Advanced Portfolio Optimization

Mean-variance portfolio optimization with covariance using w.dot(Σ @ w)
Published

February 8, 2026

1 Introduction

This tutorial builds a complete Markowitz mean-variance portfolio optimization model using Optyx. You’ll see how VectorVariable, math-like w.dot(Σ @ w) syntax, and Parameter work together for realistic portfolio problems.

By the end, you’ll be able to:

  • Model portfolio variance using quadratic forms
  • Implement efficient frontiers
  • Use Parameters for fast sensitivity analysis
  • Handle real-world constraints (sector limits, turnover)

2 The Markowitz Model

Harry Markowitz’s insight: investors care about both return (higher is better) and risk (lower is better). The mean-variance model:

\[ \begin{aligned} \text{minimize} \quad & \mathbf{w}^\top \mathbf{\Sigma} \mathbf{w} \\ \text{subject to} \quad & \mathbf{w}^\top \boldsymbol{\mu} \geq r_{\text{target}} \\ & \mathbf{w}^\top \mathbf{1} = 1 \\ & \mathbf{w} \geq 0 \end{aligned} \]

Where: - \(\mathbf{w}\) = portfolio weights (decision variables) - \(\boldsymbol{\mu}\) = expected returns vector - \(\mathbf{\Sigma}\) = covariance matrix - \(r_{\text{target}}\) = minimum acceptable return

3 Problem Setup

import numpy as np
np.random.seed(42)

# 10 assets
n_assets = 10
asset_names = [f"Asset_{i}" for i in range(n_assets)]

# Expected annual returns (5% to 15%)
expected_returns = np.linspace(0.05, 0.15, n_assets)

# Generate realistic covariance matrix
# Start with random factors, then create covariance
n_factors = 3
factor_loadings = np.random.randn(n_assets, n_factors) * 0.1
specific_var = np.diag(np.random.uniform(0.01, 0.04, n_assets))
covariance = factor_loadings @ factor_loadings.T + specific_var

# Ensure positive definiteness
covariance = (covariance + covariance.T) / 2
min_eig = np.min(np.linalg.eigvalsh(covariance))
if min_eig < 0:
    covariance += (-min_eig + 0.001) * np.eye(n_assets)

# Display statistics
print("Asset Statistics:")
print("-" * 50)
volatility = np.sqrt(np.diag(covariance))
for i in range(n_assets):
    print(f"{asset_names[i]}: return={expected_returns[i]:.1%}, vol={volatility[i]:.1%}")
Asset Statistics:
--------------------------------------------------
Asset_0: return=5.0%, vol=19.3%
Asset_1: return=6.1%, vol=21.8%
Asset_2: return=7.2%, vol=21.6%
Asset_3: return=8.3%, vol=17.9%
Asset_4: return=9.4%, vol=27.9%
Asset_5: return=10.6%, vol=22.7%
Asset_6: return=11.7%, vol=26.0%
Asset_7: return=12.8%, vol=22.5%
Asset_8: return=13.9%, vol=18.9%
Asset_9: return=15.0%, vol=17.7%

4 Basic Model with VectorVariable

from optyx import VectorVariable, Problem

# Portfolio weights
weights = VectorVariable("w", n_assets, lb=0, ub=1)

# Expected portfolio return: μᵀw
portfolio_return = expected_returns @ weights

# Portfolio variance: w · (Σw) = wᵀΣw
portfolio_variance = weights.dot(covariance @ weights)

print(f"Return expression type: {type(portfolio_return).__name__}")
print(f"Variance expression type: {type(portfolio_variance).__name__}")
Return expression type: LinearCombination
Variance expression type: QuadraticForm
NoteMath-like Syntax

w.dot(Σ @ w) reads as “w dot (Σ times w)” = wᵀΣw. Gradients are computed analytically: ∇ variance = 2Σw. No loop over n² terms needed.

5 Minimum Variance Portfolio

Find the portfolio with the lowest risk, requiring at least 8% return:

target_return = 0.08

solution = (
    Problem("min_variance")
    .minimize(portfolio_variance)
    .subject_to(weights.sum().eq(1))           # Fully invested
    .subject_to(portfolio_return >= target_return)  # Target return
    .solve()
)

print("=" * 60)
print("MINIMUM VARIANCE PORTFOLIO")
print("=" * 60)
print(f"Status: {solution.status}")
print(f"Solve time: {solution.solve_time*1000:.2f} ms")
print()

# Extract weights
opt_weights = np.array([solution[f'w[{i}]'] for i in range(n_assets)])
opt_return = expected_returns @ opt_weights
opt_vol = np.sqrt(opt_weights @ covariance @ opt_weights)

print(f"Portfolio Return: {opt_return:.2%}")
print(f"Portfolio Volatility: {opt_vol:.2%}")
print(f"Sharpe Ratio (rf=2%): {(opt_return - 0.02) / opt_vol:.3f}")
print()

print("Asset Allocation:")
for i in range(n_assets):
    if opt_weights[i] > 0.01:
        print(f"  {asset_names[i]}: {opt_weights[i]:.1%}")
============================================================
MINIMUM VARIANCE PORTFOLIO
============================================================
Status: SolverStatus.OPTIMAL
Solve time: 1.90 ms

Portfolio Return: 10.22%
Portfolio Volatility: 6.18%
Sharpe Ratio (rf=2%): 1.330

Asset Allocation:
  Asset_0: 12.5%
  Asset_1: 3.3%
  Asset_2: 19.9%
  Asset_3: 5.4%
  Asset_5: 6.5%
  Asset_6: 19.1%
  Asset_7: 9.0%
  Asset_8: 18.7%
  Asset_9: 5.6%

6 Efficient Frontier

The efficient frontier shows the best risk-return tradeoff. Let’s trace it by varying the target return:

from optyx import VectorVariable, Problem

def solve_for_target(target_return):
    """Solve minimum variance for a given target return."""
    w = VectorVariable("w", n_assets, lb=0, ub=1)
    port_ret = expected_returns @ w
    port_var = w.dot(covariance @ w)  # wᵀΣw
    
    sol = (
        Problem()
        .minimize(port_var)
        .subject_to(w.sum().eq(1))
        .subject_to(port_ret >= target_return)
        .solve()
    )
    
    if sol.status.value == "optimal":
        weights = np.array([sol[f'w[{i}]'] for i in range(n_assets)])
        ret = expected_returns @ weights
        vol = np.sqrt(weights @ covariance @ weights)
        return ret, vol, weights
    return None

# Trace the frontier
targets = np.linspace(0.05, 0.14, 20)
frontier = []

for target in targets:
    result = solve_for_target(target)
    if result:
        frontier.append(result)

# Display results
print("Efficient Frontier (Risk-Return Tradeoff):")
print("-" * 50)
print(f"{'Return':>10} {'Volatility':>12} {'Sharpe':>10}")
print("-" * 50)
for ret, vol, _ in frontier[::3]:  # Show every 3rd point
    sharpe = (ret - 0.02) / vol
    print(f"{ret:>10.2%} {vol:>12.2%} {sharpe:>10.3f}")
Efficient Frontier (Risk-Return Tradeoff):
--------------------------------------------------
    Return   Volatility     Sharpe
--------------------------------------------------
    10.22%        6.18%      1.330
    10.22%        6.18%      1.330
    10.22%        6.18%      1.330
    10.22%        6.18%      1.330
    10.68%        6.24%      1.393
    12.11%        7.32%      1.381
    13.53%        9.72%      1.186

7 Using Parameters for Fast Re-solves

When exploring scenarios, rebuilding the problem each time is wasteful. Use Parameter to update values without recompilation:

from optyx import VectorVariable, Parameter, Problem

# Create problem once
w = VectorVariable("w", n_assets, lb=0, ub=1)
target_param = Parameter("target", value=0.08)

port_ret = expected_returns @ w
port_var = w.dot(covariance @ w)  # wᵀΣw

problem = (
    Problem("parametric_portfolio")
    .minimize(port_var)
    .subject_to(w.sum().eq(1))
    .subject_to(port_ret >= target_param)
)

# Solve for different targets (much faster after first solve)
import time

targets = [0.06, 0.08, 0.10, 0.12]
results = []

print("Parametric Solves:")
print("-" * 50)
for i, target in enumerate(targets):
    target_param.set(target)
    
    start = time.perf_counter()
    sol = problem.solve()
    elapsed = time.perf_counter() - start
    
    weights = np.array([sol[f'w[{i}]'] for i in range(n_assets)])
    vol = np.sqrt(weights @ covariance @ weights)
    
    solve_type = "cold" if i == 0 else "warm"
    print(f"Target {target:.0%}: vol={vol:.2%}, time={elapsed*1000:.2f}ms ({solve_type})")
Parametric Solves:
--------------------------------------------------
Target 6%: vol=6.17%, time=1129.14ms (cold)
Target 8%: vol=6.17%, time=1088.92ms (warm)
Target 10%: vol=6.17%, time=826.10ms (warm)
Target 12%: vol=7.20%, time=820.47ms (warm)
TipParameter Performance

After the first solve, subsequent solves with different parameter values reuse the compiled problem structure. This can be 10-100x faster for complex problems.

8 Real-World Constraints

8.1 Sector Exposure Limits

Limit exposure to any single sector:

# Sector definitions (3 assets each in Tech, Energy, Finance; 1 in Other)
sectors = {
    "Tech": [0, 1, 2],
    "Energy": [3, 4, 5],
    "Finance": [6, 7, 8],
    "Other": [9]
}

w = VectorVariable("w", n_assets, lb=0, ub=1)
port_var = w.dot(covariance @ w)  # wᵀΣw
port_ret = expected_returns @ w

problem = (
    Problem("sector_constrained")
    .minimize(port_var)
    .subject_to(w.sum().eq(1))
    .subject_to(port_ret >= 0.09)
)

# Sector limits: max 40% in any sector
for sector_name, indices in sectors.items():
    sector_weight = sum(w[i] for i in indices)
    problem = problem.subject_to(sector_weight <= 0.40)

solution = problem.solve()

# Display by sector
print("Sector-Constrained Portfolio:")
print("-" * 50)
for sector_name, indices in sectors.items():
    sector_total = sum(solution[f'w[{i}]'] for i in indices)
    print(f"{sector_name}: {sector_total:.1%}")
Sector-Constrained Portfolio:
--------------------------------------------------
Tech: 35.9%
Energy: 16.5%
Finance: 40.0%
Other: 7.6%

8.2 Position Limits

Limit individual positions:

# Maximum 20% in any single asset, minimum 2% if invested
w = VectorVariable("w", n_assets, lb=0, ub=0.20)

# Note: minimum position constraints require binary variables or careful formulation
# For simplicity, we just use upper bounds here

solution = (
    Problem("position_limited")
    .minimize(w.dot(covariance @ w))  # wᵀΣw
    .subject_to(w.sum().eq(1))
    .subject_to(expected_returns @ w >= 0.08)
    .solve()
)

print("Position-Limited Portfolio:")
print("-" * 40)
weights = np.array([solution[f'w[{i}]'] for i in range(n_assets)])
for i in range(n_assets):
    if weights[i] > 0.01:
        print(f"  {asset_names[i]}: {weights[i]:.1%}")

print(f"\nMax position: {weights.max():.1%}")
print(f"Positions > 1%: {np.sum(weights > 0.01)}")
Position-Limited Portfolio:
----------------------------------------
  Asset_0: 12.5%
  Asset_1: 3.3%
  Asset_2: 19.9%
  Asset_3: 5.4%
  Asset_5: 6.5%
  Asset_6: 19.1%
  Asset_7: 9.0%
  Asset_8: 18.7%
  Asset_9: 5.6%

Max position: 19.9%
Positions > 1%: 9

9 Maximum Sharpe Ratio Portfolio

The tangency portfolio maximizes risk-adjusted return. This is a more complex problem requiring a different formulation:

from optyx import VectorVariable, Problem

# Risk-free rate
rf = 0.02

# For max Sharpe, we minimize variance for unit excess return
# This is equivalent to the tangency portfolio
w = VectorVariable("w", n_assets, lb=0)  # No upper bound initially

# Excess return must equal 1 (we'll rescale later)
excess_returns = expected_returns - rf
port_excess = excess_returns @ w

solution = (
    Problem("max_sharpe")
    .minimize(w.dot(covariance @ w))  # wᵀΣw
    .subject_to(port_excess.eq(1))  # Normalize excess return to 1
    .solve()
)

# Rescale to sum to 1
raw_weights = np.array([solution[f'w[{i}]'] for i in range(n_assets)])
tangency_weights = raw_weights / raw_weights.sum()

# Compute metrics
tang_return = expected_returns @ tangency_weights
tang_vol = np.sqrt(tangency_weights @ covariance @ tangency_weights)
tang_sharpe = (tang_return - rf) / tang_vol

print("=" * 60)
print("MAXIMUM SHARPE RATIO (TANGENCY) PORTFOLIO")
print("=" * 60)
print(f"Expected Return: {tang_return:.2%}")
print(f"Volatility: {tang_vol:.2%}")
print(f"Sharpe Ratio: {tang_sharpe:.3f}")
print()
print("Allocation:")
for i in range(n_assets):
    if tangency_weights[i] > 0.01:
        print(f"  {asset_names[i]}: {tangency_weights[i]:.1%}")
============================================================
MAXIMUM SHARPE RATIO (TANGENCY) PORTFOLIO
============================================================
Expected Return: 11.29%
Volatility: 6.54%
Sharpe Ratio: 1.420

Allocation:
  Asset_0: 6.0%
  Asset_2: 20.2%
  Asset_3: 2.3%
  Asset_5: 5.6%
  Asset_6: 20.7%
  Asset_7: 9.6%
  Asset_8: 21.4%
  Asset_9: 14.3%

10 Comparing Portfolios

Let’s compare all our portfolios:

# Define portfolios
portfolios = {
    "Equal Weight": np.ones(n_assets) / n_assets,
    "Min Variance (8% target)": np.array([solution[f'w[{i}]'] for i in range(n_assets)]),
    "Max Sharpe": tangency_weights,
}

# Add min variance solution
w_minvar = VectorVariable("w", n_assets, lb=0, ub=1)
sol_minvar = (
    Problem()
    .minimize(w_minvar.dot(covariance @ w_minvar))  # wᵀΣw
    .subject_to(w_minvar.sum().eq(1))
    .subject_to(expected_returns @ w_minvar >= 0.08)
    .solve()
)
portfolios["Min Variance"] = np.array([sol_minvar[f'w[{i}]'] for i in range(n_assets)])

print("Portfolio Comparison:")
print("=" * 60)
print(f"{'Portfolio':<25} {'Return':>10} {'Vol':>10} {'Sharpe':>10}")
print("-" * 60)

for name, weights in portfolios.items():
    ret = expected_returns @ weights
    vol = np.sqrt(weights @ covariance @ weights)
    sharpe = (ret - rf) / vol
    print(f"{name:<25} {ret:>10.2%} {vol:>10.2%} {sharpe:>10.3f}")
Portfolio Comparison:
============================================================
Portfolio                     Return        Vol     Sharpe
------------------------------------------------------------
Equal Weight                  10.00%      8.01%      0.999
Min Variance (8% target)     121.53%     70.43%      1.697
Max Sharpe                    11.29%      6.54%      1.420
Min Variance                  10.22%      6.18%      1.330

11 Performance: VectorVariable vs Loop-Based

Let’s compare the performance of VectorVariable approach vs traditional loops:

import time

def old_style_portfolio(n, returns, cov, target):
    """Build portfolio with loop-based variables."""
    from optyx import Variable, Problem
    
    # Create variables one by one
    w = [Variable(f"w_{i}", lb=0, ub=1) for i in range(n)]
    
    # Portfolio return via loop
    port_ret = sum(w[i] * returns[i] for i in range(n))
    
    # Portfolio variance via double loop (O(n²))
    port_var = sum(
        w[i] * cov[i, j] * w[j] 
        for i in range(n) 
        for j in range(n)
    )
    
    # Budget constraint via loop
    budget = sum(w)
    
    problem = (
        Problem()
        .minimize(port_var)
        .subject_to(budget.eq(1))
        .subject_to(port_ret >= target)
    )
    
    return problem.solve()


def new_style_portfolio(n, returns, cov, target):
    """Build portfolio with VectorVariable."""
    from optyx import VectorVariable, Problem
    
    w = VectorVariable("w", n, lb=0, ub=1)
    port_ret = returns @ w
    port_var = w.dot(cov @ w)  # wᵀΣw
    
    problem = (
        Problem()
        .minimize(port_var)
        .subject_to(w.sum().eq(1))
        .subject_to(port_ret >= target)
    )
    
    return problem.solve()


# Benchmark
n = 10
print("Build + Solve Time Comparison (10 assets):")
print("-" * 50)

# Old style
start = time.perf_counter()
sol_old = old_style_portfolio(n, expected_returns, covariance, 0.08)
old_time = time.perf_counter() - start

# New style
start = time.perf_counter()
sol_new = new_style_portfolio(n, expected_returns, covariance, 0.08)
new_time = time.perf_counter() - start

print(f"Loop-based:    {old_time*1000:.2f} ms")
print(f"VectorVariable: {new_time*1000:.2f} ms")
print(f"Speedup: {old_time/new_time:.1f}x")
Build + Solve Time Comparison (10 assets):
--------------------------------------------------
Loop-based:    1281.67 ms
VectorVariable: 1.89 ms
Speedup: 679.4x

12 Summary

This tutorial demonstrated:

Feature Benefit
VectorVariable Clean syntax for many variables
w.dot(Σ @ w) Math-like quadratic form with O(1) gradient
Parameter Fast re-solves for scenarios
Vector operations No explicit loops needed

Key takeaways:

  1. Use math-like syntax w.dot(Σ @ w) for variance—reads as wᵀΣw and computes gradients analytically
  2. Use Parameter when exploring different target returns or risk tolerances
  3. Vector syntax makes constraints like w.sum().eq(1) readable and maintainable

13 Next Steps