STOCHASTIC

the Kelly fraction (which maximizes the long run growth rate) actually makes
the growth rate zero plus the risk free rate; see Ziemba (2003) for more
discussion and the proof. Indeed, this is one explanation for part of the demise
of Long Term Capital Management in 1998 from over betting. Thorp and I are
proponents of the Kelly and fractional Kelly approach and observe that many of
the world's greatest investors like Warren Buffett, John Maynard Keynes, Bill
Benter (in horseracing in Hong Kong) and Thorp himself used such strategies. I
personally consulted for six such individuals who turned zero into hundreds of
millions or even billions (in the case of Jim Simons of Renaissance who made
US$1.4 billion just in 2005). Ziemba (2005) and MacLean and Ziemba (2006)
study many of these investors and Thorp (2006) discusses his use of the Kelly
approach and that of Buffett, who acts as if he was a full Kelly bettor.
Samuelson (see his article in Section 2 of Part V), however, is not a log
utility supporter. His objections are recorded in the conclusion to his article
and in Samuelson (1979). He argues that maximizing the geometric mean rather
than the arithmetic mean maximizes expected utility only for log utility. Indeed
it can be argued that log is the most risky utility function one should ever
consider in the short run, since growth decreases and risk increases for any
convex risk measure with higher than log utility wagers. Donald Hausch and
I did a simulation (see Ziemba and Hausch, 1986), which is reproduced in
MacLean and Ziemba (2006) to understand this better. We take an investor
who bets $1000 with log and half Kelly (-w 1 ) seven hundred times with five
possible wagers with probability of winning 0.19 to 0.57 corresponding to 1-1,
2-1, ..., 5-1 odds with a 14% expected value advantage. The bets are independent.
There are 1000 trials. In 166 of the 1000 trials, the final wealth for log is
greater than 100 times the initial $1000. With half Kelly it is only once this
large. But half Kelly provides higher probability of being ahead, etc. So there is
a growth-security tradeoff. However, it is possible to make 700 independent bets
all with a 14% advantage and still lose 98% of one's wealth. Half Kelly is not
much better. You can still lose 86% of your initial wealth. So the conclusion is
that log is short term risky and wins you the most money long term. The long
term can be very long, however; see Thorp (2006) for some discussion and
calculations. One compromise (see MacLean et al, 2004) is to choose at
discrete intervals the fractional Kelly that would keep you above a wealth path
with high probability. This I have just implemented in a London hedge fund but
with a convex penalty for falling below the path. Log is certainly the most
interesting and controversial utility function for investment. It is rarely taught,
however, in most university investment courses. Despite their original publish-
XXll PREFACE AND BRIEF NOTES TO THE 2006 EDITION