Naive Diversification vs Optimisation: 10 Menus Tested

Key takeaways

• Across 10 asset menus and 259 out-of-sample months to July 2026, the sample mean-variance optimiser beat equal weights on 2 menus and lost on 8.
• Equal weights produced an annualised Sharpe ratio between 0.48 and 0.65 on every menu tested. The mean-variance rule ranged from -0.35 to 0.84.
• The long-only minimum-variance rule beat equal weights on all 10 menus, but only one of those gaps was statistically significant, with a p-value of 0.029.
• Equal weights traded 1.2% to 3.4% of the portfolio a month. The mean-variance rule traded 107.6 times the portfolio a month on the 30-industry menu.
• Average pairwise correlation ran from 0.55 to 0.92 across the ten menus, which is all the raw material any optimiser had to work with.

The short answer: even splits beat the textbook optimiser on eight menus out of ten

You have probably met both claims. One says mean-variance optimisation is the mathematically correct way to set portfolio weights. The other says you may as well divide your money evenly across whatever you own. They can't both be right, so here is what happened when they were run against each other on data that postdates the argument.

Over 259 months from January 2005 to July 2026, across 10 different menus of assets, the sample mean-variance rule beat equal weighting on 2 menus and lost on 8. On the menu of book-to-market deciles it scored an annualised Sharpe ratio of -0.35 against 0.60 for the even split. Equal weighting never did brilliantly and never did badly: its Sharpe ratio sat between 0.48 and 0.65 on all ten menus.

That is roughly what Victor DeMiguel, Lorenzo Garlappi and Raman Uppal reported about naive diversification in 2009, and nothing in the mechanism has aged. But there is a second finding here that their paper stated and most summaries of it drop. One optimiser did beat the even split, on all ten menus. It's the one that refuses to estimate expected returns at all.

What DeMiguel, Garlappi and Uppal actually found

Their paper circulated in June 2006 under the title 1/N and reached the Review of Financial Studies in 2009 as Optimal Versus Naive Diversification. It tested fourteen models of optimal portfolio choice against a single rule: put 1/N of your money into each of the N things on the menu. Across seven datasets, they wrote, "none is consistently better than the 1/N rule in terms of Sharpe ratio, certainty-equivalent return, or turnover."

The reason is worth a minute, because it isn't a criticism of the maths. An optimiser needs two inputs: expected returns for each asset, and a covariance matrix describing how they move together. Both have to be estimated from past data. Both are estimated badly, and the expected returns are estimated far worse.

Their Industry dataset, ten US industry portfolios plus the US market, makes the point cleanly. The even split scored a monthly Sharpe ratio of 0.1353. The sample mean-variance rule scored -0.0363 on the same data. On their four-factor dataset the mean-variance portfolio scored 0.5364 in sample, where the optimiser is handed the answers in advance. Out of sample the same rule scored -0.0031, against 0.1753 for the even split.

You can watch it happen in the weights. Their international dataset holds eight country indices plus a World index, from January 1970 to July 2001. On that data the in-sample optimal weight on the World index was -505%. Run the same rule out of sample and the weight on that one index ranged as high as 256,461%. An expected return that is wrong by a fraction of a percentage point can move an optimiser's weights by thousands of percentage points, because the optimiser treats an estimate as a fact.

They then calculated how much history would fix it. For a portfolio of 25 assets, they find the sample mean-variance rule would need an estimation window of more than 3,000 months to beat the even split on average. For 50 assets, more than 6,000 months. The window normally used, and the one used here, is 120 months.

How this test was built, and what it reproduces

Every series here comes from the Kenneth French Data Library, which is where DeMiguel and his co-authors got five of their seven datasets. The files run to July 2026. Ten menus were assembled from them: US industry portfolios at 10, 17, 30 and 48 assets; size and book-to-market portfolios at 6 and 25; deciles sorted on momentum, book-to-market and size; and a five-way split of world equity markets into North America, Europe, Japan, Asia Pacific excluding Japan, and emerging markets.

The procedure copies theirs. At each month end, the previous 120 months of excess returns over the one-month Treasury bill are used to estimate a mean vector and a covariance matrix. Four rules then set the weights held for the following month, and the window rolls forward one month. That yields 259 out-of-sample months per menu, from January 2005 to July 2026.

None of this is the same question as equal weight vs market cap, which compares two ways of weighting one index over a much longer run. Here the menu is fixed and the weighting rule is what changes. The four rules are equal weights; sample mean-variance, which uses both estimates; minimum variance, which throws the expected returns away and uses only the covariance matrix; and minimum variance with short selling banned, so every weight sits between zero and one.

As a check on the code, the same routine was run on their Industry dataset over their own window, July 1963 to November 2004. It returns a monthly Sharpe ratio of 0.1353 for the even split, which is their published figure to four decimal places. Minimum variance comes out at 0.1537 against their 0.1554, and the constrained version at 0.1445 against their 0.1425. The unconstrained mean-variance rule comes out at -0.0186 against their -0.0363: the same conclusion, a different decimal. French has revised the underlying industry returns more than once since 2004, so an exact match on every row was never available.

Naive diversification against three optimisers, menu by menu

Sharpe ratios below are annualised, which means the monthly figure multiplied by the square root of 12. Higher is better, and the column to measure everything against is the first one.

The chart at the top of this page plots one column of that table against another: the mean-variance rule's annualised Sharpe ratio minus equal weighting's, one bar per menu. Eight of the ten bars point down.

The rule that needs expected returns is the one that breaks

The two menus where the mean-variance rule won are the two built from size and book-to-market sorts. It scored 0.84 against 0.55 on the 6-portfolio version and 0.73 against 0.53 on the 25-portfolio version. That is the same corner of the data where the 2006 paper found optimisation doing best: on its four-factor dataset, built from twenty size and book-to-market portfolios, the constrained minimum-variance rule scored 0.3580 against 0.1753 for the even split. Two tests two decades apart, and the same sort of menu favours the optimiser.

Neither win survives a significance test. The p-values on those two differences, computed the way the paper computes them, are 0.206 and 0.438. Two wins out of ten, both of which could be luck.

The losses are less ambiguous. On book-to-market deciles the mean-variance rule scored -0.35 against 0.60, with a p-value of 0.001. On 17 US industries it scored -0.07 against 0.59, p-value 0.022. Those two differences are real by any conventional standard, and they run the wrong way for the optimiser.

Then there is what it cost to run. Equal weights traded between 1.2% and 3.4% of the portfolio each month, which is just the drift being corrected. The mean-variance rule traded 107.6 times the portfolio a month on the 30-industry menu. That is not a portfolio, it's a machine for generating commission. The 2006 paper found the same thing on its own data: turnover of 0.0216 a month for the even split, and 607,479.61 times that figure for the mean-variance rule on the industry menu.

Throw the expected returns away and the optimiser starts winning

Minimum variance ignores expected returns entirely. It asks only which combination of these assets has wobbled least, and it needs one estimate rather than two. On this data it beat equal weights on 8 of the 10 menus. Add a ban on short selling, so that no weight can go negative and none can exceed the whole portfolio, and it beat equal weights on all 10.

The long-only version scored between 0.55 and 0.74 across the ten menus, against 0.48 to 0.65 for equal weights. It was the best of the four rules on the 48-industry menu, at 0.62 against 0.58, and on the size deciles, at 0.68 against 0.52. Its turnover topped out at 8.7% of the portfolio a month.

Before that reads as a clean win, look at the significance tests. Only one of those ten differences clears the conventional significance threshold: the 6-portfolio size and value menu, with a p-value of 0.029. The 25-portfolio version comes close at 0.065. On the other eight menus, the constrained optimiser's edge is indistinguishable from noise. Winning ten times out of ten by an amount you can't measure is a weaker claim than it looks, though it is still a pattern, and it points the same way every time.

The 2006 paper reached the same place. Its authors wrote that the results suggest "it may be best to ignore the data on expected returns, but still exploit the correlation structure between assets to reduce risk with the constraints helping to reduce the effect of the error in estimating the covariance matrix."

The correlation matrix shows how little there was to optimise

Here is the correlation matrix for the menu that spreads widest geographically, the five world regions, measured across the same 259 months.

The lowest number in that matrix is 0.67, between North America and Japan. The highest, ignoring the diagonal, is 0.92, between Asia Pacific excluding Japan and emerging markets. Average pairwise correlation across the ten menus ran from 0.55 on the 48-industry menu to 0.92 on size deciles.

That range is the whole story. An optimiser earns its keep by finding assets that offset each other. When North America and Europe move together at 0.85, there is very little offsetting available, and the optimiser spends its estimation error budget chasing differences that are mostly noise. This is also why correlation instability matters more than the level: a matrix estimated on the last ten years is a forecast, and it is the input the winning rule here depends on completely.

It is worth being clear that none of these menus contains a bond, a commodity or a property fund. Where risk contribution is genuinely lopsided across asset classes, the correlation matrix carries more information and an optimiser has more to find.

Turnover is where the equal weight portfolio is untouchable

Every figure above ignores trading costs, because the moment you model them the comparison stops being close. An even split rebalanced monthly trades a few percent of the portfolio. The long-only minimum-variance rule trades a few percent more. The unconstrained rules trade multiples of the portfolio, every month, forever.

Costs scale with turnover, and they fall on the optimised portfolios in rough proportion to how much estimation error they are acting on. A rule that trades 107.6 times the portfolio a month would have to be enormously right to survive the spread. On this evidence it wasn't right at all.

The strongest case against reading this as a win for even splits

The serious objection is that this tests a straw man. Mark Kritzman, Sebastien Page and David Turkington made it directly in the Financial Analysts Journal in 2010, in a paper called In Defense of Optimization: The Fallacy of 1/N. Their argument is that "the 1/N approach" wins only because the optimiser is fed expected returns estimated from rolling short-term samples, an approach that "often yields implausible expectations". Use longer-term samples or "naively contrived yet plausible assumptions" instead, they report, and optimised portfolios "outperform equally weighted portfolios out of sample".

That objection applies squarely to the test above. The mean-variance column uses a 120-month sample mean as its forecast of each asset's return, which nobody managing money would actually do. It is the textbook implementation, and the textbook implementation is what the 2006 paper set out to examine, but it is not the best case for optimisation.

The minimum-variance columns are the better test of the idea, and they support the objection rather than this article's headline. The 2006 paper also listed the conditions under which optimisation should win: a long estimation window, a true efficient-portfolio Sharpe ratio well above the even split's, and a small number of assets. Only the last of the three picks out the 6-portfolio size and value menu, which is the smallest equity menu here and the one where the constrained optimiser's edge was significant. The second condition runs the other way. The efficient portfolio estimated on the whole sample scores 0.98 annualised on that menu, against 1.99 on the 48-industry menu, where the mean-variance rule then lost by a wide margin out of sample. A high in-sample Sharpe ratio turns out to be a symptom of having too many assets to fit, not a sign that optimisation has something real to find.

A reader who concludes from all this that optimisation works, provided you stop pretending to forecast returns, is reading the same numbers correctly. The case for naive diversification here is narrower than the slogan: it survives against the textbook optimiser, not against every optimiser.

What this test cannot tell you

It's a backtest of naive diversification against three specific optimisers, not a forecast, and one path through history at that. The 259 months from January 2005 to July 2026 contain one banking crisis, one pandemic and one inflation shock, and a different 259 months could rank these rules differently.

Every menu is equities. There are no bonds, no cash beyond the Treasury bill used to compute excess returns, no commodities and no property. That matters, because the case for optimisation is strongest where assets have genuinely different risk, and this test never puts it there. Nine of the ten menus are also US-only, and the tenth is a regional split of world equity markets rather than a portfolio a UK investor would hold. How many stocks is diversified is a related question this data cannot answer either, because every item on every menu is already a diversified portfolio.

Trading costs, taxes and bid-offer spreads are all set to zero. That flatters the high-turnover rules, which is to say it flatters the ones that lost. The estimation window is fixed at 120 months throughout, and a shorter or longer one would move the results. Only four rules were tested here, against fourteen in the original paper.

One more caveat sits in the data itself. The French library switched from the legacy CRSP file format to Flat File Format 2.0 for its US research returns with the January 2025 release, which changed how monthly returns are compounded. The last stretch of this sample is built slightly differently from the rest, and the paper being replicated used the older format throughout.

What would change the conclusion

A genuine forecast of expected returns would change it, and that is the Kritzman objection stated as a condition. Nothing here tests an optimiser fed a considered view of what assets should return. It tests one fed a ten-year average, and shows that a ten-year average is not a view.

A menu of genuinely different assets would change it. Every correlation in the matrix above sits between 0.67 and 0.92. Put a government bond fund and an equity fund on the same menu and that number falls a long way, which is the situation where the covariance matrix carries real information rather than mostly noise.

And a much longer stationary history would change it, though nobody has one. The 3,000-month figure is not a rhetorical flourish; it is the arithmetic of how fast an estimate of a mean converges when the thing being estimated is that noisy.

The number to watch isn't the Sharpe ratio of any rule. It's the average pairwise correlation of what you actually hold, because that one figure decides whether there is anything for an optimiser to find. LedgerTouch computes it continuously; a spreadsheet computes it once a year. Across the ten menus here it never fell below 0.55, and on that evidence the even split's advantage was never about return. It was that it has nothing to estimate.