Kelly Allocation Across Strategies

The Kelly criterion is usually introduced with a single coin-flip bet: bet a fraction of your bankroll proportional to your edge, and you maximize the long-run compound growth rate of your money. A multi-strategy book faces the same underlying question but with several bets running at once — how much should each strategy get, given that each has its own edge and risk, and given that some of them move together?

Kelly allocation across strategies extends the single-bet formula to a portfolio, producing a capital split across sleeves that maximizes expected long-run growth rather than expected single-period return.

The single-strategy formula, then the extension

For a single bet with edge and odds known, the classic Kelly fraction is:

$$f^* = \frac{\mu}{\sigma^2}$$

In words: bet a fraction of capital equal to the strategy's expected return divided by its variance — a strategy with a bigger edge relative to its risk gets a bigger allocation, and a strategy that is all risk and no edge gets zero. Extended to multiple strategies with a covariance matrix capturing how they move together, the vector of optimal allocations becomes:

$$\mathbf{f}^* = \Sigma^{-1} \boldsymbol{\mu}$$

In words: invert the strategies' covariance matrix and multiply it by their expected returns — this automatically shrinks the allocation to any pair of strategies that are highly correlated with each other, because betting heavily on both of them isn't really two independent bets, it's closer to one bet twice as large.

Worked example

A firm has two uncorrelated sleeves. Sleeve A has an expected annual return of 8% and volatility of 16% (variance 0.0256); sleeve B has an expected return of 4% and volatility of 8% (variance 0.0064). Full Kelly sizing for each, treating them independently since they're uncorrelated, gives $f_A = 0.08/0.0256 \approx 3.1$ and $f_B = 0.04/0.0064 \approx 6.3$ — both above 1, meaning full Kelly would call for leverage on both sleeves. The firm instead runs at one-quarter Kelly, allocating $f_A/4 \approx 0.78$ and $f_B/4 \approx 1.6$ times each sleeve's baseline capital — well below the mathematically "optimal" growth-maximizing size, in exchange for materially smaller expected drawdowns.

What this means in practice

The covariance term in the multi-strategy formula is doing real work: two strategies that look attractive individually can still get a combined Kelly allocation smaller than the sum of their standalone allocations if they're correlated, because the formula recognizes that correlated bets don't diversify each other the way uncorrelated ones do. This is the mathematical version of the same lesson as strategy crowding — a portfolio of similar bets behaves like one bigger bet, not several independent ones.