# The Kelly Criterion — Where Did I Go Wrong?

Suppose the following simple/basic investment scenario:

• I have $100$USD in my bank account as a starting point (will increase/decrease as I invest).
• There are $1,000$ different investments that I'm planning to invest my money in.
• The investments are buy-sell fashioned. So I buy something low, and sell it higher.
• All my planned $1,000$ investments are executed in series (not in parallel). For example, I move to another investment only after I have sold the previous investment (hopefully sold with profit).
• All of the $1,000$ investments are independent (e.g. probability of winning/losing, or ratio of profit, are independent of how I perform with other investments).
• Probability that I will win in any of the $1,000$ investments is $0.54$ (i.e. I'm slightly more likely to win than to lose). I selected $0.54$ arbitrarily as an example, so we will stick to it for now.
• The profit rate for each of the $1,000$ is $0.3$. E.g. if I invest $5$ in any investment among the $1,000$, and I happen to win, then my revenue will be $5 + 5 \times 0.3 = 5 + 1.5 = 6.5$.

Then, the ultimate question is: how much of the total money in my bank account (at the time of investing) should I invest in an investment among the $1,000$ that I described above?

As per Wikipedia, the Kelly Criterion seems to suggest the following equation: $$f^* = \frac{bp-q}{b}$$ where:

• $f^*$ is the optimal ratio of my total money that I should invest in an investment,
• $b = 0.3$ is the profit rate if I win an investment,
• $p=0.54$ is the probability of me winning an investment,
• and $q=1-p=0.46$ is the probability of me losing an investment.

Now, if I understand the Kelly Criteron correctly, and I plug the numbers in, I get: $$f^* = \frac{0.3 \times 0.54 - 0.46}{0.3} = -0.99333$$

E.g. if I'm about to invest for the first investment (i.e. when my total money is $100$ USD), I should invest with $100 \times -0.99333 = -99.333$ USD! This makes no sense to me at all.

My attempt using a simulation code in Python, that simulates investment rates exhaustively from $0$ up to $1$ in increments of $0.05$, strongly disagrees with the $-0.99333$ above, by suggesting that $0.15$ is the optimal investment rate.

INPUT (assumptions):
* total money in bank: 100 USD.
* total number of investment projects: 1000.
* each investment has a probability 0.54 that it will be profitable.
* each investment has a profit rate of 0.3 (if successful).
* investments will be in series one by one (not parallel).

SIMULATION:
when investmenting 0% of total money, total money changed from 100  to 100.00 by the end of the journey.
when investmenting 5% of total money, total money changed from 100  to 534109.86 by the end of the journey.
when investmenting 10% of total money, total money changed from 100  to 107780173.52 by the end of the journey.
when investmenting 15% of total money, total money changed from 100  to 879409208.49 by the end of the journey.
when investmenting 20% of total money, total money changed from 100  to 291742331.78 by the end of the journey.
when investmenting 25% of total money, total money changed from 100  to 3717762.05 by the end of the journey.
when investmenting 30% of total money, total money changed from 100  to 1608.50 by the end of the journey.
when investmenting 35% of total money, total money changed from 100  to 0.02 by the end of the journey.
when investmenting 40% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 45% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 50% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 55% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 60% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 65% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 70% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 75% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 80% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 85% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 90% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 95% of total money, total money changed from 100  to 0.00 by the end of the journey.
when investmenting 100% of total money, total money changed from 100  to 0.00 by the end of the journey.

RESULT: best investment ratio is 0.15.


My question: Where did I go wrong?

If your net win is only $0.3$ times the bet, this is a terrible investment (even the expected value, let alone logarithmic utility that Kelly maximizes at each step is negative). This should be intuitive - you win only a bit over half of the times while the potential win is a small fraction of the potential loss. Or, the expected net win when betting $x$: $$p\,b\,x + q\,(-x) = -0.298\,x.$$
Looking at the Python code, it seems that you actually add $1.3$ times the bet to the bankroll when one wins (so if you invest $100$, you get the $100$ back and an additional $30$ rather than $100$ and additional $130$).
When entering $b=1.3$ into the Kelly formula quoted in the question, I get $0.19$ which is somewhat consistent with the simulation result.
(However, note that the Kelly formula is based on either assuming logarithmic subjective utility or asymptotic results, it is unclear (to me) how probable it should be is that the Kelly bet produces the best result in an experiment with $N=1000$ bets).
• Weird. Wiki says that $b$ USD is what I get as a profit, on top of my $1$ USD investment. But what you say essentially defines the profit as "everything I get back, including profit and investment". Does this mean that the wiki page is wrong? Or am I misunderstanding something again? – caveman Jun 29 '17 at 18:00