Kelly Criterion and Execution Cost Models

Worked Solution

How to Think About It: This question is really about the gap between theory and practice in portfolio management. Kelly tells you the theoretically optimal bet size to maximize long-run wealth growth -- but it assumes you can execute at the prices you want, with no friction. In reality, execution costs eat into your edge, and the bigger you size, the more they eat. A senior quant thinks about Kelly and execution costs as two sides of the same coin: your optimal size depends on your edge, your uncertainty about that edge, AND your cost of expressing it.

Key Insight: Kelly sizing and execution cost modeling are not separate topics -- they are coupled. Your true optimal position size is the Kelly fraction adjusted downward for the impact costs of getting into and out of the position.

The Method:

Part 1: Kelly Criterion

For a single bet with win probability $p$ and payoff odds $b:1$, the Kelly fraction is:

$f^{*} = \frac{bp - (1 - p)}{b} = \frac{p(b + 1) - 1}{b}$

This maximizes $E[\log W]$, the expected log-wealth, which is equivalent to maximizing the long-run geometric growth rate.

For a single asset with normally distributed returns (mean $\mu$, variance $\sigma^2$, risk-free rate $r_f$):

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

For a portfolio of $n$ assets with expected excess return vector $\boldsymbol{\mu} - r_f \mathbf{1}$ and covariance matrix $\Sigma$:

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

Notice this is exactly the mean-variance tangency portfolio, just with a different normalization. Kelly and Markowitz are the same math -- Kelly just tells you how much leverage to use.

Part 2: Why Fractional Kelly?

Full Kelly maximizes long-run growth but with enormous volatility along the way. The drawdowns are brutal -- a full Kelly bettor has a 50% chance of a 50% drawdown at some point. Practically:

• Your estimates of $\mu$ and $\Sigma$ are noisy. Kelly is highly sensitive to estimation error -- if you overestimate your edge by 2x, you overbet by 2x.
• Half-Kelly ($f^{*}/2$) gives you 75% of the growth rate with substantially less variance and drawdown risk.
• Most funds use something in the range of 0.2 to 0.5 times Kelly, depending on confidence in their edge estimates.

Part 3: Slippage and Market Impact Models

*Linear Impact Model:*

$\Delta P = \lambda \cdot Q$

where $Q$ is order size and $\lambda$ is the Kyle lambda (price impact per unit traded). Simple, tractable, good for small orders. But it overestimates impact for very small orders and underestimates it for large ones.

*Square-Root Model:*

$\text{Impact} \approx \sigma \cdot \sqrt{\frac{Q}{V}}$

where $\sigma$ is daily volatility and $V$ is average daily volume. This is the empirical workhorse -- decades of data across markets confirm the square-root scaling. It captures the concavity of impact: doubling your order size less than doubles your impact.

*Almgren-Chriss Framework:*

This adds a time dimension. You are not just asking "how much does my trade move the price?" but "how should I split my order across time to minimize total cost?" The total cost has two components:

• Temporary impact -- price displacement that decays after you stop trading
• Permanent impact -- information content of your trade that shifts the equilibrium price

The optimal execution schedule balances impact cost (trade slowly) against timing risk (trade quickly before the price moves against you). The solution is a deterministic trading trajectory that minimizes $E[\text{cost}] + \lambda \cdot \text{Var}[\text{cost}]$.

When to use which: - Linear: quick back-of-envelope calculations, small orders, academic models - Square-root: realistic cost estimation for institutional orders, pre-trade analytics - Almgren-Chriss: optimal execution scheduling, VWAP/TWAP algorithm design, balancing urgency vs. cost

Part 4: Interaction Between Kelly and Execution Costs

Execution costs reduce your effective edge. If your expected return is $\mu$ but executing costs you $c(f)$ as a function of position size, your net edge is $\mu - c(f)$. With square-root impact, costs scale as $\sqrt{f}$, so the adjusted Kelly fraction solves:

$f^{*}_{\text{net}} = \frac{\mu - c(f^{*}_{\text{net}}) - r_f}{\sigma^2}$

This is smaller than the frictionless Kelly. The practical upshot: capacity constraints are real. Even if your signal is strong, there is a maximum size beyond which execution costs eat your entire edge.

Practical Considerations:

• Estimation error in $\mu$ matters far more than in $\sigma$ for Kelly sizing -- a small bias in expected returns leads to large overbetting
• Impact models need calibration to your specific market and order type -- equity impact is very different from futures or FX
• Temporary vs. permanent impact matters for how you think about round-trip costs: if most impact is temporary, you can recover some cost by being patient on the exit
• In practice, most systematic funds compute Kelly as a theoretical upper bound and then apply a discount factor (typically 0.25-0.5x) that implicitly accounts for estimation error, execution costs, and model risk

Answer: The Kelly criterion gives the growth-optimal bet size: $f^{*} = (\mu - r_f)/\sigma^2$ for a single asset, $\mathbf{f}^{*} = \Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ for a portfolio. In practice, fractional Kelly (0.25-0.5x) is used because of parameter uncertainty and drawdown risk. Execution costs are modeled via linear impact (simple, small orders), square-root impact (empirically validated for institutional sizes), or Almgren-Chriss (optimal scheduling over time). These costs reduce effective edge and lower the optimal position size, creating a natural capacity constraint on any strategy.