Position Sizing¶

How portfolio weights are determined given a factor signal.

The Four Methods¶

Quant101 provides four sizing methods, all producing a weight DataFrame with columns (date, ticker, weight):

Equal Weight¶

\[w_i = \begin{cases} +1/N & \text{if stock } i \text{ is in the long leg} \\ -1/N & \text{if stock } i \text{ is in the short leg} \end{cases}\]

The baseline. No information beyond rank ordering is used.

Inverse Volatility¶

\[w_i \propto \frac{1}{\sigma_i}\]

Lower-volatility stocks get higher weight. Risk-parity inspired — equalizes risk contribution rather than capital contribution.

Volatility Target¶

\[w_i = \frac{\sigma_{\text{target}}}{\sigma_i}\]

Each position is sized so its dollar volatility matches a target. Useful when you want a specific portfolio volatility profile.

Signal-Weighted¶

\[w_i \propto \frac{\text{direction}_i \times |z_i|}{\sigma_i^2}\]

where:

\(z_i\) = cross-sectional z-score of the factor signal (conviction)
\(\sigma_i^2\) = rolling variance of historical returns (risk)
Normalized to \(\sum|w_i| = \text{max\_leverage}\) (default 1.0)

High-conviction, low-risk stocks get larger positions.

The Kelly Lesson¶

Critical Bug: Why 'Half-Kelly' Was Wrong

This method was originally called "Half-Kelly sizing." It took three successive bug fixes and an ablation study to understand why that name (and mental model) was fundamentally misleading.

What Happened¶

We implemented \(w_i \propto \mu_i / \sigma_i^2\) (classic Kelly) for cross-sectional portfolio allocation. Results:

Config	Sharpe
Equal-Weight + Daily	+0.814 (baseline)
Equal-Weight + Weekly	+0.417
Signal-Weighted + Daily	+0.137
Signal-Weighted + Weekly	−0.393

Signal-weighted sizing cost 0.68 Sharpe compared to equal-weight — the single largest source of degradation.

The Three Bugs¶

Normalization destroyed Kelly's answer. Normalizing \(\sum|w| = 1\) discards the leverage signal, which is the whole point of Kelly.
\(\mu_i\) from returns ignored the factor. Historical mean return over 60 days is pure noise (SNR ≈ 0.02). All configs produced identical Sharpe = 0.343 regardless of factor chosen.
Historical \(\mu\) is stealth momentum. A 60-day rolling mean injects momentum bias that conflicts with a mean-reversion signal like BBIBOLL.

Root Cause¶

Kelly criterion solves: "What fraction of my bankroll should I wager on a bet with known edge \(\mu\) and variance \(\sigma^2\)?"

\[f^* = \frac{\mu}{\sigma^2}\]

This is for a single repeated bet with independent outcomes. We misused it for cross-sectional allocation across correlated assets.

	Kelly's Problem	Our Misuse
Question	How much total capital to deploy?	How to split capital across N stocks?
Input	Edge and variance of one bet	Cross-sectional factor signal
Output	Portfolio leverage \(f^*\)	Relative stock weights
Regime	Independent repeated bets	Correlated concurrent positions

The correct multi-asset generalization is multivariate Kelly:

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

where \(\Sigma\) is the covariance matrix. Our per-stock \(\mu_i / \sigma_i^2\) ignores correlations entirely.

Why \(z / \sigma^2\) Hurts Mean-Reversion¶

For BBIBOLL, the strongest alpha comes from high-volatility stocks that have deviated far from their bands. But \(w_i \propto z_i / \sigma_i^2\) penalizes large \(\sigma_i\) quadratically. The stocks with the highest conviction (\(|z_i|\)) also have the highest \(\sigma_i\) — the penalty overwhelms the signal.

Resolution¶

Action	Status
Renamed `size_half_kelly` → `size_signal_weighted`	✅
Docstring states "Not true Kelly"	✅
Formula kept as conviction × inverse-variance	✅
True Kelly (portfolio leverage) deferred to future module	Planned

Takeaway¶

Interview One-Liner

"Kelly tells you how much to bet, not how to split the bet. Using scalar Kelly for cross-sectional allocation ignores correlations and concentrates into low-vol names — exactly the wrong thing for a volatility-driven alpha."

Full version: docs/latex/quant_lab.tex — Entry 4.