# Formula for Calculating Portfolio Beta

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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.

Example Beta Calculation

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%

Eric Bank

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:

Example of Portfolio Beta Calculation

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

Eric Bank

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.

Example Beta Calculation

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%

Eric Bank