Chapter 7: Proportional Play and the Kelly Betting
System
Proportional Play and Kelly’s criterion: Investing in the stock market is, in effect,
making a series of bets. Contrary to bets in a casino though, one would generally believe
that the stock market is on average rising, so we are making a series of superfair bets.
In order to make things concrete we imagine that we invest in the stock market by
buying (idealized) option contracts. With such a contract you may pay a certain amount
b to have the possibility to sell a certain stock at a certain price at a given date. If the
market stock price is below your option price you will buy it on the market and sell it
immediately at a profit. if the stock price is above your option price then you do nothing
and you lose the amount you paid for your option. We will make it even simpler by
assuming your option cost $1 (this your bet), and you make a profit γ with probability p
and lose your bet with probability 1 − p.
P ( Win $γ) = p P ( Lose $1) = q = (1 − p)
If the game is superfair then we assume
Expected gain E[W ] = γp − q > 0 .
Informed with our experience with the gambler’s ruin we recall that is subfair game
the best strategy is bold play, that is invest everything at each step. But if the game is
superfair then our probability to win a certain amount gets larger if we make small bets.
But if we buy options that are sold on a fixed time window and making very small bets
won’t make you much money or only very slowly. To
Proportional Play = Invest a proportion 0 ≤ f ≤ 1 of your fortune at each
bet.
Let us know computed your fortune after n bets. If you start with a fortune X0 then
you bet f X0 . If you lose your fortune is X0 (1 − f ) and if you win your fortune will be
1
X0 (1 − f ) (what you did not bet) plus f X0 (your original bet) plus γf X0 (your gain). So
we have
X0 + γf X0 if you win X0 (1 + γf ) with probability p
X1 = =
X0 − f X0 if you lose X0 (1 − f ) with probability q = 1 − p
So at each step your fortune is multiplied by a random factor either 1 + γf or 1 − f , so
we define the random variable
Q = (1 + γf ) with probability p and Q = (1 − f ) with probability q.
To find our fortune at time N we take
Q1 , Q2 , · · · QN independent identically distributed random variables
all with the same distribution as Q. Then we have
Fortune after N bets is XN = QN QN −1 · · · Q2 Q1 X0
Since we make a series of super fair bets we expect that our fortune to fluctuate but
to increase exponentially
XN ∼ X0 exp αN .
and so α represents the rate at which our fortune increases. We can compute the long
run value of α by using the law of large numbers. Indeed we have
1 1
α ∼ log(XN /X0 ) = [log(Q1 ) + log(Q2 ) + · · · log(QN )]
N N
and so by Chebyshev
1
[log(Q1 ) + log(Q2 ) + · · · log(QN )] −→ E[log(Q)] ≡ α
N
and we have
var(log(Q))
P X0 eN (α−) ≤ XN ≤ X0 eN (α+) ≥ 1 −
N 2
To find the best asset allocation we seek to maximize α = E[log(Q)].
2
Optimal proportional play ⇐⇒ maximize E[log(Q)]
We have
E[log(Q)] = p log(1 + γf ) + q log(1 − f )
Differentiating with respect to f we find
d
0 = E[log(Q)]
df
d
= p log(1 + γf ) + q log(1 − f )
df
pγ q
= −
1 + γf 1−f
(1)
and so we have
pγ(1 − f ) = q(1 + γf )
pγ − q = f γ(p + q) = f γ
Finally we obtain
pγ − q Expected gain
Optimal f : f∗ = = Kelly0 s formula
γ gain
For example if your bet results in a payout of $10 with probability 1/4 you should bet
∗ 10 41 − 43 7
f = = = 0.175
10 40
of your fortune on each bet. Your fortune, on the long run, will grow at the rate of
1 7 3 7
α= log(1 + 10 ) + log(1 − ) = 0.047
4 40 4 40
Horse races: In the same spirit we can wonder what is the optimal strategy if we are
betting on a horse race. We suppose there are m horses, and the betting odds are such
that if we pay $1 on on horse i then we earn γi of horse i wins. How should we bet to
maximize our fortune in the long run? In order not to ever lose all our money it makes
sense to split your money and bet on every horse. That is
3
Proportional Play = Bet a proportion bi of your fortune on horse i.
In order to find the optimal proportion we assume that horse i has probability pi to
win. How do we find pi is the interesting part of the problem. One possible way to
estimate pi is to trust the wisdom of the crowd and to use as pi the proportion of players
which bets on horse i. This is the idea behind betting markets where one assume that if
many people bet real money on a certain outcome, then their collective knowledge will
produce the ”true” probability. This is a version of the efficient market hypothesis which
asserts that stock always trade at their fair values. Or in the context of sport betting you
may want to believe that a knowledgeable bettor will be able to find the right value of pi
by the virtue of his accumulated knowledge.
In any case, as in the Kelly betting system above we try to maximize the growth rate.
If we bet bi on horse i and wins we earn γi bi so if we start with a fortune X after betting
our fortune will be Qb X where Qb is a random variable with
P {Qb = bi γi } = pi
and we need to find
m
X
max
P E[ln(Qb )] = max
P pi ln(bi γi )
bi : bi =1 bi : bi =1
i=1
This can be done using Lagrange multiplier. We set
m
X m
X
F (b, λ) = pi ln(bi γi ) + λ bi .
i=1 i=1
and differentiating with respect to bi we find that the maximum should satisfy
1 pi
pi + λ = 0 or bi = −
bi λ
P
Using that i bi = 1 we find λ = −1 and thus
Optimal strategy is to bet a proportion bi = pi of your fortune.
Asset allocation in the stock market: A similar situation occurs in the stock market.
We suppose there are m stocks and the ith stock rate of return is described by the random
variable Zi
Zi = ratio of the value of stock i tomorrow and the value of stock i today
4
that is Zi = .98 it means the stock i lost two percent today.
PmTo invest our money we decide to invest a proportion bi of our fortune in stock i with
i=1 bi = 1. With this allocation our fortune after N days will be
XN = QN · · · Q1 X0
where Qi are IID copies of Q
m
X
Q= bi Zi
i=1
and X0 is the initial fortune.
In order to maximize the rate of growth of the fortune, we need to find
Xm
max
P E[log( bi Zi )]
bi ≥0 bi =1
i=1
