5.9.2 Asset-Class Allocation

Basic Prerequisites: 01 · From Zero Intuition and MPT & the Efficient Frontier.

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1. Intuition & Practical Objective

Asset-class allocation is the top of the portfolio tree: given four or five coarse building blocks - equities, government bonds, commodities, credit - decide how much capital each gets. The practical objective of this page is the strategic answer plus its disciplined tactical amendment, and the closed-form mean-variance machinery that produces both.

The building blocks behave very differently, and that difference is the raw material:

Block	Role	Typical risk	Return source
Equities	growth engine	high ($\sigma \approx 16\%$)	equity risk premium
Government bonds	deflation hedge / ballast	low ($\sigma \approx 5\%$)	term / roll-down premium + flight-to-quality
Commodities	inflation hedge	high ($\sigma \approx 18\%$)	roll yield, scarcity, inflation beta
Credit	carry engine	medium ($\sigma \approx 7\%$)	credit risk premium

The strategic allocation answers “what should I hold over the next decade?” and uses long-run expected returns and covariances. The tactical allocation answers “what should I overweight for the next quarter?” and is a small, budgeted deviation from the strategic anchor. The single most important discipline: never let tactical drift silently re-write strategic policy.

The one-sentence essence. “Strategic allocation sets the anchor from long-run risk premia; tactical allocation is a tracking-error-budgeted perturbation around it - and a mean-variance solution that leans on a single premium (here credit, 57%) is a warning, not a plan.”

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2. Mathematical Ground Truth & Derivations

The mean-variance problem. Choose weights $w$ to maximise risk-adjusted return:

$$\underset{w}{\max }{\mu }^{\top }w-\frac{\lambda }{2}{w}^{\top }\Sigma w\text{s.t.}{1}^{\top }w=1.$$

Karush–Kuhn–Tucker with the budget constaint. With Lagrangian $\mathcal{L}={\mu }^{\top }w-\frac{\lambda }{2}{w}^{\top }\Sigma w-\gamma (1^{\top }w-1)$, the first-order condition is $\mu -\lambda \Sigma w-\gamma 1=0$, giving the two-fund structure

$$w^{\star }=\frac{1}{\lambda }{\Sigma }^{-1}(\mu -\gamma 1),$$

i.e. any efficient portfolio is a combination of the min-variance portfolio and the tangency portfolio - the two-fund (separation) theorem of Tobin (1958).

Global minimum-variance portfolio (the anchor with no return view at all):

$$\boxed{w_{\text{GMV}}=\frac{{\Sigma }^{-1}1}{1^{\top }{\Sigma }^{-1}1}},{\sigma }_{\text{GMV}}=\frac{1}{\sqrt{1^{\top }{\Sigma }^{-1}1}}.$$

Tangency / maximum-Sharpe portfolio (the anchor that maximises excess return per unit of risk):

$$\boxed{w_{\tan }=\frac{{\Sigma }^{-1}(\mu -r_{f}1)}{1^{\top }{\Sigma }^{-1}(\mu -r_{f}1)}},\mathrm{SR}=\sqrt{(\mu -r_{f}1{)}^{\top }{\Sigma }^{-1}(\mu -r_{f}1)}.$$

Reality check on the inputs. Chopra & Ziemba (1993): errors in means dominate errors in covariances by roughly $20\times$ (and errors in variances by $\sim 11\times$) in the MV objective. Since $\mu$ is the least estimable input, the unconstrained tangency portfolio is the most fragile object in finance. That is why the disciplined version either shrinks $\mu$ (Bayesian priors, reverse optimisation - see Black–Litterman), shrinks $\Sigma$ (Ledoit–Wolf & RMT), constrains the weights (long-only, caps), or simply equal-weights.

Strategic vs tactical. Policy $w_{\text{SAA}}$ solves the MV problem on long-run inputs; tactical $w=w_{\text{SAA}}+\Delta w$ with the tracking-error budget $\Delta w^{\top }\Sigma \Delta w\le {\mathrm{TE}}^2$. The tactical problem is the same MV problem but on a relatively better-estimated signal set - carry, valuation, trend - and with a hard cap on the deviation. The forecast origin differs, not the machinery.

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3. Computational Implementation - asset-class MVO

Runnable closed-form allocation across the four asset classes (numpy). It shows the min-variance anchor and the unconstrained tangency portfolio, plus the concentration the latter produces.

Two lessons from the numbers: (a) the min-variance portfolio is dominated by bonds and credit with almost no commodities - risk minimisation is not diversification; (b) the tangency portfolio puts 57% in a single premium (credit) because its estimated Sharpe is highest. Unconstrained MVO concentrates; the discipline of constraints and input shrinkage (this folder’s failure-mode page) exists precisely to stop this from being mistaken for diversification.

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4. Failure Modes & First-Principles Breakdowns

1. The tangency portfolio is an estimation-error maximiser. It maximises ${\Sigma }^{-1}(\mu -r_{f}1)$, so it loads hardest on the assets with the largest estimated excess return - i.e. the largest estimation error. Best & Grauer (1991): a 1% change in a mean can move weights by 50%+.
2. Min-variance is not diversification. As above, GMV concentrates risk in the compliers (bonds, credit). Diversification of sources is a different objective from minimising variance.
3. Correlation symmetry is assumed away. MVO treats upside and downside covariance as identical; in reality downside correlations are higher, so the “efficient” portfolio is riskier when it matters than the math says (page 05).
4. Tactical drift without a budget. If $\Delta w$ is unconstrained it stops being tactical; the strategic policy gets re-written by the latest quarter’s noise.

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5. References

• Markowitz, “Portfolio Selection,” Journal of Finance 7(1):77–91, 1952
• Tobin, “Liquidity Preference as Behavior Toward Risk,” Review of Economic Studies 25(2):65–86, 1958
• Merton, “An Analytic Derivation of the Efficient Portfolio Frontier,” JFQA 7(4):1851–1872, 1972
• Chopra & Ziemba, “The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice,” JPM 19(2):6–11, 1993
• DeMiguel, Garlappi & Uppal, “Optimal Versus Naive Diversification,” RFS 22(5):1915–1953, 2009
• Ang, Asset Management (2014)

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · Factor-Based Allocation · Index Hub
• Sibling: MPT & the Efficient Frontier · Black–Litterman · Covariance Shrinkage & RMT