72 Frequently Asked Questions In Quantitative Finance
generally, if the investment has an expected return of
μand a standard deviationσμthen the expected
growth for an investment fraction offis
E[ln(1+fφ)]
which can be approximated by Taylor series
fφ−^12 f^2 φ^2 +···.
The Kelly fraction, which comes from maximizing this
expression, is therefore
f=
μ
σ^2
.
In practice, because the mean and standard deviation
are rarely known accurately, one would err on the side
of caution and bet a smaller fraction. A common choice
ishalf Kelly.
Other money management strategies are, of course,
possible, involving target wealth, probability of ruin, etc.
References and Further Reading
Kelly, JL 1956 A new interpretation of information rate.Bell
Systems Tech. J. 35 917–926
Poundstone, W 2005Fortune’s Formula. Hill & Wang
