How to Compute Beta With Covariance and Regression

Beta is a fundamental concept in financial analysis, measuring a security’s or portfolio’s systematic risk. This type of risk is inherent to the overall market and cannot be eliminated through diversification. Beta quantifies the sensitivity of an asset’s returns to movements in broader market returns, indicating how much an asset’s price tends to move relative to the market, which typically has a beta of 1.0. It is widely used in models like the Capital Asset Pricing Model (CAPM) to understand an asset’s expected return given its market risk.

Required Data and Preparation

To accurately compute beta, historical data is necessary for both the individual security or portfolio and a relevant market index. You need historical closing prices for the stock or portfolio and a broad market index. For U.S. stocks, the S&P 500 is a common benchmark.

Selecting the time period and data frequency is important for beta calculation. Professionals commonly use three to five years of historical data, often monthly or weekly. While daily data captures short-term fluctuations, monthly data is preferred to smooth noise and provide a more stable beta estimate. Consistency in data frequency is important; use the same frequency for both the stock and market index.

After gathering historical closing price data, convert prices into percentage returns. This is crucial because beta measures the relationship between returns, not prices. Calculate percentage return by taking the current closing price, subtracting the previous closing price, and dividing by the previous closing price. Perform this calculation for every period for both the stock and market index, creating two series of percentage returns. Common sources for historical financial data include news websites or brokerage platforms.

Calculating Beta Using the Covariance Formula

One method for determining beta uses covariance and variance, foundational statistical concepts. The formula is: Beta = Covariance(Stock Returns, Market Returns) / Variance(Market Returns). This highlights the relationship between how the stock and market move together, relative to the market’s movement.

Covariance measures how two variables, like stock and market returns, move in tandem. Positive covariance indicates that when one variable’s returns are above average, the other’s also tend to be. To calculate covariance, find the average return for both the stock and market. Then, for each period, subtract the average return from the actual return for both. Multiply these two differences for each period, sum the products, and divide by the number of periods minus one for a sample calculation.

Variance measures how market returns deviate from their average. To calculate market variance, determine the average market return. For each period, subtract the average market return from the actual and square the difference. Sum these squared differences and divide by the number of periods minus one. Dividing the covariance of stock and market returns by market variance yields beta.

Calculating Beta Using Regression Analysis

Regression analysis is a practical, widely used method for calculating beta, often preferred for its ease with software. This technique finds the best-fit line through a scatter plot of stock returns against market returns. The slope of this line represents beta.

Stock percentage returns are the dependent variable (Y), observed in relation to the market. Market percentage returns serve as the independent variable (X), the explanatory factor. Spreadsheet software, like Excel or Google Sheets, provides built-in functions for efficient calculation.

In Excel, use the “Data Analysis Toolpak” for linear regression. Organize stock returns in one column and market returns in another. Select “Regression,” designating stock returns as “Input Y Range” and market returns as “Input X Range.” The output includes a coefficient table, where the X Variable (market returns) coefficient is your calculated beta. Alternatively, in Excel and Google Sheets, use the SLOPE function, inputting stock returns (known_y’s) followed by market returns (known_x’s) to obtain beta. The regression output also provides an R-squared value, indicating the proportion of stock return variance explained by market movements.