Multi-Asset Kelly & Applications

Hard5–6 hrsMulti-Asset KellyCovariance MatrixMean-VariancePortfolio Optimization

Overview

Multi-asset Kelly reduces to Markowitz mean-variance optimization: the Kelly-optimal portfolio weights f* = Σ⁻¹μ maximize the expected log-wealth, which for small returns is approximately E[r] - (1/2)Var[r]. This is the same objective as mean-variance optimization with risk aversion = 1. The key practical challenge is estimating μ and Σ accurately — errors in μ are amplified by Σ⁻¹, which can create extreme positions. This is why practitioners use shrinkage estimators, factor models, and regularization.

How to Recognize

→Multiple simultaneous bets/positions with known correlations
→Portfolio construction with a signal on each asset
→Mean-variance optimization (Kelly is equivalent to Markowitz!)
→'Optimal portfolio weights given alpha and covariance'

Step-by-Step Approach

1.Multi-asset Kelly: maximize E[log(W)] over portfolio weights w
2.Continuous approximation: f* = Σ⁻¹μ where Σ = covariance matrix, μ = expected returns
3.This is EXACTLY mean-variance optimization — Kelly and Markowitz are the same
4.With return constraint: λΣ⁻¹μ for some risk aversion λ
5.Practical: use shrinkage on μ and Σ to avoid estimation error amplification

Key Formulas

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

— multi-asset Kelly weights (proportional to)

G(\mathbf{f}) = \boldsymbol{\mu}^T\mathbf{f} - \frac{1}{2}\mathbf{f}^T\Sigma\mathbf{f}

— quadratic approximation

\text{Sharpe}^2 = \boldsymbol{\mu}^T\Sigma^{-1}\boldsymbol{\mu}

— max achievable Sharpe squared

\text{Information Ratio} = \text{IC} \times \sqrt{\text{breadth}}

— Fundamental Law

Worked Examples

0Quick Example

Two assets: μ=(0.1%, 0.2%), Σ=diag(1%², 2%²), ρ=0. f*=Σ⁻¹μ=(0.1/0.0001, 0.2/0.0004)=(1000, 500) — bet more on asset 1 (better risk-adjusted return), despite lower absolute return.

Problem

You have two signals: μ₁ = 0.1%/day, μ₂ = 0.1%/day, σ₁ = σ₂ = 1%/day, correlation ρ = 0.8. How do the correlated Kelly weights compare to uncorrelated?

Solution

Uncorrelated case: f₁* = f₂* = μ/σ² = 0.001/0.0001 = 10. Each bet 10× independently.

Correlated case: Σ = [[0.0001, 0.00008],[0.00008, 0.0001]]. Σ⁻¹ = (1/det) × [[0.0001, -0.00008],[-0.00008, 0.0001]].

det = 0.0001² - 0.00008² = 10⁻⁸ - 6.4×10⁻⁹ = 3.6×10⁻⁹.

Σ⁻¹ = (1/3.6×10⁻⁹) × [[10⁻⁴, -8×10⁻⁵],[-8×10⁻⁵, 10⁻⁴]] ≈ [[27,778, -22,222],[-22,222, 27,778]].

f* = Σ⁻¹μ = [[27,778, -22,222],[-22,222, 27,778]] × [0.001, 0.001] = [5.556, 5.556].

With ρ=0.8, optimal leverage drops from 10× to 5.6× per asset! High correlation reduces diversification benefit and cuts Kelly leverage almost in half.

Answer

ρ=0: each bet 10×. ρ=0.8: each bet 5.6×. Correlation penalizes you — the same signal in two correlated assets provides less diversification and lower optimal leverage.

Common Mistakes

!
Adding independent Kelly bets: f_total = Σ f_i* only works for independent bets. For correlated bets, you must use the matrix formulation.
!
Inverting a poorly-conditioned covariance matrix. With many assets and short history, Σ⁻¹ is unstable. Always regularize or use factor models.
!
Confusing annualized vs daily parameters. μ/σ² changes by a factor of 252 between daily and annual. Always use consistent time units.

Practice Problems

Click "Show Answer" to reveal

The Fundamental Law of Active Management states IR = IC × √BR. If you have IC = 0.05 and make 500 independent bets per year, what is your expected annualized information ratio?

Hint: Just plug into the formula. IR = IC × √BR. Note this is the MAXIMUM achievable IR with perfect portfolio construction.

Back to Study Plan

Fractional Kelly & Risk of Ruin Stars & Bars / Integer Partitions