Markowitz Portfolio Optimizer

Do mean–variance portfolios that look optimal in-sample hold up out of sample?

Python
NumPy
pandas
SciPy
pytest

Result. On a 231-day holdout, the minimum-variance portfolio delivered the best risk-adjusted return (Sharpe 1.32, max drawdown −4.6%). The portfolio that was optimal in-sample, maximum Sharpe, came last on Sharpe (0.59), behind even equal weighting. Expected-return estimates are the weak link.

Question

Mean–variance optimisation is only as good as its inputs. Sample means are notoriously noisy, while covariance is estimated more reliably. I wanted to measure how much that matters in practice: estimate everything on a training window, freeze the weights, and evaluate them on data the optimiser never saw.

Data

Daily adjusted closing prices (splits and distributions reflected) from Yahoo Finance for five USD-listed ETFs chosen to span different risk exposures:

Ticker	Exposure
JETS	Airlines
VNQ	U.S. real estate
JTEK	U.S. technology leaders
EWN	Dutch equities
PFIX	Hedge against rising long-term rates

Only dates with prices for every ETF are kept; nothing is forward-filled. The saved snapshot has 771 common observations (5 Oct 2023 – 30 Oct 2026), giving 770 daily returns.

Method

Daily simple returns, annualised means and sample covariance:

R_{i,t}=\frac{P_{i,t}}{P_{i,t-1}}-1,\qquad \mu=252\,\overline{R},\qquad \Sigma=252\,\operatorname{Cov}(R).

Three long-only, fully invested portfolios ( $\sum_i w_i = 1$ , $0 \le w_i \le 1$ ) are solved with SciPy’s SLSQP:

Global minimum variance: $\min_w \; w^\top \Sigma w$
Maximum Sharpe: $\max_w \; (w^\top\mu - r_f)/\sqrt{w^\top \Sigma w}$ , using deterministic restarts because SLSQP is a local solver
Efficient frontier: minimum variance at 60 target returns

Every solution is checked for solver success, bounds, budget and target constraints before it is accepted. The risk-free rate is an illustrative 3% a year.

Efficient frontier with individual ETFs and optimised portfolios

Full-sample minimum-variance and maximum-Sharpe portfolio weights

Out-of-sample backtest

The sample is split chronologically (70/30, no shuffling): training from 6 Oct 2023 to 28 Nov 2025 (539 returns) and testing from 1 Dec 2025 to 30 Oct 2026 (231 returns). Four strategies are fixed using training data only: minimum variance, maximum Sharpe, equal weight, and the single ETF with the best training Sharpe (EWN). There is no re-optimisation during the test period.

Strategy	Total return	CAGR	Volatility	Sharpe	Max drawdown
Minimum variance	14.56%	15.98%	9.31%	1.32	−4.56%
Maximum Sharpe	11.95%	13.11%	18.72%	0.59	−11.41%
Equal weight	13.81%	15.16%	14.34%	0.85	−9.73%
Best training ETF (EWN)	15.94%	17.51%	23.11%	0.69	−13.24%

Out-of-sample cumulative wealth from $1

Out-of-sample drawdowns

Takeaways

Estimation error dominates. Maximum Sharpe leans hardest on expected-return estimates, and it had the worst out-of-sample Sharpe of the four strategies.
Covariance-only allocation was the most robust. Minimum variance ignores expected returns entirely, yet it cut volatility by about a third relative to equal weighting while earning a slightly higher return.
The highest return is not the best risk-adjusted result. EWN had the best total return but 2.5× the volatility of minimum variance and nearly three times its drawdown.

Limitations

A single holdout period with five pre-selected ETFs cannot establish statistical significance.
The backtest assumes idealised daily rebalancing with no transaction costs, slippage or taxes.
PFIX holds derivatives, so long-only ETF weights do not remove its nonlinear exposures.

Next steps

Walk-forward evaluation with periodic re-estimation, shrinkage covariance (Ledoit–Wolf), transaction-cost-aware rebalancing, and bootstrap confidence intervals for the Sharpe ratio differences.