Ten Times Worse: Chopra and Ziemba on Which Input Errors Actually Hurt

You are building a portfolio. You have limited time and limited data, and you must estimate three sets of numbers: expected returns, variances, and covariances. You cannot get them all right. Where should you spend your effort?

Vijay Chopra and William Ziemba answered this question in 1993, and their answer is one of the very few results in portfolio construction that is precise enough to be actionable and memorable enough that people actually remember it.

The problem: everyone knew errors mattered, nobody had ranked them

By the early 1990s it was well established that mean-variance optimization was fragile. Michaud had named the disease. Best and Grauer had shown how violently the weights react to changes in the inputs. But nobody had put a relative price on the three different types of error.

This is not an academic distinction. It has direct practical consequences. If errors in covariances are the most damaging, then you should invest heavily in sophisticated covariance estimation: factor models, shrinkage, high-frequency data. If errors in means are the most damaging, then you should spend all your effort on return forecasting, or, if you conclude that return forecasting is hopeless, on avoiding the need for it entirely.

Two very different research programs, two very different businesses. The answer determines which one you should be running.

The key idea via analogy: budgeting your attention

Think of yourself as a builder with three things to get right: the foundations, the walls, and the paint. Obviously you should not spread your care equally across them, because getting the foundations wrong is catastrophic while getting the paint slightly wrong is cosmetic. The useful question is not "should I be careful," it is "how much more careful should I be about the foundations than the paint?" That ratio tells you where to spend.

Chopra and Ziemba measured the ratio for portfolio inputs. Their method was direct. Take a set of assets with known true parameters. Deliberately introduce errors of a given size into one input type at a time. Build the optimal portfolio using the corrupted inputs. Then measure how much wealth, in utility terms, the investor lost by holding that portfolio instead of the truly optimal one. That loss is the cash-equivalent cost of the error.

Repeat for means, then for variances, then for covariances, and compare.

The finding

The result is the reason this paper is still cited constantly.

Errors in means are roughly ten times as damaging as errors in variances, and roughly twenty times as damaging as errors in covariances.

Ten to one. Twenty to one. These are not close calls. The three inputs are not remotely comparable in importance, and the one that dominates is the expected returns, which is exactly the one that is hardest to estimate.

And it gets worse. Chopra and Ziemba also showed that the relative damage from errors in the means is even greater at higher risk tolerances. The more aggressive the investor, the further along the frontier they sit, and the further along the frontier you go, the more the optimizer is chasing return rather than avoiding risk, which means the more it leans on the numbers you know least well. An aggressive investor with noisy return forecasts is in a very bad place.

Put the two facts side by side and the picture is grim:

• Expected returns are the input that matters most, by an order of magnitude.
• Expected returns are also the input we can estimate least well, by a wide margin. Volatility can be pinned down from a few years of daily data. Expected returns need decades or centuries of data to estimate with any precision, and by then the world has changed.

The most important input is the one we are worst at. That single sentence explains most of the last thirty years of portfolio construction research.

Why it mattered

• It told the field where to spend its effort. This paper is the reason so much energy went into shrinking, constraining, and Bayesian-ing the expected returns: Jorion's Bayes-Stein estimator, Black-Litterman, robust optimization. Every one of them is a response to the fact that means are where the damage is.
• It justified the entire risk-based investing movement. If the means are hopeless, why not build a portfolio that does not use them at all? Minimum variance, risk parity, equal risk contribution, maximum diversification and hierarchical risk parity are all in the family of methods that simply refuse to estimate expected returns. Chopra and Ziemba are the reason that refusal is a defensible position rather than an abdication.
• It explains why equal weighting is so hard to beat. DeMiguel, Garlappi and Uppal's finding that 1/N holds its own against fourteen sophisticated models is much less mysterious once you accept this result. The sophisticated models are buying a small theoretical improvement at the cost of leaning heavily on the input that destroys portfolios when it is wrong.
• It gives a clean answer to a common interview question. "Which input to a mean-variance optimizer matters most, and by how much?" is a question with a right answer, and this is it.

The honest limitations

• The exact ratios are setup-dependent. The ten-to-one and twenty-to-one figures come from a particular simulation with a particular universe, a particular investor utility, and a particular way of scaling errors. The precise multipliers should not be treated as universal constants of nature. The ordering, however, is robust, and that is the part that matters.
• It measures the loss to a mean-variance investor. The framework assumes the investor's objective is mean-variance utility. An investor who cares about drawdowns or tail risk has a different loss function, and the ranking could shift.
• It does not tell you what to do about it. The paper diagnoses. It does not prescribe. Knowing that means matter most does not make means any easier to forecast, and the conclusion that most people draw, "so do not use them," is a reasonable response but not one the paper itself establishes as optimal.
• Errors are treated as independent perturbations. In reality, errors in your estimated means and covariances are correlated with each other, since they come from the same data sample. That interaction is not captured.
• The result can be over-read. "Covariances barely matter" is not the lesson. Covariance errors matter less relative to return errors, but risk-based strategies that use only covariances live or die on that input, and for them the estimation problem is very real indeed.

The one-line takeaway

Chopra and Ziemba measured what your input mistakes actually cost you and found that errors in expected returns hurt about ten times more than errors in variances and twenty times more than errors in covariances, which means the single most important number in portfolio construction is also the one you have almost no ability to estimate, and that awkward fact underlies nearly every modern approach to building portfolios.