The paper analyzes the allocation of a given initial capital between a risk-free and a risky alternative. Typically, the risky alternative is the investment into the stock market or into a stock market index. Under expected utility the optimal fraction a(*) to invest into the stock market depends on the initial capital, on the distribution of stock returns, on the planning horizon, and of course on the von Neumann/Morgenstern utility function. Moreover, the optimal a(*) can only be evaluated by numerical integration. In order to get explicit formulas and to avoid the problematic assessment of the utility function NEU (non expected utility) approaches axe discussed. The maxmin and the minmax regret criterion select only corner solutions (i.e. a(*) = 0 or a(*) = 1). The following Kataoka variant of these criteria is considered: Fix a (small) probability alpha and discard all the extremal events (which have althogether the probability alpha) from the planning procedure; i.e. define the worst case by exclusion of these extremal events. Obviously, this idea is also the basis of the well-known value-at-risk approach. The optimal fraction a(*) is no longer a corner solution. Moreover, it allows explicit formulas. These are studied in the Black/Scholes world (i.e. normally distributed log returns). Under realistic parameter values a(*) increases with the length of the planning horizon.
