03: Portfolio Optimization (QP)

Problem Class	Solver	Key APIs	Links
Convex Quadratic Programming (QP)	HiGHS (default)	m.continuous(), m.constraint(), c, minimize, dot, QuadExpr	Open in Playground Source Code

The Markowitz mean-variance portfolio optimization problem is the foundation of modern quantitative finance: allocate capital across risky assets to minimize overall portfolio variance (risk) subject to achieving a minimum required expected return.

INTERACTIVE EXAMPLE · 5 ASSETS CONVEX QP (QuadExpr)

Portfolio Allocation Optimizer (Markowitz QP)

Adjust the target expected return and click '▶ Run Optimization' to solve the 5-asset Quadratic Program.

Target Expected Return: 7.5%

HiGHS QP · 0.0 ms · Min Variance: 0.00880

US Tech Equity

25.0% E[R]=12.0%

Global Index

30.0% E[R]=8.5%

Real Estate (REIT)

15.0% E[R]=7.0%

Gold (Hedge)

10.0% E[R]=5.5%

Gov Bonds

20.0% E[R]=3.5%

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Mathematical Formulation

Let $w_i \ge 0$ denote the allocation weight for asset $i$:

$$\begin{aligned} \text{Minimize} \quad & \frac{1}{2} \mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w} & (\text{Portfolio Variance}) \\ \text{Subject to} \quad & \sum_{i=1}^n w_i = 1 & (\text{Full Capital Budget}) \\ & \sum_{i=1}^n r_i w_i \ge R_{\text{target}} & (\text{Minimum Target Return}) \\ & 0 \le w_i \le 1 & (\text{Long-Only Weights}) \end{aligned}$$

where $\boldsymbol{\Sigma}$ is the positive semi-definite asset return covariance matrix.

Implementation in WaSHI

import { Model, QuadExpr, sum, dot, c, minimize } from "@ohtaman/washi";

async function solvePortfolio() {
  const m = new Model("PortfolioQP");

  const assets = ["Tech", "Bonds", "GlobalEq", "RealEstate", "Gold"];
  const expectedReturns = [0.12, 0.04, 0.09, 0.07, 0.05];
  const targetReturn = 0.085; // 8.5% minimum expected return

  // 5x5 Covariance Matrix
  const cov = [
    [0.040, 0.002, 0.025, 0.015, 0.005],
    [0.002, 0.005, 0.003, 0.002, 0.001],
    [0.025, 0.003, 0.030, 0.018, 0.008],
    [0.015, 0.002, 0.018, 0.025, 0.004],
    [0.005, 0.001, 0.008, 0.004, 0.020],
  ];

  // Allocation weights w_i in [0, 1]
  const w = m.continuous(assets, { lb: 0, ub: 1 });

  // 1. Budget constraint: sum(w_i) == 1.0 (100%)
  m.constraint(c`${sum(assets, (a) => w[a])} == 1.0`, "budget_limit");

  // 2. Expected return constraint: sum(r_i * w_i) >= 8.5%
  const retMap = Object.fromEntries(assets.map((a, i) => [a, expectedReturns[i]]));
  m.constraint(c`${dot(retMap, w)} >= ${targetReturn}`, "target_return");

  // 3. Quadratic Objective: minimize risk w^T * Sigma * w
  let risk = new QuadExpr();
  for (let i = 0; i < assets.length; i++) {
    for (let j = 0; j < assets.length; j++) {
      if (cov[i][j] !== 0) {
        risk = risk.add(w[assets[i]].mul(w[assets[j]]).mul(cov[i][j]));
      }
    }
  }
  m.objective(minimize(risk));

  // Solve natively with HiGHS Convex QP engine
  const status = await m.solve();

  console.log(`Status: ${status}`);
  console.log(`Variance: ${m.objectiveValue?.toFixed(6)}`);
  console.log(`Volatility (Std Dev): ${((Math.sqrt(m.objectiveValue || 0)) * 100).toFixed(2)}%`);

  assets.forEach((name) => {
    console.log(`  ${name.padEnd(10)}: ${((w[name].value ?? 0) * 100).toFixed(2)}%`);
  });
}

solvePortfolio();

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