Beta, correlation and regression

Key takeaways

• Covariance measures how two return series move together; correlation rescales it to lie between minus 1 and plus 1.
• Beta is the covariance of the asset with the market divided by the variance of the market, which is exactly the slope of an ordinary least squares regression of asset returns on market returns.
• Alpha is the regression intercept: the average return not explained by market exposure. It is an estimate with a standard error, and is usually far less certain than beta.
• R squared is the share of the asset's variance explained by the market, and in a one factor regression equals the correlation squared.
• The CAPM turns beta into a required return: risk free rate plus beta times the equity risk premium.

Covariance and correlation

Covariance measures whether two series tend to be above their means at the same time. For each period, multiply the asset's deviation from its mean by the market's deviation from its mean; average those products (dividing by n minus 1 for a sample). A positive covariance means the two usually move in the same direction, a negative one means they usually move in opposite directions, and a value near zero means there is no linear relationship.

Covariance is hard to read on its own because its units are the product of the two series' units, percent squared for returns, and its size depends on how volatile each series is. Correlation fixes that by dividing the covariance by the product of the two standard deviations. The result is a pure number between minus 1 and plus 1. A correlation of 0.87 says the two series move together closely; it says nothing about how much one moves when the other does.

That last point is the most common confusion in this area. Correlation measures how tightly two series line up; beta measures how much one moves per unit move in the other. A small cap stock can have a correlation of 0.6 with the index and a beta of 1.5, while a utility can have a correlation of 0.6 and a beta of 0.5. The difference is volatility: beta equals correlation multiplied by the ratio of the asset's volatility to the market's.

Beta as a regression slope

Beta is the slope of the line that best fits the asset's returns plotted against the market's. 'Best' in ordinary least squares (OLS) means the line that minimises the sum of squared vertical distances between each observation and the line. Solving that minimisation gives a neat result: the slope is the covariance of the asset with the market divided by the variance of the market. So beta, as a risk concept, and the OLS slope, as a statistical one, are the same number.

A beta of 1 means the asset tends to move one for one with the market. A beta above 1 amplifies market moves, typical of cyclical, highly leveraged or small companies; a beta below 1 dampens them, typical of utilities and consumer staples. A beta of zero does not mean an asset is riskless, only that its risk is unrelated to the market. A market neutral hedge fund aims for a beta near zero precisely so that its returns reflect stock selection rather than market direction.

Practical beta estimates are sensitive to choices that have nothing to do with the company: the index used as the market, the return frequency (daily, weekly, monthly), and the window (commonly two years of weekly or five years of monthly data). Data vendors also publish 'adjusted' betas that shrink the raw estimate toward 1, commonly two thirds raw plus one third times 1, on the evidence that extreme betas tend to drift back toward the average over time.

Residuals, R squared and the split of risk

Once the line is fitted, each observation splits into the part the market explains (alpha plus beta times the market return) and a residual. R squared is one minus the ratio of the residual sum of squares to the total sum of squares of the asset's returns around their mean. It is the share of the asset's variance explained by the market, and in a regression with one explanatory variable it equals the correlation squared.

The same split applies to risk. Total variance equals systematic variance (beta squared times the market variance) plus idiosyncratic variance (the variance of the residuals). Systematic risk cannot be diversified away by holding more stocks, because every stock shares it; idiosyncratic risk largely can. That distinction is the foundation of the CAPM and of the portfolio construction topic that follows.

Single stocks typically have R squared values against a broad index somewhere between 0.2 and 0.6 on monthly data; diversified equity funds often exceed 0.9. The 0.75 in this example is high for a single stock, which is a feature of the small made up sample. For a hedge fund, a high R squared against an equity index says most of its returns are market exposure, which should be priced as cheap beta rather than expensive alpha.

The CAPM: from beta to required return

The capital asset pricing model says that in equilibrium investors are rewarded only for systematic risk, because idiosyncratic risk can be diversified away at no cost. The expected return on any asset is therefore the risk free rate plus its beta times the market's expected excess return, the equity risk premium (ERP). Plotted against beta, this relationship is a straight line called the security market line.

In practice the CAPM is used in two directions. Forward, it produces a cost of equity for a discounted cash flow valuation or a hurdle rate for an investment committee: plug in today's risk free rate, an estimate of the ERP and the asset's beta. Backward, it defines alpha: if a manager's realised return beats what the CAPM predicts for the beta taken, the excess is Jensen's alpha, which is the regression intercept when the regression is run on excess returns.

The model's empirical record is mixed. Low beta stocks have historically earned more than the CAPM predicts and high beta stocks less, and factors such as size, value and momentum explain returns that beta does not. Practitioners therefore treat the CAPM as a disciplined starting point for a cost of equity rather than a precise forecast, and use multi factor regressions when evaluating managers.

How much to trust the estimate

Regression output always comes with standard errors, and they are not optional reading. The standard error of beta is the residual standard deviation divided by the square root of the sum of squared market deviations. The standard error of alpha is larger relative to its size, because alpha is a small number estimated from noisy data. The t statistic, the estimate divided by its standard error, says how many standard errors the estimate sits away from zero.

For stock S, beta has a t statistic of about 3.5, which with four degrees of freedom (six observations minus two estimated parameters) clears the 5 percent critical value of 2.776. Alpha has a t statistic of about 0.24, indistinguishable from zero. The six months tell us, weakly, that S has market exposure. They tell us nothing at all about whether S outperforms.

This asymmetry holds for real managers too. Betas can be estimated reasonably well from a few years of data; alphas generally cannot. A manager with a true alpha of 2 percent a year and 8 percent of residual volatility needs, by the same arithmetic, many years of data before the alpha is statistically distinguishable from luck. That is why serious allocators combine the regression with a qualitative view of the process that generates the returns.

import numpy as np

# Illustrative monthly returns, as decimals
m = np.array([2, -1, 3, -2, 4, 0]) / 100
s = np.array([3, -1, 2, -4, 6, 3]) / 100

beta = np.cov(s, m)[0, 1] / np.var(m, ddof=1)
alpha = s.mean() - beta * m.mean()
corr = np.corrcoef(s, m)[0, 1]

print(f"beta        {beta:.4f}")
print(f"alpha       {alpha:.4%} per month")
print(f"correlation {corr:.4f}")
print(f"R squared   {corr ** 2:.4f}")

beta        1.2857
alpha       0.2143% per month
correlation 0.8675
R squared   0.7526

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