The risk of outliving your money (or shortfall) with low risk, low return investments is very often more serious than the risk of losing money on high risk investments, until quite late in life. A stochastic process model incorporating mortality tables for men and women of retirement age, random rates of return and fixed initial wealth and desired level of consumption provides the analytical tool. A simulation using Canadian mortality tables and rates of return shows that almost all retirees should invest some of their wealth in equity, and for many the optimal allocation is 70–100% equity. The risk of shortfall is surprisingly high for a reasonable range of values of the variables, especially for an allocation of 100% in treasury bills. Women face much greater risk of shortfall than men. The analytical model also permits calculation of the distribution of the bequest and hence allows an individual to trade off changes in shortfall risk against changes in the expected bequest to the heirs.