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# Calculation and interpretation of beta coefficient

A market portfolio consists of all assets in all asset classes. Creating such a portfolio is impractical. An alternative approach is to start with the broad market index and add assets based on their expected return, variance, correlation coefficient, etc. as long as such addition has a significant impact on the performance. Such a process requires estimates of only systematic risk because unsystematic risk can be diversified away.

$$R_i = R_f × (1- β_i) + β_i × R_m+e_i$$

Since the systematic risk of an asset depends on the covariance between asset returns and market returns, which can be shown to be equal to the product of beta coefficient of asset i and market variance.

In other words, the beta coefficient can be calculated using the following relationship:

$$Cov(R_i,R_m)= {β_i}{σ_m^2}$$

$$β_i=\frac{Cov(R_i,R_m)}{σ_m^2}$$

We know that the covariance equals the correlation coefficient multiplied by the standard deviation of the asset and the market, hence we can write the following expression for the beta coefficient:

$$Beta\ Coefficient = Correlation\ Coefficient × \frac{σ_a}{σ_m}$$

## Interpretation of beta

Beta is a measure of how sensitive an asset’s return is to the market as a whole. It captures the systematic risk, the component of risk which cannot be eliminated by diversification. The market has a beta of 1 and the risk-free asset has a beta of zero.

Most stocks in developed markets, stock indexes, and sectors have a high correlation coefficient which shows that they are exposed to similar economic and market factors.

## Estimating beta using regression analysis

Beta is often estimated directly from historical returns on an asset compared with the market by using regression analysis. In such a model, beta equals the slope of the regression line. However, beta accuracy is a subject of considerable debate. Selecting a shorter period may provide a better indication of a firm’s current level of risk, but they may be prone to special events that occur during that short period.

Once we know an asset’s beta coefficient, risk-free rate and return on the broad market, we can estimate the expected return on a stock using the capital asset pricing model.

## Test

If the correlation between a stock of the broad market is 0.9, the variance of the stock returns is 2% and that of the market returns is 1.9%, the beta coefficient would be:

1. 0.947
2. 0.923
3. 0.874

B is correct. We first need to convert the variances to standard deviation by taking the square roots. Beta can be calculated as a correlation coefficient (0.9) multiplied by standard deviation of the stock return (14.1%) divided by the standard deviation of the market (13.8%).

If an analyst is concerned with the predictive capacity of the beta coefficient, he should estimate it based on:

1. Last 12-month data
2. Last 3-year data
3. Last 10-year data