Drew Tilley

Research notes on modeling complexity and decision systems under uncertainty.

Artifact 01 Specification

Time-Path Sensitivity of Minimum-Variance Portfolio Optimization

Objective

Demonstrate that portfolio optimization outcomes are highly sensitive to plausible alternative time paths drawn from the same risk model, even in the absence of structural breaks or extreme events.

This artifact evaluates fragility under model-consistent uncertainty, not predictive power.

Universe

The asset universe is fixed and intentionally small to ensure transparency and reproducibility using public data.

Assets (11): AAPL, MSFT, NVDA, META, JPM, CAT, XOM, JNJ, PG, KO, AMT

Risk Model is a linear factor model with the following structure:

ri,t=βimktftmkt+βisectorftsector+ϵi,t

Factors

Market factor: SPY

Sector factors: Corresponding SPDR sector ETFs for each asset

Factor loadings and residuals are estimated once using a fixed in-sample window.

Optimization Objective

Minimum Variance Portfolio

Subject to:

  • Long-only weights
  • Fully invested portfolio
  • No leverage
  • No turnover or transaction cost modeling

The optimization objective and constraints are fixed across all runs.

Canonical Perturbation

Alternative Time-Path Bootstrap

Alternative market histories are generated by resampling the time index with replacement.

For each bootstrap draw:

  1. A sequence of time indices is sampled with replacement from the original estimation window.
  2. The same resampled index sequence is applied jointly to:
    • Market factor returns
    • Sector factor returns
    • Asset-level residuals
  3. Returns are reconstructed using fixed factor loadings.
  4. Number of Draws: N bootstrap draws (fixed prior to analysis)

This procedure preserves cross-sectional dependence and the model’s factor structure while altering temporal ordering. All results are fully reproducible via random seed

Outputs

Figure 1 — Cumulative Return Path Overlay (Optimized Portfolio)

  • Cumulative returns for each bootstrap draw
  • Median path highlighted
  • Original historical path shown for reference

Purpose: Visualize dispersion across plausible time paths.

Figure 2 — Terminal Return Distribution

  • Distribution of final cumulative returns
  • Comparison between: Minimum-variance optimized portfolio, Equal-weight portfolio baseline

Purpose: Highlight dispersion and tail sensitivity without tail events.

Figure 3 — Weight Dispersion Heatmap

  • Portfolio weights across bootstrap draws
  • Assets on x-axis, bootstrap draws on y-axis

Purpose: Expose instability and concentration in optimized allocations.

Summary Table

Metrics reported for both portfolios:

  • Median terminal return
  • Interquartile range
  • Worst decile outcome
  • In-sample Sharpe
  • Median out-of-path Sharpe

Interpretation Guidance

If materially different outcomes arise under equally plausible histories drawn from the same model, confidence in any single optimized solution should be tempered.

Reproducibility

All data sources are public. All randomness is seed-controlled. No live feeds or proprietary inputs are used.