Optimization methods in finance

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1.3. FINANCIAL MATHEMATICS 17
Markowitz’ portfolio optimization problem, also called the mean-variance
optimization (MVO) problem, can be formulated in three different but equivalent ways. One formulation results in the problem of finding a minimum
variance portfolio of the securities 1 to n that yields at least a target value
R of expected return. Mathematically, this formulation produces a convex
quadratic programming problem:
minx x
T x = 1
T x ≥ R
x ≥ 0,
(1.15)
where e is an n-dimensional vector all of which components are equal to
1. The first constraint indicates that the proportions xi should sum to 1.
The second constraint indicates that the expected return is no less than the
target value and, as we discussed above, the objective function corresponds
to the total variance of the portfolio. Nonnegativity constraints on xi are
introduced to rule out short sales (selling a security that you do not have).
Note that the matrix Q is positive semidefinite since x
TQx, the variance of
the portfolio, must be nonnegative for every portfolio (feasible or not) x.
As an alternative to problem (1.15), we may choose to maximize the
expected return of a portfolio while limiting the variance of its return. Or,
we can maximize a risk-adjusted expected return which is defined as the
expected return minus a multiple of the variance. These two formulations
are essentially equivalent to (1.15) as we will see in Chapter 8.
The model (1.15) is rather versatile. For example, if short sales are permitted on some or all of the securities, then this can be incorporated into
the model simply by removing the nonnegativity constraint on the corresponding variables. If regulations or investor preferences limit the amount
of investment in a subset of the securities, the model can be augmented with
a linear constraint to reflect such a limit. In principle, any linear constraint
can be added to the model without making it significantly harder to solve.
Asset allocation problems have the same mathematical structure as portfolio selection problems. In these problems the objective is not to choose
a portfolio of stocks (or other securities) but to determine the optimal investment among a set of asset classes. Examples of asset classes are large
capitalization stocks, small capitalization stocks, foreign stocks, government
bonds, corporate bonds, etc. There are many mutual funds focusing on
specific asset classes and one can therefore conveniently invest in these asset classes by purchasing the relevant mutual funds. After estimating the
expected returns, variances, and covariances for different asset classes, one
can formulate a QP identical to (1.15) and obtain efficient portfolios of these
asset classes.
A different strategy for portfolio selection is to try to mirror the movements of a broad market population using a significantly smaller number of
securities. Such a portfolio is called an index fund. No effort is made to
identify mispriced securities. The assumption is that the market is efficient
and therefore no superior risk-adjusted returns can be achieved by stock

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