System · Research
Research Papers
Portfolio sizing — papers, implementations and lessons.
Optimal Portfolio Size under Parameter Uncertainty
Find the optimal number of assets before optimizing
Optimal Granularity for Portfolio Choice
Group assets before optimizing — the best of both 1/N and MV
Risk Budgeting Portfolios
Allocate risk by contribution — natural diversification
ML-Based Risk Allocation
LSTM volatility forecasting plus differentiable risk budgeting
Portfolio Diversification with Varying Investor Abilities
The optimal size depends on the investor's skill level
LTCM Sizing Failure — A Case Study
How excessive leverage and concentration caused the collapse
Optimal Portfolio Size under Parameter Uncertainty
Vanderveken, Lassance & Vrins · UCLouvain LIDAM LFIN 2024/04
The optimal number of assets N comes from a trade-off between two opposing forces: more assets give more investment opportunities, but more parameters to estimate raise estimation risk and worsen out-of-sample performance.
KEY FINDING
Optimal N ≈ Nsample / 2 for the robust 2F/3F combination rules under typical equity correlations
PROCESS (3 STAGES)
Choose strategy → Compute optimal N by out-of-sample expected utility → Select N assets
portfolio-simulation: optimal_portfolio_size() implements the empirical N search via walk-forward splits — see the interactive page at /quant/papers/optimal-size.
Optimal Granularity for Portfolio Choice
Branger, Lucivjanska & Weissensteiner · J. Financial Econometrics (2019)
Grouping strategy: cluster assets into equally-weighted groups, then optimize over the groups. It beats both full optimization and 1/N when the group size is optimal.
FULL MV
High estimation risk → worse than 1/N
1/N RULE
No optimization → robust but sub-optimal
GROUPING
Best of both: middle risk + middle benefit
portfolio-simulation: grouping_strategy() implements hierarchical clustering + intra-group optimization — see the interactive page at /quant/papers/granularity.
Risk Budgeting Portfolios
da Costa, Pesenti & Targino · arXiv:2302.01196
Allocates risk (not capital) across assets. Each asset contributes a predefined fraction of the portfolio's total risk. The canonical case — risk parity — gives each asset an equal share of the risk.
MV vs RISK BUDGETING
MV needs expected returns (hard to estimate). RB only needs covariance (more stable).
ALGORITHM
Iterative Proportional Fitting (IPF) — converges in <100 iterations for 100+ assets
portfolio-simulation: risk_budgeting_weights() for risk parity · run_strategy(risk_guard=True) for enforced de-risking from the observatory's signals — see the interactive page at /quant/papers/risk-budgeting.
LTCM Sizing Failure — A Case Study
US President's Working Group on Financial Markets (1999)
In August 1998, LTCM lost 44% of its equity in a single month and needed a $3.6 billion bailout from 14 banks organized by the Federal Reserve.
~25-30:1 leverage
~$4.7B equity → ~$125B in assets (peak). A 2.5% adverse move wiped out the equity.
~5% of some US futures markets
~$500B in futures — about 5% of certain U.S. futures contracts. Positions too large to exit without moving prices against themselves.
Hidden concentration
All positions were relative-value (geographic diversification is irrelevant when correlations → 1).
Lesson: LTCM's own partners admitted
"Reducing positions to 1/4 of their size might have let us survive." — Sizing was the problem.
portfolio-simulation: ltcm_sizing_diagnosis() returns a dict mapping the failures to papers and actions in the code — see the interactive page at /quant/papers/ltcm.
Portfolio Diversification with Varying Investor Abilities
James & Menzies · arXiv:2311.06519
Whether to diversify or concentrate depends on who is investing. An investor with a genuine edge should concentrate; one without an edge is better off spreading risk widely.
LOW SKILL → DIVERSIFY
Optimal N sits near the ~100-stock upper bound — the portfolio behaves close to 1/N.
HIGH SKILL → CONCENTRATE
Optimal N drops to ~10 — fewer, higher-confidence bets dominate.
portfolio-simulation: optimal_portfolio_size() bounds N before rebalancing (the strategy argument picks the combination rule) — see the interactive page at /quant/papers/varying-abilities.
ML-Based Risk Allocation
Scientific Reports (2025) · LSTM volatility forecasting + differentiable risk budgeting
A machine-learning architecture that couples an LSTM volatility forecaster with differentiable risk-budgeting layers and regime-switching. It learns the risk profile and re-allocates as volatility shifts across regimes.
OOS SHARPE
1.38 out-of-sample (2017-2022)
VS RISK PARITY
+55% over traditional risk parity (0.89)
VS DEEP LEARNING
+23% over deep-learning benchmarks
portfolio-simulation: backend endpoint /quant-api/api/papers/ml-risk-allocation returns the regime-conditional Sharpe comparison — see the interactive page at /quant/papers/ml-risk.