- 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.