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<reference>
Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

FE670 Algorithmic Trading Strategies
Lecture 6. Portfolio Optimization: Basic Theory and Practice

Steve Yang

Stevens Institute of Technology
10/03/2013

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Outline
1

Mean-Variance Analysis: Overview

2

Classical Framework for Mean-Variance Optimization

3

Mean-Variance Optimization with a Risk-Free Asset

4

Portfolio Constraints Commonly Used in Practice

5

Portfolio Optimization with Other Risk Measures

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Mean-Variance Analysis: Overview
- Markowitz’s starting point is that of a rational investor
who, at time t, decides what portfolio of investments to
hold for a time horizon of δt.
- The investor makes decisions on the gains and losses he
will make at time t + δt.
- At time t + δt, the investor will reconsider the situation
and decide anew.
* This one-period framework is often referred to as myopic
or short-sighted behavior. In general, a myopic investor’s
behavior is suboptimal in comparison to an investor who
makes investment decisions based upon multiple periods
ahead.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Markowitz reasoned that investors should decide on the basis
of a trade off between risk and expected return. Expected
return of a security is defined as the expected price change
plus any additional income over the time horizon considered,
such as dividend payments, divided by the beginning price of
the security.
- He suggested that risk should be measured by the variance of
returns - the average squared deviation around the expected
return.
* Markowitz’s mean-variance framework does not assume joint
normality of security returns. However, the mean-variance
approach is consistent with two different starting points:
(1) expected utility maximization under certain assumptions.
(2) the assumption that security returns are jointly normally
distributed.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Exhibit 8.1 provides a graphical illustration of the efficient
frontier.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- The set of all possible portfolios that can be constructed are
called feasible set. The set of all mean-variance efficient
portfolios, for different desired returns, is called the efficient
frontier.
- Moreover, Markowitz argued that for any given level of
expected return, a rational investor would choose the portfolio
with minimum variance from amongst the set of all possible
portfolios.
- Therefore, the efficient frontier provides the best possible
trade-off between expected return and risk. The portfolio at
point II is often referred to as the global minimum variance
portfolio (GMV), as it is the portfolio on the efficient frontier
with the smallest variance.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Exhibit 8.2 shows the investment process often referred to
as mean-variance optimization or theory of portfolio
selection.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Classical Framework for Mean-Variance Optimization
- Suppose an investor has to choose a portfolio comprised of N
risky assets. The investor’s choice is embodied in an N-vector
w = (w1 , w2 , ..., wN )0 of weights, where each weight i
represents the fraction of the i-th asset held in the portfolio
N
X

wi = 1

i=1

For now, we permit short selling, which means that
weights can be negative.
- Suppose the assets’ returns R = (R1 , R2 , ..., RN )0 have
expected returns µ = (µ1 , µ2 , ..., µN )0 and covariance
matrix given by


σ11
 ..
Σ= .
σN1

...
..
.
...


σ1N
.. 
. 
σNN

where σij denotes the covariance between asset i and j.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Under these assumptions, the return of a portfolio with
weights w is a random variable Rp = w0 R with expected
return and variance given by
µp = w 0 µ
σp2 = w0 Σw
For now, we simply assume that expected returns, µ, and
their covariance matrix, Σ, are given.
- To calculate the weights fro one possible pair, we choose a
targeted mean return, µ0 , Following Markowitz, the investor’s
problem is constrained minimization problem:
1
min w0 Σw
w 2
s. t.
µ0 = w0 µ, w0 l0 = 1, l0 = [1, 1, ..., 1]

- We refer to this version of the classical mean-variance
optimization problem as the risk minimization formulation.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- This problem is a quadratic optimization problem with
equality constraints with the solution given by
w = λΣ−1 l + γΣ−1 µ
where
C − µ0 B
µ0 A − B
λ=
,γ =
∆
∆
0 −1
0 −1
0 −1
A = l Σ l, B = l Σ µ, C = µ Σ µ
- It is easy to see that
σ02 = w0 Σw
Aµ20 − 2Bµ0 + C
=
∆

* In extensions involving only so-called equality constraints,
finding the optimum portfolio reduces to solving a set of
linear equations. For formulations involving inequality
constraints, analytical solutions are not available, and
numerical optimization needs to be applied.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- The mean-variance optimization problem has several
alternative but equivalent formulations that are very useful in
practical applications.
- First, we can choose a certain level of targeted portfolio risk,
say σ0 , and then maximize the expected return of the
portfolio (expected return maximization formulation):


max w 0 µ
w

s. t.
σ02 = w0 Σw, w0 l0 = 1, l0 = [1, 1, ..., 1]
- Alternatively, we can explicitly model the trade-off between
risk and return in the objective function using a risk-aversion
coefficient λ (risk-aversion formulation):


1 0
0
max w µ − λ w Σw
w
2
s. t.
w0 l0 = 1, l0 = [1, 1, ..., 1]

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Mean-Variance Optimization with a Risk-Free Asset
- Assume that there is risk-free asset, with risk-free return
denoted by Rf and that the investor is able to borrow and
lend at this rate. The investor has to choose a combination of
the N risky assets plus the risk-free asset.
- The weights wR0 = (wR1 , wR2 , ..., wRN ) do not have to sum to
1 as the remaining part (1 − w0 l) is the investment in the
risk-free asset.
- The portfolio’s expected return and variance are
µp = wR0 µ + (1 − wR0 l)Rf
σp2 = wR0 ΣwR
because the risk-free asset has zero variance and is
uncorrelated with the risky assets.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- The investor’s objective is again for a targeted level of
expected portfolio return, µ0 , to choose allocations by solving
a quadratic optimization problem
min wR0 ΣwR
wR

subject to
µ0 = wR0 µ + (1 − wR0 l)Rf
- The optimal portfolio weights are given by
wR = C Σ−1 (µ − Rf l)
where
µ0 − R f
C=
(µ − Rf l)0 Σ−1 (µ − Rf l)
Therefore, with a risk-free asset, all minimum variance
portfolios are a combination of the risk-free asset and a given
risky portfolio. The risky portfolio is often called tangency
portfolio or market portfolio.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Exhibit 8.3 Capital Market Line and the Markowitz
Efficient Frontier.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- In Exhibit 8.3 every combination of the risk-free asset and the
market portfolio M is shown on the line drawn from the
vertical axis at the risk-free rate tangent to the Markowitz
efficient frontier. All the portfolios on the line are feasible for
the investor to construct. The line from the risk-free rate that
is tangent to the efficient frontier of risky assets is called the
Capital Market Line (CML).
We observe that with the exception of the market portfolio,
the minimum variance portfolios that are a combination of the
market portfolio and the risk-free asset are superior to the
portfolio on the Markowitz efficient frontier for the same level
of risk.
- With the introduction of the risk-free asset, we can now say
that an investor will select a portfolio on the CML that
represents a combination of borrowing or lending at the
risk-free rate and the market portfolio. This important
property is called separation.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Portfolio to the left of the market portfolio represent
combinations of risky assets and the risk-free asset.
- Portfolio to the right of the market portfolio include purchases
of risky assets made with funds borrowed at the risk-free rate.
Such a portfolio is called a leveraged portfolio because it
involves the used of borrowed funds.
- Practical portfolio construction is normally broken down into
at least two steps:
1

Asset allocation: Decide how to allocate the investor’s wealth
between the risk-free security and the set of risky securities

2

Risky portfolio construction: Decide how to distribute the risky
portion of the investment among the set of risky securities.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Deriving the Capital Market Line
- We can drive a formula for the CML algebraically. Based on
the assumption of homogeneous expectations regarding the
inputs in the portfolio construction process, all investors can
create an efficient portfolio consisting of wf placed in the
risk-free asset and wM in the market portfolio.
- Thus, wf + wM = 1. As the expected return of the portfolio,
E (Rp ), is equal to the weighted average of the expected
returns of the two assets, we have
E (Rp ) = wf Rf + wM E (RM )
Since we know that wf = 1 − wM , we can rewrite E (Rp ) as
E (Rp ) = (1 − wM )Rf + wM E (RM )
which can be simplified to
E (Rp ) = Rf + wM (E (RM ) − Rf )

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Deriving the Capital Market Line
- Since the return of the risk-free asset and the return of the
market portfolio are uncorrelated and the variance of the
risk-free asset is equal to zero, the variance of the portfolio is
given by
2
σp2 = var (Rp ) = wf2 var (Rf ) + wM
var (RM ) + 2wf wM cov (Rf , RM )
2
2 2
= wM
var (RM ) = wM
σM

In other words, the variance of the portfolio is represented by
the weighted variance of the market portfolio. We can write
σp
wM =
σM
If we substitute the preceding result and rearrange terms, we
get the explicit expression for the CML


E (RM ) − Rf
σp ← risk premium
E (Rp ) = Rf +
σM

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- The numerator of the bracketed expression is the expected
return from investing in the market beyond the risk-free
return. It is a measure of the reward for holding the risky
market portfolio rather than the risk-free asset.
- The slope of the CML, measures the reward per unit of
market risk. Since the CML represents the return offered to
compensate for a perceived level of risk, each point on the
CML is a balanced market condition, or equilibrium.
- The slope of the CML determines the additional return
needed to compensate for a unit change in risk, which is why
it is also referred to as the equilibrium market price of risk.
E (Rp ) = Rf + Market price of risk × Quality of risk

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Portfolio Constraints Commonly Used in Practice
- Institutional features and investment policy decisions often
lead to more complicated constraints and portfolio
management objectives than those present in the classical
format.
[Notations: w0 - the current portfolio weights; w - the
targeted portfolio weights; x = w − w0 - the amount to be
traded.
- Linear and Quadratic Constraints Will discuss some of the
more commonly used ones.
- Long-only Constraints When short-selling is not allowed, we
require that w ≥ 0. This is a frequently used constraint, as
many funds and institutional investors are prohibited from
selling stocks short.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Turnover Constraints High portfolio turnover can result in
large transaction costs that make portfolio re-balancing
inefficient. The most common turnover constraints limit
turnover on each individual asset
|xi | ≤ Ui or

X

|xi | ≤ Uportfolio

i∈I

- Holding Constraints A well-diversified portfolio should
not exhibit large concentrations in any specific assets,
industries, sectors, or countries. Maximal holdings in an
individual asset can be controlled by the constraint
Li ≤ wi ≤ Ui

where Li and Ui are vectors representing the lower and
upper bounds of the holdings of asset i. To constrain the
exposure to a set Ii , we can have
Li ≤

X
j∈Ii

wj ≤ Ui

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Risk Factor Constraints In practice, it is very common for
portfolio managers to use factor models to control for
different risk exposures to risk factors. Let us assume that
security returns have a factor structure with K risk factors:
Ri = αi +

K
X

βik Fk + i

k=1

where Fk , k = 1, ..., K are the K factors common to all
the securities, βik is the sensitivity of the i-th security to
the k-th factor, and i is the noise for the i-th security.
To limit exposure to the k-th risk factor, we impose
N
X

βik wi ≤ Uk

i=1

where Uk denotes maximum exposure allowed. To
construct a portfolio that is neutral to the k-th risk factor
N
X
i=1

βik wi = 0

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Benchmark Exposure and Tracking Error Constraints
Many portfolio managers are faced with the objective of
managing their portfolio relative to a benchmark. A portfolio
manager might choose to limit the deviations of the portfolio
weights from the benchmark weights:
||w − wb || ≤ M or

X

(wj − wbj ) ≤ Mi

j∈Ii

where wb is the benchmark weights, and Ii is a specific
industry.
- However, the most commonly used metric to measure the
deviation from the benchmark is the tracking error:
TEVp = var (Rp − Rb ) = var (w0 R − wb0 R) = (w − wb )0 Σ(w − wb )

where Σ is the covariance matrix of the asset returns. To
limit the tracking error, we will have
2
(w − wb )0 Σ(w − wb ) ≤ σTE

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- General Linear and Quadratic Constraints The constraints
described in this section are all linear or quadratic, that is they
can be cast either as
Aw w ≤ dw or Ax x ≤ dx or Ab (w − wb ) ≤ db

or as
w 0 Qw w ≤ q w
x0 Qx x ≤ qx
(w − wb )0 Qb (w − wb ) ≤ qb

These types of constraints can be dealt with directly
within the quadratic programming framework, and there
are very efficient algorithms available that are capable of
solving practical portfolio optimization problems with
thousands of assets in a matter of seconds.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Combinatorial and Integer Constraints The following
binary decision variable is useful in describing some
combinatorial and integer constraints:

δi =

1, if wi 6= 0
0, if wi = 0

where wi denotes the portfolio weight of the i-th asset.
- Cardinality Constraints A portfolio manager might want
to restrict the number of assets allowed in a portfolio.
The cardinality constraint takes the form
N
X

δi = K

i=1

where K is a positive integer significantly less than the
number of assets in the investment universe, N.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Minimum Holding and Transaction Size Constraints The
classical mean-variance optimization problem often results in a
few large and many small positions. In practice, due to
transaction costs and other ticket charges, small holdings are
undesirable. In order to eliminate small holdings, threshold
constraints are often used
|wi | ≥ Lwi δi , i = 1, ..., N

where Lwi is the smallest holding size allowed for asset i.
- This approach can be used to eliminate small trades,
because the fixed costs related to trading each individual
security. In practice, few portfolio managers go to the
extent of including constraints of this type in their
optimization framework. Instead, a standard
mean-variance optimization problem is solved and then, in
a post-optimization step, generated portfolio weights or
trades that are smaller than a certain threshold are
eliminated.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Round Lot Constraints In reality, securities are transacted in
multiples of a minimum transaction lots, or rounds (e.g. 100
or 500 shares). In order to model transaction round lots
explicitly in the optimization problem, portfolio weights can be
represented as
wi = zi · fi , i = 1, ..., N

where fi is a fraction of portfolio wealth and zi is an
integer number of round lots.
- In applying round lot constraints, the budget constraint
N
X

wi = 1

i=1

may not be exactly satisfied.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

- Round Lot Constraints To accommodate this situation, the
budget constraint is relaxed with undershoot and overshoot
variables, − ≥ 0 and + ≥ 0, so that
N
X

wi + − − + = 1

i=1

subject to
0

z Λl + 

−

+

0

−  = 1, l = [1, ..., 1]

where λ and γ are parameters chosen by the portfolio
manager.
- Normally, the inclusion of round lot constraints to the
mean-variance optimization problem only produces a small
increase in risk for a pre-specified expected return.
Furthermore, the portfolios obtained in this manner
cannot be obtained by simply rounding the portfolio
weights from a standard mean-variance optimization to
the nearest round lot.

Mean-Variance Analysis: Overview Classical Framework for Mean-Variance Optimization Mean-Variance Optimization with a Risk-F

Portfolio Optimization with Other Risk Measures
- Many risks and undesirable scenarios faced by a portfolio
manager cannot be captured solely by the variance of the
portfolio. Consequently, especially in cases of significant
non-normality, the classical mean-variance approach will not
be a satisfactory portfolio allocation model.
- Since about the mid-1990s, considerable thought and
innovation in the financial industry have been directed toward
creating a management of risk and its measurement, and
toward improving the management of risk in financial
portfolios.
- The race for inventing the best risk measure for a given
situation or portfolio is still ongoing, and the choice to some
extent remains an art. We distinguish between two types of
risk measures: (1) dispersion and (2) downside measures.
</reference>

<statements>
1. Risk is measured by variance of portfolio returns, given a vector of expected returns and a covariance matrix; optimization trades off expected return versus variance subject to constraints.
2. Variance (or standard deviation) of portfolio returns is the canonical risk measure; risk is determined by the covariance matrix \(\Sigma\) and weights \(w\) through \(w^\top \Sigma w\).
3. Markowitz’s framework is consistent with either (a) quadratic utility over mean and variance or (b) joint normality of returns; in practice, it assumes the mean vector and covariance matrix of future returns are known or well‑estimated, which is rarely true.
4. Optimization problem: Choose weights \(w\) to minimize variance for a target expected return, or maximize expected return for a given variance, or maximize mean-variance utility \(w^\top \mu - \gamma w^\top \Sigma w\), subject to budget and possibly other constraints.
5. Mean–Variance: Risk measure / handling: Variance via covariance matrix; can extend to downside/tail measures in variants.
6. Mean–Variance: Allocation behavior: Single‑period optimization; static weights chosen to trade off mean vs variance.
7. Mean–Variance: Main strengths: Simple, tractable; foundational; clear geometry of efficient frontier.
</statements>

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