Northwestern University
Marciano Siniscalchi
Fall 2009
Econ 3310
MEANVARIANCE ANALYSIS AND
PORTFOLIO CHOICE WITH MANY ASSETS
1.
Mean is Good, Variance is Bad (or is it?)
Portfolio choice with a single risky asset reduces to a simple tradeoﬀ: stocks (generally) yield a
higher expected return than bonds, but they are risky. A riskaverse investor will put some money
in stocks, but her optimal portfolio will have to balance these two considerations.
A common measure of risk is the
variance
of returns. Recall that, if
X
is a random variable with
probability distribution (
x
1
,p
1
;
...
;
x
n
,p
n
), its variance is
Var[
X
] =
n
X
i
=1
p
i
(
x
i

E[
X
])
2
= E[
X
2
]

(E[
X
])
2
(if the ﬁrst equality is the deﬁnition of variance; the second is a wellknown and useful identity;
make sure you see why it is true). Also recall that the
standard deviation
of a r.v. is the square root
of its variance: thus, in particular, StDev [
X
] =
p
Var[
X
]. Similar deﬁnitions hold for continuous
random variables.
The idea is that, if returns vary a lot around their expectation, they are intuitively “risky”. Note
however that one might argue that upside variability are not so bad, so one should really only
take downside variability into account; it is indeed possible to do so, but the math is ugly, so this
consideration is typically disregarded.
In general, the DM’s preferences among risky assets, or random variables in general, are
not
solely a function of their means and variances. For instance, if the probability distribution of
X
is (100
,
9
32
; 200
,
14
32
; 300
,
9
32
) and that of
Y
is (50
,
1
8
; 200
,
3
4
; 350
,
1
8
), then E[
X
] = E[
Y
] = 200 and
Var[
X
] = Var [
Y
] = 75, but a DM with utility
u
(
x
) =
√
x
will strictly prefer
X
to
Y
, as will a DM
with power utility
u
(
x
) =
x
1

γ
1

γ
with e.g.
γ
= 3.
However, the idea that “expected return is good, variance is bad” is so appealing (and, it
turns out, simple) that people have tried to provide justiﬁcations for analyzing portfolio choice
solely in terms of these two statistics. A “local” rationale for this is provided by the Arrow
approximation, which we considered last time. Recall that the (insurance) risk premium
π
(
W,
˜
±
) for
a small risk ˜
±
with mean 0 is approximately
1
2
A
(
W
)Var[˜
±
], where
A
(
w
) is the ArrowPratt measure
of risk aversion. Also remember that the certainty equivalent and risk premium are related by the
equation
W

π
(
W,
˜
±
) =
c
W
+˜
±
. If we now reinterpret
W
as the expectation of a random variable
˜
W
representing the ﬁnal wealth of an investor who purchases a risky portfolio with Var[
˜
W
] = Var[˜
±
],
we can write
c
˜
W
= E[
˜
W
]

1
2
A
(
W
)Var[
˜
W
] :
that is, if the variance of
˜
W
is small, the certainty equivalent of terminal wealth is approximately
equal to its expectation minus a constant times its variance. Since ranking random variables by
their certainty equivalent is by deﬁnition the same as ranking them according to their expected
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