Background: Mean-Variance Portfolio Optimization

"Don't put all your eggs in one basket—but calculate exactly how many to put in each."

Core Idea

Problem: Given N assets, how to allocate capital to maximize returns and minimize risk?

Markowitz's Insight: Returns can be weighted averaged, but risk isn't simply additive—correlation determines the effectiveness of diversification.

Basic Math

Portfolio Return

Expected Portfolio Return = Sum(wi x mui)

Where:
  wi = weight of asset i
  mui = expected return of asset i

Example: Two assets

AAPL: weight 60%, expected return 12%
MSFT: weight 40%, expected return 10%

Expected Portfolio Return = 0.6 x 12% + 0.4 x 10% = 11.2%

Portfolio Risk

Portfolio Variance = Sum_i Sum_j (wi x wj x sigma_ij)

Where:
  sigma_ij = covariance between assets i and j
  sigma_ii = variance of asset i

Simplified form for two assets:

sigma_p^2 = w1^2*sigma1^2 + w2^2*sigma2^2 + 2*w1*w2*rho*sigma1*sigma2

Where:
  rho = correlation coefficient between the two assets

The Power of Correlation

Example: Two assets, each with 50% weight

• Asset A: volatility 20%
• Asset B: volatility 20%

Key Insight: Lower correlation means better diversification.

Efficient Frontier

Definition: The set of portfolios that achieve the highest expected return for a given level of risk.

Return
  |                    *------- Highest return point
  |                 ***
  |              ***
  |           ***    <- Efficient Frontier
  |        ***
  |     ***
  |  ***------------- Minimum variance point
  |
  +--------------------> Risk

Points below the efficient frontier are "inefficient":
Same risk could achieve higher return

Computing the Efficient Frontier

Optimization Problem:

Maximize: Portfolio Return = w'mu
Subject to:
  1. Portfolio Risk = sqrt(w'Sigma*w) <= sigma_target
  2. Sum(wi) = 1 (weights sum to 1)
  3. wi >= 0 (optional: no short selling)

Practical Calculation Example

Three-asset portfolio: AAPL, MSFT, GOOGL

Input Data:

Expected Returns (annualized):
  AAPL: 15%
  MSFT: 12%
  GOOGL: 18%

Volatility (annualized):
  AAPL: 25%
  MSFT: 20%
  GOOGL: 30%

Correlation Matrix:
       AAPL  MSFT  GOOGL
AAPL   1.0   0.7   0.6
MSFT   0.7   1.0   0.5
GOOGL  0.6   0.5   1.0

Minimum Variance Portfolio:

Weights: AAPL 25%, MSFT 55%, GOOGL 20%
Return: 13.5%
Volatility: 17.2%

Maximum Sharpe Portfolio (assuming risk-free rate 2%):

Weights: AAPL 30%, MSFT 30%, GOOGL 40%
Return: 15.6%
Volatility: 20.1%
Sharpe: 0.68

Common Optimization Objectives

Real-World Implementation Issues

Issue 1: Estimation Error

Theory requires: Precise expected returns and covariance matrix

Reality: Estimated from historical data, with large errors

Estimation Error Impact:

Return estimation error → Large weight fluctuations
  - Historical 5-year AAPL return 18%
  - But future could be 10% or 25%
  - Small return prediction changes → Dramatic weight changes

Covariance estimation error → Unstable correlations
  - Normal periods: AAPL-MSFT correlation 0.6
  - Crisis periods: Correlation spikes to 0.9
  - Diversification effect disappears

Issue 2: Extreme Weights

Unconstrained optimization often produces extreme results:

Theoretical optimum:
  Asset A: +250% (long)
  Asset B: -150% (short)

Problems:
  - High leverage risk
  - Short selling costs
  - Liquidity constraints

Solution: Add constraints

Common constraints:
  - 0 <= wi <= 30% (single asset cap)
  - Sum(wi) = 1 (fully invested)
  - wi >= 0 (no short selling)

Issue 3: High Turnover

Optimization results are sensitive to inputs, each reoptimization may produce large rebalancing:

This month optimal: AAPL 40%, MSFT 30%, GOOGL 30%
Next month optimal: AAPL 20%, MSFT 50%, GOOGL 30%

Turnover: |40-20| + |30-50| + |30-30| = 40%
Cost: 40% x 0.2% x 2 = 0.16%

Annualized cost can consume most of the excess returns

Solutions:

1. Turnover penalty: Objective function - lambda x Turnover
2. Only rebalance when deviation exceeds threshold
3. Use more stable estimation methods

Improvement Methods

1. Shrinkage Estimation

"Shrink" sample estimates toward more stable priors:

Shrunk Covariance = alpha x Sample Covariance + (1-alpha) x Structured Estimate

Common structured estimates:
  - Diagonal matrix (assume uncorrelated)
  - Single-factor model
  - Equal correlation model

2. Black-Litterman Model

Combines market equilibrium with subjective views:

Inputs:
  1. Market equilibrium returns (implied from market cap weights)
  2. Investor views (e.g., "I believe AAPL will outperform MSFT by 3%")
  3. View confidence levels

Output:
  Adjusted expected returns → More stable weights

3. Risk Parity

Don't predict returns, just balance risk contributions:

Goal: Each asset contributes equally to portfolio risk

Three-asset example:
  Total risk = 15%
  Each asset contribution = 5%

Result: Low volatility assets get higher weights, high volatility assets get lower weights

Multi-Agent Perspective

In a multi-agent architecture, portfolio optimization can be applied as follows:

Signal Agents (multiple)
  |
  +-- Agent A: Output AAPL expected return
  +-- Agent B: Output MSFT expected return
  +-- Agent C: Output GOOGL expected return
       |
       v
Portfolio Agent (optimizer)
  |
  +-- Input: Return predictions from each Agent
  +-- Estimate covariance matrix
  +-- Execute mean-variance optimization
  +-- Output: Target weights
       |
       v
Risk Agent
  |
  +-- Check if weights violate risk limits
  +-- Check if turnover is too high
  +-- Adjust or reject proposed weights

Common Misconceptions

Misconception 1: Historically optimal portfolio will be optimal in the future

Wrong. Optimization is the perfect overfitting tool:

• Over-reliance on historical data noise
• Historical correlations can change
• Expected return estimates are unreliable

Misconception 2: More assets means better diversification

There's an upper limit. Marginal benefits diminish:

Number of assets vs diversification effect:
  2 -> 10: Significant risk reduction
  10 -> 30: Moderate effect
  30 -> 100: Limited effect, increased complexity

Misconception 3: Covariance matrix is stable

Dangerous assumption. Covariance changes dramatically across regimes:

• Normal periods: Diversification works
• Crisis periods: Correlations approach 1, diversification fails

Practical Recommendations

1. Start Simple

Starting choices:
- Equal weight (1/N): Robust, no estimates needed
- Risk parity: No return prediction needed
- Minimum variance: Only covariance estimate needed

2. Add Reasonable Constraints

Recommended constraints:
- Single asset weight <= 30%
- No short selling (unless you have a clear short strategy)
- Turnover penalty

3. Rebalance Periodically

Rebalancing strategies:
- Fixed period: Monthly/Quarterly
- Threshold trigger: When deviation from target weight >5%
- Combination: Threshold trigger + minimum interval

Summary