Portfolio Optimisation with Modern Portfolio Theory

Most investors diversify by instinct: “don’t put all your eggs in one basket.” In 1952, Harry Markowitz turned that instinct into mathematics. Modern Portfolio Theory (MPT) shows how to combine assets so that, for any level of risk, you earn the highest possible expected return—and how to find the specific portfolios that do it.

This guide builds MPT step by step and lets you experiment with two interactive labs. It continues from Basics of Risk and Return, which covers variance, covariance and the Sharpe ratio.

What Modern Portfolio Theory says

MPT rests on one powerful observation: a portfolio’s risk is not the average of its assets’ risks. Because assets do not move in perfect lockstep, combining them can cut risk without cutting expected return.

Assumptions to keep in mind

• Investors care only about expected return and variance (returns roughly normal, or quadratic preferences).
• Investors are risk-averse: for the same return, they prefer less risk.
• Inputs are known for a single period; markets are frictionless (no taxes or costs).
• Unlimited borrowing and lending at the risk-free rate (for the Capital Market Line).

None holds perfectly—section 10 covers what goes wrong in practice—but the framework remains the backbone of asset allocation, from pension funds to robo-advisers.

Diversification: the free lunch

Total risk splits into systematic risk, driven by the economy and markets, and specific risk, unique to each company. Diversification erodes specific risk; systematic risk remains. That is why, in equilibrium, investors are paid for bearing market risk (beta), not for bearing risk they could have diversified away.

Two-asset portfolios

Start with equity A (expected return 12%, volatility 18%) and bonds B (7%, 6%), correlation 0.1, risk-free rate 6.5%.

Notice the surprise: adding 10% equity to an all-bond portfolio raises return and lowers risk. The all-bond portfolio is inefficient.

Minimum-variance and maximum-Sharpe weights

Multi-asset portfolios

With more assets, the same logic applies in matrix form. Our running example uses four asset classes with illustrative long-term assumptions:

An equal-weighted portfolio (25% each) returns 10.00% with 8.34% volatility—a Sharpe ratio of 0.420. Can optimisation do better? With n assets you need n expected returns, n variances and n(n−1)/2 correlations: 14 inputs for 4 assets, but 1,325 for 50. That is where estimation error creeps in.

The efficient frontier

Every dot is a feasible portfolio. A portfolio is efficient if no other portfolio offers a higher expected return for the same risk, or lower risk for the same return. The set of efficient portfolios forms the upper-left boundary—the efficient frontier.

• Everything below the frontier is dominated: move up (more return) or left (less risk) for free.
• The frontier starts at the global minimum-variance portfolio; the curve below it is inefficient.
• Which point on the frontier is “best” depends on your risk tolerance—until we add a risk-free asset.

Optimisation 1: the minimum-variance portfolio

The GMV needs no expected-return estimates—only the covariance matrix. That makes it popular in practice, because expected returns are the hardest input to forecast.

The GMV volatility (4.73%) is lower than even the safest single asset, bonds (5%), thanks to small diversifying positions in equity and gold. Here all weights are positive, so the long-only and unconstrained answers coincide.

Optimisation 2: maximum Sharpe ratio and the tangency portfolio

Compared with equal weights (Sharpe 0.420) the gain looks modest; compared with GMV (0.208) it is large. Gold earns a big weight not because of its return but because its negative correlation with Indian equity makes it an excellent diversifier.

The Capital Market Line and two-fund separation

Once a risk-free asset exists, the best risky portfolio is the same for everyone: the tangency portfolio. Investors simply choose how much to hold in it versus the risk-free asset—the separation theorem (Tobin, 1958). The line joining them, the Capital Market Line, dominates the curved frontier everywhere except at the tangency point.

Excel Solver and Python

Excel Solver: maximum Sharpe in six steps

1. Enter expected returns (μ) in a column and the covariance matrix Σ in a 4×4 block (use COVARIANCE.S on return history, or σᵢσⱼρᵢⱼ).
2. Put starting weights (25% each) in a column w.
3. Portfolio return: =SUMPRODUCT(w, mu). Portfolio variance: =MMULT(TRANSPOSE(w), MMULT(Sigma, w)); volatility = SQRT(variance).
4. Sharpe: =(return − rf) / volatility.
5. Solver → Set objective: Sharpe cell → Max; By changing: weights; Constraints: SUM(w) = 1 and w ≥ 0; method GRG Nonlinear.
6. For the GMV, change the objective to variance → Min. For the frontier, add a constraint return = target and repeat for several targets.

import numpy as np
from scipy.optimize import minimize

mu  = np.array([0.13, 0.11, 0.07, 0.09])          # expected returns
vol = np.array([0.18, 0.16, 0.05, 0.15])
rho = np.array([[1, .5, .1, -.1],
                [.5, 1, 0,  0 ],
                [.1, 0, 1,  .2],
                [-.1, 0, .2, 1 ]])
Sigma = np.outer(vol, vol) * rho
rf = 0.065

def stats(w):
    ret = w @ mu
    sd  = np.sqrt(w @ Sigma @ w)
    return ret, sd, (ret - rf) / sd

cons   = {"type": "eq", "fun": lambda w: w.sum() - 1}
bounds = [(0, 1)] * len(mu)                       # long-only
w0     = np.full(len(mu), 1 / len(mu))

gmv = minimize(lambda w: w @ Sigma @ w, w0, bounds=bounds, constraints=cons).x
tan = minimize(lambda w: -stats(w)[2],  w0, bounds=bounds, constraints=cons).x

print("GMV     ", gmv.round(3), stats(gmv))  # ~[0.013 0.080 0.864 0.043]
print("Tangency", tan.round(3), stats(tan))  # ~[0.398 0.175 0.136 0.291]

# closed forms when short selling is allowed
one   = np.ones(len(mu)); Si = np.linalg.inv(Sigma)
w_gmv = Si @ one / (one @ Si @ one)
w_tan = Si @ (mu - rf) / (one @ Si @ (mu - rf))

Limitations—and how practitioners respond

Alternatives built on the same foundations include risk parity (equalise each asset’s risk contribution), maximum diversification and hierarchical risk parity. Each trades some theoretical optimality for robustness.

Frequently asked questions