1. LOG OPTIMAL PORTFOLIO: KELLY CRITERION Xiaoying Liu Advisor: Prof. Philip H. Dybvig

2. Content  What is Kelly Criterion?  Good and bad properties of Kelly Criterion  Proof: Limitation of Kelly Criterion

3. A Gamble Problem

4. Kelly Criterion  Kelly (1956) defined “exponential rate of growth”, or “long run growth rate” as  The link between Kelly rule and log utility

5. Good Properties  Maximizing E(logW) asymptotically maximizes the rate of asset growth.  The absolute amount bet is monotone increasing in wealth.  Never risks ruin  Has an optimal myopic policy

6. Bad Properties  Kelly criterion can be very risky in the short term.  Kelly criterion is limited to use when risk aversion is equal to one  What if risk aversion is not equal to one

7. The Optimization Problem

8.  Value function for the optimal strategy V(t, w) where Conjecture that with h(T) = 1 We get Therefore,

9.  Value function for the optimal strategy V(t, w) where Conjecture that with and We get g(t) = g(T) = 1 and Therefore, The Optimization Problem -U(Wt) = log(Wt)

10. The Optimization Problem

14. Summary  Kelly criterion is linked to the logarithmic utility, and thus implicitly assume that the relative risk averse is always equal to one. Investors having different relative risk averse are not optimized by Kelly.