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, and Table 1 names its parts.

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 window, 2020 to 2025. The first panel of Figure 1 is NVDA’s cloud, with a slope of 1.80.

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 (Table 2).

GS and XOM swing by the same amount relative to the market (1.58 each) and have quite different betas (1.18 against 0.81), because GS follows the market more tightly (0.75 against 0.51). 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: half for NVDA, two percent for gold. The rest is $\varepsilon$.

Two worked cases that catch people (Table 3):

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

Table 2 is one number per ticker for the whole window. Estimate it on a one-year window and slide the window along, and the second panel of Figure 1 appears: NVDA’s beta ran from 1.34 to 2.92, XOM’s from 0.15 to 1.33, gold’s from −0.09 to +0.35. Beta depends on the window, the frequency (daily, weekly, monthly give different answers) and on the market itself changing. Report it with the window it came from, and expect a hedge built on last year’s beta to be off this year. The standard error of the full-window NVDA beta is 0.048, so the 1.80 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 (Table 4):

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 15.3% a year over 2020 to 2025 and a 2% risk-free rate (Table 5):

(Computed from the unrounded beta and market return, so redoing a row from the rounded figures shown can differ by 0.1.)

Read GS’s row carefully. It made 26.2% a year, eleven points more than the market, and its alpha is only +8.5%: with a beta of 1.18 it should have made 17.7% just for the exposure it carried. XOM runs the other way: 13.7% a year, below the market’s 15.3%, and still a small positive alpha of +1.0%, because a beta of 0.81 asked for only 12.7%. Beating cash, or even beating the market, is not alpha if beta explains it, and trailing the market is not the absence of 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 of Table 6 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; Table 7 does the same sum for a $50M position:

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

Real run. Hold NVDA through 2024, a year SPY rose 24.9%. NVDA’s beta estimated on the four years before was 1.72 (Table 8).

In an up year the hedges cost money (the third panel of Figure 1): the dollar hedge gave back 23 points and still left a beta of 1.66. The beta hedge gave back 40 points and still left a beta of 0.94, not zero, because NVDA’s beta in 2024 was 2.66, far above the 1.72 estimated on the four years before, so a hedge sized on the old beta cancelled little more than half the exposure. That is the last trap in Table 9, in numbers. What survived, +74.0%, is mostly NVDA’s own story, which in 2024 was the AI run. 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. Romero and Balch’s example (chapter 7) is long XOM against short SPY; with XOM’s beta at 0.77 in their data, it needs about 56% long and 44% short by dollars, since $0.77 \times 0.56 \approx 0.44$. On this series’ data XOM’s beta is 0.81, close to theirs, so the split is about 55% long and 45% short, since $0.81 \times 0.55 \approx 0.45$.

Traps

The code

NVDA’s beta and alpha, the rolling beta, the CAPM line and the hedge size. It reads the blog’s price file from the examples repo, so it runs as pasted (pip install pandas). m holds SPY’s daily returns (the market) and y NVDA’s.

import pandas as pd

URL = "https://raw.githubusercontent.com/ankurtrapasiya/examples/main/data/prices.csv"
prices = pd.read_csv(URL, index_col="date", parse_dates=True)
r = (prices / prices.shift(1) - 1).dropna()     # daily returns, one column per ticker
m, y = r["SPY"], r["NVDA"]                       # the market, and the stock

beta = y.cov(m) / m.var()                        # the slope
alpha_daily = y.mean() - beta * m.mean()         # the intercept
alpha_year = alpha_daily * 252
rho, ratio = y.corr(m), y.std() / m.std()        # correlation, and how much bigger its swings are
print(f"beta {beta:.2f} = {rho:.2f} x {ratio:.2f}, alpha {alpha_year:+.1%} a year")   # 1.80 = 0.70 x 2.56, +45.4%

rolling_beta = y.rolling(252).cov(m) / m.rolling(252).var()   # one-year window, slid daily
print(f"rolling beta from {rolling_beta.min():.2f} to {rolling_beta.max():.2f}")      # 1.34 to 2.92

rm = (1 + m).prod() ** (252 / len(m)) - 1        # SPY's compound yearly return
required = 0.02 + beta * (rm - 0.02)             # CAPM, with a 2% risk-free rate
print(f"market {rm:.1%}, required {required:.1%}")                                     # 15.3%, 25.8%

hedge = 10e6 * 1.3 / 1.0                         # SPY to short against $10M at beta 1.3
print(f"hedge ${hedge / 1e6:.0f}M")                                                    # $13M

It prints beta 1.80 as $0.70 \times 2.56$ and the +45.4% alpha, the rolling beta’s range (1.34 to 2.92), SPY’s 15.3% with the 25.8% required, and the $13M hedge. y.cov(m) / m.var() is the slope of the regression line, the same number the linear regression post’s formula gives.

The script behind all the numbers

The tables, including the 2024 hedge, come from examples/beta-is-a-slope/beta.py. To run it, save it next to the shared helpers market_data.py and chartstyle.py and the price file data/prices.csv, keeping the folders as in the links, then run uv run examples/beta-is-a-slope/beta.py.

Summary

Table 10 collects the formulas.

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, and the beta-neutral XOM against SPY pair
• 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