LOG OPTIMAL PORTFOLIO:
KELLY CRITERION
Xiaoying Liu
Advisor: Prof. Philip H. Dybvig
Content



What is Kelly Criterion?
Good and bad properties of Kelly Criterion
Proof: Limitation of Kelly Criterion
A Gamble Problem



Given an initial stake 𝑊0
At the beginning, we will choose the fraction bet f,
for each round afterwards
In order to get the maximal stake at the end of the
gamble, what f should we choose?
Kelly Criterion


Kelly (1956) defined “exponential rate of growth”,
or “long run growth rate” as
The link between Kelly rule and log utility
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
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
The Optimization Problem

Objective:
𝑊𝑡 1−𝑅
1−𝑅

U Wt =

U Wt = 𝑙𝑜𝑔 (𝑊𝑡)
The Optimization Problem
- U Wt =

𝑊𝑡 1−𝑅
1−𝑅
Value function for the optimal strategy V(t, w)
where
Conjecture that
with h(T) = 1
We get
Therefore,
The Optimization Problem
-U(Wt) = log(Wt)

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



Assume we invest a fraction f in the stock
𝑊𝑡 =
𝑘𝑓 𝑡 𝑓
𝑊0 𝑒 𝑠𝑡
, by Ito’s lemma, we get
1
𝑘𝑓 = 𝑟 1 − 𝑓 − 𝑓(𝑓 − 1)𝜎 2
2
The Optimization Problem


Given a random variable c, the certainty equivalent
of c is the constant amount CE(c) that yields the
same utility as c, i.e.,
u(CE(c)) = E[u(c)]
Here,
𝑓
𝑊𝑡1−𝑅
(𝑊0 𝑒 𝑘𝑓𝑡 𝑆𝑡 )1−𝑅
E u Wt = 𝐸
= 𝐸[
]
1−𝑅
1−𝑅
The Optimization Problem

Since Wt is
E u Wt
𝑊𝑡1−𝑅
log normal,
is also log normal, we get
1−𝑅
1
1
1−𝑅 𝑙𝑜𝑔𝑤0+𝑘𝑓 𝑡+𝑓 𝜇− 𝜎2 𝑡 −log 1−𝑅 + (1−𝑅)2 𝜎2 𝑡
=𝑒
2

Then we get

CE 𝑊𝑡 =
And we already know
1
1
𝑘𝑓 𝑡+𝑓 𝜇−2𝜎2 𝑡+2(1−𝑅)𝜎2 𝑡
𝑊0 𝑒
1
2
𝑘𝑓 = 𝑟 1 − 𝑓 − 𝑓(𝑓 − 1)𝜎 2 and f =

So CE 𝑊𝑡 = 𝑊0 𝑒
[𝑟+
2
𝜇−𝑟 2
]𝑡
2𝑅𝜎2
𝜇−𝑟
𝑅𝜎2
The Optimization Problem

E[u(Wt)] = E(logWt)



CE 𝑊𝑡 = 𝑊0 𝑒
CE 𝑊𝑡 = 𝑊0 𝑒
1
2
𝑘𝑓 𝑡+𝑓 𝜇− 𝜎2 𝑡
[𝑟+
𝜇−𝑟 2
]𝑡
2𝜎2
with f =
𝜇−𝑟
𝜎2
When risk aversion gets really high, to reach the
same utility, initial wealth would be a big number
R
= 3, by using log optimal strategy, you need to invest
$600 to get the same utility compared to $100 by
using the optimal strategy
 R = 10, $1200 V.S. $100
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.