Position Sizing & Leverage: Kelly Criterion Strategy › Risk Budgeting and Capital Allocation

Real portfolios contain multiple strategies, each with its own expected return, volatility, and correlations with other strategies. How should capital be allocated across strategies to maximize overall portfolio performance? This question extends Kelly's single-bet framework to the multi-dimensional setting that characterizes real trading operations.

Consider a portfolio of $n$ strategies with return vector $\mathbf{r} = (r_1, \ldots, r_n)^T$, expected return vector $\boldsymbol{\mu} = (\mu_1, \ldots, \mu_n)^T$, and covariance matrix $\boldsymbol{\Sigma}$. Let $\mathbf{w} = (w_1, \ldots, w_n)^T$ be the weight, or capital allocation, vector. The notation uses vectors and matrices because the interactions between strategies, captured by the covariance matrix, fundamentally affect optimal allocation.

The portfolio expected return is $\mu_p = \mathbf{w}^T \boldsymbol{\mu}$, which is simply the weighted average of individual strategy returns. The portfolio variance is $\sigma_p^2 = \mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}$, which accounts for individual strategy variances and all pairwise covariances. When strategies are negatively correlated, diversification reduces portfolio variance below the weighted average of individual variances.

As we discussed in Modern Portfolio Theory and Mean-Variance Optimization, the maximum Sharpe ratio portfolio solves:

$$\max_{\mathbf{w}} \frac{\mathbf{w}^T \boldsymbol{\mu}}{\sqrt{\mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}}}$$

where:

• $\mathbf{w}$: portfolio weight vector
• $\boldsymbol{\mu}$: expected return vector
• $\boldsymbol{\Sigma}$: covariance matrix of returns

This optimization seeks the portfolio with the best risk-adjusted return, balancing expected return in the numerator against risk in the denominator. For unconstrained optimization, which allows both leverage and short selling, the solution has a closed form:

$$\mathbf{w}^* \propto \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu}$$

where:

• $\mathbf{w}^*$: optimal weight vector
• $\boldsymbol{\Sigma}^{-1}$: inverse covariance matrix
• $\boldsymbol{\mu}$: expected return vector

This is the multi-asset generalization of the Kelly criterion. Each strategy receives weight proportional to its expected return, adjusted by the inverse covariance matrix, which accounts for both individual volatility and correlations. The inverse covariance matrix effectively adjusts weights downward for volatile strategies and for strategies that are highly correlated with others, because these provide less diversification benefit.

When strategies are independent, meaning they have zero correlation with each other, the covariance matrix is diagonal: $\boldsymbol{\Sigma} = \text{diag}(\sigma_1^2, \ldots, \sigma_n^2)$. The inverse of a diagonal matrix is simply the diagonal matrix of reciprocals. In this special case, the optimal weights simplify dramatically to:

$$w_i^* \propto \frac{\mu_i}{\sigma_i^2}$$

where:

• $w_i^*$: optimal weight for strategy $i$
• $\mu_i$: expected return of strategy $i$
• $\sigma_i^2$: variance of strategy $i$

This formula states that each strategy should be sized according to its individual Kelly criterion, independent of other strategies. This independence is powerful because it means you can optimize each strategy separately without worrying about interactions. The total portfolio leverage is then simply the sum of individual leverages.

For correlated strategies, the picture is more complex. Positive correlation between strategies reduces diversification benefits, and the optimal allocation accounts for this by reducing weights on highly correlated strategies. Intuitively, having two highly correlated strategies is almost like having twice the position in a single strategy, which may violate risk limits even when individual position sizes appear reasonable.

An alternative to return-based allocation is risk budgeting, where we allocate a "risk budget" to each strategy rather than capital directly. This approach is particularly useful when expected returns are uncertain but risk estimates are more reliable. In practice, volatilities and correlations tend to be more persistent and easier to estimate than expected returns, making risk budgeting less dependent on noisy expected-return estimates.

In a risk budgeting framework, we specify target risk contributions $b_i$ (summing to 1) and find weights such that:

$$\frac{\text{TRC}_i}{\sigma_p} = b_i$$

where:

• $\text{TRC}_i$: total risk contribution of strategy $i$
• $\sigma_p$: portfolio volatility
• $b_i$: target risk contribution proportion for strategy $i$

This constraint says that strategy $i$ should contribute fraction $b_i$ of total portfolio risk. Combining this with the definition of TRC leads to solving:

$$w_i (\boldsymbol{\Sigma} \mathbf{w})_i = b_i \cdot \sigma_p^2 \quad \text{for all } i$$

where:

• $w_i$: weight of strategy $i$
• $(\boldsymbol{\Sigma} \mathbf{w})_i$: $i$-th element of the vector $\boldsymbol{\Sigma} \mathbf{w}$ (marginal covariance)
• $b_i$: target risk budget
• $\sigma_p^2$: portfolio variance

This system of equations is nonlinear in the weights and typically requires numerical optimization to solve. The most common special case is equal risk contribution, where $b_i = 1/n$ for all strategies. This equal risk contribution constraint is the foundation of risk parity approaches, which have gained substantial popularity in institutional investing.