Mean-Variance Optimization (MVO) is the most common quantitative approach to asset allocation. It assumes investors are risk-averse and seek to maximize utility.
The MVO Objective Function: The goal is to maximize the utility (\(U_m\)) of asset mix \(m\):
\[U_m = E(R_m) - 0.005 \times \lambda \times \sigma^2_m\]
Where: * \(E(R_m)\) = Expected return of the mix (in %). * \(\sigma^2_m\) = Variance of the mix (in %). * \(\lambda\) = Risk Aversion Coefficient (Higher \(\lambda\) \(\rightarrow\) greater penalty for risk). * \(\lambda\) ranges from 1 (aggressive) to 10 (conservative). Moderate \(\approx\) 4.
Constraints: * Budget Constraint: Sum of weights must equal 1 (\(\sum w_i = 1\)). * Non-Negativity Constraint: Weights cannot be negative (\(w_i \ge 0\)) unless short selling is allowed.
Criticisms of MVO
The Economic Balance Sheet includes financial and non-financial assets (Human Capital, Real Estate, Pension Wealth).
The Global Market Portfolio (GMP) represents the supply of all risky assets.
Monte Carlo Simulation addresses the single-period limitation of MVO.
Scenario Analysis: * Stress tests the portfolio against specific historical or hypothetical events (e.g., “What if 2008 happens again?” or “What if interest rates rise 2%?”). * Useful for evaluating Tail Risk (losses in extreme markets) which MVO normal distributions underestimate.
Liquidity affects asset allocation in three ways:
Risk Budgeting allocates the total permissible risk (like a currency) to different assets or managers.
Marginal Contribution to Total Risk (MCTR): * The change in portfolio risk for a small change in an asset’s weight. * * \[MCTR_i = \beta_i \times \sigma_p\] * (Beta of asset \(i\) to portfolio \(\times\) Portfolio volatility).
Optimal Risk Budgeting: * Allocations are optimal when the ratio of Excess Return to MCTR is equal for all assets. \[\frac{E(R_i) - R_f}{MCTR_i} = \text{Constant for all assets}\]
Factor-Based Asset Allocation allocates to sources of systematic risk (Factors) rather than asset classes.
Used by DB Plans, Insurers, and Banks where the primary goal is paying liabilities.
1. Surplus Optimization * Applies MVO to the Surplus (Assets - Liabilities). * Objective: Maximize Surplus Return net of Surplus Volatility penalty. \[U_{surplus} = E(R_s) - 0.005 \times \lambda \times \sigma^2(R_s)\] * Risk-Free Asset: The asset that best mimics the liability (Hedging Asset).
2. Hedging/Return-Seeking Portfolio (Two-Portfolio Approach) * Portfolio 1 (Hedging): Fully hedges the liability (e.g., cash flow matching, immunization). * Portfolio 2 (Return-Seeking): Invests the surplus in risky assets to generate growth. * Requirement: Requires a funded status \(\ge\) 100% (Surplus > 0) to work purely. If underfunded, the hedging portfolio is partial.
3. Integrated Asset-Liability Management * Jointly optimizes assets and liabilities. * Common for Banks/Insurers who can adjust the liability side (e.g., issuing different insurance products) based on asset opportunities. * Often uses multi-period models.
Goals-Based Investing (GBI) breaks the portfolio into sub-portfolios (mental accounts), each funding a specific goal.
Process: 1. Identify Goals: Categorize by urgency (Essential vs. Aspirational) and horizon. 2. Construct Sub-Portfolios: * Essential Goals (e.g., Living expenses): High probability of success required (e.g., 95-99%). Allocation: Low risk (Cash, Bonds). * Aspirational Goals (e.g., Vacation home, Legacy): Lower probability of success accepted. Allocation: High risk (Equities, PE). 3. Summation: The total AA is the weighted sum of the sub-portfolios.
Advantages: Aligning investments with psychological needs; better client discipline. Disadvantages: May not be mean-variance efficient in the aggregate.
1. 120 Minus Age: * Rule: Equity % = 120 - Age. * Critique: Simple, considers horizon (human capital decline), but ignores individual risk tolerance and wealth levels.
2. 60/40 Split: * Rule: 60% Equity / 40% Bond. * Critique: Historical standard, but ignores valuation/correlation changes. Not liability-aware.
3. Endowment Model (Yale Model): * Focus: High allocation to Alternatives (PE, Hedge Funds, Real Assets) to capture illiquidity premiums. * Critique: Requires long horizon, high expertise, and access to top managers. Hard to replicate for smaller investors.
4. Risk Parity: * Rule: Each asset class contributes equally to total portfolio risk. * Mechanism: Because bonds have lower volatility than stocks, Risk Parity uses leverage to boost bond exposure/risk up to equity levels. * Critique: Relies on leverage (costly/risky); assumes historical correlations hold.
5. 1/N Rule: * Rule: Equal weight to all assets (\(1/N\)). * Critique: Naive diversification. Performance depends entirely on the menu of assets chosen (e.g., if 3 of 4 assets are equity funds, it’s 75% equity).
Rebalancing restores strategic weights, enforcing a “buy low, sell high” discipline. It primarily reduces risk (drift).
Methods: * Calendar: Rebalance periodically (e.g., quarterly). * Range-Based (Corridor): Rebalance when weight deviates by \(\pm X\%\).
Optimal Corridor Width Factors:
| Factor | Change | Impact on Corridor Width | Reasoning |
|---|---|---|---|
| Transaction Costs | Higher | Wider | Trade less frequently to save costs. |
| Risk Tolerance | Higher | Wider | Less sensitive to risk drift. |
| Correlation | Higher | Wider | Asset moves with the portfolio, drift is slower. |
| Volatility | Higher | Narrower | High vol assets drift fast; need tight control to manage risk. |
| Taxes | Higher | Wider | Avoid realizing gains. |
Conflict: High volatility implies narrower bands (risk control), but also implies higher transaction costs (trading cost). Investors must balance these competing objectives.