5. Estimation error

Level 7: Linear algebra and portfolio optimization

The economist Richard Michaud called the Markowitz optimizer an "error maximizer", and the phrase has become famous in quantitative management.

It is not a rhetorical exaggeration, but a fairly literal description of what happens when you feed it estimated data.

The assumption nobody can meet

Everything seen so far in this level implicitly assumes that you know two things exactly:

• The expected return of each asset.
• The complete covariance matrix.

In practice you know neither. You estimate them from historical data.

And that estimate comes with a margin of error that Markowitz, as originally formulated, has no way of recognizing or handling cautiously.

Why the optimizer maximizes the errors

Michaud's idea is that the optimizer does not distinguish between two very different situations:

• A high expected return because the asset genuinely deserves it.
• A high expected return because, through pure statistical noise in the historical sample used, that asset happened to have a good run.

In both cases the optimizer assigns it a disproportionate weight. And it does so precisely because its job is to exploit to the maximum any difference it sees in the data you gave it.

The optimizer does not know which of those differences are signal and which are noise.

Expected returns are the worst estimated of all

The problem is more severe for expected returns than for volatilities and correlations.

The reason relates to something you already saw in the inferential statistics level. The standard error of an estimated mean shrinks in proportion to the square root of the sample size, not linearly.

Think about what that implies for an individual stock, whose typical volatility is around 20-30% per year. To estimate its annual expected return with reasonable precision you would need decades or even centuries of stable historical data, something that neither exists nor would be useful if it did.

Volatility and correlation, by contrast, are estimated with relative reliability using much shorter data windows.

The reason for that difference is specific. They do not depend on getting the average direction of the movement right, only its magnitude and its relationship with other assets, something that reveals itself through many more useful observations even within the same short time window.

The uncomfortable asymmetry, and the jump from one portfolio to another

This asymmetry has an uncomfortable practical consequence. The part of the Markowitz problem that contributes most to the instability of the results, the expected returns, is precisely the part that can be estimated worst.

Look at it with two assets that are highly correlated with each other. Small differences in the expected return assigned to them can make the optimizer jump from a solution where one weighs 90% and the other 0%, to the opposite solution, on a minimal change in the input.

This is not a failure of the algorithm: it is a mathematically correct consequence of a problem badly conditioned by unreliable inputs.

The optimizer is doing exactly what it is asked to do. What happens is that what it is asked to do rests on numbers we barely know with any precision.

The second problem: N against T

There is a second, distinct but related problem with the covariance matrix. It appears when the number of assets (N) approaches or exceeds the number of available historical observations (T).

Imagine you try to estimate a covariance matrix for 500 assets using only 250 trading days, fewer observations than assets to estimate:

• The resulting matrix becomes statistically unstable.
• In the extreme case where T is smaller than N, it is not even mathematically invertible, a property the optimizer needs in order to solve the problem.

And even if you have enough data for it to be technically invertible, with N and T of similar magnitude the estimated matrix contains a great deal of noise.

That noise does not stay inside the matrix. It translates into extreme, counterintuitive portfolio weights that are highly unstable from one month to the next, even though the true underlying risk has barely changed.

The practical solutions, none of them perfect

Faced with these problems, quantitative management has developed several families of practical solutions, none of them perfect.

Shrinkage consists of "shrinking" the individual estimates toward a more stable and less noisy value. For example, toward the average across all assets, or toward a simplified structure for the matrix, accepting some bias in exchange for much less variance in the estimate.

Imposing additional constraints on the weights, such as maximum limits per asset or per sector, acts in practice as a way of restraining the optimizer so that it does not take estimation errors to extremes.

The Black-Litterman model, which you will see mentioned in professional practice although we will not go into its formulation here, combines three ingredients:

• The historical estimates.
• The market expectations implicit in current prices.
• The manager's explicit views.

Its purpose is to avoid starting from expected returns based solely on the past.

And, as you will see in the next module, there is one radical and surprisingly competitive option: to stop estimating expected returns altogether.