204 Optimizing Optimization
Section 9.3 will detail the optimization technique. Two aspects relevant for prac-
tical applications are the stochastic nature of the solutions obtained from TA,
and of course, some diagnostics for the algorithm. These issues are discussed in
Sections 9.4 and 9.5, respectively. Section 9.6 concludes.
9.2 Portfolio optimization problems
We assume that there are nA risky assets available, with current prices col-
lected in a vector p 0. We are endowed with an initial wealth v 0 , and intend
to select a portfolio x of the given assets. We can thus write down a budget
constraint:
vxp^00 (9.1)^
The vector x stores the number of shares or contracts, i.e., integer numbers.
If we need portfolio weights, we divide both sides of Equation (9.1) by v 0.
The chosen portfolio is held for one period, from now (time 0) to time T.
End-of-period wealth is given by:
vxpTT ’where the vector p T holds the asset prices at T. Since these prices are not
known at the time when the portfolio is formed, v T will be a random variable,
following some unknown distribution. It is often convenient to rescale v T to a
return r T , i.e.:
r
v
T v
T
019.2.1 Risk and reward
The first step in an optimization model is to characterize a given portfolio
in terms of its desirability, thus we want to map a given portfolio into a real
number. This mapping is called an objective function, or selection criterion.
The great insight of Markowitz was that such a function cannot consist of just
the expected profits. A common approach is hence to define two properties,
“ reward ” and “ risk, ” that describe the portfolio, and to trade them off against
each other. Markowitz identified these properties with the mean and variance
of returns, respectively.
With our optimization technique, we are not bound to this choice. In our expo-
sition here, our objective functions will be general ratios of risk and reward to
be minimized. (We could equivalently maximize reward – risk ratios.) Ratios have
the advantage of being easy to communicate and interpret ( Stoyanov, Rachev, &
Fabozzi, 2007 ). Even though, numerically, linear combinations are often more