Capital Growth: Theory and Practice∗

In capital accumulation under uncertainty, a decision-maker must determine how much capital to invest in riskless and risky investment opportunities over time. The investment strategy yields a stream of capital, with investment decisions made so that the dynamic distribution of wealth has desirable properties. The distribution of accumulated capital to a fixed point in time and the distribution of the first passage time to a fixed level of accumulated capital are variables controlled by the investment decisions. An investment strategy which has many attractive and some not attractive properties is the growth optimal strategy, where the expected logarithm of wealth is maximized. This strategy is also referred to as the Kelly strategy It maximizes the rate of growth of accumulated capital.. With the Kelly strategy, the first passage time to arbitrary large wealth targets is minimized, and the probability of reaching those targets is maximized. However, the strategy is very aggressive since the Arrow-Pratt risk aversion index is essentially zero. Hence, the chances of losing a substantial portion of wealth are very high, particularly if the estimates of the returns distribution are in error. In the time domain, the chances are high ∗This research has been supported by the Natural Sciences and Engineering Research Council of Canada †School of Business Administration, Dalhousie University, Halifax, NS, Canada B3H 3J5 ‡Sauder School of Business, University of British Columbia,Vancouver, BC, Canada V6T 1Z2 and Visiting Professor of Finance, Sloan School of Management, MIT, Cambridge, MA 02138