Mean-Variance Portfolio Analysis
Ilya Dergunov
Theory of Finance
Higher School of Economics
Module 3, 2023
Outline
Outline
This Lecture
Mean-Variance Analysis: Nrisky assets
Mean-Variance Analysis: Nrisky assets + risk-free asset
Mean-Variance Portfolio Optimization
Ilya Dergunov 1
MV: NRisky Assets
Mean-Variance Analysis: NRisky Assets
Mean-Variance Portfolio Optimization
Ilya Dergunov 2
MV: NRisky Assets
The Mean-Variance Portfolio Problem
Suppose there are Nrisky assets available for investment
let µdenote the vector of asset expected returns and Σ the
variance-covariance matrix of returns.
Then, for any portfolio w, expected return and portfolio
variance are given by
µp=w′µ σ2
p=w′Σw
The minimum-variance portfolio with expected return µpis
the solution w(µp) to
min
w
1
2w′Σw
s.t.1′w= 1
µ′w=µp
Mean-Variance Portfolio Optimization
Ilya Dergunov 3
MV: NRisky Assets
The Mean-Variance Portfolio Problem (cont.)
To solve this problem, set up the Lagrangian
L=1
2w′Σw+λ1(µp−µ′w) + λ2(1 −1′w)
The FOCs are
∂L
∂w= Σw−λ1µ−λ21= 0
∂L
∂λ1
=µp−µ′w= 0
∂L
∂λ2
= 1 −1′w= 0
Hence, all minimum-variance portfolios are of the form
w=λ1Σ−1µ′+λ2Σ−11
Mean-Variance Portfolio Optimization
Ilya Dergunov 4
MV: NRisky Assets
The Mean-Variance Portfolio Problem (cont.)
FOC already presents interesting result
Mean-variance efficient portfolio is constructed on the basis of
two mutual funds, based on Σ−11and Σ−1µ′
The first mutual fund based on Σ−11will have the intuition of
being the global minimum variance portfolio
The second fund focuses instead on obtaining the highest
expected return per unit of standard deviation
⇒entire m-v frontier is spanned by these two intuitive mutual
funds
Mean-Variance Portfolio Optimization
Ilya Dergunov 5
MV: NRisky Assets
The Mean-Variance Portfolio Problem (cont.)
In order to determine the constants λ1and λ2, just use the
two constraints and require that they be satisfied
For the first constraint, we have
µ′w=µ′(λ2Σ−11+λ1Σ−1µ′) = µp
or
