Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 6
Content:
• Mean-variance utility maximization
• Variance minimization with target mean return
• Tangency portfolio
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 𝑼 =
1
𝛾
∑−𝟏 𝝁
The weight for the risk-free asset is equal to 1 − 𝒘 𝑼′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 𝑼 =
1
2𝛾
𝝁′∑−𝟏
𝝁
Mean-variance utility maximization
What if we want to invest everything in the risky assets?
𝒘 must sum up to 1.
Therefore, we have to solve the following constrained
optimization problem:
max
𝒘
𝒘′𝝁 −
𝛾
2
𝒘′Ʃ𝒘
subject to:
𝒘′𝟏 = 1
We solve it with the method of Lagrange multipliers.
Mean-variance utility maximization
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′𝝁 −
𝛾
2
𝒘′Ʃ𝒘 + 𝜆 1 − 𝒘′𝟏
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.
The conditions involve two simultaneous equations, as we
have to compute the partial derivative both with respect to
𝒘 and to 𝜆.
Mean-variance utility maximization
𝜕𝐿
𝜕𝒘
= 𝝁 −
2𝛾
2
Ʃ𝒘 − 𝜆𝟏 = 𝝁 − 𝛾Ʃ𝒘 − 𝜆𝟏 = 𝟎
𝜕𝐿
𝜕𝜆
= 1 − 𝒘′ 𝟏 = 0
We start by solving the first equation for 𝒘:
𝛾Ʃ𝒘 = 𝝁 − 𝜆𝟏
𝒘 = 𝛾Ʃ −1
𝝁 − 𝜆𝟏
Now we can plug this into the second equation:
1 − 𝛾Ʃ −1
𝝁 − 𝜆𝟏 ′
𝟏 = 0
Mean-variance utility maximization
Remember that 𝑨𝑩 ′ = 𝑩′𝑨′, and therefore we have:
𝝁 − 𝜆𝟏 ′ 𝛾Ʃ −1 ′
𝟏 = 1
(𝝁 − 𝜆𝟏)′
Ʃ−𝟏
𝛾
′
𝟏 = 1
Ʃ−𝟏 is a symmetric matrix, so its transpose is still Ʃ−𝟏
(𝝁′ − 𝜆𝟏′)
Ʃ−𝟏
𝛾
𝟏 = 1
(𝝁′ − 𝜆𝟏′)Ʃ−𝟏 𝟏 = 𝛾
𝝁′Ʃ−𝟏 𝟏 − 𝜆𝟏′Ʃ−𝟏 𝟏 = 𝛾
Mean-variance utility maximization
So, we arrive at an expression for 𝛾:
𝜆 =
𝝁′
Ʃ−𝟏
𝟏 − 𝛾
𝟏′Ʃ−𝟏 𝟏
We plug it in the first equation
𝒘 = 𝛾Ʃ −1(𝝁 − 𝜆𝟏)
obtaining the solution to the optimization problem:
𝒘 𝑼∗ =
Ʃ−𝟏
𝛾
𝝁 +
𝛾 − 𝝁′Ʃ−𝟏 𝟏
𝟏′Ʃ−𝟏 𝟏
𝟏
Mean-variance utility maximization
Given a certain 𝝁 and Ʃ, if 𝛾 changes:
• optimal weights without a risk-free asset, 𝒘 𝑼∗, change
without keeping their proportions
• optimal weights with a risk-free asset, 𝒘 𝑼, scale up (if 𝛾
decreases) or down (if 𝛾 increases), keeping proportions
With a risk-free asset, the only thing changing with different
values of 𝛾 is how much wealth is allocated in risky assets.
For example, if 𝛾 = 3 we have 𝒘 𝑼
′
= 0.2 0.4 0.3 , with
𝛾 = 6 we will have 𝒘 𝑼
′
= 0.1 0.2 0.15 . All the weights
for risky assets were cut in half, and the amount invested in
the risk-free asset doubled.
Mean-variance utility maximization
By computing optimized portfolios with different 𝛾 the
efficient frontier, i.e., the set of portfolios with the
highest mean-variance utility for each given value of 𝛾
Variance minimization with target mean return
• 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 𝑅𝑒
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.
Variance minimization with target mean return
It is possible (and preferable) to work with excess returns
instead of explicitly including the risk-free asset in the
asset menu. In this case 𝑅𝑒 is the desired excess return,
and the problem simplifies to:
min
𝒘
𝒘′
∑𝒘
subject to:
𝒘′ 𝝁 = 𝑅𝑒
The Lagrangian function is:
𝐿 𝒘, 𝜆 = 𝒘′Ʃ𝒘 + 𝜆 𝑅𝑒 − 𝒘′ 𝝁
Variance minimization with target mean return
We can multiply the first term by 0.5, which does not
alter the result (minimizing half the variance is equivalent
to minimizing the variance). So, the Lagrangian becomes:
𝐿 𝒘, 𝜆 =
1
2
𝒘′
Ʃ𝒘 + 𝜆 𝑅𝑒 − 𝒘′
𝝁
The first order conditions are:
𝜕𝐿
𝜕𝒘
= Ʃ𝒘 − 𝜆𝝁 = 𝟎
𝜕𝐿
𝜕𝜆
= 𝑅𝑒 − 𝒘′ 𝝁 = 0
Variance minimization with target mean return
We start by solving for 𝒘 in the first equation:
𝒘 = 𝜆Ʃ−𝟏
𝝁
Then we plug it into the second equation and solve for 𝜆:
𝑅𝑒 = 𝒘′ 𝝁
𝑅𝑒 = 𝜆Ʃ−𝟏 𝝁
′
𝝁 = 𝜆𝝁′Ʃ−𝟏 𝝁
𝜆 =
𝑅𝑒
𝝁′Ʃ−𝟏 𝝁
We can now substitute this back in the first equation and
we get the solution we wanted:
𝒘 𝒎𝒗 =
𝑅𝑒
𝝁′Ʃ−𝟏 𝝁
Ʃ−𝟏
𝝁
Variance minimization with target mean return
Obviously, it is also possible to specify a given level of
variance and maximize the mean return:
max
𝒘
𝒘′ 𝝁
subject to:
𝒘′Ʃ𝒘 = 𝜎2
We write the Lagrangian and the first order conditions:
𝐿 𝒘, 𝜆 = 𝒘′
𝝁 + 𝜆 𝜎2
− 𝒘′
Ʃ𝒘
𝜕𝐿
𝜕𝒘
= 𝝁 − 2𝜆Ʃ𝒘 = 𝟎
𝜕𝐿
𝜕𝜆
= 𝜎2 − 𝒘′Ʃ𝒘 = 0
Variance minimization with target mean return
We solve the first equation for 𝒘:
2𝜆Ʃ𝒘 = 𝝁
𝒘 =
Ʃ−𝟏 𝝁
2𝜆
Now we plug into the second equation:
𝜎2 = 𝒘′Ʃ𝒘
𝜎2 =
Ʃ−𝟏 𝝁
2𝜆
′
Ʃ
Ʃ−𝟏 𝝁
2𝜆
=
𝝁′Ʃ−𝟏
2𝜆
Ʃ
Ʃ−𝟏 𝝁
2𝜆
𝜎2 =
𝝁′𝑰
2𝜆
Ʃ−𝟏
𝝁
2𝜆
=
𝝁′Ʃ−𝟏
𝝁
4𝜆2
Variance minimization with target mean return
𝜎 =
𝝁′Ʃ−𝟏 𝝁
2𝜆
𝜆 =
𝝁′Ʃ−𝟏 𝝁
2𝜎
We replace this in the first equation to get the solution:
𝒘 =
Ʃ−𝟏 𝝁
2𝜆
=
Ʃ−𝟏 𝝁
2
𝝁′Ʃ−𝟏 𝝁
2𝜎
=
Ʃ−𝟏 𝝁
𝝁′Ʃ−𝟏 𝝁
𝜎 =
𝜎
𝝁′Ʃ−𝟏 𝝁
Ʃ−𝟏
𝝁
These are two equivalent optimization problems.
However, it is more intuitive and more common to specify
the desired mean and minimize the variance.
Variance minimization with target mean return
We can again add the additional constraint of weights for
the risky assets summing up to 1:
min
𝒘
𝒘′Ʃ𝒘
subject to:
𝒘′ 𝝁 = 𝑅𝑒
𝒘′
𝟏 = 1
As there are two constraints, there are two Lagrange
multipliers:
𝐿 𝒘, 𝜆1, 𝜆2 =
1
2
𝒘′
Ʃ𝒘 + 𝜆1 𝑅𝑒 − 𝒘′
𝝁 + 𝜆2 1 − 𝒘′
𝟏
Variance minimization with target mean return
The first order conditions are:
𝜕𝐿
𝜕𝒘
= Ʃ𝒘 − 𝜆1 𝝁 − 𝜆2 𝟏 = 𝟎
𝜕𝐿
𝜕𝜆1
= 𝑅𝑒 − 𝒘′ 𝝁 = 0
𝜕𝐿
𝜕𝜆2
= 1 − 𝒘′ 𝟏 = 0
We solve the first equation for 𝒘:
Ʃ𝒘 = 𝜆1 𝝁 + 𝜆2 𝟏
𝒘 = Ʃ−𝟏 𝜆1 𝝁 + 𝜆2 𝟏 = 𝜆1Ʃ−𝟏 𝝁 + 𝜆2Ʃ−𝟏 𝟏
Variance minimization with target mean return
We need to get a formula for 𝜆1 and one for 𝜆2 that do
not contain each other among their terms.
Notice that if we pre-multiply each side by 𝝁′ we get
𝝁′𝒘 = 𝜆1 𝝁′Ʃ−𝟏 𝝁 + 𝜆2 𝝁′Ʃ−𝟏 𝟏
𝑅𝑒 = 𝜆1 𝝁′Ʃ−𝟏
𝝁 + 𝜆2 𝝁′Ʃ−𝟏
𝟏
Likewise, if we pre-multiply each side by 𝟏′ we get
𝟏′𝒘 = 𝜆1 𝟏′Ʃ−𝟏 𝝁 + 𝜆2 𝟏′Ʃ−𝟏 𝟏
1 = 𝜆1 𝟏′Ʃ−𝟏 𝝁 + 𝜆2 𝟏′Ʃ−𝟏 𝟏
Variance minimization with target mean return
Therefore, we get a system of two equations:
𝑅𝑒 = 𝜆1 𝝁′Ʃ−𝟏 𝝁 + 𝜆2 𝝁′Ʃ−𝟏 𝟏
1 = 𝜆1 𝟏′Ʃ−𝟏
𝝁 + 𝜆2 𝟏′Ʃ−𝟏
𝟏
𝝁′Ʃ−𝟏
𝝁, 𝝁′
Ʃ−𝟏
𝟏, 𝟏′Ʃ−𝟏
𝝁 and 𝟏′
Ʃ−𝟏
𝟏 are scalars.
We name them as
𝐴 = 𝝁′Ʃ−𝟏
𝝁 𝐵 = 𝝁′
Ʃ−𝟏
𝟏 = 𝟏′
Ʃ−𝟏
𝝁 𝐶 = 𝟏′
Ʃ−𝟏
𝟏
So the system of two equations is:
𝜆1 𝐴 + 𝜆2 𝐵 = 𝑅𝑒
𝜆1 𝐵 + 𝜆2 𝐶 = 1
Variance minimization with target mean return
In matrix form the system is
𝐴 𝐵
𝐵 𝐶
𝜆1
𝜆2
=
𝑅𝑒
1
We solve the system for 𝜆1 and 𝜆2:
𝜆1
𝜆2
=
𝐴 𝐵
𝐵 𝐶
−1
𝑅𝑒
1
The inverse of 2 × 2 matrix 𝑴 =
𝑎 𝑏
𝑐 𝑑
is :
𝑴−𝟏 =
1
𝑎𝑑 − 𝑏𝑐
𝑑 −𝑏
−𝑐 𝑎
Variance minimization with target mean return
Hence, our system becomes
𝜆1
𝜆2
=
1
𝐴𝐶 − 𝐵2
𝐶 −𝐵
−𝐵 𝐴
𝑅𝑒
1
𝜆1
𝜆2
=
1
𝐴𝐶 − 𝐵2
𝐶𝑅𝑒 − 𝐵
−𝐵𝑅𝑒 + 𝐴
which are the formulas we were looking for:
𝜆1 =
𝐶𝑅𝑒 − 𝐵
𝐴𝐶 − 𝐵2
𝜆2 =
𝐴 − 𝐵𝑅𝑒
𝐴𝐶 − 𝐵2
Variance minimization with target mean return
Finally we plug these terms back in the first equation:
𝒘 = 𝜆1Ʃ−𝟏 𝝁 + 𝜆2Ʃ−𝟏 𝟏
𝒘 =
𝐶𝑅𝑒 − 𝐵
𝐴𝐶 − 𝐵2
Ʃ−𝟏 𝝁 +
𝐴 − 𝐵𝑅𝑒
𝐴𝐶 − 𝐵2
Ʃ−𝟏 𝟏
𝒘 𝒎𝒗∗ = Ʃ−𝟏
𝐶𝑅𝑒 − 𝐵
𝐴𝐶 − 𝐵2
𝝁 +
𝐴 − 𝐵𝑅𝑒
𝐴𝐶 − 𝐵2
𝟏
where 𝐴 = 𝝁′Ʃ−𝟏 𝝁, 𝐵 = 𝟏′Ʃ−𝟏 𝝁 and 𝐶 = 𝟏′Ʃ−𝟏 𝟏.
Variance minimization with target mean return
Given a certain 𝝁 and Ʃ, if the target return changes:
• optimal weights without a risk-free asset, 𝒘 𝒎𝒗∗,
change without keeping their proportions
• optimal weights with a risk-free asset, 𝒘 𝒎𝒘, scale up
(if 𝑅𝑒 increases) or down (if 𝑅𝑒 decreases), keeping
their proportions
With a risk-free asset, the only thing changing with
different values of 𝑅𝑒 is how much wealth is allocated
in risky assets.
Variance minimization with target mean return
Minimizing the portfolio variance with different Re is
another, equivalent, way to draw the efficient frontier.
Variance minimization with target mean return
The efficient allocations are those associated with a
positive slope.
Those associated with a negative slope, which are
obtained with very low or even negative 𝑅𝑒, are
frontier but inefficient allocations.
In the approach where we maximize utility given 𝛾,
those inefficient frontier allocations are obtained with
negative 𝛾 values (i.e., the investor is risk-seeking).
Tangency portfolio
There is a point at which the efficient frontiers with
and without a risk-free asset touch each other.
That corresponds to the mean and standard deviation
of the tangency portfolio: the one portfolio of risky
assets which has the highest possible Sharpe ratio.
All the points on the efficient frontier with a risk-free
asset have the same Sharpe ratio.
Therefore, the highest Sharpe ratio can always be
achieved with the optimal weights given by the
optimization procedures with a risk-free asset:
𝒘 𝑼 and 1 − 𝒘 𝑼′𝟏, or 𝒘 𝒎𝒗 and 1 − 𝒘 𝒎𝒘′𝟏.
Tangency portfolio
Two-fund separation theorem: the combination of
the tangency portfolio and a risk-free asset gives the
highest utility for a given risk-aversion level.
To compute the tangency portfolio, we have maximize
the Sharpe ratio with full investment in risky assets.
Working with excess returns, the problem is:
max
𝒘
𝒘′𝝁
𝒘′Ʃ𝒘
subject to:
𝒘′ 𝟏 = 1
Tangency portfolio
Notice that 𝒘 𝑼 (and 𝒘 𝒎𝒗) are the tangency portfolio
weights scaled according to the risk preferences of
the investor.
Therefore, we can simply take the formula for 𝒘 𝑼 and
divide it by its own sum:
𝒘 𝒕𝒂𝒏 =
1
𝛾
Ʃ−𝟏 𝝁
𝟏′
1
𝛾
Ʃ−𝟏 𝝁
=
Ʃ−𝟏 𝝁
𝟏′Ʃ−𝟏 𝝁
This portfolio is plotted in purple in the next graph.
Tangency portfolio