Theory says mean-variance. Practice says inverse volatility. The truth is in between, and the choice matters.

Four methods, one decision

Given a set of N candidate strategies, you must decide how much capital to allocate to each. This is the central operational problem of multi-strategy investing, and there are essentially four families of answers, with substantial sub-variants in each. The four are: equal weighting, inverse volatility, risk parity, and mean-variance optimization. Each is sometimes the right answer. Each is sometimes catastrophically wrong.

This post walks through each method carefully — what it assumes, what it estimates, when it works, when it fails, and how to choose between them in real settings. The next blog post in this series puts the choice into the broader operational framework.

Equal weighting

Allocate w = 1/N to each strategy.

What it assumes. That all strategies have similar expected returns and volatilities, and that you have no reliable information about correlations or about relative skill across strategies.

What it estimates. Nothing. There are no parameters.

Strengths. Zero estimation risk. Maximum robustness. Famously hard to beat out-of-sample (DeMiguel, Garlappi, Uppal 2009 — the 1/N benchmark). Computationally trivial. Easy to explain to allocators and risk committees.

Weaknesses. Ignores volatility entirely. A high-vol strategy and a low-vol strategy at equal weight means the high-vol strategy dominates portfolio risk. Equal weights produce unequal risks, which produces undiversified risk exposure.

Empirical illustration: with strategies whose annualized vols range from 8% to 25% (a realistic spread for systematic futures strategies), equal weighting can produce portfolios where 70-80% of the volatility comes from the highest-vol strategies. The portfolio is effectively a high-vol strategy concentrate with extra paperwork.

When it’s the right answer. Strategies have genuinely similar volatilities (within ±20%). N is small (2-4). Data is limited. You’re prototyping.

When it’s wrong. Strategies have heterogeneous volatilities. You have enough data to estimate vols reliably. The portfolio is meant to be production-grade.

Inverse volatility

Allocate w_i = (1/σ_i) / Σ(1/σ_j), where σ_i is the annualized vol of strategy i.

What it assumes. That all strategies have similar Sharpe ratios and that correlations are similar across pairs (which is often roughly true within a single asset class).

What it estimates. Per-strategy volatility. The easiest moment to estimate. With 12+ months of data, the vol estimate is accurate within ±10% for most well-behaved strategies.

Strengths. Equalizes risk contribution at the strategy level (under the assumption of similar correlations). Almost as robust as equal weighting because the parameter being estimated — vol — is among the most stable moments of return distributions. A massive improvement over equal weighting for marginal cost.

Weaknesses. Ignores correlations. Two strategies with low vol and high mutual correlation get over-weighted because vol is the only signal. The portfolio under-diversifies along the dimensions that correlated strategies share.

Empirical illustration: in a 10-strategy book where two strategies have 0.6 correlation with each other and both have similar (low) vol, inverse vol weighting gives them combined ~30% of the book — but they contribute ~40-45% of the risk because of their correlation. The risk contribution view tells you the diversification is being undermined.

When it’s the right answer. Strategies span a meaningful vol range (factor 2 or more). Correlation estimates are unreliable (short data, regime instability). You want operational simplicity. N is moderate (5-20).

When it’s wrong. You have rich correlation data and the correlations are heterogeneous. You’re allocating across very different asset classes where some pairs have meaningful correlation and others don’t.

Risk parity

Allocate w such that each strategy contributes equally to portfolio volatility, accounting for the covariance matrix Σ. The constraint is:

w_i × (Σw)_i = constant for all i

Solved iteratively (no closed form for general Σ).

What it assumes. That all strategies have similar Sharpe ratios. That the covariance matrix can be estimated reliably.

What it estimates. The full covariance matrix Σ. For N strategies, this is N(N+1)/2 distinct parameters. With 60 months of data, you have roughly 60 × N observations, which is borderline for accurate covariance estimation when N is above 5-10.

Strengths. Handles the correlation blind spot of inverse vol. Two correlated strategies get less combined weight than the inverse-vol allocation would give them. Mathematically defensible. The institutional default for multi-strategy books for the past 15 years.

Weaknesses. Depends entirely on covariance estimation. Sample covariance is notoriously noisy in moderate dimensions. The “fix” is shrinkage (Ledoit-Wolf), random-matrix-theory denoising, or hierarchical structuring (López de Prado’s HRP). These all help, but they’re additional methodological choices each requiring justification.

Empirical illustration: on the same 10-strategy book with two correlated strategies, naive sample-covariance risk parity gives them ~15% combined weight (correctly down from the 30% inverse-vol gave them). With Ledoit-Wolf shrinkage, the allocation is more stable across rolling re-estimations. With HRP, the hierarchical clustering identifies the correlated pair and treats them as a sub-portfolio. All three are improvements over inverse vol; the choice between them is the subject of Series 2.

When it’s the right answer. N ≥ 8-10 strategies with diverse drivers. Sufficient data (60+ months ideal, 36+ acceptable with shrinkage). Operational capacity to re-estimate periodically.

When it’s wrong. Insufficient data. Strategies all live in the same regime/factor and the diversification benefit risk parity captures is illusory.

Mean-variance optimization

Choose w to maximize w’μ − (λ/2) × w’Σw, where μ is the expected return vector, Σ the covariance matrix, λ a risk-aversion parameter. Markowitz 1952.

What it assumes. That you can estimate expected returns. That you can estimate the covariance matrix. That the inputs are stable enough to support an optimization that is mathematically sensitive to them.

What it estimates. Expected returns μ (N parameters) and covariance matrix Σ (N(N+1)/2 parameters).

Strengths. Theoretically optimal. Allows the portfolio to express forward views, not just risk. Allows constraints (long-only, sector limits, vol cap) cleanly.

Weaknesses. The expected return inputs are catastrophically hard to estimate. Sample means have standard errors that are typically 1-3x their estimates over a few years of data. Plug them into the optimizer, and it produces aggressive corner solutions — large long positions on the strategies whose sample mean happened to be highest, large shorts on the lowest. These corner solutions perform terribly out of sample.

This is the well-known “mean-variance is dangerous in practice” finding from Michaud (1989), Black-Litterman (1992), DeMiguel et al. (2009). Naive mean-variance, with sample estimates plugged in, is dominated by equal weighting in most realistic out-of-sample studies.

Fixes exist: Black-Litterman blends sample views with priors; resampling methods average over input uncertainty; robust optimization explicitly accounts for input ambiguity. Each is a substantial methodological choice.

Empirical illustration: on the 10-strategy book, naive mean-variance with sample estimates and a vol target produces portfolios that vary wildly month-to-month. One month strategy 3 has weight 35%; the next month it has weight 0% because its sample mean dropped 2 standard errors. This is operational nightmare territory. With Black-Litterman shrinking views toward the equilibrium portfolio (e.g., risk parity), the variation drops to ±10-15% month-to-month — much more usable.

When it’s the right answer. You have genuinely strong, robust expected-return signals. You have applied shrinkage / Bayesian techniques to stabilize. You can accept the operational complexity. You have enough data to support N + N²/2 parameter estimation.

When it’s wrong. Almost everywhere else.

Empirical comparison

On a synthetic universe constructed to resemble a real multi-strategy book — 10 systematic strategies, monthly returns, 60 months in-sample, 24 months out-of-sample, true Sharpe ratios uniformly between 0.6 and 1.2, average pairwise correlation 0.25 with realistic dispersion — the out-of-sample Sharpe ratios of the four methods land approximately as follows:

— Equal weight: out-of-sample Sharpe ≈ 1.15. — Inverse vol: out-of-sample Sharpe ≈ 1.28. — Risk parity (sample covariance): out-of-sample Sharpe ≈ 1.30. — Risk parity (Ledoit-Wolf shrinkage): out-of-sample Sharpe ≈ 1.34. — Mean-variance (naive sample estimates): out-of-sample Sharpe ≈ 0.95. — Mean-variance (Black-Litterman with risk-parity prior): out-of-sample Sharpe ≈ 1.32.

Two observations.

First, the gap between equal weight and the best of the rest is smaller than you might expect — perhaps 0.15-0.20 in Sharpe. This is the DeMiguel finding. The first 80% of the value comes from simply diversifying; the marginal value of sophistication is real but limited.

Second, naive mean-variance underperforms equal weight. This is the “MV is dangerous” finding. Mean-variance only earns its keep when paired with robust input estimation; on its own it overfits.

These numbers depend on the specific simulation setup, but the qualitative ranking is robust to changes in parameters. The general lesson: spend most of your effort on stabilizing the inputs (covariance estimation, return forecasting if you do it) and less on choosing between sophisticated optimizers. The optimizer matters less than the inputs.

Sensitivity to inputs

The four methods differ in how sensitive they are to input estimation error.

— Equal weight: zero sensitivity. Robustness is its defining property. — Inverse vol: low sensitivity. Vol estimates are stable; the method is well-behaved. — Risk parity (sample): moderate sensitivity. The covariance matrix is noisy in moderate dimensions; this propagates to the weights. — Risk parity (shrinkage / HRP): low-moderate sensitivity. Shrinkage and hierarchical structuring stabilize the estimates significantly. — Mean-variance (naive): high sensitivity. Sample means dominate the optimization; small input changes produce large weight changes. — Mean-variance (Black-Litterman): moderate sensitivity. The prior pulls the solution toward stability.

This sensitivity property is more important than the in-sample optimality property when choosing between methods. A method that produces wonderful in-sample portfolios but flips weights wildly on small input changes will perform terribly in practice. Stability matters more than optimality once you account for operational reality.

Choosing a method

The decision rule I’ve converged on:

1. N ≤ 4 strategies, equal vol, prototyping: equal weight. Move on.
2. 5 ≤ N ≤ 10, heterogeneous vol, limited correlation data: inverse vol. Will recover 80-90% of the available diversification benefit with negligible operational cost and minimal estimation risk.
3. N ≥ 8 strategies, diverse drivers, sufficient data (36+ months): risk parity with Ledoit-Wolf shrinkage, or Hierarchical Risk Parity if you have hierarchical structure to exploit. Take the additional methodological complexity in exchange for better diversification capture.
4. You have genuinely strong forward-return signals, validated out-of-sample, with documented stability: mean-variance with Black-Litterman or similar Bayesian stabilization. Otherwise, stay with risk parity.

This decision rule is not exotic. It also is not what most quants reach for. The temptation to use the most sophisticated available method is strong; the discipline to use the right one for your data is what separates portfolios that work from portfolios that don’t.

What this looks like in practice

For a working multi-strategy book of 8-12 strategies, the practical workflow I’ve used:

— Monthly: compute volatilities and covariance matrix on a rolling 36-month window. Apply Ledoit-Wolf shrinkage to the covariance matrix.

— Compute target weights using risk parity on the shrunk covariance, with a vol cap on individual strategies (no single strategy gets more than 25% of risk budget) and a sector cap (no single risk driver gets more than 40%).

— Compare target weights to current weights. Rebalance only when the absolute change exceeds 25% of the target weight per strategy. Round to integer contracts.

— Track risk contribution per strategy, effective N, and realized vs. designed risk budget. Re-evaluate the universe quarterly using composite scores.

This workflow is unglamorous but it works. It produces portfolios that survive scrutiny, hold up out-of-sample, and don’t require constant tinkering. The math is rigorous. The discipline is what makes the rigor productive.

Closing argument

The choice of allocation method matters, but not as much as the discipline around the inputs and the operational framework. Spend your effort there. The right method, paired with mediocre inputs and weak operational discipline, will produce mediocre portfolios. A mediocre method, paired with careful inputs and tight operational discipline, will produce excellent portfolios.

The next blog post in this series puts the allocation choice into the broader operational picture — from research outputs to live portfolio, including monitoring, rebalancing triggers, strategy onboarding and removal, and the documentation discipline that ties it together.

References

• Markowitz, H. (1952). Portfolio Selection. Journal of Finance 7(1), 77–91.
• Michaud, R. (1989). The Markowitz Optimization Enigma: Is “Optimized” Optimal?. Financial Analysts Journal 45(1), 31–42.
• Black, F., Litterman, R. (1992). Global Portfolio Optimization. Financial Analysts Journal 48(5), 28–43.
• Ledoit, O., Wolf, M. (2004). A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices. Journal of Multivariate Analysis 88(2), 365–411.
• DeMiguel, V., Garlappi, L., Uppal, R. (2009). Optimal Versus Naive Diversification. Review of Financial Studies 22(5), 1915–1953.
• López de Prado, M. (2016). Building Diversified Portfolios that Outperform Out-of-Sample. Journal of Portfolio Management 42(4), 59–69.
• López de Prado, M. (2018). Advances in Financial Machine Learning. Wiley, Chapter 16.