MATLAB FINANCIAL DERIVATIVES TOOLBOX Instrukcja Użytkownika Pobierz pdf (Strona 114)

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with )r,rcov(s

jiij

= . Additionally, for the portfolio optimization problem, it

is required that the sum of the assets weights should be added to one:

with

N to be a column vector with unity elements.

From the above it is obvious that the portfolio optimization problem can be

solved via the minimization of a quadratic volatility function subjected to

linear constrains, one related with the portfolio expected return and another

that assures that all available funds are invested. There are several

convenient mathematical methods for solving this problem. In here, in first

place, it is illustrated the one that is called the method of the Langrangean

multiplier that allows the solution of the general portfolio problem with short

sales with the requirement that the constraints must be in equality form.

When you take the PBA 521: Financial Theory course, you will have to solve

such kind of problems.

The example shown below is copied from the class notes of the above course

offered in 2001 by Professor Spiros Martzoukos. It is the case of three risky

assets without a risk-free rate (it is trivial to generalize it to n risky assets).

The inclusion of the risk free rate is easy; by treating the risk-free rate as a

“risky asset” with zero volatility and zero covariance with all the other assets.

The problem formulation that achieves an expected return equal to R

follows:

WOWmin

s.t.

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MATLAB FINANCIAL DERIVATIVES TOOLBOX Instrukcja Użytkownika Strona 114