Formula for Calculating Portfolio Beta

An asset's beta is a metric describing how likely it is to change in value as broader financial markets change. You can measure the beta of an individual asset, such as a stock, with this formula: ‌Asset Beta = Covariance/Variance‌.

Covariance is the joint variability of two variables, such as an individual stock and a market index. Variance measures how far a set of values spreads out from the average value. Its square root is the standard deviation. In symbols, the formula is:

‌β = Cov(r_i, r_m) / Var(r_m)‌

where:

Cov(r_i, r_m) = [∑(r_i- r_iA) (r_m - r_mA)] / (n-1)

Var(r_m) = [∑(r_m - r_mA)²] / (n-1)

‌r_i‌ = expected return on an individual asset

‌r_m‌ = average expected return on the market portfolio

‌A‌ = the average return of the asset or portfolio

‌n‌ = number of observations

The return on an asset or portfolio equals ((opening price – closing price)/opening price).

You can use Microsoft Excel or Google Sheets for calculating beta. You can calculate a stock's beta by constructing arrays of prices (usually daily closing prices) and stock returns ((closing stock price – previous closing)/previous closing) of the stock and an overall market index for a set number of consecutive observations.

For example, you could create a table where each row contains the daily closing price, the returns for stock X and the stock market returns of the S&P 500 benchmark index. According to the U.S. Securities and Exchange Commission, the S&P 500 index represents a sample of leading companies in leading industries within the U.S. economy.

The spreadsheet returns a covariance of 0.002629 and a variance of 0.001504. Dividing the variance into the covariance gives a beta of 1.748.

To calculate the beta of a portfolio you have through a brokerage, you sum the weighted betas of each stock inside the portfolio. The portfolio beta formula is:

‌βₚ = W₁β₁+W₂ β₂+W₃ β ₃…W_n β_n‌

where:

‌β_p‌ = portfolio beta

‌n‌ = asset number

‌W_n‌ = weight of asset n

‌β_n ‌ ‌=‌ beta of asset n

In this example, the allocation of the portfolio consists of stocks X, Y and Z, with the indicated values and betas:

The 1.093 value resulting from the weighted beta calculation indicates a positively correlated portfolio with slightly higher volatility than the market.

As with individual assets, understanding a portfolio's risk and possible returns is key to making informed investment decisions. Measurements such as portfolio beta provide data that can help an investor reshape a portfolio in changing economic times or ensure that a group of investments has a high probability of performing as expected.

Aswath Damodaran explains in a New York University paper that the Capital Asset Pricing Model (CAPM) postulates a linear relationship between the expected return for an asset and its beta – the slope of a linear regression. When an asset has a beta of zero, it changes independently of fluctuations in the market. A positive value indicates that an asset rises when the markets rise and falls when markets fall. Conversely, a negative beta indicates that an asset moves in the opposite direction of the financial market.

The market has a beta of 1.0, so an asset or portfolio with a high beta (greater than 1.0) is more volatile than the market, while it would be less volatile if the beta of a stock is under 1.0. Low-beta stocks are only weakly correlated to the market.

Portfolio beta shows how closely a portfolio follows more general market trends. This allows an investor to take advantage of projected market growth or hedge against market downtrends through diversification into negative-beta assets that will move contrary to a market. Portfolio beta measures also provide a reference point for comparing projected returns for markets, individual assets and portfolios.