Testing Robust Kelly Exposure Against an Equal-Risk Baseline

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# How should a low-sample event-driven allocator learn current exposure when robust Kelly loses to equal risk on a chronological holdout?












I am building an event-driven capital allocator for three trading strategies. This is a portfolio-optimization and validation question, not a request for investment advice.

This question follows an earlier question about pricing capital reserved for stochastic future signals. A new experiment appears to have resolved that particular issue, but it has isolated a different one: estimating the appropriate current exposure.

### Portfolio objectives

Signals arrive asynchronously, multiple signals can be available simultaneously, and accepted positions lock capital until they close.

The allocator must make three related decisions:

- How much available capital should be invested now?

- How much capacity should be reserved for signals that may arrive later?

- How should currently invested capital be divided among simultaneous signals, while generally favoring signals with higher posterior expected value?

The intended objective is long-run geometric growth, subject to stop-loss, gross-exposure, per-trade, per-model, and portfolio-level constraints.

For each current signal $i$, I have:

- a posterior predictive return distribution $R_{i,t}$;

- a stop-loss risk estimate $s_{i,t}$;

- an estimated holding period;

- the model that generated the signal.

A useful decomposition of the allocation decision is

$$ x_{i,t} = \alpha_t c_t w_{i,t}, $$

where

$$ 0 \leq \alpha_t \leq 1, \qquad w_t \in \Delta. $$

Here, $c_t$ is currently available capital, $\alpha_t$ is the fraction invested now, and $w_t$ determines how that exposure is distributed among current signals.

This separates two questions that may need different estimators:

- sizing: estimate $\alpha_t$;

- cross-sectional allocation: estimate $w_t$.

### Current optimizer

The current implementation uses a robust expected-log optimizer with Bayesian-bootstrap return worlds and common random numbers:

$$ \max_{\alpha_t,w_t} \inf_{P \in \mathcal{U}_t} \mathbb{E}_P \left[ \log\left( 1+\alpha_t w_t^\top R_t \right) \right], $$

subject to the portfolio constraints.

I previously added a simulated continuation-value term for future signal arrivals. The continuation multiplier is now restricted to the grid

$$ G = \{0,\ 0.10,\ 0.25,\ 0.50,\ 1.00\}. $$

Its marginal value is estimated using paired finite differences on the same replay paths, followed by a lower confidence bound and shrinkage toward zero.

The allocator also has an expected-value tilt that can move the current weights away from equal stop risk without changing the total exposure selected by the optimizer.

### Data and validation

I have approximately 18 months of data and roughly 800 reconstructed signal observations across three strategies. The effective sample is smaller because signals and holding periods overlap.

The latest search used:

- 3,000 policy configurations;

- three chronological tuning folds;

- five Bayesian-bootstrap return worlds;

- 250 scenarios per decision during screening;

- three independent seeds for the finalists;

- 1,000 scenarios per decision during final validation;

- a fixed final six-month chronological holdout not used to select the portfolio policy.

For transparency, the three strategies were selected beforehand, but an eligibility sanity check still used the complete 18-month window. Therefore, the final period is untouched by the portfolio-policy search, but it is not a completely virgin model-selection holdout.

### New empirical result

The selected policy chose:

- continuation-value weight: 0;

- expected-value tilt: 0;

- expanding calibration;

- prior strength: 0;

- uncertainty discount: 0.

Therefore, the search rejected both simulated future capacity and the learned expected-value tilt.

The selected current weights reduced to equal stop-risk weights at the exposure chosen by the robust optimizer.

The remaining difference between the challenger and the equal-risk benchmark is primarily:

- the total current exposure;

- which signals are accepted or rejected.

Using identical signals, stops, initial capital, and execution assumptions:

| Period | Robust optimizer | Equal-risk benchmark | Challenger / benchmark |
| Internal walk-forward validation | USD 2.059 million | USD 22.457 million | 9.2% |
| Final chronological holdout | USD 6.741 million | USD 9.755 million | 69.1% |

On the final holdout:

- robust optimizer maximum drawdown: 55.91%;

- equal-risk maximum drawdown: 56.31%;

- official signals: 285;

- robust optimizer accepted signals: 144.

Thus, the robust optimizer sacrificed about 31% of terminal wealth on the holdout for only a 0.4 percentage-point reduction in maximum drawdown.

The absolute wealth values should not be interpreted as deployable capacity estimates. They result from mechanically compounding historical trade returns and do not establish that equivalent liquidity or market impact would be available.

The paired comparison between the allocation rules is the relevant diagnostic.

Another important detail is that the current selection procedure searches only within the robust-optimizer family. The equal-risk benchmark is diagnostic and is not allowed to replace the challenger, even when it performs substantially better.

### Main question

What statistically defensible, low-sample method should be used to estimate the total current-exposure decision $\alpha_t$, while allowing a simple equal-risk policy to remain the champion unless a more complex allocator demonstrates genuine incremental value?

In particular, would one of the following approaches be more appropriate?

#### 1. Cross-validated fractional Kelly

First estimate the unconstrained robust-Kelly allocation $x_t^K$, then learn only a scalar fraction:

$$ x_t(f) = f x_t^K, \qquad f \in [0,1]. $$

The scalar $f$ could be selected using nested chronological validation and a coarse grid.

#### 2. Shrinkage toward the equal-risk baseline

Define

$$ x_t(\gamma) = (1-\gamma)x_t^{ER} + \gamma x_t^K, \qquad \gamma \in [0,1], $$

where $x_t^{ER}$ is the equal stop-risk allocation and $x_t^K$ is the robust-Kelly allocation.

The allocator would remain close to equal risk unless the data consistently support moving toward Kelly.

#### 3. Baseline-relative optimization

Evaluate policies using paired log-growth differences:

$$ d_t = \Delta \log W_t^{\mathrm{candidate}} - \Delta \log W_t^{ER}, $$

and promote a candidate only if a block-bootstrap or chronological lower confidence bound for its cumulative relative log growth is positive.

#### 4. Online expert aggregation

Treat equal risk, fractional Kelly, and the robust optimizer as competing experts and update their mixture weights sequentially using only information available at each event.

I am particularly interested in how to distinguish between:

- a misspecified return distribution;

- excessive conservatism in robust Kelly;

- a sizing problem in $\alpha_t$;

- an accept/reject threshold problem;

- ordinary estimation noise caused by the limited effective sample.

A useful answer would ideally describe:

- the exposure estimator;

- how the equal-risk benchmark should enter the feasible policy class or promotion rule;

- an appropriate nested chronological validation procedure;

- a statistically meaningful paired test against the baseline;

- how many independent event blocks or forward observations would be needed before permitting live deployment.

I am looking for a simple, auditable statistical solution appropriate for a few hundred dependent event decisions, not a neural-network or reinforcement-learning approach.