Beta and alpha: how much of the market you carry, and what is left

Take a stock’s daily returns and the market’s daily returns over the same days, one dot per day, market across and stock up. Fit a straight line through the cloud. That line is the whole post.

beta, the slopethe stock's returnvolatility, the spread of returns

$$\good{r_{\text{stock}}} = \alpha + \key{\beta}\,r_{\text{market}} + \varepsilon$$

This is linear regression with the market as the one feature. The market here is SPY, the S&P 500 fund, and the data is the series’ usual five years, 2019 to 2023.

Beta: the slope

Beta is covariance with the market divided by the market’s variance, and it has a second form that is easier to think with:

$$\key{\beta} = \frac{\text{Cov}(r_{\text{stock}}, r_{\text{market}})}{\text{Var}(r_{\text{market}})} = \rho \times \frac{\risk{\sigma_{\text{stock}}}}{\risk{\sigma_{\text{market}}}}$$

Beta is how tightly the stock follows the market (the correlation $\rho$) times how much bigger its swings are (the ratio of the two volatilities). Both parts matter, and they can pull in opposite directions.

XOM is more volatile than JPM relative to the market (1.63 against 1.52) and has the lower beta (0.90 against 1.11), because it follows the market less tightly. Gold swings a fair amount and has a beta of nearly zero, because it does not follow the market at all. High volatility does not mean high beta. The last column, $R^2$, is the share of the stock’s daily variation the market explains: two thirds for AAPL, one percent for gold. The rest is $\varepsilon$.

Two worked cases that catch people:

Beta is linear in weights, so a portfolio’s beta is the weighted average of its holdings’ betas: 60% at beta 1.4 and 40% at beta 0.5 is $0.84 + 0.20 = 1.04$. Cash has beta 0. A short position has a negative weight, so it subtracts beta.

Beta moves

The table above is one number per ticker for five years. Estimate it on a one-year window and slide the window along, and the second panel of the chart appears: AAPL’s beta ran from 1.11 to 1.59, XOM’s from 0.35 to 1.33, gold’s from −0.31 to +0.31. Beta depends on the window, the frequency (daily, weekly, monthly give different answers) and the regime. Report it with the window it came from, and expect a hedge built on last year’s beta to be a little off this year. The standard error of the five-year AAPL beta is 0.026, so the 1.24 is well pinned down; the year-to-year wandering is real change, not estimation noise.

Alpha: what is left

The intercept is what the stock returned on average on a day the market was flat, scaled up to a year by multiplying by 252:

The capital asset pricing model turns the slope into a required return. The return you deserve for carrying beta is the risk-free rate plus beta times the market’s extra return over it; alpha, in this form, is what you actually got minus that.

$$\text{required return} = R_f + \key{\beta}\,(R_m - R_f)$$ $$\alpha = \good{R_{\text{actual}}} - \big[R_f + \key{\beta}\,(R_m - R_f)\big]$$

With SPY’s compound return of 15.6% a year over 2019 to 2023 and a 2% risk-free rate:

Read JPM’s row carefully. It made 14.8% a year, nearly as much as the market, and its alpha is negative: with a beta of 1.11 it should have made 17.1% just for the exposure it carried. Beating cash, or even nearly matching the market, is not alpha if beta explains it. The model’s claim is that, across all stocks together, alpha averages zero: the market is the sum of its parts, so every positive alpha is someone else’s negative one. SPY itself has beta 1 and alpha 0 by construction.

The last row is the usual disappointment: an estimated alpha needs to be about twice its standard error before it means anything, and most are not.

In daily work the risk-free term is often dropped, because a day’s risk-free return is a few thousandths of a percent: $r_{\text{stock}} = \key{\beta}\,r_{\text{market}} + \alpha$, with beta the slope and alpha the intercept, exactly the regression at the top.

Hedging cancels beta-dollars, not dollars

Suppose you hold $10M of a stock with beta 1.3 and want no market exposure. Shorting $10M of SPY leaves you with $10 \times 1.3 - 10 \times 1.0 = 3$ beta-dollars of exposure. The hedge that cancels is $13M of SPY:

$$\key{\text{hedge}} = \text{dollars held} \times \frac{\beta_{\text{held}}}{\beta_{\text{hedge}}}$$

Real run. Hold AAPL through 2022, a year SPY fell 18.2%. AAPL’s beta estimated on the three years before was 1.22.

The dollar hedge removed most of the damage and left a beta of 0.30, which in a down year still cost four points. The beta hedge took the exposure to 0.08 and left only AAPL’s own story, which in 2022 was slightly negative. Being beta neutral is not being safe: the leftover $\varepsilon$ is the company-specific risk, and it is the thing you chose to hold. A pair that is long one stock and short another with weights that cancel the beta-dollars is the basic hedge-fund trade: long XOM at beta 0.77 against short SPY at beta 1.0 needs about 56% long and 44% short by dollars, since $0.77 \times 0.56 \approx 0.44$.

Traps

The code

import numpy as np
import pandas as pd

prices = pd.read_csv("prices.csv", index_col="date", parse_dates=True)
r = (prices / prices.shift(1) - 1).dropna()
m, y = r["SPY"], r["AAPL"]

beta = y.cov(m) / m.var()                       # 1.24
alpha_daily = y.mean() - beta * m.mean()        # the intercept
alpha_year = alpha_daily * 252                  # +17.1%
rho, ratio = y.corr(m), y.std() / m.std()       # 0.80 and 1.54: rho * ratio == beta

rolling_beta = y.rolling(252).cov(m) / m.rolling(252).var()   # one-year window, slid daily

rm = (1 + m).prod() ** (252 / len(m)) - 1       # SPY's compound annual return, 15.6%
required = 0.02 + beta * (rm - 0.02)            # CAPM: 18.8%
hedge = 10e6 * 1.3 / 1.0                        # $13M of SPY against $10M at beta 1.3

Summary

Next

Portfolio optimisation: with expected returns, volatilities and correlations in hand, choose the weights.

References

• William F. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk”, The Journal of Finance 19(3), 425–442, 1964
• Philip J. Romero and Tucker Balch, What Hedge Funds Really Do: An Introduction to Portfolio Management (Business Expert Press, 2014), chapter 7: CAPM as a regression line, beta-neutral pairs
• James, Witten, Hastie, Tibshirani and Taylor, An Introduction to Statistical Learning, with Applications in Python (Springer, 2023), section 3.1: the standard error and t-statistic of a slope