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?