# Kelly criterion for multiple gambles

Kelly says you should invest $x\%$ of your bankroll in a gamble:

$$x = \frac{pE-1}{p-1}$$

where $p$ is the probability of winning and $E$ is the expected payoff multiplier if you win (i.e. $E$ times how much you bet).

But lets say you have two options to invest in: One that has high risk and high payoff potential and another one that's absolutely safe, but only pays very little (e.g., FDIC insured bank account).

Is there a formula for this case? I was thinking to subtract the yield of the absolutely safe instrument from the expected payoff of the high risk venture and then just using the formula above.

EDIT: I found the formula: [pE-(1+r)]/[E-(1+r)] where r is the interest rate

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I think, the correct Formula is: $$x = \frac{pE-1}{E-1}$$ But what brings me to this post: Is E really an expected value? Or the exact value, when you win? If it is the expected value: Where can I find a derivation for this? – CCommander Nov 6 '13 at 12:30

You may be interested in looking into a portfolio approach. One classical extension of the Kelly Criterion is maximizing the geometric mean return of the portfolio (as opposed to maximizing the Sharpe ratio which is in vogue today).

Given two assets with mean return vector $\mathbf{\mu}$ (i.e., $\mu_i$ is the mean return of asset $i$) and covariance matrix $\Sigma$ we can formulate the problem of maximizing the geometric mean return of the portfolio $x$ by solving $$\max \left \{ \ln (1+x^T\mu) - \frac{x^T\Sigma x}{(1+x^T\mu)^2} \quad \right \}$$ subject to the constraints $\sum_{i} x_i = 1$ and $x_i \geq 0$ for all $i$. The vector $x$ denotes the fraction of your budget that you should invest in asset $i$.

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Mathematically, the Kelly criterion is equivalent to the maximization of the logarithmic utility. Thus you can expand log(terminalWealth) to the Taylor series and maximize its expectation.

Have a look at my Paper: Kelly criterion for multivariate portfolios: a model-free approach http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2259133

In my paper you will find both theory and numerical examples with source code in R and CUDA.

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Are you sure, that the only way to solve it is by using brute-force optimization? You've mentioned in that paper, that there is an analytic solution, if we can assume zero correlation between outcomes. – Adam Ryczkowski May 19 '14 at 17:07
@Adam Ryczkowski In my paper there are both analytical and numerical solutions. Moreover, I do not assume zero correlation between outcomes (what I assume is that the outcomes are serially uncorrelated: to put it simply, the outcomes yesterday tell you nothing about the outcomes today). My numerical method is definitely not the best one, however, it is very straightforward and it does work :) – user156718 Jun 12 '14 at 12:23

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