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Financial Economics (ECON 369)
- Using the CAPM assumptions, we need to focus on the representative or the average investor, who maximizes the mean-variance utility of the following
U (rp, σ^2 p ) = rp −
Aσ^2 p.
This representative investor constructs her portfolio composed of the mar- ket risky portfolio (with weight α) and the risk-free asset (with weight 1 − α). The market portfolio has risky return rM with expected return rM and variance σ^2 M ; the risk-free asset has return rf with certainty. The market risky portfolio is composed of wi share in risky asset i (i = 1 : n), whose risky return is ri, expected return is ri. The variance-covariance matrix of the n risky assets are:
Cov(r 1 , r 1 ) Cov(r 1 , r 2 ) · · · Cov(r 1 , rn) Cov(r 2 , r 1 ) Cov(r 2 , r 2 ) · · · Cov(r 2 , rn) · · · · · · · · · · · · Cov(rn, r 1 ) Cov(rn, r 2 ) · · · Cov(rn, rn)
or
σ^21 σ 12 · · · σ 1 n σ 21 σ^22 · · · σ 2 n · · · · · · · · · · · · σn 1 σn 2 · · · σ^2 n
by using the Greek notation.
(a) By fixing the weight α, w 2 , · · · , and wn, what’s the optimal condition for market risky portfolio in the asset 1, w 1? Hint: by isolating all the terms that involve w 1 from the objection function, we have:
−αw 1 rf + αw 1 r 1 −
Aα^2
w^21 σ^21 + 2w 1 w 2 σ 12 + · · · + 2w 1 wnσ 1 n
Answer: Since
rp = (1 − α)rf + αrM = (1 − α)rf + α(w 1 r 1 + w 2 r 2 + · · · + wnrn); rp = (1 − α)rf + α(w 1 r 1 + w 2 r 2 + · · · + wnrn); Var(rp) = α^2
w^21 σ 12 + w 1 w 2 σ 12 + · · · w 1 wnσ 1 n + · · · + wnw 1 σn 1 + wnw 2 σn 2 + · · · w^2 nσ^2 n
we can rewrite the maximization problem as: max α,w 1 ,w 2 ,··· ,wn (1 − α)rf + α(w 1 r 1 + w 2 r 2 + · · · + wnrn) −
1 2
Aα^2
w^21 σ^21 + w 1 w 2 σ 12 + · · · w 1 wnσ 1 n + · · · + wnw 1 σn 1 + wnw 2 σn 2 + · · · w n^2 σ n^2
Subject to a constraint that w 1 + w 2 + · · · + wn = 1. Replacing (1 − α)rf by [1 − α(w 1 + w 2 + · · · + wn)]rf , we can embed the constraint into the object function. To find the optimal condition regarding w 1 , let’s isolate all terms that regards w 1 in the object function:
−αw 1 rf + αw 1 r 1 −
Aα^2
w^21 σ^21 + 2w 1 w 2 σ 12 + · · · + 2w 1 wnσ 1 n
the optimal w 1 is found by setting the first order derivative of the above (part of) the object function with respect to w 1 as zero.
(b) By solving the optimal condition in part a), we find another way to express α, the optimal share invested in the market risky portfolio. Compare this result to the previous one where
α =
rM − rf Aσ^2 M
Derive the CAPM result for asset 1. Answer: From the previous part, we have −αrf + αr 1 − Aα^2
w 1 σ 12 + w 2 σ 12 + · · · + wnσ 1 n
(r 1 − rf )α = Aα^2
w 1 σ^21 + w 2 σ 12 + · · · + wnσ 1 n
r 1 − rf = Aα [w 1 Cov(r 1 , r 1 ) + w 2 Cov(r 1 , r 2 ) + · · · + wnCov(r 1 , rn)] =⇒ r 1 − rf = Aα [Cov(r 1 , w 1 r 1 ) + Cov(r 1 , w 2 r 2 ) + · · · + Cov(r 1 , wnrn)] =⇒ r 1 − rf = AαCov(r 1 , w 1 r 1 + w 2 r 2 + · · · + wnrn) =⇒ r 1 − rf = AαCov(r 1 , rM ) =⇒
α =
r 1 − rf ACov(r 1 , rM )
Compare the above result with
α =
rM − rf Aσ^2 M
we have r 1 − rf ACov(r 1 , rM )
rM − rf Aσ M^2
r 1 − rf =
Cov(r 1 , rM ) σ^2 M
(rM − rf ).
