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How is the Kelly criteria formula modified when the unused capital accrues interest ?

Let us assume you have $\$1000$ and this generates interest compounded at $e^{(rt)}$

For example, you can bet \$300 and then put the \$700 in the bank to generate interest. The unused portion of the total bankroll gathers interest. This is applicable for long-term bets where interest becomes a bigger variable.

cannot find any info on google about how to modify the formula

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Let's say you have two assets which have returns given by random variables $R_1$ and $R_2$. Let $w_1$ and $w_2$ be portfolio weights on assets 1 and 2 respectively.

Your portfolio return $R_p$ is given by: $$ 1 + R_p = w_1 (1 + R_1) + w_2 (1 + R_2)$$

The Kelly criteria is simply the the growth optimal portfolio specialized to a setting with binary bets and 0 risk free rate. Maximizing the expected log return results in a growth optimal portfolio, that is, you solve:

\begin{equation} \begin{array}{*2{>{\displaystyle}r}} \mbox{maximize (over $w_1, w_2$)} & \operatorname{E}[\log (1 + R_p)] \\ \mbox{subject to} & w_1 + w_2 = 1 \\ & R_p = w_1 R_1 + w_2 R_2 \end{array} \end{equation}

Your problem (as I understand what you're trying to do) is just to specialize this further to the case where $R_1$ is a constant $r_f$ and $R_2$ is a binary random variable that takes the value $b$ with probability $p$ and $-1$ with probability $1-p$.

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If you're modeling the bank as being risk free, then you simply reduce the payoff by the interest. For instance, suppose you have a bet that either doubles what you put at risk, or loses all of it. You also have the option of 10% risk free interest.

Going by the terms used here: https://en.wikipedia.org/wiki/Kelly_criterion the unmodified $b$ in this example is 1: for every 100 dollars you put at risk, you win 100 additional dollars if the bet pays off. However, you are risking 100 present-dollars for 100 future-dollars. And 100 present-dollars are worth 110 future-dollars. Thus, you are risking 110 future-dollars for 100 future-dollars, which means that your modified $b$ is .909 (this is calculated by dividing 100 by 110).

So in summary, if you're using the formula given in the wikipedia article, and $b$ is the odds before taking interest into account, then you should use $b'= be^{-rt}$ for a risk-free bank account.

If the investment is not risk-free, then you need to model this as simultaneous Kelly bets. One question that deals with that is here: https://math.stackexchange.com/questions/2358037/kelly-criterion-for-simultaneous-independent-bets .

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