Copyright © 2011-2014 *airxc.com*, *airxcell.com*

**Table of Contents**

This chapter presents a quick overview of the underlying theoretical math concepts used by the
portfolio optimization procedures implemented in the
Portfolio (15) dynamic
form. Both methodologies on which are built the optimization algorithms are presented, namely
the *Markowitz Portfolio Theory* and the *Mean-CVaR Portfolio Theory*.

Most of the concepts presented here are compiled from several reference books on this specific field such as [10] and [11]. Moreover, the following documents available online provide interesting additional insights: [7], [10] and [3].

In financial engineering, we often work with *asset returns* instead of
*asset prices*. Given a financial instrument's price
*S _{t}* where

The global return of a portfolio of assets *R _{t}*, is defined as
the sum of the weighted return of each asset, expressed as follows:

with *w _{ti}* being the weight of asset

The concepts presented in this section are largely inspired from [6], [7] and [10].

In this section we define the original mean-variance portfolio optimization
problem and several related problems. The problem of minimizing the covariance
risk for a given target return with optional box and group constraints
is a quadratic programming problem with linear constraints. We call this problem
*minimum risk mean-variance portfolio*.

The opposite case, fixing the risk and
maximizing the return, has a linear objective function with quadratic constraints.
We call this programming problem the *maximum return mean-
variance* portfolio problem. This problem is much more complex than
the previous one.

If we have even more complex constraints, i.e. nonlinear constraints,
we need a new class of solvers. This is called *non-linear constrained*
portfolio problem. This allows us to handle the case of linear and quadratic
objective functions with non-linear constraints. In all three cases we speak of
Markowitz' portfolio optimization problem, although they require different
classes of solvers with increasing complexity.

Following Markowitz we define the problem of portfolio selection as follows:

The formula expresses that we minimize the variance - covariance risk
σ̄
, where the matrix
Σ̂
is an estimate of the covariance of the assets. The
vector
ω
denotes the individual investments subject to the condition
ω^{T} 1 = 1
that the available capital is fully invested. The expected or target return
r̄
is expressed by the condition
ω^{T} μ̂ = r̄
, where the p-dimensional vector
μ̂
estimates the expected mean of the assets.

Markowitz' portfolio modell has a unique solution:

where

with

The corresponding standard deviation
σ̄
for the optimal portfolio with
weights
ω^{*}
is

The locus of this set in the {
σ̄
,
r̄
}-space are hyperbolas. The set inside the
hyperbola is the *feasible set* of mean/standard deviation portfolios, and the
borders are the *efficient frontier* (upper border), and the minimum variance
locus (lower border). Here, *r*^{*} is the return of the minimum variance
portfolio.

The point with the smallest risk on the efficient frontier is called the global minimum variance portfolio, MVP. The MVP represents just the minimum risk point on the efficient frontier.The set of weight is:

In contrast to minimum risk portfolios, where we minimize the risk for a given target return, maximum return portfolios work the opposite way: Maximize the return for a given target risk.

Note that now we are concerned with a linear programming problem and quadratic constraints. This can be solved in the Portfolio optimization (15) dynamic form using the available linear solver (15.4.5).

The concepts presented in this section are largely inspired from [2], [3] and [10].

In this chapter we formulate and solve the mean-CVaR portfolio model, where covariance risk is now replaced by the conditional Value at Risk as the risk measure. In contrast to the mean-variance portfolio optimization problem, we no longer assume the restriction consisting in the set of assets to have a multivariate elliptically contoured distribution.

We consider a portfolio of assets with random returns. We denote the portfolio
vector of weights with
ω
and the random events by the vector *r*.
Let *f* (
ω
, *r*)
denote the loss function when we choose the portfolio W from a set X of
feasible portfolios and let *r* be the realization of the random events.We assume
that the random vector *r* has a probability density function denoted by
*p*(*r*).
For a fixed decision vector
ω
, we compute the cumulative distribution function
of the loss associated with that vector
ω.

Then, for a given confidence level α, the VaR_{α} associated with portfolio W is
given as

Similarly, we define the CVaR_{α} associated with portfolio W

We then define the problem of mean-CVaR portfolio selection as follows:

In general, minimizing
CVaR_{α} and VaR_{α} are not equivalent. Since the definition
of CVaR_{α} involves the VaR_{α} function explicitly, it is difficult
to optimize or work with this function. Instead, we consider the following simpler
auxiliary function:

Alternatively, we can write *F*_{α}(ω, γ) as follows:

where z^{+} = max(z, 0). This final function of γ has the following important
properties that make it useful for the computation of CVaR_{α} and VaR_{α}:

- F
_{α}(ω, γ) is a convex function of γ, - VaR
_{α}(ω) is a minimizer of F_{α}(ω, γ), - the minimum value of the function F
_{α}(ω, γ)) is CVaR_{α}(ω).

As a consequence, we deduce that CVaR_{α} can be optimized via optimization
of the function F_{α}(ω, γ) with respect to the weights w and VaR g. If the loss
function *f (ω, r)* is a convex function of the portfolio variables w,
then F_{α}(ω, γ)
is also a convex function of ω. In this case, provided the feasible portfolio set
ω is also convex, the optimization problems are smooth convex optimization
problems that can be solved using well-known optimization techniques for
such problems.

Figure Figure 17.1, “Efficient mean-variance frontier [8]” shows the *Efficient mean-variance frontier*
with the *global minimum variance* portfolio, the global minimum Value at Risk (5%) portfolio and the
global minimum Conditional Value at Risk (5%) portfolio. The efficient frontiers under the various measures, are the
subset of boundaries above the corresponding minimum global risk portfolios. We see that under 5% VaR and 5%
CVaR the set of efficient portfolios is reduced with respect to the variance.

Often it is not possible or desirable to compute/determine the joint density
function *p(r)* of the random events in our formulation. Instead, we may
have a number of scenarios, say r_{s} for s = 1, ... , *S*, which may represent
some historical values of the returns. In this case, we obtain the following approximation to the function
F_{α}(ω, γ) by using the empirical distribution of
the random returns based on the available scenarios:

Now, the problem min_{ω∈W} CVaR_{α}(ω)
can be approximated by

To solve this optimization problem, we introduce artificial variables *z _{s}* to
replace

Note that the constraints
*z _{s}* ≥

Figure Figure 17.2, “Mean-VaR(5%)-boundary [8]” shows the *Mean-VaR(5%)-boundary* with the
*global minimum variance*
portfolio. Portfolios on the mean-VaR(5%)-boundary between the global minimum VaR(
5%) portfolio and the global minimum variance portfolio, are mean-variance
efficient. The VaR constraint (vertical line) could force mean-variance
investors with high variance to reduce the variance, and mean-variance investors
with low variance to increase the variance, in order to be on the left
side of the VaR constraint.

In the case that
*f(ω, r _{s}) - γ* is linear in
ω
, all the expressions