Kelly Portfolio Weights with Correlated Assets

Summary

The document asks how to extend the Kelly criterion, which seeks to maximize long-run growth, to a portfolio of correlated stocks. It points to a multivariate treatment in an investment science text and summarizes a constrained quadratic optimization model for a portfolio without leverage or short positions. The objective combines the risk-free return, each asset’s expected excess return weighted by its allocation, and a variance penalty built from the full covariance matrix.

The covariance terms are central: they account for how asset returns move together, so portfolio weights are not chosen independently from each other. The stated constraints keep each allocation between zero and one and limit total risky investment to the available capital. The document cites a derivation and mentions numerical optimization software as a way to solve the problem. It does not provide the derivation, data requirements, or evidence about realized performance, and its specific constraints exclude leveraged and short portfolios; the resulting allocation also depends on the expected returns and covariance estimates supplied to the model.

Key ideas

• The multivariate Kelly problem can account for correlated assets through their return covariance matrix.
• The described growth objective rewards expected excess returns and penalizes portfolio variance.
• The example imposes no leverage and no short selling through allocation bounds and a total investment constraint.
• Optimization software can solve the resulting constrained quadratic program.
• Kelly weights depend on estimated expected returns and covariances, and the stated setup does not cover leverage or short positions.

Tags

# Kelly Criterion in correlated stocks


# Kelly Criterion in correlated stocks












I would like to ask if there exist any mathematical proof or model which addresses how the Kelly criterion can be applied to find portfolio weights when the stocks are correlated.

## Answer by Enrico Schumann (score 2)

https://quant.stackexchange.com/a/54342

Luenberger's book has a discussion on growth-optimal (i.e. Kelly) portfolios, also for the multivariate case with correlated assets.

```
@BOOK{Luenberger1998,
  title        = {Investment Science},
  publisher    = {Oxford University Press},
  year         = 1998,
  author       = {David G. Luenberger}
}
```

## Answer by hubbs5 (score 1)

https://quant.stackexchange.com/a/70753

This post provides a model for the Kelly Criterion under no leverage and no short constraints, and yields the following quadratic program:

$$\max_f g = r + \sum_{i=1}^n f_i(\mu_i - r) - \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n f_i f_j \hat{\Sigma}_{ij}$$

$$\textrm{s.t.} \sum_{i=1}^n f_i \leq 1$$

$$f_i \in [0, 1]$$

They also have code examples in Python showing how to solve it with Pyomo and IPOpt.

They cite this paper, which looks like it has a derivation of this model.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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