MATLAB FINANCIAL DERIVATIVES TOOLBOX Uživatelský manuál stáhnout pdf (Strana 113)

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8.3 Portfolio Optimization

Many times, a researcher needs to create a portfolio of securities that for a

desire level of return/profit, it bears the least/minimum risk. The portfolio

optimization problem has been defined since Markowitz (1952) who assumed

that the risk associated with a portfolio can be measured with variance.

The researcher objective is do define via an optimization programming

methodology the weights that each of the alternative assets should have in

the portfolio. By construction, the portfolio return is linear w.r.t to the

weights of the assets and it is given from the following relation:

eW)r(E

where r

is the weighted average return of the portfolio and E(.) is the

expectation operation. The weight representation of each asset in the

portfolio is given by W and the expected return of each asset is given by e as

follows:













and













where n is the number of available assets. Contrary, the volatility of the

portfolio w.r.t the weights is a high nonlinear function given by the following

relation:

WOWs

where

s represents the weighted average volatility (standard deviation) of

the portfolio, and O is the assets’ variance-covariance matrix:

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MATLAB FINANCIAL DERIVATIVES TOOLBOX Uživatelský manuál Strana 113