joellewis/asset-allocation

Asset Allocation

Core Concepts

Strategic Asset Allocation (SAA)

The long-term policy portfolio based on an investor's risk tolerance, return objectives, time horizon, and constraints. SAA determines the baseline target weights (e.g., 60% equity / 30% bonds / 10% alternatives) and is the dominant driver of long-term portfolio returns. SAA should be revisited when investor circumstances change, not in response to market movements.

Tactical Asset Allocation (TAA)

Short-to-medium-term deviations from the SAA based on market views, valuations, or momentum signals. TAA requires a disciplined process to avoid becoming ad hoc market timing. Key considerations:

• Define allowable deviation bands (e.g., +/- 10% from SAA)
• Have a clear signal framework (valuation, momentum, macro)
• Set reversion rules: when to return to SAA weights

Mean-Variance Optimization (MVO)

Markowitz's framework for finding optimal portfolio weights that maximize risk-adjusted return:

max w'*mu - (lambda/2) * w'Sigmaw

subject to: sum(w_i) = 1, w_i >= 0 (if long-only), and any additional constraints.

Where:

• w = weight vector
• mu = expected return vector
• Sigma = covariance matrix
• lambda = risk aversion parameter

MVO requires three inputs: expected returns, the covariance matrix, and risk aversion. The solution is highly sensitive to expected return inputs.

Black-Litterman Model

Combines market equilibrium returns with investor views to produce more stable, intuitive portfolio weights. Two-step process:

Step 1 — Implied Equilibrium Returns: Pi = lambda * Sigma * w_mkt

where w_mkt is the market-capitalization weight vector, lambda is the risk aversion parameter, and Sigma is the covariance matrix. These are the returns the market implicitly expects given current prices.

Step 2 — Blending with Views: E(R) = [(tau*Sigma)^(-1) + P'*Omega^(-1)P]^(-1) * [(tauSigma)^(-1)*Pi + P'*Omega^(-1)*Q]

where:

• tau = scalar (uncertainty of equilibrium, typically 0.025-0.05)
• P = pick matrix (identifies assets in each view)
• Q = view vector (expected returns from views)
• Omega = diagonal matrix of view uncertainties

The result is a posterior expected return vector that tilts away from equilibrium toward the investor's views, proportional to confidence.

Risk Parity

Equalizes the risk contribution from each asset (or factor) rather than equalizing capital allocation:

RC_i = w_i * (Sigma*w)_i / sigma_p

Set RC_i = RC_j for all i, j.

In a simple two-asset case with no correlation: w_i is proportional to 1/sigma_i

Risk parity portfolios allocate more capital to lower-volatility assets (typically bonds) and often require leverage to achieve competitive return targets.

Glide Path

An age-based or time-based allocation that systematically shifts from growth assets to defensive assets as the investor ages or the target date approaches:

Common rule of thumb: Equity % = 110 - Age

Target-date fund glide paths typically:

• Start at 90% equity for young investors
• Decrease by ~1-2% per year
• Reach 30-40% equity at retirement
• Continue to "through" allocation post-retirement

Core-Satellite

A hybrid approach combining:

• Core (60-80%): Low-cost, broadly diversified index funds or ETFs
• Satellites (20-40%): Active strategies, factor tilts, alternatives, or concentrated positions

This structure captures the market return efficiently (core) while allowing alpha generation or specific exposures (satellites).

Asset-Liability Matching

For investors with defined liabilities (pensions, insurance, endowments with spending rules):

• Match asset duration and cash flows to liability duration and timing
• Surplus optimization: optimize the portfolio relative to liabilities, not absolute return
• Liability-driven investing (LDI): hedge liability risk with duration-matched bonds, invest surplus in return-seeking assets

Key Formulas

| Formula | Expression | Use Case | |---------|-----------|----------| | MVO Objective | max w'*mu - (lambda/2)w'Sigmaw | Optimal portfolio weights | | Equilibrium Returns | Pi = lambda * Sigma * w_mkt | Black-Litterman starting point | | BL Posterior | E(R) = [(tauSigma)^(-1) + P'*Omega^(-1)P]^(-1) * [(tauSigma)^(-1)*Pi + P'*Omega^(-1)Q] | Blended expected returns | | Risk Contribution | RC_i = w_i * (Sigmaw)_i / sigma_p | Risk parity target | | Risk Parity Condition | RC_i = RC_j for all i, j | Equal risk contribution | | Glide Path Rule | Equity % = 110 - Age | Age-based allocation |

Worked Examples

Example 1: Three-Asset Mean-Variance Optimization

Given:

• Assets: US Equity (mu=8%, sigma=16%), Int'l Equity (mu=7%, sigma=18%), US Bonds (mu=3%, sigma=4%)
• Correlations: US/Intl Equity = 0.75, US Equity/Bonds = 0.10, Intl Equity/Bonds = 0.05
• Risk aversion: lambda = 4
• Constraints: long-only, fully invested

Calculate: Optimal weights

Solution:

Covariance matrix:

• Cov(US,US) = 0.16^2 = 0.0256
• Cov(Intl,Intl) = 0.18^2 = 0.0324
• Cov(Bond,Bond) = 0.04^2 = 0.0016
• Cov(US,Intl) = 0.75 * 0.16 * 0.18 = 0.0216
• Cov(US,Bond) = 0.10 * 0.16 * 0.04 = 0.00064
• Cov(Intl,Bond) = 0.05 * 0.18 * 0.04 = 0.00036

MVO with lambda=4 (solving numerically or via quadratic programming):

Optimal weights (long-only):

• US Equity: 51.9%
• Int'l Equity: 0%
• US Bonds: 48.1%

Portfolio: expected return = 5.60%, volatility = 8.71%

Note: International equity is driven to zero — it is highly correlated with US equity (0.75) but has a lower expected return, so the optimizer sees no reason to hold it. This is classic MVO behavior: small input differences produce corner solutions. Adding a maximum-weight or minimum-allocation constraint would force diversification. The high bond allocation reflects the heavy variance penalty (lambda=4); reducing lambda shifts toward equities.

Example 2: Black-Litterman with a Relative View

Given: The same three assets and covariance matrix as Example 1.

• Market-cap weights: US Equity 55%, Int'l Equity 30%, US Bonds 15%
• Risk aversion lambda = 2.5, tau = 0.05
• Investor view: Int'l Equity will outperform US Bonds by 3% (view uncertainty Omega = [0.001]; lower = higher confidence)

Calculate: Equilibrium and posterior expected returns

Solution:

Step 1 — Equilibrium returns, Pi = lambda × Sigma × w_mkt:

• US Equity: 5.16%
• Int'l Equity: 5.41%
• US Bonds: 0.18%

Step 2 — View specification: P = [0, 1, -1], Q = [3%].

The equilibrium already implies Int'l beats Bonds by 5.23%, so a 3% view is bearish relative to equilibrium. Applying the Black-Litterman posterior formula:

• US Equity: 4.28% (pulled down via its 0.75 correlation with Int'l)
• Int'l Equity: 4.07% (down from 5.41%)
• US Bonds: 0.23% (up slightly)

The posterior tilts returns toward the view in proportion to confidence. Fed into MVO, these returns shift weights away from equities and toward bonds relative to market-cap weights — moderately, avoiding the extreme corner solutions that raw MVO produces (compare Example 1). Note that views are always evaluated relative to what equilibrium already implies, not in isolation.

Common Pitfalls

• MVO is highly sensitive to expected return inputs and has been called an "error maximizer" — small changes in returns produce large changes in weights
• Unconstrained MVO often produces extreme, concentrated positions — always add constraints (long-only, max weight, turnover limits)
• Black-Litterman requires the analyst to specify confidence in views (Omega), which is itself uncertain
• Risk parity portfolios require leverage to achieve equity-like returns, introducing borrowing costs and leverage risk
• Ignoring implementation costs: transaction costs, bid-ask spreads, and taxes can significantly erode theoretical optimal returns
• Ignoring liquidity constraints: some asset classes (private equity, real estate) cannot be rebalanced quickly
• Glide paths assume a generic investor — individual circumstances may require customization
• Over-reliance on historical covariance matrices that may not reflect future relationships

Cross-References

• historical-risk: volatility and correlation inputs for mean-variance optimization
• forward-risk: expected return forecasts and scenario analysis for portfolio optimization
• diversification: diversification principles underpin all allocation frameworks
• bet-sizing: position sizing within the allocated asset classes
• rebalancing: maintaining allocation targets over time
• quantitative-valuation: valuation signals can inform TAA decisions

Running the Script

uv run scripts/asset_allocation.py            # run the demo (uses PEP 723 inline deps)
uv run scripts/asset_allocation.py --verify   # check demo outputs against the worked examples (exit 1 on mismatch)
python3 scripts/asset_allocation.py            # alternative (requires: pip install numpy scipy)

The demo prints the calculations covered above; its values match the worked examples in this skill. Run --help for a list of the classes and functions. For programmatic use, import the module rather than running it — the demo only executes under python asset_allocation.py.