Act as if you have log utility and with probability 1 your wealth will converge to infinity.

Sergiu Hart presented this paper at Northwestern last week.  Suppose you are going to be presented an infinite sequence of gambles.  Each has positive expected return but also a positive probability of a loss.  You have to decide which gambles to accept and which gambles to reject. You can also invest purchase fractions of gambles: exposing yourself to some share \alpha of its returns. Your wealth accumulates (or depreciates) along the way as you accept gambles and absorb their realized returns.

Here is a simple investment strategy that guarantees infinite wealth.  First, for every gamble g that appears you calculate the wealth level such that an investor with that as his current wealth and who has logarithmic utility for final wealth would be just indifferent between accepting and rejecting the gamble.  Let’s call that critical wealth level R(g).  In particular, such an investor strictly prefers to accept g if his wealth is higher than R(g) and strictly prefers to reject it if his wealth is below that level.

Next, when your wealth level is actually W and you are presented gamble g, you find the maximum share of the gamble that an investor with logarithmic utility would be willing to take.  In particular, you determine the share of g such that the critical wealth level R(\alpha g) of the resulting gamble \alpha g is exactly W. Now the sure-thing strategy for your hedge fund is the following:  purchase the share \alpha of the gamble g, realize its returns, wait for next gamble, repeat.

If you follow this rule then no matter what sequence of gambles appears you will never go bankrupt and your wealth will converge to infinity. What’s more, this is in some sense the most aggressive investment strategy you can take without running the risk of going bankrupt.  Foster and Hart show that any investor that is willing to accept some gambles g at wealth levels W below the critical wealth level R(g) there is a sequence of gambles that will drive that investor to bankruptcy.  (This last result assumes that the investor is using a “scale free” investment strategy, one whose acceptance decisions scale proportionally with wealth.  That’s an unappealing assumption but there is a convincing version of the result without this assumption.)

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