What the model needs to know

The frontier itself is uncontroversial arithmetic. The inputs are the problem, and the size of the problem can be measured with a published table of real returns rather than argued about.

Counting the inputs

How badly the first input is known

The published table gives 61 annual returns for the index with dividends, from 1965 to 2025. Their arithmetic mean is 11.95 percent and their standard deviation is 16.93 points. The standard error of the mean is the standard deviation divided by the square root of 61, which is 2.17.

A 95 percent range around the mean is therefore roughly 7.7 percent to 16.2 percent. After sixty-one years of the most-studied series in finance, the expected annual return is known to within about plus or minus four points.

The report's own figures, and mine

The same table states a compounded annual gain of 10.5 percent for the index over 1965 to 2025 and an overall gain of 46,061 percent over 1964 to 2025. Those are the report's printed figures.

The mean, standard deviation and standard error above are computed here from the 61 annual percentages the table prints, which is why the 11.95 percent arithmetic mean sits above the 10.5 percent compounded figure. The two answer different questions, and the difference between them is the subject of its own lesson in the financial mathematics course.

The errors do not cancel

There is a natural reply to all this: the estimates are noisy, but the errors go in both directions and should wash out. They do not, and the reason is what the optimiser does with them.

An optimiser is a machine for finding the extremes of its inputs. The holding whose expected return was overestimated and whose correlation was underestimated is exactly the holding it puts the most money into. Noise in equals concentrated position out, and the process systematically selects for the inputs that are most wrong in the most useful direction.

That is why a portfolio built by optimising noisy estimates often ends up in a handful of holdings that a person looking at the list would reject on sight, and why practitioners who use these models constrain them heavily: caps per holding, caps per sector, or shrinking the estimates towards a common value before the model ever sees them.

What to do with a model you cannot feed

Use it for what it teaches rather than for what it outputs. The formula shows that co-movement matters more than individual volatility, that adding holdings has diminishing returns, and that the benefit depends on an estimate nobody has.

Then size positions by what a wrong thesis costs, which is a judgement about a business and does not need 230 estimated parameters.

Check yourself