 2 4% 16%

Compute the gradientβVand interceptαVof the line of best fit.

Definition 5.3

We call

βV=Cov(KV,KM)

σM^2

thebeta factorof the given portfolio or individual security.