5.9 Multi-Asset & Factor Allocation

Basic Prerequisites: Mean–Variance Optimization & the Efficient Frontier and Factor Investing & Factor Timing. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

“Multi-asset & factor allocation” is the discipline of deciding what to hold at the top of the portfolio tree - not which 500 stocks, but how much equity, how much bonds, how much commodities, how much credit, and - the modern refinement - how much value, momentum, carry, and defensive exposure regardless of the label attached to the vehicle. It is the bridge between Pillar 1 (where the return sources are identified) and Pillar 5’s optimizer (where they are held).

The practical objective of this folder is a lookup hub plus a walked path: the formulas you need to size an asset-class or factor allocation, the carry identities that price each building block, and the failure modes that make the whole exercise fragile in a crisis.

Two framings, one decision:

1. Asset-class allocation - the traditional split across equities / bonds / commodities / credit. The building blocks are labels, and the covariance matrix of the labels is the object of study. A 60/40 portfolio looks balanced in dollars and is overwhelmingly equity risk in reality (≈98% here - see §3).
2. Factor-based allocation - the modern view (Ang 2014). Asset classes are bundles of factors; two “different” assets can be the same bet, and two “same” assets can be different bets. Allocating to factor premia directly decorrelates the portfolio far more effectively than allocating to labels: the diversified factor portfolio here carries a diversification ratio of 1.98 against 1.46 for the asset-class portfolio.

The one-sentence essence. “Allocate across return sources, not across names - a diversified asset-class portfolio is usually a concentrated factor bet, and a diversified factor portfolio is what you thought you owned.”

2. Mathematical Ground Truth & Derivations

Notation. $w\in {\mathbb{R}}^N$ weights (fractions of capital, $1^{\top }w=1$), $\Sigma$ the covariance matrix, ${\sigma }_i=\sqrt{{\Sigma }_{ii}}$, $\mu$ expected returns, $r_f$ the risk-free rate, $B$ the $N\times K$ factor-loading matrix, $f$ the $K$ factor returns, $\epsilon$ idiosyncratic noise.

The three decompositions that matter.

• Capital vs risk. Euler’s theorem on the degree-1 homogeneous $\sigma (w)$ splits total risk exactly: $\sigma (w)={\sum }_i{w}_i\partial \sigma /\partial w_i$. This is why a "$60/40$" portfolio is a $97.75\%$-equity bet: dollars and risk are different units.
• Asset vs factor. ${\Sigma }_r=B{\Sigma }_f{B}^{\top }+D$ separates systematic co-movement (through $B,{\Sigma }_f$) from idiosyncratic noise $D$. A cross-asset correlation of $0.2$ can be entirely a single shared factor; once you allocate to the factors, the residual correlations vanish.
• Strategic vs tactical. Strategic allocation fixes a long-horizon policy weight $w_{\text{SAA}}$ from long-run $\mu ,\Sigma$; tactical allocation overlays a deviation $\Delta w$ driven by short-horizon signals (valuation, carry, trend, regime), subject to a tracking-error budget $\Delta w^{\top }\Sigma \Delta w\le {\text{TE}}^2$.

3. Computational Implementation - the hub engine

One self-contained snippet (numpy) that computes every headline number in the table: portfolio volatility, the diversification ratio, risk shares, and the factor-vs-asset contrast.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s full failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Diversification is regime-dependent. Cross-asset correlations rise precisely when you need them low; the equal-weight portfolio’s volatility jumps $1.19\times$ from calm to crisis and its effective number of bets collapses ($3.17\to 2.20$).
2. Factor crowding. Factors are traded - a popular factor gets crowded, its Sharpe decays, and its correlation to everything else rises (the bridge to Pillar 1’s crowding page).
3. Estimation error is the dominant term. A mean-variance optimizer fed 24 months of noisy history posts a worse out-of-sample Sharpe ($0.14$) than naive $1/N$ ($0.30$), at roughly $8\times$ the volatility - MVO is an estimation-error maximizer.
4. Carry is not free. The carry premium is compensation for crash risk; carry strategies have a famously negative skew (the “carry crash”).

5. References

• Ang, Andrew: Asset Management: A Systematic Approach to Factor Investing (Oxford University Press, 2014)
• Meucci, Attilio: Risk and Asset Allocation (Springer Finance, 2005)
• Ilmanen, Antti: Expected Returns: An Investor’s Guide to Harvesting Market Rewards (Wiley, 2011)
• Qian, Hua & Sorensen: Quantitative Equity Portfolio Management (Chapman & Hall/CRC, 2007)
• Grinold & Kahn: Active Portfolio Management (2nd ed., 2000)

6. Connected Graph Bridges

• Sub-pages (in-folder): 01 · From Zero Intuition · 02 · Asset-Class Allocation · 03 · Factor-Based Allocation · 04 · Carry & Styles · 05 · Failure Modes · 06 · Advanced Extensions
• Sibling topics (Pillar 5): Black–Litterman Bayesian Allocation (turning views into strategic weights) · Risk Parity & ERC (the all-weather multi-asset idea) · MPT & the Efficient Frontier · Covariance Shrinkage & RMT · Hierarchical Risk Parity
• Cross-pillar: Factor Investing & Factor Timing (the factor zoo, crowding, timing) · Regime Detection (the input to page 06)
• Foundations: Linear Algebra & Matrices · Calculus & Convex Optimization · Statistics & Inference

Beginner: start at 01 · Practitioner: start at 05