Beta and alpha: how closely a fund moves with the market

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Beta shows how strongly a fund moves on average when the market moves: a beta of 1.2 means that it gains about 12% when the market rises by 10%, and loses about 12% when the market falls by 10%. Alpha is the return that cannot be explained by this market risk. Both come from a simple regression.

How to calculate beta

Equivalently, beta is the slope of the regression line when you plot the fund's returns against the market's. The intercept of this line is the alpha per period, and the goodness of fit is the R²: the share of the fund's fluctuations that the market explains.

Six months are just for illustration. In practice, you use 36 to 60 monthly returns so that outliers in individual months do not determine the result.

Where beta comes from: the CAPM

Beta comes from the capital asset pricing model, developed by Sharpe (1964) and, independently, by Lintner (1965). The idea: a stock's own, specific risk can be almost entirely eliminated through diversification, but market risk cannot. In equilibrium, therefore, only market risk is rewarded with a premium, and beta measures precisely this market risk.

If you plot expected returns against beta, the model says that all investments lie on a straight line, the security market line. More on the theory and its limits in the article CAPM.

Interpreting beta correctly

Beta is only as good as the benchmark index and the R²: a gold miners ETF with a beta of 0.8 and an R² of 0.2 moves mainly for reasons other than the stock market. That is why Portfolio Lab's look-through always shows the R² and the t-statistic next to the beta of each position.

A low beta does not automatically mean low risk either. A commodity ETF can have a beta close to zero and still be more volatile than the stock market. Its risk is simply of a different kind. Beta measures the link to the market, not total fluctuation. Conversely, if you know the beta of your overall portfolio, you know roughly how it will react in a stock market crash. A portfolio with a beta of 0.6 loses about 18% on average when the market falls by 30%, more or less in any individual case.

Upside and downside capture

A single beta assumes that a fund moves equally strongly with rising and falling markets. The capture ratios separate the two: upside capture compares the fund's average return in months when the market rose with the market's own return in those months, and downside capture does the same for falling months. Ideally, upside capture is high and downside capture low.

Alpha: return above the market risk

Jensen's alpha is the return above what the fund should have delivered given its beta. Jensen (1968) introduced it to evaluate mutual funds, with a sobering result: the funds in his sample could not beat the market on average, let alone after costs. A positive alpha can be skill, but it can also be luck or a factor that the market alone does not capture, such as a tilt towards small or cheap companies. The Fama-French factor regression takes several factors into account at once; Carhart (1997) showed that seemingly persistent fund success largely disappears once momentum is included.

Fama and French (2010) found for US equity funds that only a few funds achieve enough alpha to cover their costs; the distribution of alphas is largely consistent with luck. An alpha from three or five years is therefore rarely a reliable sign of skill.

For advanced readers: how reliable is a beta?

Beta and alpha are estimates with standard errors. The standard error of beta gets smaller with more observations, more market fluctuation and a better model (smaller residuals ε):

In the example, this gives a standard error of 0.09 and a t-statistic of around 13. With only six months, that is a lucky consequence of the very high R². For alpha, the situation is much worse: even over ten years, an alpha of 1% a year with a tracking error of 4% can hardly be distinguished statistically from zero (t-statistic ≈ 1 / 4 × √10 = 0.79).

Betas also change over time. Very high and very low estimated betas are often closer to 1 in the next period; data providers therefore frequently shrink them towards 1 (“adjusted beta”, for example two thirds of the estimated beta plus one third times 1). And finally, everything depends on the market index chosen: Roll (1977) pointed out that the true market portfolio of all assets cannot be observed. Every test of the CAPM is also a test of the index.

What the research shows

The CAPM is elegant but empirically weak. Fama and French (2004) summarise that the security market line is too flat in the data: low-beta stocks returned more, and high-beta stocks less, than the model predicts. Black (1972) had offered an explanation early on: investors who cannot borrow at the risk-free rate turn to high-beta assets for more return and drive up their prices. Frazzini and Pedersen (2014) confirm this for US stocks, 20 international stock markets, bonds and futures: high beta went hand in hand with low alpha. This gave rise to the betting-against-beta strategy; see low-volatility anomaly.

Treynor ratio

The Treynor ratio is the sister of the Sharpe ratio: instead of dividing by total volatility, it divides only by market risk. That makes sense for a building block in a broadly diversified portfolio, whose own specific risk is diversified away. A fund with an 8% return, a 2% interest rate and a beta of 1.2 has a Treynor ratio of (8 − 2) / 1.2 = 5.00%: that is the excess return per unit of market risk. The market itself has a beta of 1 by definition, so its Treynor ratio is simply its excess return.

How Portfolio Lab calculates it

In the Backtest, Portfolio Lab calculates beta, alpha and R² against the chosen benchmark over all shared periods (at least six). Beta is the covariance divided by the variance of the period returns. Alpha p.a. is the portfolio's mean excess return minus beta times the benchmark's mean excess return, annualised with 12 or 252 respectively; excess returns are measured against the risk-free rate for the period. R² is the square of the correlation. Upside and downside capture compare the geometric means in periods with a rising or falling benchmark, and the Treynor ratio divides the annualised mean excess return by beta. In the Look-through, the funds' equity beta comes from 60 monthly returns against a world equity index. More in the metrics methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.