Investigation of investment strategies of portfolio management

The paper presents mixed portfolio of n risk assets and one riskless asset. Management of theportfolio consists of two stages. The first stage is finding risk structure of portfolio. The secondstage is distribution capital between risk portfolio and riskless one. Paper deals with analysis ofseveral investment strategies. They have different risk structure of portfolio and different forms ofmanagement. Risk structures were found as solving next optimization task The objective functionto be maximized is1 ( )2 p p U x E V = ƒ − ,where ( ) U x - the utility of portfolio, p E - the expected return of the portfolio, p V - the varianceof the portfolio return, ¦ - the risk tolerance for the portfolio.The optimization task is maximizing ( ) U x provided that11niix== .If ¦ - n vector of the expected return of securities, C - matrix of covariance between the returns,then the solution is1 h D x Dh D hw wƒ ⎡ ⎤ ƒ= +ƒ ƒ− ⎢ ⎥⎣ ⎦,where 1 D C− = , [ ] 11 1 eƒ = … , w h Dh ƒ = . If ¦ =0, we have portfolio 1 x Dhw= , this portfolio isnamed minimum-variance portfolio. The objective function in this case is 12 p V .The risk structure, corresponding maximum the Sharpe ratio, is founded as solving next optimizationtaskmax max px x pE r x rx Cxƒƒ− ƒ −=ƒ.The second stage - distribution of capital is realized with using Pontriagin principle maximum.Two forms of management are investigated, programming management and feed backmanagement.

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