Asset Allocation & Risk Management Handbook

Part 00

Three questions, not one

Nearly every argument about investing comes from blending three separate questions into one. Separate them and each has its own set of frameworks, and the reliability of the evidence for each set varies enormously.

Part 01

Allocation theory: from Markowitz to admitting failure

Markowitz (1952) — the right idea, with inputs that do not exist

Markowitz's contribution was to stop viewing a portfolio as a collection of good tickers, and start viewing it as a single statistical object with two properties: expected return and variance. The key point: portfolio risk is not the weighted average of the components' risks — it is governed by the covariance structure.

The limit of diversification, for an equal-weight portfolio of n assets, with average variance σ̄² and average covariance c̄:

Idiosyncratic risk is eliminated at a rate of 1/n; the average covariance is the floor that cannot be broken. This is why adding the 30th, 40th, 50th ticker is nearly meaningless, and why the diversification that is worth having is diversification across different return drivers, not within a single asset class.

The problem: the number of parameters

The model needs n(n+3)/2 parameters. With 50 assets that is 1,325 numbers, of which 1,225 are covariances. And the optimal solution contains Σ⁻¹ — the inverse matrix amplifies every input error into enormous divergence in the output weights.

Chopra & Ziemba (1993) — the most useful number in this entire document

They measured the damage from error in each type of input. The result (at a risk-tolerance level of 50), expressed as a cash-equivalent loss:

The thing hardest to estimate is also the thing that does the most damage when it is wrong. That is the entire problem of this field in one sentence.

DeMiguel, Garlappi & Uppal (2009) — 1/N beats 13 sophisticated models

They compared 14 portfolio models across 7 datasets, including mean-variance, Bayes-Stein shrinkage, minimum-variance and constrained variants. The result: "no model consistently outperforms the 1/N rule, whether measured by Sharpe ratio, certainty-equivalent return, or turnover."

Read it precisely: DGU do not prove that 1/N is optimal. They prove that estimation error is so large that it eats the entire theoretical benefit of optimization. 1/N is merely the benchmark with zero estimation error — and 13 sophisticated methods could not clear that benchmark.

A notable rebuttal: Kritzman, Page & Turkington (2010) argue that DGU indict only one particular way of choosing inputs — using a 5-year moving average as the expected return, which is nearly pure noise. They found: "any reasonable set of expected returns, even chosen by arbitrary judgment, beats 1/N after optimization." Note the asymmetry of position: DGU was published in the Review of Financial Studies; Kritzman et al. in a practitioner journal, and the authors work at a firm that sells optimization services.

Black-Litterman — the lesson that needs no linear algebra

Black & Litterman's diagnosis: MPT breaks not because the optimization is wrong, but because the prior is wrong. The fix is to invert the problem — take market-cap weights as the starting point, assume the market sits at CAPM equilibrium, then solve backward for the set of expected returns that make those weights optimal.

For the individual investor, the value lies in the principle, not the formula: start from the market portfolio, and deviate from it only exactly where you truly hold a view, deviating in proportion to your confidence. If you hold no views, the model returns exactly the market portfolio — meaning: just buy the market and ignore the model.

Risk parity — beautiful math, but not the leveraged version

Qian's opening observation: the 60/40 portfolio is called "balanced" because capital is split 60/40, but capital and risk are two different things. Because equities are roughly three times as volatile as bonds, 60/40 takes about 90–95% of its risk from the equity leg.

Qian verified this with actual losses: over 1976–2009, across the 32 months in which the portfolio lost more than 3%, equities contributed on average 97% of the loss, bonds 3%.

The Maillard–Roncalli–Teiletche theorem: σ_minvar ≤ σ_ERC ≤ σ_equal. ERC is a genuine middle ground between min-variance and equal weighting.

Robustness ranking — what survives out of sample

A note on 1/N: it is a strong benchmark within a single asset class, and a weak benchmark across asset classes. Splitting equally between equities and Treasury bills is not a sensible portfolio — that is Qian's objection to 60/40 stated another way.

Part 02

Model portfolios: what has actually been tested

These are portfolios with names, fixed ratios, and a track record. They are useful because the ratios are public and require no estimation — but nearly every performance figure carries a serious sample bias you need to know first.

Golden Butterfly: 40% US equity, comprising 20% large-cap blend + 20% small-cap value.

2022: the test the whole "four economic quadrants" family failed

This is the most important empirical fact in this section.

Portfolios designed to survive every economic regime recorded their worst losses in history in exactly the inflationary regime they were built to resist. The leveraged version lost more than the S&P 500 itself.

Taleb's Barbell — the right structure, but both legs have problems

The proposal: about 85–90% in absolutely safe instruments (Treasury bills) and 10–15% in extremely speculative, maximally convex bets. Nothing in the middle — because the middle is precisely where risk is unmeasurable and models fail.

The core argument is cutting the left tail, not optimizing the expectation. With 90% in cash, you cannot lose more than 10% — a known, bounded, calculable loss — while the other 10% has unbounded upside. Taleb: "The first step toward antifragility is to reduce the downside, not to increase the upside."

Part 03

Position sizing and risk management

This is the most certain and least disputed math in the entire document. It is also the part that most determines real-world outcomes — because unlike Part 01, the formulas here do not require you to predict anything about the future correctly.

Kelly — and why no one uses full Kelly

The growth function is a downward parabola, peaking exactly at f*. Setting f = c·f*, the fraction of excess growth retained is (2c − c²):

The growth function is flat near the peak, while the risk function is linear. Cutting risk in half costs only a quarter of the edge. Probability of drawdown, per Thorp: P(V ≤ x·V₀) = x^(2/c − 1).

Risk of ruin — exponential in the inverse of position size

Halving risk does not halve the probability of ruin — it squares it. This is the real mathematical reason for the 1–2% rule, not generic caution. This function falls off a cliff.

The math of drawdown — a convex function, and that is the whole problem

Loss is linear in D, but recovery is convex — the function D/(1−D) has a pole at D = 1. This is why blocking the left tail is worth more than chasing the right tail, and why sizing to survive dominates sizing to grow.

Volatility drag — the most important formula for highly volatile assets

This is not a behavioral bias, not a fee, not a side effect — it is the deterministic penalty that multiplicative dynamics impose on any volatile process. And it grows with the square of volatility:

Correlation in a crisis — the exact numbers

Longin & Solnik (2001) is the reference study, and it is far more precise than the folk saying "correlations run to 1":

Three corrections to the common understanding:

• It is direction, not volatility. The claim that "correlation rises during volatile periods" is contaminated by a well-known statistical artifact: conditioning on large absolute returns automatically inflates the sample correlation even for a stationary normal distribution. Anyone citing that conclusion from a rolling-window chart has most likely measured exactly that artifact.
• Asymmetry. Co-movement on the upside actually decreases at the extremes. Diversification works well in rising markets — precisely when you do not need it.
• Linear correlation is the wrong statistic for this question. The Gaussian copula has tail dependence of 0 for any ρ < 1 — this is the mathematical root of the 2008 CDO disaster. The model did not underestimate tail correlation; it assumed tail correlation does not exist.

Application: do not size positions on a correlation matrix estimated during a calm period. Rerun the allocation with all correlations forced to 0.7–0.9 and check whether it survives. And distinguish return diversification (works most of the time, a genuine free lunch) from risk diversification (fails exactly when you need it, and is partly an illusion).

Vol targeting, stop-loss, VaR — three tools, three levels of evidence

Part 04

Rebalancing: less important than you think, and more dangerous than you think

Vanguard: there is no optimal frequency

The reference study (Jaconetti, Kinniry & Zilbering, 2010), a 60/40 portfolio, over 1926–2009:

Vanguard's own conclusion, and it is more modest than it is usually quoted: "there is no optimal frequency or threshold" — the difference in risk-adjusted return across every strategy tested is not meaningfully significant. The recommendation to "monitor annually or semi-annually and rebalance at a 5% threshold" is a cost-minimizing heuristic, not an optimum.

Note the last row: not rebalancing gives a higher return (9.1%) — because equities beat bonds for 84 straight years — but with markedly higher volatility, and the equity weight drifting up to 84%. Rebalancing is a risk-control tool, not a return-boosting tool.

Swedroe's 5/25 rule

"Rebalance only when an asset class's weight changes by more than 5 absolute percentage points or 25% relative to the target — whichever is smaller."

The most misread word is "smaller" — take the tighter threshold, not the wider one. The crossover point is exactly 20%:

What few say: rebalancing is shorting a straddle

Rattray, Granger, Harvey & Van Hemert (2019) offer the sharpest framing of the problem:

Three cases where rebalancing hurts

1. A large and persistent long-run return gap. Continuous rebalancing keeps pulling money out of the winning asset. The "never rebalance" row in the Vanguard table is the proof: +0.5–0.6%/yr just from equities beating bonds.
2. A strongly trending market. Rebalancing is essentially shorting momentum. Because cross-sectional momentum is strongest over the 3–12 month horizon, monthly rebalancing is the maximally anti-momentum choice; a cycle of one year or longer automatically avoids most of this bleed without any extra mechanism.
3. An asset that does not recover. The math of the diversification bonus assumes the assets are drawn from a stable distribution. It says nothing about an asset heading to zero. Rebalancing into a dying asset is a scheduled capital-destruction machine.

How large is the rebalancing bonus — the real number

The Bernstein (1996) and Willenbrock (2011) formulas give the same result:

The real magnitude: a few dozen basis points, not a few percent. Bernstein measured 0.49%/yr for a 50/50 equity/bond split over 1926–1994. Willenbrock measured 1.12% for 50/50 over 2000–2009 — a decade that is close to a best case.

Set it against the cost: a single rebalance in a taxable account can cost 1.19% of portfolio value immediately. Against a bonus of 30–100 basis points per year, that takes many years to pay back.

Cost order — exhaust the cheap methods before the expensive ones: (1) direct new contributions toward the underweight asset; (2) route dividends and interest to the underweight asset instead of reinvesting in place; (3) withdraw from the overweight asset; (4) rebalance in tax-free/tax-deferred accounts first; (5) harvest losses; (6) only as a last resort, sell to raise cash.

Part 05

Crypto: the numbers, and who is paying for them

The published allocation recommendations

A third, independent line of evidence, arriving from an entirely different direction: Kelly with conservative assumptions gives 10%, multiplied by ¼–½ Kelly (the standard adjustment for parameter uncertainty) → 2.5–5%. Three unrelated methods all point to the same place.

Drawdown history — four times in fifteen years

Glassnode records the historical norm: "bear-market bottoms are established at drawdowns of −75% to −84% from the peak, lasting from 260 days (2019–20) to 410 days (2015)."

Ethereum is deeper in every cycle: −94% in 2018 (from $1,431 down to $81.30), −79.5% in 2022. In 2022 ETH even fell below the peak of the 2018 cycle — that is, wiping out the gains of an entire cycle. The ordering BTC < ETH < large altcoins < small altcoins holds in every cycle.

Correlation changed in 2020 — in the worst possible direction

• Bitcoin is not "digital gold" in the correlation data. Its correlation with gold averages 0.10 over a decade with no trend, while its correlation with equities has risen structurally. On the evidence, BTC behaves like a high-beta risk asset sensitive to liquidity, not a monetary hedge. NYDIG sums it up: "gold is a real-rate hedge, while bitcoin has evolved into a liquidity barometer."
• The average correlation hides the correlation during a collapse. IMF: "spillovers between crypto and equity markets tend to rise during periods of financial volatility." BlackRock itself concedes: "unstable correlation with equities during risk-off periods weakens the diversification argument."
• Nearly every allocation study in the table above uses the full-sample average correlation, thereby embedding the low-correlation regime prior to 2020. Rerunning on post-2020 data only would yield a lower optimal weight — no study doing this could be found. This is a genuine gap in the published literature.

Altcoins: strong evidence on direction, weak on magnitude

The oft-cited numbers — "80% of altcoins never revisit their old high", "7 of every 10 coins that once entered the top 100 have vanished" — have no original study behind them. Do not cite them.

What can be verified: CoinGecko (04/2026) records 13.4 million of 25.2 million tokens as "dead" — a rate of 53.2%. But 86% of the dead fall in 2025 and are a memecoin-launchpad phenomenon, so this figure says nothing useful about whether SOL or LINK will revisit their highs.

Stablecoins: an unsecured private debt

Part 06

Reduced to usable rules

If you strip away all the formulas and keep only what the evidence can actually support, this is what remains.

On allocation

1. Do not run an optimizer on your own estimates. Error in expected returns does ~11× the damage of error in covariances, and you would need 250 years of data for optimization to beat equal weighting.
2. Prefer methods that need no return forecast: cap-weighted indices, equal weighting within an asset class, inverse volatility, ERC. These are the ones that survive out of sample over 70 years.
3. Diversify across return drivers, not across tickers. The average covariance is the floor that cannot be broken — adding the 30th ticker within the same asset class does almost nothing.
4. Do not trust any backtest that starts in 1970 with a high gold weight, or any portfolio designed on the very dataset used to show it off.

On position sizing — the part more important than allocation

1. Always bet below your point estimate. Betting 50% too little costs 25% of the edge; betting 100% too much costs 100% of the edge; betting further beyond that is near-certain ruin. The error is symmetric, the consequences are not.
2. Risk of ruin is exponential in the inverse of position size. Going from 5% to 2% per trade does not halve risk — it drops it from 1.8% to 0.005%.
3. Count risk by correlation, not by number of positions. Ten 2% trades in correlated tickers is one 20% bet. This is the most common failure mode of every personal risk-management system.
4. At high volatility, drag eats the return. At σ = 75%, an asset must earn ~28%/yr arithmetically just to break even when compounded. A positive expected return can still yield negative compound growth.
5. Size to survive, do not size to grow. The growth function is concave and bounded; the ruin function is not.

On operations

1. Rebalancing frequency barely matters — Vanguard tested 84 years and found no meaningful difference. Pick once a year because it is cheapest and least anti-momentum, then stop thinking about it.
2. Rebalancing is risk control, not return enhancement. The real bonus is a few dozen basis points; a single taxable rebalance can cost more than many years of that bonus.
3. Never rebalance into an asset that can go to zero. The math of the bonus assumes a stable distribution; it does not apply to something that is dying.
4. Use new contributions to adjust weights before considering selling anything.

On crypto

1. 1–3% if you want a number that does not depend on forecasting correctly. That is where the only two frameworks needing no return forecast both point, and fractional Kelly on conservative assumptions also lands at 2.5–5%.
2. Size by risk contribution, not by capital. 1% of capital ≈ 2% of portfolio risk; 2% of capital ≈ 5% of risk.
3. Size so that a total 100% loss is still not an event in your life plan. This is the final check and it overrides every calculation above.
4. Bitcoin is not a hedge. Correlation with gold is 0.10 over a decade; correlation with equities has risen structurally after 2020, and rises further during a crisis.
5. No stablecoin is cash. USDC does almost everything right and still lost its peg by 5% within hours because its partner bank failed.
6. Futures are not a savings vehicle. A position that can be liquidated to zero is not a reserve asset by any definition.