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Michael Brenndoerfer

Modern Portfolio Theory and Mean-Variance Optimization › Practical Implementation with Simulated Return Data

Michael BrenndoerferDecember 13, 20255 min read

Part 9 of 15Portfolio Theory and Investment Analysis

Chapter 46Modern Portfolio Theory and Mean-Variance Optimization › Practical Implementation with Simulated Return Data

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Let's apply mean-variance optimization to a seeded five-asset Gaussian return sample with a specified covariance structure. The 1,260 observations represent five simulated trading years, not historical market observations.

In[26]:

Code

import numpy as np

# Simulated daily returns for 5 assets (252 trading days x 5 simulated years)
np.random.seed(42)
n_days = 252 * 5
n_assets = 5
asset_names = ["Large Cap", "Small Cap", "Intl Stocks", "Bonds", "REITs"]

# True parameters for simulation
true_annual_returns = np.array([0.10, 0.12, 0.08, 0.04, 0.09])
true_annual_vols = np.array([0.16, 0.22, 0.18, 0.04, 0.20])

# Correlation structure
true_corr = np.array(
    [
        [1.00, 0.75, 0.65, 0.10, 0.55],
        [0.75, 1.00, 0.60, 0.05, 0.50],
        [0.65, 0.60, 1.00, 0.15, 0.45],
        [0.10, 0.05, 0.15, 1.00, 0.20],
        [0.55, 0.50, 0.45, 0.20, 1.00],
    ]
)

# Convert to daily parameters
daily_returns_mean = true_annual_returns / 252
daily_vols = true_annual_vols / np.sqrt(252)
true_cov_daily = np.diag(daily_vols) @ true_corr @ np.diag(daily_vols)

# Generate returns using Cholesky decomposition
L = np.linalg.cholesky(true_cov_daily)
z = np.random.randn(n_days, n_assets)
returns = daily_returns_mean + z @ L.T

In[27]:

Code

# Estimate annualized parameters from simulated daily returns (IID arithmetic scaling)
estimated_mu = returns.mean(axis=0) * 252  # Annualize
estimated_cov = np.cov(returns.T) * 252  # Annualize

# Compute the long-only minimum-variance curve, including its lower branch
frontier_ret, frontier_risk, frontier_w = efficient_frontier(
    estimated_mu, estimated_cov, n_points=100, allow_short=False
)

# Select the best sampled Sharpe-ratio portfolio (approximate tangent), rf = 2%
rf = 0.02
sharpe_ratios = (frontier_ret - rf) / frontier_risk
max_sharpe_idx = np.argmax(sharpe_ratios)
tangent_weights = frontier_w[max_sharpe_idx]
tangent_return = frontier_ret[max_sharpe_idx]
tangent_risk = frontier_risk[max_sharpe_idx]
max_sharpe_ratio = sharpe_ratios[max_sharpe_idx]

# Calculate Capital Allocation Line (CAL)
cml_risks = np.linspace(0, tangent_risk * 1.5, 50)
cml_returns = rf + max_sharpe_ratio * cml_risks

Out[28]:

Visualization

Simulated-return risk-return chart with five labeled assets, a long-only minimum-variance curve, an approximate tangent star near 16.8% risk and 13.5% return, and a dashed capital-allocation line from the 2% risk-free point. — Minimum-variance curve estimated from seeded simulated returns. The star marks the highest-Sharpe portfolio among 100 target-return grid points, an approximation to continuous tangency. The dashed capital-allocation line combines this fund with a 2% risk-free asset; its extension assumes frictionless borrowing and lending at that same rate.

This chart uses simulated returns. The dashed capital-allocation line combines the 2% risk-free asset with the best Sharpe portfolio found on the finite grid. It approximates the tangency opportunity set in the stated mean-variance model; extending beyond the risky fund assumes borrowing at the same risk-free rate, without trading frictions or additional leverage constraints.

Out[29]:

Console

Approximate Tangent Portfolio (Best Sampled Sharpe Ratio):
--------------------------------------------------
  Large Cap      :  14.49%
  Small Cap      :   0.00%
  Intl Stocks    :   0.00%
  Bonds          :   7.02%
  REITs          :  78.50%
--------------------------------------------------
  Expected Return: 13.45%
  Standard Deviation: 16.79%
  Sharpe Ratio: 0.682

Out[30]:

Visualization

Donut chart showing the tangent portfolio allocated 78.5% to REITs, 14.5% to Large Cap, and 7.0% to Bonds. — Allocation of the approximate maximum-Sharpe portfolio in this simulated example. The selected long-only grid solution allocates about 78.5% to REITs, 14.5% to Large Cap, and 7.0% to Bonds; Small Cap and International Stocks receive zero weight.

The selected portfolio has the highest Sharpe ratio among the sampled targets. In the frictionless mean-variance model with a suitable tangency portfolio and unrestricted borrowing and lending at the same risk-free rate, efficient allocations combine that risky fund with the risk-free asset. Actual funding limits, objectives, and exposures can change this conclusion. The next chapter connects portfolio choice to CAPM equilibrium.