5.9.1 Multi-Asset & Factor Allocation from Zero

Basic Prerequisites: none - this page needs no prior portfolio theory. Linear algebra helps with the notation later.

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1. Intuition & Practical Objective

This page builds the why of multi-asset and factor allocation with no prior knowledge needed. The objective is one idea: you diversify by owning return sources whose bad days do not coincide - and the labels “stocks” and “bonds” hide how few sources a typical portfolio really owns.

Start with the dumbest question: why hold anything besides the best-performing asset? Because no one knows which asset that is next period, and because combining two imperfectly-correlated bets gives a portfolio whose wobble is less than the average of its parts. That surplus - the gap between the average of the individual volatilities and the portfolio’s actual volatility - is the whole point of allocation. We call it the diversification ratio:

$$\mathrm{DR}(w)=\frac{{\sum }_i{w}_i{\sigma }_i}{\sqrt{w^{\top }\Sigma w}}.$$

For an equal-weight four-asset portfolio of equities, bonds, commodities and credit, the average leg wobbles $11.5\%$ but the portfolio only wobbles $7.9\%$ - a $\mathrm{DR}=1.46$. The $3.6$ percentage points of “wobble” that vanished were paid for by owning things that move differently.

Three steps:

1. Diversification is a correlation property, not a count. Owning ten assets that all fall together is one bet. The relevant question is not “how many names?” but “how many independent sources?” - measured by the effective number of bets, not the number of tickers.
2. Labels lie; factors are the truth. When you hold a global equity fund and an emerging-market equity fund, you own one bet (a market factor) twice. When you hold a value fund and a momentum fund, you own two genuinely different bets that often move in opposite directions. Allocation across factors decorrelates far more than allocation across labels - the factor portfolio’s $\mathrm{DR}=1.98$ versus $1.46$ for the asset portfolio (page 03).
3. Tactical is a perturbation, not a new plan. The long-horizon “policy” portfolio (strategic allocation) is the anchor; tactical tilts are small, budgeted deviations from it. Confusing the two - re-optimising the strategic plan every month on fresh noise - is estimation error dressed up as agility.

A clean number to carry through the folder: a 60/40 stock/bond portfolio looks balanced but gives 97.75% of its risk to the equity leg. That single number is why the “all-weather” / risk-parity multi-asset idea exists (see Risk Parity & ERC): dollars and risk are different units.

The one-sentence essence. “Allocation is the search for return sources whose bad days do not coincide; the more you allocate to factors, the closer the portfolio’s risk gets to the sum of genuinely independent pieces.”

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2. Mathematical Ground Truth & Derivations

The diversification engine. With weights $w$, covariance $\Sigma$, the portfolio volatility is

$$\sigma (w)=\sqrt{w^{\top }\Sigma w}.$$

Because $\sigma$ is homogeneous of degree 1, Euler’s theorem gives the exact risk-contribution decomposition

$$\sigma (w)=\sum _i{w}_i\frac{\partial \sigma }{\partial w_i}=\sum _i\frac{w_i(\Sigma w{)}_i}{\sigma (w)}=\sum _i{\mathrm{RC}}_i,$$

so each asset’s share of risk is ${\mathrm{RC}}_i/\sigma (w)$. For a well-diversified portfolio these shares are roughly equal; when one asset dominates, the portfolio is a single bet wearing several costumes.

Why mixing works. Write the two-asset case explicitly. With weights $w,1-w$ and volatilities ${\sigma }_1,{\sigma }_2$, correlation $\rho$,

$${\sigma }_p^2=w^2{\sigma }_1^2+(1-w{)}^2{\sigma }_2^2+2w(1-w)\rho {\sigma }_1{\sigma }_2.$$

The first two terms are the “average” (what you’d get with $\rho =1$); the cross term is the diversification dividend. It grows as $\rho$ falls below $1$ and vanishes - or turns into a penalty - as $\rho \to 1$.

From assets to factors. The linear factor model writes returns as

$$r=\alpha +Bf+\epsilon ,\mathbb{E}[\epsilon ]=0,\mathrm{Cov}(f,\epsilon )=0,$$

which implies the covariance decomposition

$${\Sigma }_r=B{\Sigma }_f{B}^{\top }+D,D=\mathrm{diag}({\sigma }_{\epsilon ,1}^2,\dots ).$$

Allocate to factors with weights $w_f$; the induced asset exposure is $w_{\text{asset}}=B{w}_f$, and the portfolio variance through the factors is ${\sigma }_f^2=w_f^{\top }{\Sigma }_f{w}_f$ (plus the small idiosyncratic $w_{\text{asset}}^{\top }D{w}_{\text{asset}}$ if the specific risk is not diversified away). Because ${\Sigma }_f$ has small off-diagonal entries while the asset correlations come mostly from a shared load on $f$, the factor portfolio’s diversification ratio is substantially higher.

The strategic/tactical decomposition. Fix policy weights $w_{\text{SAA}}$ from long-run moments; the tactical overlay is a deviation $\Delta w$ with

$$w=w_{\text{SAA}}+\Delta w,\Delta w^{\top }\Sigma \Delta w\le {\mathrm{TE}}^2,1^{\top }\Delta w=0.$$

The tracking-error budget $\mathrm{TE}$ is the only honest constraint on how much the tactical process may override the strategic plan; without it, tactical deviations silently become the strategic portfolio.

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3. Computational Implementation - capital weight vs risk weight

This is the single most convincing way to see the whole idea. Stdlib + numpy. It computes the diversification ratio and the risk shares for the equal-weight portfolio, then exposes the 60/40 illusion.

Read the risk shares: under equal dollars, equities and commodities together carry 83% of the risk and bonds carry 2.4%. The portfolio is diversified by label and concentrated by risk. That is the seed of every failure mode in 05 · Failure Modes.

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4. Failure Modes & First-Principles Breakdowns

1. Diversification lives in the correlations, which are unstable. The $\rho$ in the two-asset formula is an estimate and it moves; in a crisis it heads toward $1$, and the diversification dividend in the cross term shrinks. A portfolio “diversified” at $\rho =0.2$ can be a single bet at $\rho =0.85$ (page 05).
2. Counting assets instead of sources. N assets with one shared factor is one bet. Believing otherwise is the most common naive error and it survives every backtest run on calm data.
3. The 60/40 illusion. Capital balance $\ne$ risk balance. Anything that leaves ~90%+ of risk in one leg is one asset with decoration - see Risk Parity · 01.
4. Strategic drift. Letting tactical tilts accumulate (no tracking-error budget, no rebalancing) means the strategic plan is rewritten by noise.

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5. References

• Ang, Asset Management: A Systematic Approach to Factor Investing (2014)
• Ilmanen, Expected Returns (2011)
• Markowitz, “Portfolio Selection,” Journal of Finance 7(1):77–91, 1952
• Qian, “Risk Parity Portfolios: Efficient Portfolios Through True Diversification,” PanAgora, 2005

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6. Connected Graph Bridges

• Base: MPT & the Efficient Frontier · Linear Algebra & Matrices
• Continue: 02 · Asset-Class Allocation · Index Hub
• Sibling: Risk Parity & ERC · Factor Investing & Timing