Portfolio allocation based on expected profit and loss measures

We formulate the portfolio allocation problem from a trading point of view, allowing both long and short positions and taking trading and interest rate costs into account. Expressions are derived for the portfolio profit or loss (PL) that may result over a holding period. The expected profit (EP) and the expected loss (EL) are taken as measures of reward and risk. Optimal portfolios are considered to be allocations that maximize EP subject to EL being below a specified fraction of the EP. Simple expressions are shown for the reward and risk contributions of individual stocks to the portfolio expected PL. This optimal portfolio approach is referred to as the EP–EL method, and it is compared with a method based on maximal expected PL subject to controlled volatility measured by expected absolute PL deviation. The calculations required for these optimal portfolios are formulated as linear programming problems. Extensive results based on the market trading data of 12 stocks are provided to illustrate the properties of the EP–EL method. Among others, this leads to allocations that simultaneously maximize EP and EL while keeping the latter below an acceptable fraction of the former.