Beta coefficient is a measure of the systematic risk of a security or a portfolio compared with the market as a whole. It is widely used in portfolio theory and namely in capital asset pricing model (CAPM) and security market line (SML). Beta shows whether the volatility of return of a given security is higher or lower than market return volatility.
Beta coefficient is estimated by regression analysis. In general terms, it can be calculated as follows:
where Cov(Ra, RM) is a covariance between the return of a given security and market return, and Var(RM) is a variance of market return.
The formula above can be modified and written as follows:
|β =||N||(ki - k)(pi - p)|
|N||(pi - p)|
where ki is the actual return of a security in time period i, k is the expected return of a security, pi is the actual return of a market portfolio in time period i, and p is the expected return of a market portfolio.
If there is a finite data set of N values of security and portfolio actual return, the beta coefficient can be estimated using the formula above.
The beta of a portfolio is a weighted average of all beta coefficients of its constituent securities.
|βP =||N||wi × βi|
|i = 1|
where wi is the proportion of a given security in a portfolio, βi is the beta coefficient of a given security, and N is the number of securities in a portfolio.
Assume there is Portfolio XYZ consisting of three stocks in the following proportions:
The beta coefficient of Portfolio XYZ is 1.0625.
βXYZ = 0.4 × 0.85 + 0.35 × 1.1 + 0.25 × 1.35 = 1.0625
The interpretation of the key values of beta is shown below.
Let’s assume that Stock A and the market demonstrated the following return over the last 5 years:
|Return of Stock A||%||8.75||11.50||6.25||1.25||9.50|
The expected return of Stock A is 7.45%, and the expected market return is 6.25%.
E(RA) = (8.75 + 11.50 + 6.25 + 1.25 + 9.50) ÷ 5 = 7.45%
E(RM) = (6.50 + 7.75 + 5.25 + 3.50 + 8.25) ÷ 5 = 6.25%
The beta coefficient of Stock A is 1.93.
βA = [(8.75 - 7.45) × (6.50 - 6.25) + (11.50 - 7.45) × (7.75 - 6.25) + (6.25 - 7.45) × (5.25 - 6.25) + (1.75 - 7.45) × (3.50 - 6.25) + (9.50 - 7.45) × (8.25 - 6.25)] ÷ [(6.50 - 6.25)2 + (7.75 - 6.25)2 + (5.25 - 6.25)2 + (3.50 - 6.25)2 + (8.25 - 6.25)2] = 1.93