Kelly Criterion

Kelly Criterion is used to guide an investor to take more risk when investments are winning and cut risk when investments returns is deteriorating.

Kelly Formula is used to calculate optimal capital allocation between different investments and the optimal leverage of a portfolio.

Kelly Criterion can be used in either discrete finance or continuous finance applications.

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Discrete Finance
The formula for Kelly Criterion is

Kelly Formula

Kelly Criterion Example

Assuming the following scenario: Winning trades = 30, Total trades = 60, Win $ = $272,000 , Loss $ = $148,000
W = (30 winning / 60 total traders) =0.50
R = 272,000 / 148,000 = 1.84
Kelly % = 22.79%

Continuous Finance

An investor has several investments with their own expected returns and standard deviations.He wants to allocate capital optimally among different investments and find his portfolio’s overall leverage (ratio of size of portfolio to his account equity).

The objective of the optimization problem is to maximize long-term wealth which is equivalent of maximizing the long-term compounded growth rate g of the portfolio. The objective means that ruin (equity going to zero) must be avoided.

It is assumed that all trading profits are reinvested, and therefore it is levered, compounded growth rate that is of ultimate importance.

The model’s assumption is that probability distribution of returns of each of investment i is a Normal Distribution (Gaussian) with a fixed mean mi and standard deviation si (returns are net financing costs, hence they are excess returns)

This is a rough approximation since big losses in financial markets occur with higher frequencies than Gaussian probability distributions predict.

Let optimal fractions of equity allocate of each of your n strategies by column vector F: F* = (f1*, f2*, …. fn*)T

Given optimization objecting and gaussian assumption the optimal allocation is given by

Kelly Formula Continuous 2

Returns are one-period, simple (uncompounded) unlevered returns. If strategy is long $1 of stock A and short $1 of stock B and made $0.1 profit, m is 0.05

If we assume strategies are statistically independent, the covariance matrix becomes diagonal matrix, with diagonal elements equal to variance of individual strategies.

Kelly Formula Continuous

This is Kelly formula and it gives optimal leverage one should employ for particular trading strategy.

Because of uncertainties in parameter estimations, and because return distributions are not Guassian, traders prefer to cut recommended leverage to half-kelly for safety.

The maximum compounded growth rate by adopting capita allocation and leverage is:

Kelly Formula Continuous 3

S = Sharpe ratio of portfolio

Kelly formula requires you to adjust your capital allocation as your equity changes so that it remains optimal. Assume a portfolio loses 10%. The portfolio drops and hence Kelly’s criterion dictates to reduce portfolio by optimal leverage amount.

The continuous updating of capital allocation should occur at least once at end of each day. You should also update F* by recalculating most recent trailing mean return and standard deviation.

The look back period depends on average holding period of strategy. For positions of 1-2 days use look back periods of 6 months.

A short look back period allows for reduction of exposure to strategies that have been losing their performance. The frequency of update must be done daily F*.