Alpha and Benchmark Selection

Why this matters

"This fund beat the market" is a common claim in investing and an uninformative one. A fund can beat the market by being riskier than the market in a rising year, by being measured against the wrong market, or by luck. Alpha is the attempt to measure what's left when the first of those is removed. This lesson shows how to calculate it, how little a typical record can prove, and how the benchmark can decide the answer before any arithmetic is done.

What alpha is

The last lesson described beta as the amount of market risk an investment carries. In the capital asset pricing model, an investment's expected return is the risk-free rate plus its beta times the market's premium over that rate. Alpha is the difference between what it earned and that figure.

Alpha = Actual return − (Risk-free rate + Beta × (Market return − Risk-free rate))

A positive alpha means the investment earned more than its market risk explains. A negative alpha means it earned less. An index fund has a beta of 1 and an alpha of about zero, less its costs.

Average yearly return, smallest tenth of US companies	17.8%
Average yearly return, S&P 500	11.9%
Average yearly return, 3-month Treasury bills	3.4%
Beta of small companies against the S&P 500	1.41
Return their beta predicts: 3.4% + 1.41 × (11.9% − 3.4%)	15.3%
Alpha: 17.8% − 15.3%	2.5%

Calculated for this lesson from Aswath Damodaran's yearly data.¹ Small companies beat the S&P 500 by 5.9 points a year on average. More than half of that, 3.4 points, is the reward for carrying 1.41 times the market's risk. What's left is an alpha of about 2.5% a year. These are arithmetic averages before costs, and the cost of trading the smallest companies isn't in them.

The example shows the first use of alpha: it stops risk being mistaken for skill. A fund with a beta of 1.4 will beat the market handsomely in good years without any skill at all, and will lose by the same margin in bad ones.

Skill or luck?

An alpha is an average of noisy numbers, so it has a standard error, like the expected return in Expected Return and Standard Deviation. The noise that matters here is the part of the investment's returns that the market doesn't explain. For small companies that part has a standard deviation of about 26%, so over 98 years the standard error of the alpha is about 26% ÷ √98 = 2.7%.

The alpha is 2.5% and its standard error is 2.7%. The usual test asks for an estimate at least two standard errors from zero. This one is less than one. After 98 years, the data can't rule out that small companies have no alpha at all. The record supports that reading in other ways too: they beat the S&P 500 in 49 of the 98 years, exactly half, and from 1984 through 2025 they compounded at 9.0% a year against the S&P 500's 11.7%.¹

The same arithmetic applies to any fund manager. The number of years needed for an alpha to stand two standard errors clear of zero is:

Years needed = (2 × Standard deviation of the unexplained return ÷ Alpha)²

True alpha	Unexplained return varies by	Years needed
2% a year	4% a year	16
2% a year	6% a year	36
1% a year	6% a year	144
4% a year	10% a year	25

A manager who really is good enough to add 2% a year, while straying from the benchmark by a moderate 6% a year, needs 36 years of results before the evidence is solid. Three-year and five-year records can't show much either way.

That doesn't mean skill doesn't exist. It means past returns alone can rarely prove it, which is a reason to look at how a manager invests, what it costs, and whether the approach makes sense, and not only at the record.

The rest of this lesson is for members

Still to come in this lesson: What the evidence says about alpha in general, The benchmark decides the answer, Two related measures, Questions for any claim of outperformance, the key takeaways, and 2 quiz questions with explanations.

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