On the optimization of a CAPM portfolio using lower partial moments as measure of risk and using the possibility of safeguarding its loss

Using a portfolio built from bonds (investment without volatility) and shares (investment with volatility) corresponding to the CAPM we calculate the possible loss of this portfolio. The loss is measured by a so-called lower partial moment of the rate of return of the portfolio. Using this loss, we optimize the composition of the portfolio with respect to this loss. Also we investigate the optimization of the portfolio when the loss can be underwritten by an insurance. Concerning the premium of this insurance contract, we show that when the premium is defined inadequate, e.g. proportional to the investment or proportional to the amount of investment in shares, the optimal portfolio consists only of investment in shares. When the premium is defined more suitable, e.g. proportional to the loss, the optimal portfolio is built by an investment in bonds and shares.