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D Generalized capital asset princing model
Our proof of the generalized capital asset princing model is similar to the usual demontration of the CAPM.
Let us consider an efficient portfolio P. It necessarily satisfies equation (105) in appendix B :
459
∂ρα
(w
∂wi
∗ 1, · · · , w ∗ N) = λ1 (µ(i) − µ0), i ∈ {1, · · · , N}. (131)
Let us now choose any portfolio R made only of risky assets and let us denote by wi(R) its weights. We
can thus write
N
i=1
wi(R) · ∂ρα
(w
∂wi
∗ 1, · · · , w ∗ N) = λ1
N
wi(R) · (µ(i) − µ0), (132)
i=1
= λ1 (µR − µ0). (133)
We can apply this last relation to the market portfolio Π, because it is only composed of risky assets (as
proved in appendix B). This leads to wi(R) = w ∗ i and µR = µΠ, so that
N
i=1
which, by the homogeneity of the risk measures ρα, yields
Substituting equation (131) into (135) allows us to obtain
where
w ∗ i · ∂ρα
(w
∂wi
∗ 1, · · · , w ∗ N) = λ1 (µΠ − µ0), (134)
α · ρα(w ∗ 1, · · · , w ∗ N) = λ1 (µΠ − µ0) . (135)
µj − µ0 = β j α · (µΠ − µ0), (136)
β j α =
∂
1
ln ρα α
∂wj
, (137)
calculated at the point {w∗ 1 , · · · , w∗ N }. Expression (135) with (137) provides our CAPM, generalized with
respect to the risk measures ρα.
In the case where ρα denotes the variance, the second-order centered moment is equal to the second-order
cumulant and reads
Since
we find
C2 = w ∗ 1 · Var[X1] + 2w ∗ 1w ∗ 2 · Cov(X1, X2) + w ∗ 2 · Var[X2], (138)
= Var[Π] . (139)
1 ∂C2
·
2 ∂w1
= w ∗ 1 · Var[X1] + w ∗ 2 · Cov(X1, X2) , (140)
= Cov(X1, Π), (141)
β = Cov(X1, XΠ)
Var[XΠ]
which is the standard result of the CAPM derived from the mean-variance theory.
35
, (142)