We study the decision theory of a maximally risk-averse investor ---

one whose objective, in the face of stochastic uncertainties, is to

minimize the probability of ever going broke. With a view to

developing the mathematical basics of such a theory, we start with a

very simple model and obtain the following results: a characterization

of best play by investors; an explanation of why poor and rich players

may have different best strategies; an explanation of why

expectation-maximization is not necessarily the best strategy even for

rich players. For computation of optimal play, we show how to apply

the Value Iteration method, and prove a bound on its convergence

rate.