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(ri, rm) / Var(rm)
where:
Cov(ri, rm) = [∑(ri- riA) (rm - rmA)] / (n-1)
Var(rm) = [∑(rm - rmA)2] / (n-1)
ri = expected return on an individual asset
rm = 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).
Example Calculation for the Beta of an Asset
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.
Stock X | Stock X | S&P 500 | S&P 500 | |
|---|---|---|---|---|
| Observation | Close | Return | Close | Return |
| 1 | $10.20 | 3,845.53 | ||
| 2 | $10.35 | 1.47% | 3,899.22 | 1.40% |
| 3 | $9.98 | -3.57% | 3,814.57 | -2.17% |
| 4 | $10.45 | 4.74% | 3,965.69 | 3.96% |
| 5 | $10.69 | 2.30% | 3,789.55 | -4.44% |
| 6 | $11.22 | 4.94% | 3,954.56 | 4.35% |
| 7 | $12.28 | 9.47% | 3,920.44 | -0.86% |
| 8 | $11.30 | -7.99% | 3,968.50 | 1.23% |
| 9 | $9.54 | -15.59% | 4,009.30 | 1.03% |
| 10 | $9.09 | -4.71% | 3,956.17 | -1.33% |
| 11 | $12.84 | 41.21% | 3,870.60 | -2.16% |
| 12 | $10.22 | -20.35% | 3,788.88 | -2.11% |
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.
Example Calculation for the Beta of an Investment Portfolio
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…Wn βn
where:
βp = portfolio beta
n = asset number
Wn = weight of asset n
βn = 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:
Weighted | ||||
|---|---|---|---|---|
| Stock | Value | Weight | Beta | Beta |
| X | $10,894 | 32.98% | 1.748 | 0.577 |
| Y | $14,099 | 42.69% | 0.897 | 0.383 |
| Z | $8,037 | 24.33% | 0.551 | 0.134 |
| Total | $33,030 | 100.00% | 1.093 |
The 1.093 value resulting from the weighted beta calculation indicates a positively correlated portfolio with slightly higher volatility than the market.
Interpreting Beta Values
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.
Stock X | Stock X | S&P 500 | S&P 500 | |
|---|---|---|---|---|
| Observation | Close | Return | Close | Return |
| 1 | $10.20 | 3,845.53 | ||
| 2 | $10.35 | 1.47% | 3,899.22 | 1.40% |
| 3 | $9.98 | -3.57% | 3,814.57 | -2.17% |
| 4 | $10.45 | 4.74% | 3,965.69 | 3.96% |
| 5 | $10.69 | 2.30% | 3,789.55 | -4.44% |
| 6 | $11.22 | 4.94% | 3,954.56 | 4.35% |
| 7 | $12.28 | 9.47% | 3,920.44 | -0.86% |
| 8 | $11.30 | -7.99% | 3,968.50 | 1.23% |
| 9 | $9.54 | -15.59% | 4,009.30 | 1.03% |
| 10 | $9.09 | -4.71% | 3,956.17 | -1.33% |
| 11 | $12.84 | 41.21% | 3,870.60 | -2.16% |
| 12 | $10.22 | -20.35% | 3,788.88 | -2.11% |
