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Deriving the Discrete Kelly Bet from Expected Log Wealth

Article Quant Q&A · Author: randomwalker
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Summary

The document compares a two-outcome Kelly formula, expressed using win and loss payoffs, with the approximation that relates the optimal fraction to expected return divided by return variance. It asks why the variance term does not appear to match the payoff product in the discrete formula. The response derives the discrete result by maximizing expected log wealth for a bet with an upside payoff and a downside payoff. Setting the derivative to zero yields the familiar edge-over-odds expression, with the payoff product in the denominator.

This derivation explains the exact log-utility optimization behind the discrete formula; it does not show that the variance-based expression is generally equivalent. That expression is typically an approximation under particular conditions, so a direct substitution of the two-outcome variance need not reproduce the exact discrete optimum. The source gives an algebraic argument rather than empirical evidence and does not discuss practical issues such as estimation error, repeated bets, or constraints on leverage.

Key ideas

  • The discrete Kelly fraction follows by maximizing expected logarithmic wealth across win and loss outcomes.
  • Setting the derivative of expected log wealth to zero produces the edge-over-odds formula.
  • The payoff product in the denominator arises from the algebra of the exact optimization.
  • The variance-based expression should not automatically be treated as an exact equivalent of the discrete formula.

Tags

Position sizingRisk managementStatistics
Full text
# Kelly criterion: reconciliate discrete and continuous case


# Kelly criterion: reconciliate discrete and continuous case












The Kelly criterion for the discrete case is: $$\tag{1}f=\frac{pW-(1-p)L}{LW}$$ where $f$ is the fraction of wealth to invest, $p$ the probability of winning, $(1-p)$ the probability of losing, $W$ the return if win, $-L$ the return if loss.

The continuous case is: $$\tag{2}f=\frac{E[r]}{V(r)}$$ where $E[r]$ is the expected return, and $V(r)$ is the variance of returns.

I am trying to derive the discrete case from the continuous case. So, the expected return $E[r]$ when the distribution of returns is discrete (it takes two values $W$ and $-L$) is actually equal to $pW-(1-p)L$, which is the numerator in the discrete case. So far so good. However, $V(r)$ should be equal to $pq(W-L)^2$, which is quite different from $LW$ in formula 1. Why is it so?

## Answer by demully (score 0)

https://quant.stackexchange.com/a/69977

So imagine you specified your discrete outcomes by virtue of a classic "stop-loss" and "target" price level. You buy/sell and hold, until either critical level is met, creating your discrete set of outcomes.

Assuming: p is the probability of hitting the upside barrier before the downside equivalent U is the upside payoff D is the downside payoff L is your leverage, ie your Kelly bet.

Your expected log wealth (ELW) becomes ELW = p * ln(1+LU) - (1-p) * ln(1+LD) I want to maximise DPW with respect to L.

The derivative of ELW wrt L = pU/(1+LU) - (1-p)D/(1+LD) Equals zero at a maximum, so the two parts above must equate

Do the algebra, and you will end up with: L = (p*U - (1-p)*D) / (U * D)

Which is "edge over odds" in the traditional formulation. Why U*D represents "odds" is maybe not intuitive. Just divide everything above by D so U is an x:1 odds bet. Which gives you the classic Kelly discrete formula.

hope this explains to your intuition and satisfaction, DEM

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.