# Beta Coefficient

**Definition**

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.

**Formula**

Beta coefficient is estimated by regression analysis. In general terms, it can be calculated as follows:

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:

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.

**Beta**** of portfolio**

The beta of a portfolio is a weighted average of all beta coefficients of its constituent securities.

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:

- 40% of Stock X with β
_{A }= 0.85 - 35% of Stock Y with β
_{A }= 1.1 - 25% of Stock Z with β
_{A }= 1.35

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

**Interpretation**

The interpretation of the key values of beta is shown below.

**β****< 0.**Return of a security drives in the opposite direction from the market return. A negative value is very rare for long positions but is normal for short positions.**β****= 0.**There is no correlation between a security return and the market return. For example, zero beta coefficient has fixed income securities because the return doesn’t depend on market return movements. Another example is cash under the condition of zero inflation because its value doesn’t change over time unlike market return.**0 <****β****<1.**Return of a security moves in the same direction as market return, but its volatility is less than market volatility.**β****= 1.**The security return and market return move in the same direction and have equal volatility.**β****>1.**The return of a security moves in the same direction as the market and has higher volatility than the market return.

**Example**

Let’s assume that Stock A and the market demonstrated the following return over the last 5 years:

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