A gambler on an oil rig can listen to baseball games on the radio and place bets after knowing the outcome. When the radio breaks, he is only 60% sure of the outcome. The Kelly criterion provides that he should bet 20% of his bankroll in this situation to maximize expected gain per game. More generally, the Kelly fraction recommends betting a portion based on the probability of winning, probability of losing, and odds to maximize long-term expected returns when an edge exists between one's beliefs and the offered odds.

1. Kelly Criterion (intro)
Pf Quant
15 Jun 2018

2. A group of roughnecks on an oil rig love to bet on the outcome
of baseball games between two imaginary teams I just made
up:
Team 1) The Stars Team 2) The Stripes
Both teams are equally skilled, so
the rig’s bookie pays even-money,
or, equivalently, 2:1 odds, on either
team winning a game.
Baseball games are played during a 12-hour shift, so the games are
recorded. Everyone watches the recorded games together and places
their bets before the tape is started.

3. ...but a gambler on the rig has a
handheld transistor radio!
Gambler can pick up the results of the game as it happens, during the
shift, and then place a bet later when everyone gathers to watch the tape!
In theory, Gambler would hazard all of his
bankroll on every game since, with this ‘side
information’, the probability of winning is 100%.
𝑉𝑁 = 2 𝑁
𝑉0
Starting with bankroll V0, the gambler’s
expected bankroll V after N games is:
Because he’s certain to win,
Gambler simply bets his entire
bankroll on the outcome of each
game, every time.

4. ...but after a few games (and wins),
the radio is damaged.
Stars?
60%
Stripes?
40%
Because he now stands a non-zero
chance of losing everything if he bets his
bankroll and heard wrong, he now only
wants to bet a fraction of his bankroll.
Gambler is pretty sure he can
understand which team wins…
but not certain.
What fraction should he bet?

5. What fraction should he bet?
𝑉𝑁 = 1 + 𝑓 𝑊 1 − 𝑓 𝐿 𝑉0
Every time Gambler wins a bet, he multiplies his bankroll by (1 + f).
𝑉𝑁 = 1 + 𝑓 𝑉𝑁−1
𝑉𝑁 = 1 + 𝑓 1 + 𝑓 1 + 𝑓 . . . 𝑉𝑁−...
...which is to say:
...and so on...
Such that if he wins W times, his new bankroll is:
𝑉 𝑊 = 1 + 𝑓 𝑊
𝑉0
Therefore, after N games of W wins and L losses, our gambler’s
bankroll may be written:
Question: Does the order of wins and losses matter, or is the
equation above always right, regardless of order?
𝑉𝐿 = 1 + 𝑓 𝐿
𝑉0
(Same thing for losses.)

6. No, order does not matter to the end result,
although it does to the path.
1: Win; -1: Loss – 10 bets: 6 wins and 4 losses f: 0.5
Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5
Bet 1 1 1 1 1 -1
Bet 2 -1 1 1 1 -1
Bet 3 1 1 1 1 -1
Bet 4 -1 -1 1 -1 -1
Bet 5 -1 1 1 -1 1
Bet 6 1 -1 -1 -1 1
Bet 7 1 -1 -1 -1 1
Bet 8 1 1 -1 1 1
Bet 9 -1 1 1 1 1
Bet 10 1 -1 -1 1 1
starting cash: $100.00 $100.00 $100.00 $100.00 $100.00
Bet 1 $150.00 $150.00 $150.00 $150.00 $50.00
Bet 2 $75.00 $225.00 $225.00 $225.00 $25.00
Bet 3 $112.50 $337.50 $337.50 $337.50 $12.50
Bet 4 $56.25 $168.75 $506.25 $168.75 $6.25
Bet 5 $28.13 $253.13 $759.38 $84.38 $9.38
Bet 6 $42.19 $126.56 $379.69 $42.19 $14.06
Bet 7 $63.28 $63.28 $189.84 $21.09 $21.09
Bet 8 $94.92 $94.92 $94.92 $31.64 $31.64
Bet 9 $47.46 $142.38 $142.38 $47.46 $47.46
Bet 10 $71.19 $71.19 $71.19 $71.19 $71.19
if you don’t believe me, consider:
Randomly
typed in 6
wins and 4
losses for
these
columns. Try
your own!

7. 𝑉𝑁 ,
Now we have a formula for The Expected Value of Capital After N
Events, to which we’re going to give the symbol
We don’t really know which baseball game will be a win or a loss
for Gambler, although we can estimate the estimate the fraction of
Ws and Ls we expect to occur over N games.
→ For which, in this case, we have found
𝑉𝑁 = 1 + 𝑓 𝑊
1 − 𝑓 𝐿
𝑉0.
What Gambler is truly interested in is his expected return per
event; in other words, the quantity
𝐺 = 𝑙𝑖𝑚
𝑁→∞
1
𝑁
𝑙𝑜𝑔2
𝑉𝑁
𝑉0
Why log
base 2?

8. Base 2 log because:
𝑉𝑁 = 2 𝑁 𝑉0.
In the perfect case (complete certainty), the gambler has
𝐺 = 𝑙𝑖𝑚
𝑁→∞
1
𝑁
𝑙𝑜𝑔2
𝑉𝑁
𝑉0
= 1.
Therefore,
In other words, it will take you 1 event to double your wealth.
This doesn’t NEED to be base 2, but knowing how many events it
takes to double your wealth is very handy and easy to understand!!!

9. 𝐺 = 𝑙𝑖𝑚
𝑁→∞
𝑊
𝑁
𝑙𝑜𝑔2 1 + 𝑓 +
𝐿
𝑁
𝑙𝑜𝑔2 1 − 𝑓
Putting this definition of expected return per event…
...together with the expected value of capital after N events…
yields:
𝑉𝑁 = 1 + 𝑓 𝑊
1 − 𝑓 𝐿
𝑉0
𝐺 = 𝑙𝑖𝑚
𝑁→∞
1
𝑁
𝑙𝑜𝑔2
𝑉𝑁
𝑉0

10. So if we make a couple of simplifying definitions:
𝑙𝑖𝑚
𝑁→∞
𝑊
𝑁
= 𝑞
𝑙𝑖𝑚
𝑁→∞
𝐿
𝑁
= 𝑝
𝑙𝑖𝑚
𝑁→∞
𝑊
𝑁
= 𝑞
The long-term probability that gambler WINS
...and ditto for LOSSES...
𝐺 = 𝑞𝑙𝑜𝑔2 1 + 𝑓 + 𝑝𝑙𝑜𝑔2 1 − 𝑓
Then we can rewrite G in a much friendlier form:
Expected return per event

11. Pause for a second...
𝑝
𝑞
: LOSSES.
: WINS.
Keep that in mind.

12. 𝐺 = 𝑞𝑙𝑜𝑔2 1 + 𝑓 + 𝑝𝑙𝑜𝑔2 1 − 𝑓
Gambler simply wants to maximize his expected return per
baseball game (event).
In other words, he’s happy if he can find some fmax that
maximizes G in the equation from last slide:

13. 𝑙𝑖𝑚
𝑁→∞
𝑊
𝑁
= 𝑞
Lemma: The maximum value with respect to Yi of a quantity of
the form Z = ∑Xi logYi is obtained by using Yi s:
𝑌𝑖,𝑚𝑎𝑥 =
𝑌𝑖
𝑋𝑖
𝑋𝑖.
𝑌𝑖 = 1 + 𝑓 , 1 − 𝑓 ;
To use this handy fact, we simply vectorize G from its original form:
𝑋𝑖 = 𝑞, 𝑝
𝐺 = 𝑞𝑙𝑜𝑔2 1 + 𝑓 + 𝑝𝑙𝑜𝑔2 1 − 𝑓
… to a vector form
𝐺 = 1 + 𝑓 , 1 − 𝑓 ∗ 𝑞, 𝑝
… and make
substitutions
Lets find Gmax.

14. From the substitution definitions
𝑙𝑖𝑚
𝑁→∞
𝑊
𝑁
= 𝑞
𝑌𝑖,𝑚𝑎𝑥 =
𝑌𝑖
𝑋𝑖
𝑋𝑖 =
2
1
𝑞, 𝑝 .
Or, more simply:
𝑌𝑖 = 1 + 𝑓 , 1 − 𝑓 ; 𝑋𝑖 = 𝑞, 𝑝 ,
It’s clear that
𝑌𝑖,𝑚𝑎𝑥 = 2 𝑞, 𝑝

15. Therefore G is maximized by the fraction that makes the substitution
sentence true:
𝐺 𝑚𝑎𝑥 = 1 + 𝑝𝑙𝑜𝑔 𝑝 + 𝑞𝑙𝑜𝑔 𝑞
𝑓𝑚𝑎𝑥 = 2𝑞 − 1 = 1 − 2𝑝
...and betting with this fraction will maximize G at
(In units of bits/event)

16. Returning to the Gambler -
Stars?
60%
Stripes?
40%
Now that he’s only 60% sure of
winning a particular bet, his
Kelly fraction is 20%
And his expected gain / event is about
0.029 bits/event, which means it’ll
take him about 35 games to double
his wealth.
But that’s not QUITE the
whole story...

20. Generalized Kelly Fraction:
𝑓 𝑏𝑒𝑠𝑡
=
𝑏𝑝 − 𝑞
𝑏
=
𝑝 𝑏 + 1 − 1
𝑏
𝑓 ∗
=
𝑏𝑞 − 𝑝
𝑏
=
𝑞 𝑏 + 1
𝑏
Wherein:
 f*: how much you should wager
 b: odds ratio (e.g, “2” if 2:1)
 q: probability of winning
 p: probability of losing
 1 = p + q, always
NOTE: In the literature, not everyone chooses to follow Kelly’s
original convention of p: losing, q: winning, so always make
sure you know what the author means by ‘p’ or ‘q’.

21. edge ≡ 𝑏𝑝 − 𝑞
Point 1: The Edge
Let’s consider what this inequality is really saying.
The Odds Ratio b can be thought of as an engineering parameter that has units of [$
Returned if Successful] / [$ Wagered].
q can be thought of as an engineering parameter that has units of [# of total
wins] / [# of total replications] if you, in a large number of identical parallel
universes, were to observe the outcome of a large number of bets.
Ditto for p: a parameter that has units of [# of total losses] / [# of total
replications] for a sufficiently large sampling of events.
𝑏 > 𝑞 𝑝
The “edge” is the special name given to the numerator of the Kelly fraction.
It’s important to note that – unless you can take a short position – the Kelly fraction
tells us to place a bet ONLY when the edge is positive; equivalently, only when the
following inequality is true:

22. edge ≡ 𝑏𝑝 − 𝑞;
Point 1: The Edge
With the observations from last slide, let’s do some dimensional analysis:
𝑏 > 𝑞 𝑝
Bet when
𝑏 > 𝑞 𝑝
Has units of: $ Returned if Win
$ Hazarded
$ Returned if Win
$ Hazarded
>
# Wins
# Losses
The dollar payout ratio if
Event happens
-> given to you by BOOKIE
Your own private assessment of
the win-loss ratio
-> comes from YOU
Conclusion:
Kelly says to only take bets in which you believe the bookie has,
mistakenly, underestimated the odds.

23. 𝑓 𝑏𝑒𝑠𝑡
=
𝐸 − 𝑟
𝑠𝑖𝑔𝑚𝑎2
Point 2:
For a stock having expected return E, expected
volatility sigma^2, and risk free rate r:

24. ...but wait… that’s familiar…
𝑓 𝑏𝑒𝑠𝑡 =
𝐸 − 𝑟
𝑠𝑖𝑔𝑚𝑎2
𝑆 =
𝐸 − 𝑟
𝑠𝑖𝑔𝑚𝑎2
...and from Markowitz:
(From last page)
S: Sharpe Ratio, E: Expectecd Return, r: risk-free rate,
sigma^2: expected volatility
Slope of the security market line
for an optimized portfolio:

25. Not an accident.
Tune in next week :)