Mean-Variance Portfolio Analysis
Ilya Dergunov
Theory of Finance
Higher School of Economics
Module 3, 2023
Outline
Outline
This Lecture
Application of expected utility theory to the financial problem
of choosing a portfolio
Consider the problem of combining risky assets using the
classic mean-variance analysis of Markowitz (1952)
Mean-Variance Portfolio Optimization
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Preliminaries
Justifying mean-variance analysis
How can we justify mean-variance analysis?
⇒where the objective function maximized by investors has only
two arguments, namely expected return and variance of the
portfolio?
Consider Taylor series around E[W]
E[u(W)] = u(E[W]) + 1
2u′′(E[W])E[(W−E[W])2] + r3,
where
r3=
∞
X
n=3
1
n!u(n)(E[W])E[(W−E[W])n]
Therefore, any expected utility maximizer with utility over
wealth, cares about mean and variance of wealth
AND about a remainder term that captures all the higher
moments (weighted by the higher derivatives of u(·))
Mean-Variance Portfolio Optimization
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Preliminaries
Justifying mean-variance analysis (Cont.)
Investors like higher expected returns (as long as utility is
increasing, i.e. u′(·)>0)
Dislike variance (as long as investors are risk-averse or utility
is strictly concave, i.e. u′′(·)<0)
Mean-variance analysis is fine as long as we can kill the r3
term
One obvious way to achieve this is to assume quadratic utility
as then u(n)(·) = 0 for n>2
But quadratic utility is not monotonically increasing thus
exhibiting satiation and that it violates DARA
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Preliminaries
Justifying mean-variance analysis (Cont.)
Another justification for mean-variance analysis is that returns
are normally distributed.
⇒In that case the remainder term r3is fully characterized by
mean and variance
It’s easy to show that people like mean and dislike variance in
that case
If returns are normally distributed, (the return on) a portfolio
invested in those assets and hence (the return on) future
wealth will also be normally distributed
Remark: even if these conditions are not satisfied, one can still do
mean-variance analysis for purposes of illustration, intuition or data
representation or description.
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Example with Different Utility Functions
Quadratic Utility
Case of quadratic utility, which has a special importance in
finance theory, because it implies mean-variance preferences
u( ˜w) = −1
2( ˜w−η)2=−1
2η2+η˜w−1
2˜w2
Specifically, the investor’s expected utility is, ignoring the
additive constant
ηE[ ˜w]−1
2E[ ˜w2] = ηE[ ˜w]−1
2E2[ ˜w]−1
2var( ˜w),
where var( ˜w) denotes the variance of ˜w
Thus, preferences over gambles depend only on their means
and variances when an investor has quadratic utility
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CARA Utility
Choosing Risk Exposure
An investor with initial wealth W
Can invest in a safe asset with return Rfor a risky asset with
return Rf+ ˜x
˜x, the excess return on the risky asset over the safe asset,
need not have a zero mean.
