Fractional Kelly Sizing Under Parameter Uncertainty & Out-of-Sample Brier Decomposition

Q: Why does full Kelly sizing underperform when probability estimates contain noise?

The logarithmic bankroll growth curve g(f) is strictly concave around the true optimal fraction f*. When an agent substitutes a noisy probability estimate p_hat = p + epsilon into the Kelly formula, estimation errors create symmetric upward and downward bet-sizing variance, but asymmetric concave penalties: overbetting beyond 2f* drives expected geometric growth negative even on positive-EV propositions.

Q: How much drawdown risk does quarter-Kelly (c = 0.25) eliminate compared to full Kelly?

Under continuous-time Kelly diffusion approximations, scaling the Kelly stake by fraction c in (0, 1) reduces portfolio return variance to c^2 of full-Kelly variance and bounds the probability of bankroll falling to fraction x of peak to x^(2/c - 1). Moving from full Kelly (c = 1.0) to quarter-Kelly (c = 0.25) compresses variance by 93.75% and reduces the analytical probability of a 50% drawdown from 50.0% to 0.78%.

Q: What do Reliability, Resolution, and Uncertainty measure in the Murphy Brier decomposition?

Murphy (1973) decomposes Brier Score into BS = REL - RES + UNC. Reliability (REL, lower is better) measures calibration error—how closely binned forecast probabilities match realized outcome frequencies. Resolution (RES, higher is better) measures discrimination—how strongly conditional outcome frequencies separate from the overall base rate. Uncertainty (UNC) captures intrinsic base-rate entropy independent of the model.

SECTION 01

The Asymmetry of Estimation Error: Why Full Kelly Overbets

For a binary event contract paying net fractional profit $b = O - 1 > 0$ per unit staked on success (probability $p$) and losing unit stake on failure ($q = 1 - p$), expected logarithmic capital growth per trial as a function of bankroll fraction $f \in [0, 1)$ is:

$$g(f) = p \ln(1 + b f) + q \ln(1 - f)$$

Setting $g'(f) = 0$ yields the classic Kelly (1956) fraction $f^* = \frac{b p - q}{b} = p - \frac{q}{b}$. Textbook treatments implicitly assume $p$ is an known physical constant. In real prediction and event derivatives markets, a consensus swarm produces an estimator $\hat{p} = p + \epsilon$ with estimation variance $\operatorname{Var}(\epsilon) = \sigma_p^2 > 0$.

Substituting $\hat{p}$ directly into the Kelly rule produces a random stake $\hat{f} = f^* + \frac{b + 1}{b}\epsilon$. Expanding realized conditional log-growth $g(\hat{f})$ to second order around $f^*$ (where $g'(f^*) = 0$ and $g''(f^*) = -\frac{b^2}{p q \cdot (O)^2} < 0$ near small edges) and taking expectation over $\epsilon$ reveals a strict Jensen’s inequality penalty:

$$\mathbb{E}_{\epsilon}\!\big[g(\hat{f})\big] \approx g(f^*) - \frac{1}{2}\,\big|g''(f^*)\big| \left(\frac{b + 1}{b}\right)^2 \sigma_p^2$$

Two consequences make unfiltered full Kelly lethal in automated trading:

1. Selection Bias Conditioning on Trade Entry: Because a desk only routes orders when perceived edge $\hat{e} = b\hat{p} - \hat{q} > e_{\min}$, conditioned on entry the error term satisfies $\mathbb{E}[\epsilon \mid \text{trade}] > 0$ (the winner’s curse of signal screening). Naive Kelly therefore sizes largest precisely on the observations with the largest positive estimation errors.

2. The $2 f^*$ Ruin Cliff: Under quadratic approximation $g(f) \approx f \mu - \frac{1}{2} f^2 \sigma^2$ with $f^* = \mu / \sigma^2$, any realized bet fraction exceeding $2 f^*$ produces $g(f) < 0$—destroying compound wealth over time even though arithmetic expected return $f \mu$ remains positive.

SECTION 02

Analytical Quarter-Kelly ($c = 0.25$) Growth & Drawdown Bounds

Let $f = c \cdot f^*$ for fractional multiplier $c \in (0, 1]$. Under continuous-time diffusion approximations (Thorp, 2006; MacLean, Thorp & Ziemba, 2011), scaling Kelly by $c$ transforms asymptotic geometric drift $g(c f^*)$, portfolio variance $\operatorname{Var}(c f^*)$, and peak-to-trough drawdown probability $\mathbb{P}\!\left(\min_{t \ge 0} W_t / W_0 \le x\right)$ for $x \in (0, 1)$ according to exact power laws:

$$\frac{g(c f^*)}{g(f^*)} = c(2 - c), \qquad \frac{\operatorname{Var}(c f^*)}{\operatorname{Var}(f^*)} = c^2, \qquad \mathbb{P}\!\left(\inf_{t \ge 0} \frac{W_t}{W_{\max, t}} \le x\right) = x^{\,\frac{2}{c} - 1}$$

REGIME	MULTIPLIER $c$	CERTAIN-EDGE GROWTH	VARIANCE RATIO	PROB. OF -30% DD ($x=0.7$)	PROB. OF -50% DD ($x=0.5$)
Full Kelly	1.00	100.00%	100.00%	70.00%	50.00%
Half Kelly	0.50	75.00%	25.00%	34.30%	12.50%
Quarter Kelly (Prod)	0.25	43.75%	6.25%	8.24%	0.78%
Eighth Kelly	0.125	23.44%	1.56%	0.47%	0.003%

Starling Quantitative standardizes on Quarter-Kelly ($c = 0.25$) as its default ceiling before hard per-event bankroll caps. Even if true edge were known with zero error, $c = 0.25$ eliminates $93.75\%$ of return variance and cuts the probability of a $50\%$ peak-to-trough drawdown from a coin flip ($50.00\%$) to $0.78\%$. Once realistic parameter uncertainty $\sigma_p > 0$ is present, Quarter-Kelly frequently achieves higher realized out-of-sample geometric growth than naive full Kelly because $\hat{f}_{0.25}$ almost never crosses the $2 f^*$ wealth-destruction boundary.

SECTION 03

Bayesian Market-Prior Shrinkage & Portfolio Covariance Caps

Before computing $f^*$, Tier 02 consensus nodes shrink raw swarm probability $\hat{p}_{\text{swarm}}$ toward the Power-devigged sharp market benchmark $p_{\text{sharp}}$ (derived in Research Note RN-01) using precision weights proportional to out-of-sample log-loss / Brier precision:

$$p_{\text{post}} = \omega \, \hat{p}_{\text{swarm}} + (1 - \omega) \, p_{\text{sharp}}, \qquad \omega \in [0, 1]$$

Across simultaneous positions $\mathbf{f} = (f_1, \dots, f_M)^\top$—such as adjacent macro CPI brackets, multi-date Fed rate contracts, or correlated cross-venue event legs—independent single-asset Kelly sizing over-allocates to shared latent factors. Tier 03 risk nodes solve the constrained quadratic program:

$$\max_{\mathbf{f} \ge \mathbf{0}} \; \boldsymbol{\mu}^\top \mathbf{f} - \frac{1}{2c}\,\mathbf{f}^\top \boldsymbol{\Sigma} \mathbf{f} \quad \text{s.t.} \quad \sum_{m \in \mathcal{C}_g} f_m \le F_{\text{cluster}}, \quad \|\mathbf{f}\|_{\infty} \le F_{\max}$$

where $\boldsymbol{\Sigma}$ is the outcome covariance matrix and $\mathcal{C}_g$ groups all contracts exposed to the same underlying event resolution.

SECTION 04

Out-of-Sample Brier Score & the Murphy (1973) Decomposition

Realized PnL and Sharpe ratios over moderate sample sizes ($N < 250$ binary resolutions) suffer severe luck variance—especially on skewed tail contracts. To measure true epistemic edge independently of payout variance, every pre-trade probability $p_t \in [0, 1]$ and binary settlement $y_t \in \{0, 1\}$ written to Starling Quantitative’s SQLite flight recorder is scored out-of-sample via the Brier Score:

$$\operatorname{BS} = \frac{1}{N} \sum_{t=1}^{N} (p_t - y_t)^2$$

Partitioning historical forecasts into $K$ calibration bins $\mathcal{B}_1, \dots, \mathcal{B}_K$ of count $n_k$, bin mean forecast $\bar{p}_k = \frac{1}{n_k}\sum_{t \in \mathcal{B}_k} p_t$, bin realized frequency $\bar{o}_k = \frac{1}{n_k}\sum_{t \in \mathcal{B}_k} y_t$, and overall base rate $\bar{o} = \frac{1}{N}\sum_{t=1}^N y_t$, Allan Murphy (1973) proved the exact algebraic identity:

$$\operatorname{BS} = \underbrace{\frac{1}{N}\sum_{k=1}^{K} n_k \big(\bar{p}_k - \bar{o}_k\big)^2}_{\text{Reliability (REL)} \;\downarrow} \;-\; \underbrace{\frac{1}{N}\sum_{k=1}^{K} n_k \big(\bar{o}_k - \bar{o}\big)^2}_{\text{Resolution (RES)} \;\uparrow} \;+\; \underbrace{\bar{o}(1 - \bar{o})}_{\text{Uncertainty (UNC)}}$$

Separating $\operatorname{BS}$ into these three orthogonal terms diagnoses distinct model pathologies during nightly reflection:

1. High Reliability Error ($\operatorname{REL} \gg 0$): The strategy possesses genuine signal ($\operatorname{RES} > 0$) but is overconfident in the tails (for example, predicting $85\%$ on events settling $71\%$ of the time). Remedy: tighten Bayesian shrinkage weight $\omega$ or apply isotonic / Platt recalibration—without discarding the underlying alpha signal.

2. Collapsing Resolution ($\operatorname{RES} \to 0$): Conditional outcome frequencies no longer separate from the base rate or sharp market prior; microstructure alpha has decayed. Remedy: attenuate strategy capital allocation immediately.

SECTION 05

Discounted Thompson Sampling & Pre-Trade Circuit Breakers

Nightly Brier skill scores ($\operatorname{BSS} = 1 - \operatorname{BS}_{\text{model}} / \operatorname{BS}_{\text{sharp}}$) feed a Discounted Thompson Sampling allocator ($\lambda = 0.98$) across competing sub-strategies so capital smoothly migrates out of decaying market regimes without waiting for severe PnL drawdowns to accumulate.

Finally, quantitative calibration is backed by three non-negotiable deterministic execution circuit breakers:

• Compute-to-Alpha Ratio (CAR) Gate: Frontier multi-agent deliberation is blocked on micro-notional opportunities unless expected net dollar alpha exceeds inference cost by $\ge 10\times$, falling back to deterministic sub-millisecond reflex pricing.
• Clock Drift & Sequence Guard: Order routing halts automatically if NTP clock drift exceeds $500\text{ ms}$ or if WebSocket order-book sequence gaps remain unhealed.
• Daily Drawdown Kill Switch: Breaching the daily drawdown threshold halts automated DMA routing immediately and dispatches a critical operator alert for manual post-mortem review.

SECTION 06

Technical FAQ

Why does full Kelly sizing underperform when probability estimates contain noise?

Because expected log-wealth $g(f)$ is strictly concave around $f^*$, symmetric probability estimation errors inflict asymmetric geometric penalties. Overestimating edge pushes stakes toward or past $2 f^*$, where long-run compound growth turns negative despite positive arithmetic expectation.

How much drawdown risk does quarter-Kelly ($c = 0.25$) eliminate compared to full Kelly?

Scaling Kelly by $c = 0.25$ compresses return variance by $93.75\%$ ($c^2 = 0.0625$) and reduces the continuous-time probability of a $50\%$ bankroll drawdown ($0.5^{2/c - 1}$) from $50.00\%$ to $0.78\%$.

What do Reliability, Resolution, and Uncertainty measure in the Murphy Brier decomposition?

Reliability measures calibration accuracy across probability buckets (lower is better); Resolution measures how sharply conditional event frequencies separate from the base rate (higher is better); Uncertainty reflects intrinsic outcome variance $\bar{o}(1-\bar{o})$.

COMPANION RESEARCH NOTE // RN-01

Extracting True Implied Probabilities from Vigged Markets: Additive, Multiplicative, Power, and Shin Devigging Compared

← READ RN-01

Fractional Kelly Sizing Under Parameter Uncertainty &
Out-of-Sample Brier Decomposition.