Signal Processing - May 2017 - 18
Mean-Variance Portfolio
Mean-variance portfolios (MVPs) are the solution of the
following mean-variance optimization (convex quadratic
programming) criterion:
" J (x ip) = v 2Rp = x Tp Rx p ,,
x )ip = arg mxin
ip
n
s.t. E [R p] = / x ip E [R i] = x Tp E [R ] = re,
that we are indifferent to holding different assets given riskcompensated premia (in "Take a Bet: Source of the Excess
Return," it means we are indifferent to taking the cash or the
bet). Therefore, the market portfolio (e.g., the basket of all
stocks in the U.S. stock market) must be an efficient
portfolio e.
Note that according to the envelope theorem [22], the
Lagrange multiplier 2m e in (6) is the rate of change (the slope)
of the optimal portfolio variance v 2R e as a function of desired
expected portfolio return E [R e]; i.e.,
i =1
where / i x ip =1. Here, re is a given target level of
expected return. Such portfolios are called mean-variance
efficient portfolios. All efficient portfolios with different re's
constitute an efficient frontier.
where the subscript e represents the optimal (efficient) portfolio. Therefore,
n
/ x je v ij - m e E [R i] - z e = 0, 6i = 1g n.
2m e =
i.e., the efficient frontier is parabolic, as shown in Figure 3.
Figure 3 also shows that the efficient frontier is the boundary
of all attainable portfolios and securities (within the efficient
frontier). The portfolio B is the minimum-variance portfolio.
For an efficient portfolio A, the slope of the tangent line in
Figure 3 is
Se =
(7)
j =1
dv 2R e
,
dE [R e]
dE [R e]
= me .
v Re
dv R e
We know that an efficient portfolio satisfies (7), implying that
In matrix form, R x e - m e E [R] - z e = 0. Along with
x Te E [R] = re, and 1 T x e = 1, we have
R
0
E [R] 1 x e
T
=
E
[
R
]
0
0
m
r
e
>
H
>
H
> eH.
0 0 -z e
1T
1
n
n
j=1
j=1
/ x je v kj - m e E [R k] = / x je v ij - m e E [R i].
Multiplying both sides by x ke and then summing over k, we get
2
v R e - m e re =
n
/ x je v ij - m e E [R i].
j=1
The optimal MVP can then be obtained by solving the linear equation.
Note that the expected return of the efficient portfolio is
E [R e] = re by definition. Rearranging, we get
E [R i] - E [R e] = 1 e / x je v ij - v 2R e o .
m
n
Capital asset pricing model
Assume that market participants, or at least some of them,
are rational in the sense of mean-variance optimal (by maximizing the expected utility). Then all securities in the market
should be priced such that the market is in equilibrium. Note
e
Therefore,
E [R i] = (E [R e] - S e v R e) + S e Cov (R i, R e)
v Re
= r 0e +
MVP Efficient Frontier
[R p]
j =1
E [R e] - r 0e
v Re
Cov (R i, R e)
= r0e + b 0e ^E [R e] - r0eh,
(8)
where
M
B
r0e r
f
A
b 0e =
P
FIGURE 3. The efficient frontier of MVP.
18
2
v Re
,
(9)
and the quantity
Attainable Pool of
Risk Securities and Portfolios
O
Cov (R i, R e)
r0e _ E [R e] - S e v R e
σRP
(10)
is the intercept with zero variance and represents a risk-free asset.
The relationship (8) must hold in the market equilibrium between
the return of a single security and the efficient portfolio. Therefore,
there is no expected return reward for the part of security risk that
is uncorrelated (zero covariance) with the efficient frontier.
IEEE Signal Processing Magazine
|
May 2017
|
Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
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Signal Processing - May 2017 - 110
Signal Processing - May 2017 - 111
Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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