In this post we consider a simple way to derive/prove the “Fortune’s formula” (see [1]), also known as Kelly criterion (see [2]).
If p -denotes a probability to win a bet (or trade, etc.), O -denotes decimal odds on a winning bet, L -denotes a share of loss on a bet then the optimal portion of a bankroll/starting capital (let us denote it as OP) in percents for a bet/trade/investment, which maximize the bankroll’s rate of growth, is determined as
OP=100*(p/L-(1-p)/(O-1)).
If we denote ROI/100 or RPT/100 as R, where ROI is an average return on investments (in percents) and RPT is an average return per trade (in percents) then we can rewrite the formula above as
OP=100*(p/L-(1-p)/R).
A simple way to derive the “Fortune’s formula” is to follow these steps.
Step 1. Define a bankroll’s rate of growth -RG(x), where x is a proportion from the current bankroll to put at risk.
Step 2. Take a derivative from a logarithm of this rate: d/dx log(RG(x).
Step 3. Solve the equation d/dx log(RG(x)=0 relative to x. This gives us the optimal x, which maximize RG(x), because in all optimal x the derivative is equal to zero.
The proof.
Step 1. If we place a bet with a size equal to x*S and win then our bankroll S will increase on x*S*R and will become equal to S*(1+x*R). If we win k times then our bankroll will be S*(1+x*R)^k. If we lose than our bankroll will decrease on x*S*L and will become equal to S*(1-x*L). If we lose k times then our bankroll will be S*(1-x*L)^k. When the number of games increase, the proportion of wins will be close to p and the proportion of loses will be close to 1-p, therefore the bankroll’s rate of growth in the long run is: RG(x)=(1+R*x)^p *(1-x*L) ^(1-p).
Step 2. d/dx log(1+R*x)^p *(1-x*L) ^(1-p)=p*R/(1+R*x)-(1-p)*L/(1-x*L)
Step 3. To solve the equation d/dx log(RG(x)=0 we multiply the last expression (see Step 2) on (1-x*L)*(1+R*x), which gives us this equation: p*R*(1-x*L)-(1-p)*L*(1+R*x)=0.
To solve this equation relative to x, we move all products with x to the left side and all products without x to the right side: -R*L*x=-p*R+(1-p)*L. Now we divide both sides on -R*L and receive the final formula: x=p/L-(1-p)/R or if we want it in percents, we multiply it on 100.
If a user wants a convenient way to use this formula then she/he can use these online tools:
https://dynpass.online/tools/ff/ff_o.html -for betting/gambling,
https://dynpass.online/tools/ff/ff_r.html -for trading/investing.
Here are some examples.
Example 1.
Maria plays the “tossing coin game”. If a result will be “head” then Maria will win a bet, but if a result will be “tail” then Maria will lose the bet. A probability that a result will be “head” is 55%. What is the optimal portion of bankroll to have the fastest bankroll growth rate in the long run?
In this case O=2, ROI=L=100%, p=55%. If we put these values into the online calculator
we get OP=10%
Example 2.
Peter trades (makes short term investments in) highly volatile stocks, cryptos, futures and options. He places stop loss orders in such way that his loses not exceeds 50% of amount traded. An average ROI on his successful trades is 53%. What is the optimal OP for Peter?
We have L=50%, ROI=53%, p=53%
I we put these values into the calculator.
We get OP=17.32%
Example 3.
Sue invests in short term treasure and pearls hunting projects. If a project is successful she gets ROI=300%, if not she loses 100% of her investment. On average 40% of her projects are successful. What is the optimal OP for Sue?
We have p=40%, ROI=300%, L=100%.
I we put these values into the calculator.
We get OP=20%.
P.S. OP values, given by the Fortune’s formula (Kelly criterion) work for large numbers of bets/trades (according to the “Central Limit Theorem”). On the initial stage (when numbers of trades/bets is not large enough), volatility may be very high. To reduce high volatility, experts recommend to use values less than those given by the Fortune’s formula and gradually increase the values to the OP values given by the formula, as the number of bets/trades increase (see [3]).
References:
1. Fortune's formula : the untold story of the scientific betting system that beat the casinos and Wall Street, William Poundstone, 2005, 386 pages
2. How to Bet Using the Kelly Criterion
https://insights.matchbook.com/betting-strategy/the-kelly-criterion/
3. Revisiting the Kelly Criterion: Fractional Kelly
