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:
β = | Cov(R_{a},R_{M}) |
Var(R_{M}) |
where Cov(R_{a}, R_{M}) is a covariance between the return of a given security and market return, and Var(R_{M}) is a variance of market return.
The formula above can be modified and written as follows:
β = | N | (k_{i} - k)(p_{i} - p) |
Σ | ||
i=1 | ||
N | (p_{i} - p) | |
Σ | ||
i=1 |
where k_{i} is the actual return of a security in time period i, k is the expected return of a security, p_{i} 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.
β_{P} = | N | w_{i} × β_{i} |
Σ | ||
i = 1 |
where w_{i} is the proportion of a given security in a portfolio, β_{i} 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.
β_{XYZ} = 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:
Years | 1 | 2 | 3 | 4 | 5 | |
Return of Stock A | % | 8.75 | 11.50 | 6.25 | 1.25 | 9.50 |
Market return | % | 6.50 | 7.75 | 5.25 | 3.50 | 8.25 |
The expected return of Stock A is 7.45%, and the expected market return is 6.25%.
E(R_{A}) = (8.75 + 11.50 + 6.25 + 1.25 + 9.50) ÷ 5 = 7.45%
E(R_{M}) = (6.50 + 7.75 + 5.25 + 3.50 + 8.25) ÷ 5 = 6.25%
The beta coefficient of Stock A is 1.93.
β_{A} = [(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