Mean-Variance Optimization Out of Sample: The 1/N Test, Rerun on 21 Years of New Data

At the end of 2004, a mean-variance optimizer given ten years of returns on Ken French’s ten US industry portfolios would have recommended 39% health care, 36% energy, 12% consumer nondurables and small slices of three other industries. The backtest of that portfolio over the decade it was fitted on had a Sharpe ratio of 0.87, against 0.69 for an equal-weighted mix of all ten. Over the next 21 years, from January 2005 to August 2026, the optimized portfolio returned a Sharpe ratio of 0.63 and the equal-weighted mix 0.65.¹

Run the same optimizer on the ten years ending August 2026 and it puts 79% in technology, 17% in utilities and 4% in energy. The record suggests giving that answer little weight. In a retest of the best-known study on 21 years of data its authors never saw, no optimizer beat equal weights by a margin that could be told apart from luck, and every rule built on historical average returns trailed equal weights over the full 90 years of data.

What a mean-variance optimizer solves

Harry Markowitz’s 1952 paper made the case that a holding should be judged by what it does to the whole portfolio, and gave the arithmetic for doing it.² With portfolio weights $w$, expected returns $\mu$ and covariance matrix $\Sigma$, one standard version picks the weights that maximize expected return minus a penalty for variance:

Here $\gamma$ is risk aversion. The resulting curve of best portfolios at each risk level is the efficient frontier, and the modern portfolio theory guide covers how correlation shapes it. The mathematics is not in dispute. The trouble is that nobody knows $\mu$ or $\Sigma$. In practice the optimizer is fed estimates, usually historical averages $\hat\mu$ and a historical covariance matrix $\hat\Sigma$, and it returns the best portfolio for those estimates.

Historical average returns carry wide error bars

If annual returns are independent draws with volatility $\sigma$, the standard error of a $T$-year average is:

For an asset with 16% volatility and 20 years of data, that is 3.6 percentage points, so a measured 7% average is consistent with a true expected return anywhere from about 0% to 14%. Collecting monthly or daily data does not help. Robert Merton showed in 1980 that the precision of an estimated mean depends on the calendar span of the data, while the precision of an estimated variance improves with sampling frequency; his own conclusion was that mean estimates from realistic samples are close to useless.³ Forecasting models have not closed the gap reliably: Amit Goyal, Ivo Welch and Athanasse Zafirov reexamined 29 equity-premium predictors published after 2008 and found that more than a third had lost significance even in sample, and half of the remainder forecast poorly out of sample.⁴

This asymmetry is why the problem lands on expected returns. Vijay Chopra and William Ziemba measured the certainty-equivalent cost of errors in each input. Depending on the risk tolerance assumed, errors in means cost roughly 3 to 21 times as much as errors of the same size in variances, and more again relative to errors in covariances; the gap is smallest for the most risk-averse investors.⁵

Small input changes move the weights a long way

An optimizer looks for the largest differences between assets, so it leans hardest on the inputs with the most error. Richard Michaud called this “error maximization.”⁶ Michael Best and Robert Grauer showed analytically that a small rise in one asset’s mean can push half the holdings out of an efficient portfolio.⁷ Mark Britten-Jones found that with 20 years of data on 11 national stock markets, the sampling error in estimated efficient weights was large.⁸

Best and Grauer also found the other half of the story: under a no-short-selling constraint, the portfolio’s expected return and risk barely moved while the weights swung. Unstable weights do not always mean an expensive mistake. Two very different portfolios can be close to equally good, which also means the optimizer’s choice between them carries little information.

Re-running history when the answer is known

Real data cannot show how far an estimated optimum sits from the true one, because the true one is unknown. A simulation can. I set “true” annual excess returns over cash of 5% for US stocks (16% volatility), 5% for international stocks (17%) and 1.5% for bonds (6%), with a 0.8 correlation between the two stock markets and 0.1 between stocks and bonds. These are round illustrative numbers, not forecasts. At a risk aversion of 2.5 the true optimum is 42% US, 22% international and 36% bonds. I then drew 2,000 possible histories of monthly returns from those parameters, let the optimizer choose from each history, and scored each choice at the true parameters.¹

With 20 years of monthly history, in this setup:

• The middle 90% of the optimizer’s choices ran from 0% to 100% in stocks.
• The average choice fell short of the true optimum by 54 basis points a year of certainty-equivalent return. Equal weights (one third in each) fell short by 1.6 basis points.
• The optimizer beat equal weights in 1.3% of histories. With 100 years of history it still beat them in only 3.8%.

That result flatters equal weights, because one third in each of these three assets happens to sit close to the true optimum. Swapping the menu changes the answer. Adding gold, with 1% expected excess return, 16% volatility and no correlation to anything, makes equal weights 28 basis points worse than the optimum. The optimizer then beat equal weights in 20% of 20-year histories, 29% of 40-year histories and 55% of 100-year histories, and even at 100 years its average shortfall (32 basis points) was larger than equal weights’. How good 1/N looks depends on which assets you decide to split money across.

Two estimators designed to reduce error helped less than I expected with three or four assets. Philippe Jorion’s Bayes-Stein shrinkage of the means⁹ cut the average shortfall in the gold calibration from 129 to 92 basis points with five years of data and from 103 to 77 with ten, matched the plain estimate at 40 years and did worse at 100. Covariance shrinkage made no measurable difference, which fits its purpose: it matters when there are many assets relative to the number of observations.

Try it yourself

The simulator below runs the same experiment in your browser with 300 histories. Its default settings reproduce the script’s check run exactly: the portfolios the optimizer picked expected an average excess return of 5.4% from their own history and delivered 3.7% at the true parameters. Switch the preset to stocks, bonds and gold to see a menu where 1/N is a poor default, and slide the years of history to see how slowly the cloud of choices tightens.

The 1/N evidence and the case against it

Victor DeMiguel, Lorenzo Garlappi and Raman Uppal published the best-known test in 2009. Of 14 optimizing models across seven datasets, none was consistently better than equal weights on Sharpe ratio, certainty-equivalent return or turnover. Their calibrated analysis put the estimation window that sample-based mean-variance strategies and their extensions need to beat 1/N at around 3,000 months (250 years) for 25 assets and 6,000 months for 50, under parameters calibrated to the US stock market.¹⁰

The strongest reply came from Mark Kritzman, Sébastien Page and David Turkington. They argued that 1/N looked good because those tests estimated expected returns from short rolling samples, which produce implausible inputs, and that optimizers given longer samples or simple, plausible assumptions beat 1/N out of sample across 13 datasets.¹¹ Others found middle ground. Raymond Kan and Guofu Zhou showed that adding the minimum-variance portfolio to the optimizer’s choice set reduces the damage from estimation error,¹² and Jun Tu and Guofu Zhou found that blends of 1/N with sophisticated rules beat 1/N in most of their scenarios.¹³ Georg Pflug, Alois Pichler and David Wozabal proved that as uncertainty about the model grows, the optimal choice for a broad class of risk measures converges to 1/N.¹⁴

We reran the industry test on 21 years of new data

DeMiguel and coauthors’ industry dataset ended in November 2004. Everything since then is out of sample for their published claim and could not have influenced it. I wrote down the strategies, windows, costs and metrics before running anything, and the pre-registration is in the repository with its commit date.¹

The setup copies theirs: French’s ten value-weighted industry portfolios, returns in excess of the one-month T-bill, a rolling 120-month estimation window, and monthly rebalancing, with each month chosen using only the months before it.¹⁵ Over their window the replication gives monthly Sharpe ratios of 0.137 for 1/N and 0.151 for unconstrained minimum variance; their 2005 working paper reports 0.135 and 0.155.¹⁶ French revises his data, so a small gap is expected. The table annualizes the Sharpe ratios. The p-value tests each strategy against 1/N over the new period.

“Traded per month” counts buys and sells as a share of the portfolio. “Shrunk covariance” is Olivier Ledoit and Michael Wolf’s constant-correlation estimator.¹⁷ “Equal Sharpe ratios assumed” replaces every historical mean with the asset’s volatility times a common Sharpe ratio, a plausible-input rule in the spirit of Kritzman and coauthors. Results are gross of fund fees and taxes. Charging 10 basis points per dollar traded lowered the historical max Sharpe rule’s post-2004 figure by about 0.01, which put it just below market-cap weights.

The post-2004 results:

• No strategy beat 1/N by a statistically significant margin after 2004. The closest was long-only minimum variance at 0.73 against 0.66, with a p-value of 0.58. Twenty-one years is not enough to tell these strategies apart.
• The consistent benefit was lower volatility. Long-only minimum variance ran at 11.8% volatility against 15.0% for 1/N after 2004, and at a similar or higher Sharpe ratio in every period I tested. It needs no expected-return estimates at all.
• Plausible inputs beat historical means. The equal-Sharpe rule was at or above 1/N in all four periods (1936 to 1963, DeMiguel and coauthors’ window, after 2004, and the full sample), by 0.03 to 0.04, and never significantly. The rule using raw historical means trailed 1/N badly over 1973 to 2004 and tied it afterward.
• Unconstrained Markowitz was the worst rule in every period. With shorting allowed, the sample optimizer averaged 9.5 times gross leverage over 1973 to 2004 and a negative Sharpe ratio.
• A longer history did not rescue historical means. Estimating from all data since 1926, as Kritzman and coauthors suggest, trailed 1/N after 2004 (0.60 against 0.66).

A 60-month window, the other length DeMiguel and coauthors tried, gave the same picture: no strategy beat 1/N significantly in any period. The full output for both windows and all four periods is in the repository.

A do-it-yourself asset-class menu

Industry portfolios are not what most people allocate across, so I ran the same comparison on Aswath Damodaran’s annual returns for the S&P 500, US small caps, 10-year Treasuries and Baa corporate bonds, 1928 to 2025, with a 20-year rolling window.¹⁸ I had pre-registered gold as a fifth asset and dropped it after the first run: Damodaran’s gold series before 1972 is an administered price that barely moves, and the minimum-variance rules treated it as a riskless asset. US residents could not legally hold gold bullion for most of that period either. The run with gold is still in the repository.

The first half explains the numbers. In 1948 the max Sharpe rule looked back at 1928 to 1947, when the 10-year Treasury’s excess return had a Sharpe ratio of 0.70 against 0.24 for the S&P 500, and put 98% of the portfolio in Treasuries. From 1948 through 1981 the Treasury’s return trailed T-bills by 1.45 percentage points a year. Every optimizer that leaned on the bond market’s recent record was positioned for the regime that had just ended. From 1987 on, the optimizers roughly tied the simple mixes. With 39 observations per half, none of the 1987 to 2025 differences is significant. Over the full period the six optimizers trail 1/N with p-values between 0.09 and 0.16, and the half-and-half blend’s gap has a p-value of 0.53.

What reduces the damage

The methods that helped in the published research share one feature: they pull the optimizer toward a simple anchor.

• Long-only constraints. Ravi Jagannathan and Tongshu Ma showed that forbidding short sales acts like shrinking the largest covariance estimates, which is why the constraint improves estimated minimum-variance portfolios even when the true optimum would short.¹⁹ A 2019 note with Jinyuan Zhang corrected an omitted term in the proofs; the propositions as stated still hold.²⁰
• Covariance shrinkage. In Ledoit and Wolf’s 2003 test on roughly 900 to 1,300 NYSE and AMEX stocks, a shrunk covariance matrix produced minimum-variance portfolios with 9.55% annualized out-of-sample volatility over 1972 to 1995, against 12.37% for the sample matrix (pseudo-inverted, because there were more stocks than months).²¹ With ten industries and long-only weights in our retest, shrinkage changed almost nothing, consistent with Jagannathan and Ma: the constraint had already done most of that work.
• Norm constraints. DeMiguel, Garlappi, Francisco Nogales and Uppal capped how concentrated the weights can get, a family of rules that includes 1/N as a special case, and their rules often had higher Sharpe ratios than ten other strategies across five datasets.²²
• Equilibrium inputs. Fischer Black and Robert Litterman start from the expected returns implied by market-cap weights and adjust only where the investor has an explicit view and a stated confidence in it.²³ The result stays close to the market portfolio unless the views are strong.

How to read a backtest someone shows you

A historical portfolio comparison can make four quite different claims, and most presentations do not say which one they are making:

1. Description. This fixed mix returned X with Y volatility over these dates. Useful, and says nothing about optimality.
2. Hindsight optimum. This mix had the best Sharpe ratio over the whole period. Nobody could have held it, because choosing it required the returns it is being judged on. The 0.87 Sharpe ratio at the top of this page is this kind of number.
3. Walk-forward test. A rule chose weights each period from earlier data only. Closer to real experience, though settings tuned on the full history leak information back in.
4. Live record. The rule was fixed and then run on data that did not exist yet. The retest above is the closest thing to it here: DeMiguel and coauthors’ claim was fixed before the 2005 to 2026 data existed, though the extra strategies I tested were chosen in 2026.

The SEC’s investor bulletin on performance claims puts it directly: back-tested performance is hypothetical and does not reflect actual performance, so ask for the actual record.²⁴ The number of things tried matters as much as the method. David Bailey, Jonathan Borwein, Marcos López de Prado and Qiji Jim Zhu calculated that with five years of data, trying more than about 45 independent strategy variations almost guarantees one with an in-sample Sharpe ratio of 1 even when every variation’s true Sharpe ratio is zero.²⁵ Campbell Harvey and Yan Liu show how to discount a reported Sharpe ratio for the number of tests behind it, and note that the right discount is far larger for marginal results than for very strong ones, so a flat 50% haircut misstates both.²⁶

Before trusting an optimized backtest, check:

• whether the weights were chosen with data from the test period;
• how many asset menus, windows and objectives were tried;
• whether funds that closed are missing, and whether pre-launch years are index or simulated returns presented as fund history;
• whether moving the start date by five years changes the ranking;
• whether the winner simply holds more stock than the benchmark;
• whether trading costs and taxes are included.

Retail optimizers make the hindsight optimum easy to produce. Portfolio Visualizer’s optimizer, for example, uses historical return statistics by default, although it also accepts your own estimates and offers resampling.²⁷ The R² trap guide shows the same in-sample problem in valuation charts.

Acting on a new optimum can cost more than it gains

Suppose a taxable position is worth $100,000 with an $80,000 basis, and the whole gain is taxed at 15%. Selling it to move toward a new target costs $3,000, or 3% of the position, today and with certainty. In the three-asset simulation above, the gap between equal weights and the true optimum was 1.6 basis points a year; in the gold version it was 28 basis points, which would take more than a decade to recover a 3% tax bill before discounting. And those gaps assume you know the true optimum, which is the one thing nobody has.

New contributions, dividends and trades inside IRAs and 401(k)s move an allocation without realizing gains. The household portfolio guide covers treating every account as one portfolio, How Often Should You Rebalance covers bands and timing, and tax-loss harvesting covers offsetting gains you do realize.

When an optimizer is worth using

Optimization earns its keep when:

• the menu is a few broad asset classes with very different risk, such as stocks, bonds and cash, where covariance carries real information;
• expected returns come from something steadier than the last ten years: equilibrium weights, equal Sharpe ratios, or long-run estimates you would defend before seeing the output;
• the result is constrained to long-only weights with caps, and rounded;
• you use leverage or volatile diversifiers, where position sizing matters and the Merton share and risk parity framings apply;
• the rule is judged walk-forward against 1/N and a fixed mix.

It is a poor tool when:

• the candidates overlap, such as five US stock funds, where small historical differences become large weights;
• the inputs are ETF histories of ten years or less, which in this retest’s terms is a single 120-month window;
• the output would trigger taxable sales.

What I recommend

Do not take portfolio weights from an optimizer fed historical average returns. The research and this retest agree that those weights carry little information beyond what a simple mix already has, and on the asset-class menu they pointed the wrong way for decades at a time. Choose your stock and bond split from your horizon, your need to take risk and how much decline you can sit through, as the 60/40 vs. target-date vs. 100% stocks guide describes, and hold broad market-cap index funds within each part.

Use the frontier to understand the trade-off: how much volatility a bond allocation removes, how much a diversifier helps at different correlations. If you want an optimizer’s help, give it three to five distinct asset classes, inputs you would defend in advance, and long-only limits, and treat a recommendation of 27.4% as “somewhere around a quarter.” If the goal is lower volatility, long-only minimum variance across broad assets is the version with the best record here, and it needs no return forecast.

Summitward’s portfolio dashboard compares your current weights with max Sharpe, minimum volatility, risk parity and equal weights. Those models are fitted and scored on the same historical window, so treat the comparison as a hindsight optimum, the second kind of claim above.

Key Takeaways

• The math is right; the inputs are guesses. A mean-variance optimizer returns the best portfolio for its estimates, and a 20-year average return on a 16%-volatility asset has a standard error of 3.6 percentage points.
• Errors in expected returns cost the most. In Chopra and Ziemba’s tests, roughly 3 to 21 times as much as errors in variances, depending on risk tolerance.
• In our simulation with known parameters and 20 years of data, the optimizer’s choices spanned 0% to 100% stocks and beat equal weights in 1.3% of histories.
• In our retest on ten US industries from December 2004 to August 2026, no optimizer beat 1/N by a significant margin; long-only minimum variance cut volatility from 15.0% to 11.8% at a similar Sharpe ratio.
• 1/N is only as good as the menu. Adding a low-return asset to the list makes equal weights worse and gives the optimizer room to win.
• Ask which kind of backtest you are seeing. A hindsight optimum cannot be invested in; a walk-forward test or a live record can.

Frequently Asked Questions

What is mean-variance optimization?

A method from Harry Markowitz’s 1952 work that chooses portfolio weights to get the highest expected return for a given level of variance, using each asset’s expected return, volatility and correlations. The set of best portfolios at each risk level is the efficient frontier.

Why do optimized portfolios do worse than their backtests?

The optimizer favors assets whose historical averages were high, and part of every high average is luck. Scored at the true parameters, the portfolios in our simulation’s default run expected 5.4% from their own history and delivered 3.7%. The gap between in-sample and out-of-sample results comes mostly from estimation error in expected returns.

Does equal weighting beat optimization?

Often, but not as a rule. DeMiguel, Garlappi and Uppal found no optimizer consistently better than 1/N across seven datasets, and our retest on 2005 to 2026 data agrees. Kritzman, Page and Turkington found that optimizers with plausible long-run inputs do beat 1/N. And 1/N depends entirely on which assets you list: splitting equally across three stock funds and one bond fund is a 75/25 portfolio.

Is minimum variance a better use of an optimizer?

It avoids expected-return estimates, the noisiest input, and in our industry retest the long-only version lowered volatility by about a fifth without lowering the Sharpe ratio. It still depends on covariance estimates, and it tends to concentrate in the lowest-volatility assets, which on the asset-class menu meant holding only Treasuries from 1948 into the 1950s, at the start of the bond market’s long run of losses against T-bills.

How many years of data does an optimizer need?

More than anyone has, for historical means. DeMiguel and coauthors’ calibration put it at about 250 years for 25 assets. In our three-asset simulation, even 100 years of data left the optimizer behind equal weights in more than 96% of histories. That is why the methods that work replace historical means with priors or drop them entirely.

Should I use an online portfolio optimizer?

As a way to see trade-offs, yes. As a source of weights, only if you supply the expected returns yourself, keep it long-only with caps, and check the result against a simple mix. If the tool uses the historical average returns of the funds you typed in, its recommendation is a hindsight optimum for that period.

Related Guides

• Modern Portfolio Theory for Real Life for how correlation shapes the frontier and the marginal-fund test.
• HEDGEFUNDIE’s Excellent Adventure for an optimized leveraged portfolio and what 2022 did to it.
• Minimum Volatility and Betting Against Beta for the stock-level version of minimum variance.
• The Risk Parity Reality Check for weighting by risk contribution instead of expected return.
• Portfolio Risk Is More Than Volatility for drawdown measures and why ranking backtests by any metric overfits.
• If Ben Felix Uses One Fund, Do You Need Ten? for when adding funds stops adding diversification.

Sources

1. Author’s calculation. Script, data, pre-registration and full output: summitward-research/mean-variance-optimization.
2. Harry Markowitz, “Portfolio Selection,” Journal of Finance 7(1), 1952, 77–91. doi.org
3. Robert C. Merton, “On Estimating the Expected Return on the Market: An Exploratory Investigation,” Journal of Financial Economics 8(4), 1980, 323–361. doi.org; working paper at nber.org.
4. Amit Goyal, Ivo Welch and Athanasse Zafirov, “A Comprehensive 2022 Look at the Empirical Performance of Equity Premium Prediction,” Review of Financial Studies 37(11), 2024, 3490–3557. doi.org
5. Vijay K. Chopra and William T. Ziemba, “The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice,” Journal of Portfolio Management 19(2), 1993, 6–11. doi.org
6. Richard O. Michaud, “The Markowitz Optimization Enigma: Is ‘Optimized’ Optimal?” Financial Analysts Journal 45(1), 1989, 31–42. doi.org
7. Michael J. Best and Robert R. Grauer, “On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results,” Review of Financial Studies 4(2), 1991, 315–342. doi.org
8. Mark Britten-Jones, “The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights,” Journal of Finance 54(2), 1999, 655–671. doi.org
9. Philippe Jorion, “Bayes-Stein Estimation for Portfolio Analysis,” Journal of Financial and Quantitative Analysis 21(3), 1986, 279–292. doi.org
10. Victor DeMiguel, Lorenzo Garlappi and Raman Uppal, “Optimal Versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?” Review of Financial Studies 22(5), 2009, 1915–1953. doi.org
11. Mark Kritzman, Sébastien Page and David Turkington, “In Defense of Optimization: The Fallacy of 1/N,” Financial Analysts Journal 66(2), 2010, 31–39. doi.org
12. Raymond Kan and Guofu Zhou, “Optimal Portfolio Choice with Parameter Uncertainty,” Journal of Financial and Quantitative Analysis 42(3), 2007, 621–656. doi.org
13. Jun Tu and Guofu Zhou, “Markowitz Meets Talmud: A Combination of Sophisticated and Naive Diversification Strategies,” Journal of Financial Economics 99(1), 2011, 204–215. doi.org
14. Georg Ch. Pflug, Alois Pichler and David Wozabal, “The 1/N Investment Strategy Is Optimal Under High Model Ambiguity,” Journal of Banking & Finance 36(2), 2012, 410–417. doi.org
15. Kenneth R. French, Data Library, 10 Industry Portfolios and Fama/French 3 Factors (monthly, 202608 CRSP vintage), downloaded October 15, 2026.
16. Victor DeMiguel, Lorenzo Garlappi and Raman Uppal, “How Inefficient Is the 1/N Asset-Allocation Strategy?” working paper, December 1, 2005, Section 4.3. warwick.ac.uk
17. Olivier Ledoit and Michael Wolf, “Honey, I Shrunk the Sample Covariance Matrix,” Journal of Portfolio Management 30(4), 2004, 110–119. doi.org
18. Aswath Damodaran, Historical Returns on Stocks, Bonds and Bills, NYU Stern, annual data 1928–2025.
19. Ravi Jagannathan and Tongshu Ma, “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps,” Journal of Finance 58(4), 2003, 1651–1683. doi.org
20. Ravi Jagannathan, Tongshu Ma and Jinyuan Zhang, “A Note on ‘Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps’,” Journal of Finance 74(5), 2019, 2689–2696. doi.org
21. Olivier Ledoit and Michael Wolf, “Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection,” Journal of Empirical Finance 10(5), 2003, 603–621, Table 1. doi.org
22. Victor DeMiguel, Lorenzo Garlappi, Francisco J. Nogales and Raman Uppal, “A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms,” Management Science 55(5), 2009, 798–812. doi.org
23. Fischer Black and Robert Litterman, “Global Portfolio Optimization,” Financial Analysts Journal 48(5), 1992, 28–43. doi.org
24. SEC Office of Investor Education and Advocacy, “Investor Bulletin: Performance Claims,” September 15, 2022. investor.gov
25. David H. Bailey, Jonathan M. Borwein, Marcos López de Prado and Qiji Jim Zhu, “Pseudo-Mathematics and Financial Charlatanism: The Effects of Backtest Overfitting on Out-of-Sample Performance,” Notices of the AMS 61(5), 2014, 458–471 (preprint); see also “The Probability of Backtest Overfitting,” Journal of Computational Finance 20(4), 2017. doi.org
26. Campbell R. Harvey and Yan Liu, “Backtesting,” Journal of Portfolio Management 42(1), 2015, 12–28. duke.edu
27. Portfolio Visualizer, Portfolio Optimization and Efficient Frontier tool pages, accessed October 15, 2026.

Author disclosure

Summitward has no business relationship with Portfolio Visualizer or any fund sponsor and receives no compensation from them. Figures labeled as the author’s calculation are historical measurements or simulations under stated assumptions, not forecasts, and are gross of fund fees and taxes. Nothing here is investment advice.