Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 7
Content:
• Mean-variance utility maximization
• Mean-variance portfolio optimization
• Global minimum variance portfolio
• Portfolios that mitigate parameter uncertainty
Mean-variance utility maximization
We can model an investor’s behavior as an attempt to
maximize a utility function.
A very simple one is the linear utility function:
𝑈 𝑉 = 𝑎 + 𝑏𝑉, 𝑏 > 0
where 𝑈 𝑉 is the utility that the investor gets depending
on the value 𝑉 of the portfolio. Only the mean is considered.
Markowitz (1952) adds risk by assuming that the investor
cares both about the mean and the variance of the portfolio
returns, i.e. the investor has a mean-variance utility.
Mean-variance utility maximization
• Given a certain mean return, the utility increases as the
variance (which quantifies risk) gets lower.
• Equivalently, given a certain level of variance, the utility
increases with a higher mean return.
The decision in the trade-off between return and risk is
quantified by a risk aversion parameter 𝛾: a higher 𝛾 means
that the investor is more risk averse and will therefore
require a higher compensation for an increased risk.
All else equal, the lower 𝛾 is, the more the investor will
create a portfolio with a higher mean return but also a
higher variance.
Mean-variance utility maximization
We model such preferences with a quadratic utility function:
𝑈 𝑉 = 𝑉 −
𝛾
2
𝑉2, 𝛾 > 0
𝛾 is assumed to be positive to have a concave utility function:
Mean-variance utility maximization
• 𝛾 > 0 The investor is risk-averse
• 𝛾 = 0 The investor is indifferent to risk
• 𝛾 < 0 The investor is risk-taker (i.e., prefers more risk
with the same amount of wealth)
For a portfolio we have 𝜇 𝑃 = 𝒘′ 𝝁 = 𝑉, where 𝒘 is the
vector of portfolio weights and 𝝁 is the vector of mean
return of the single assets. Moreover, the variance is the
expected value of the squared deviation from the mean.
So, the mean-variance utility function is:
𝑈 𝒘 = 𝒘′𝝁 −
𝛾
2
𝒘′∑𝒘
Mean-variance utility maximization
Given a risk-free asset and 𝑁 risky assets with mean
returns 𝝁 and covariance matrix 𝜮, and a certain risk
aversion coefficient 𝛾, the investor wants to select the
weights 𝒘 in a way that maximizes the utility function:
max
𝒘
𝒘′𝝁 −
𝛾
2
𝒘′∑𝒘
We set the first-order condition, i.e., take the partial
derivative with respect to 𝒘 and set it equal to zero:
𝜕𝑈(𝒘)
𝜕𝒘
= 𝝁 −
2𝛾
2
∑𝒘 = 𝝁 − 𝛾∑𝒘 = 𝟎
Mean-variance utility maximization
Solving for 𝒘, we obtain the optimal weights for the risky assets:
∑𝒘 =
1
𝛾
𝝁
𝒘 𝒎𝒗 =
1
𝛾
∑−𝟏 𝝁
The weight for the risk-free asset is equal to 1 − 𝒘 𝒎𝒗′𝟏, where 𝟏
is a vector of 1 with length equal to the number of risky assets.
The resulting optimal expected utility is:
𝑈 𝒘 𝒎𝒗 =
1
2𝛾
𝝁′∑−𝟏
𝝁
Mean-variance portfolio optimization
• Choosing a value for 𝛾 might be difficult for real investors
• The value of the utility 𝑈 is not very informative
More intuitive approach: specify a desired mean return and
minimize the variance.
This is a constrained optimization problem:
min
𝒘
𝒘′∑𝒘
subject to:
𝒘′ 𝝁 + 1 − 𝒘′𝟏 𝑅𝑓 = 𝑅𝑒
𝒘′
𝝁 is the return of the risky assets, and 1 − 𝒘′𝟏 𝑅𝑓 is
the return of the risk-free asset.
Mean-variance portfolio optimization
We solve it with the method of Lagrange multipliers.
First we define the Lagrangian function, i.e., a modified
version of the objective function that incorporates the
constraint in this way:
𝐿 𝒘, 𝜆 = 𝒘′∑𝒘 + 𝜆 𝑅𝑒 − 𝒘′ 𝝁 − 1 − 𝒘′𝟏 𝑅𝑓
where 𝜆 is the Lagrange multiplier.
We can now solve an unconstrained problem instead of a
constrained one by setting the first order conditions for
the Lagrangian.
Mean-variance portfolio optimization
The conditions involve two simultaneous equations, as we
have to compute the partial derivative both with respect
to 𝒘 and to 𝜆:
𝜕𝐿
𝜕𝒘
= 2∑𝒘 − 𝜆𝝁 + 𝜆𝑅𝑓 𝟏 = 𝟎
𝜕𝐿
𝜕𝜆
= 𝑅𝑒 − 𝒘′ 𝝁 − 1 − 𝒘′ 𝟏 𝑅𝑓 = 0
Solving this system for 𝒘 gives the following closed-form
solution:
𝒘 𝒎𝒗 =
𝑅𝑒
𝝁′∑−𝟏 𝝁
∑−𝟏
𝝁
Mean-variance portfolio optimization
It is also possible (and mathematically equivalent) to
specify a given level of variance and maximize the mean
return. However, it is more intuitive and more common to
specify the desired mean and minimize the variance.
In practice, we do not need to explicitly include the riskfree
asset in the asset menu, as it is equivalent and
simpler to work with the excess returns of the risky assets.
Global minimum variance portfolio
In real applications 𝝁 and ∑ need to be estimated.
Plug-in approach: the sample estimates of the inputs are
computed from past data, and are then plugged into the
optimization problem as if they were the true values.
Sample estimates can be very imprecise, especially that of
the mean. Therefore, ignoring 𝝁 and only minimizing the
variance usually gives better results.
Therefore, the investor might want to compute the global
minimum variance portfolio (GMV), also simply called
minimum variance portfolio.
Global minimum variance portfolio
This is a constrained optimization problem, as we need the
weights of the risky assets to sum up to one, which means
that nothing is invested in the risk-free asset (otherwise,
everything would be invested in the risk-free asset):
min
𝒘
𝒘′
∑𝒘
subject to:
𝒘′𝟏 = 1
We write the Lagrangian function:
𝐿 𝒘, 𝜆 = 𝒘′
∑𝒘 + 𝜆 1 − 𝒘′𝟏
Global minimum variance portfolio
It is convenient to multiply the first term by 0.5, which
does not alter the result. So the Lagrangian becomes:
𝐿 𝒘, 𝜆 =
1
2
𝒘′∑𝒘 + 𝜆 1 − 𝒘′𝟏
The first order conditions are:
𝜕𝐿
𝜕𝒘
= ∑𝒘 − 𝜆𝟏 = 𝟎
𝜕𝐿
𝜕𝜆
= 1 − 𝒘′ 𝟏 = 0
Global minimum variance portfolio
Through some simple rearrangement we get:
𝒘 = 𝜆∑−𝟏 𝟏
𝒘′
𝟏 = 1
In the first equation we multiply both sides by 𝟏′:
𝟏′𝒘 = 𝜆𝟏′∑−𝟏
𝟏
From the second equation we know that:
𝒘′
𝟏 = 𝟏′
𝒘 = 1
Hence, the first equation becomes:
1 = 𝜆𝟏′∑−𝟏
𝟏
𝜆 =
1
𝟏′∑−𝟏 𝟏
Global minimum variance portfolio
We take this last result and replace 𝜆 in 𝒘 = 𝜆∑−𝟏 𝟏,
obtaining the closed form-solution that gives the minimum
variance portfolio weights:
𝒘 𝒗 =
1
𝟏′∑−𝟏 𝟏
∑−𝟏 𝟏
• As nothing is invested in the risk-free rate, it is
equivalent to work with returns or excess returns.
• However, working with excess returns makes it easier to
compare the results with those obtained by the meanvariance
portfolio.
Portfolios that mitigate parameter uncertainty
To further reduce the impact of estimation errors, we can
compute a long-only minimum variance portfolio.
In other words, we add a second constraint that requires
all the weights be positive. This typically improves the
results out-of-sample.
This problem does not have a closed form solution, but it
can easily be solved in R using available packages.
Portfolios that mitigate parameter uncertainty
When a value for 𝛾 is specified, another strategy that
mitigates the impact of the estimation error is the 1/N rule.
The (sample) estimates of 𝝁 and ∑ are used to optimally
allocate the wealth between the risk-free asset and the
equally weighted risky assets.
The weights for the risky assets using this strategy are:
𝒘 𝟏/𝑵 =
1
𝛾
𝟏′𝝁
𝟏′∑𝟏
𝟏
and the weight for the riskless asset is equal to 1 − 𝒘′ 𝟏/𝑵 𝟏.
Portfolios that mitigate parameter uncertainty
For example, if 𝑁 = 5 and this rule returns a weight of 0.15
for each risky asset, we equally divide 75% of our wealth
among the risky assets (i.e., 15% on each risky asset), and
then place the remaining 25% on the risk-free asset.
This rule typically performs very well in terms of meanvariance
utility.