Asset Pricing John H. Cochrane June 12, 2000

CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS
Then, for a given mean portfolio return µ, the minimum variance portfolio has variance
var (R p )= Cµ2 − 2Bµ + A
AC − B2 (60)
and is formed by portfolio weights
−1 E (Cµ − B)+1(A− Bµ)
w = Σ
(AC − B2 .
)
Equation (5.60) shows that the variance is a quadratic function of the mean. The square
root of a parabola is a hyperbola, which is why we draw hyperbolic regions in mean-standard
deviation space.
The minimum-variance portfolio is interesting in its own right. It appears as a special case
in many theorems and it appears in several test statistics. We can find it by minimizing (5.60)
over µ, giving µ min var = B/C. The weights of the minimum variance portfolio are thus
w = Σ −1 1/(1 0 Σ −1 1).
We can get to any point on the mean-variance frontier by starting with two returns on
the frontier and forming portfolios. The frontier is spanned by any two frontier returns.
Toseethisfact,noticethatwis a linear function of µ. Thus, if you take the portfolios
corresponding to any two distinct mean returns µ 1 and µ 2, the weights on a third portfolio
with mean µ 3 = λµ 1 +(1−λ)µ 2 are given by w3 = λw1 +(1−λ)w2. Derivation: To derive the solution, introduce Lagrange multipliers 2λ and 2δ on the constraints.
The first order conditions to (5.59) are then
Σw−λE − δ1 =0
We find the Lagrange multipliers from the constraints,
or
w = Σ −1 (λE + δ1). (61)
E 0 w = E 0 Σ −1 (λE + δ1) = µ
1 0 w =1 0 Σ −1 (λE + δ1) = 1
· E 0 Σ −1 E E 0 Σ −1 1
1 0 Σ −1 E 1 0 Σ −1 1
· A B
B C
¸· λ
δ
82
¸· λ
δ
¸
=
¸
=
· µ
1
¸
· µ
1
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