Not an implementation claim. These are the working members of the μ-free family from the estimation-error guide — allocation without return forecasts. This guide is what each actually optimizes and bets on, which their marketing rarely states.
The three, precisely
Minimum variance: minimize w’Σw. Pure risk minimization — concentrates in low-vol, low-correlation assets, which in equity universes means the low-volatility corner (and its factor exposure: min-var equity portfolios are, in large part, the low-vol premium in optimizer clothing). Estimation burden: Σ only, and mostly its top structure — the family’s most robust member.
Risk parity: equalize risk contributions — each asset (or asset class) contributes equally to portfolio variance, so low-vol assets get large weights. The inseparable second half: unlevered risk parity across stocks/bonds is mostly bonds and underperforms by construction, so risk parity is a leverage strategy — lever the balanced-risk portfolio to a target vol. Its historical record therefore embeds a financing-cost and bond-regime bet: decades of falling rates flattered it; the 2022 regime — stocks and bonds down together, financing costs rising — was its documented stress case, right on schedule.
Maximum diversification: maximize the diversification ratio (weighted average vol ÷ portfolio vol) — tilting toward assets whose correlation structure adds the most independent risk. Sits between min-var and equal-weight; its bet is that the correlation matrix’s structure is real and stable, the exact quantity that degrades in stress.
The unifying view: all three are answers to “what should weights depend on, if not μ?” — variance level (min-var), risk budgets (parity), or correlation structure (max-div). Each is a different implicit bet dressed as neutrality, and choosing between them is choosing which estimate you trust.
The comparison that matters
| Min variance | Risk parity | Max diversification | 1/N | |
|---|---|---|---|---|
| Estimates needed | Σ (top structure) | vols + a leverage rule | full correlation matrix | none |
| Implicit factor bet | Low-vol premium | Bond duration + financing | Correlation stability | Small-cap/equal tilt |
| Leverage | No | Essential | Optional | No |
| Stress behavior | Best of family | Financing + correlation squeeze | Correlation convergence hurts most | Transparent |
| Turnover | Moderate | Vol-tracking driven | Highest (correlation-chasing) | Lowest |
Common ways the family fails in production
- The 2022 mode (risk parity): the diversifying asset and the growth asset fall together while leverage costs rise — three assumptions failing jointly, as regime shifts do.
- Vol-tracking churn: parity and min-var re-estimate risk continuously; vol spikes force deleveraging into falling markets — the family shares volatility targeting’s procyclical sell-low mechanics.
- Correlation-matrix fragility (max-div especially): weights chase estimated correlation structure that is part noise at realistic sample sizes; stress converges correlations and deletes the claimed diversification.
- Leverage plumbing (parity): financing spreads widen and margin terms tighten exactly in the drawdowns — the urgency-convexity of costs applied to the whole portfolio.
- Backtest-era dependence: the family’s flagship long-run results were earned in a forty-year bond bull; subperiod tables are not optional here.
Questions to ask a paper in this family
- Which estimates does the scheme actually consume, and how were they computed out-of-sample?
- For parity: what financing cost is assumed, and what happens at +200 bps?
- Is 1/N (and min-var, for the fancier schemes) in the comparison?
- How did the scheme behave in stocks-bonds-correlated stress — in and out of sample?
- What’s the turnover, and who pays it?
What our scoring captures
The portfolio hub’s risk-based-allocation papers score well on rigor when they do the three honest things: subperiod (regime) tables, financing-cost accounting for levered schemes, and naive benchmarks. The recurring low-rigor pattern is the forty-year gross backtest ending before the regime that tests the scheme’s core bet — which the scoring axis penalizes as incomplete evidence, and this guide lets you name precisely.
The estimation problem underneath: optimization & estimation error → · The leverage layer: volatility targeting → · From the corpus: risk measures beyond VaR →