Beating a Benchmark: Dynamic Programming May Not Be the Right Numerical Approach
Bibliographic record
Abstract
Abstract. We analyze dynamic investment strategies for benchmark outperformance using two widely used objectives of practical interest to investors: (i) maximizing the information ratio (IR), and (ii) obtaining a favorable tracking difference (cumulative outperformance) relative to the benchmark. In the case of the tracking difference, we propose a simple and intuitive objective function based on the quadratic deviation (QD) from an elevated benchmark. In order to gain some intuition about these strategies, we provide closed-form solutions for the controls under idealized assumptions. For more realistic cases, we represent the control using a neural network (NN) and directly solve a sampled optimization problem, which approximates the original optimal stochastic control formulation. Unlike the typical approach based on dynamic programming (DP), e.g., reinforcement learning, solving the sampled optimization with an NN as a control avoids computing conditional expectations and leads to an optimization problem with a small number of variables. In addition, our NN parameter size is independent of the number of portfolio rebalancing times. Under some assumptions, we prove that a traditional DP approach results in a high-dimensional problem, whereas directly solving for the control without using DP yields a low-dimensional problem. Our analytical and numerical results illustrate that, compared with IR-optimal strategies with the same expected value of terminal wealth, the QD-optimal investment strategies result in comparatively more diversified asset allocations during certain periods of the investment time horizon.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".