# Beta Explained

Beta is a measure of systematic risk. Statistically, it depends on the degree of correlation between a security and the market.

## Calculating Beta

We begin with the single index model using realized returns constructed as follows:

$$R_i – R_f = \beta_i [R_i – R_f] + e_i$$

Which we can also formulate as:

$$R_i = (1 – \beta_i ) R_f + \beta_i × R_m + e_i$$

Systematic risk depends on the correlation between the asset and the market. Therefore, beta can be measured by examining the covariance between Ri and Rm:

$$Cov(R_i, R_m) = Cov(\beta_i × R_m + e_i , R_m)$$

$$Cov(R_i, R_m) = \beta_i Cov(R_m, R_m) + Cov(e_i , R_m)$$

$$Cov(R_i, R_m) = \beta_i \sigma_m^2 + 0$$

Note: Cov(ei , Rm) = 0 because the error term is uncorrelated with the market. By rearranging the equation to solve beta, we have:

$$\beta_i = \frac { Cov(R_i, R_m )}{ \sigma_m^2}$$

Where $$Cov(R_i, R_m) = \rho_{ i,m} \sigma_i \sigma_m$$ which, when substituted into the equation, simplifies it to $$\beta_i= \frac{\rho_{ i,m} \sigma_i } {\sigma_m}$$.

Beta provides a measurement of the sensitivity of the asset returns to the market as a whole. Aside from this, it captures the portion of the asset risk that cannot be diversified.

## Estimating Beta

The variances and correlations required to calculate beta are usually determined using the historical returns for the asset and market. A regression analysis can be performed. The analysis essentially plots the market returns on the x-axis and the security returns on the y-axis and then finds the “best fit” straight line through these points. The slope of the regression line is the measure of beta. Using return data over the prior 12 months tends to represent the security’s current level of systematic risk. However, this approach may be less accurate than a beta measured over 3 to 5 years, given that a short-term event may impact the data.

It is important to recognize that irrespective of the data period, beta is an estimate of systematic risk based on historical data and may not represent future systematic risk.

## Interpreting Beta

A positive beta indicates that the asset moves in the same direction as the market, whereas a negative beta indicates the opposite.

The beta of a risk-free asset is zero because the covariance of the risk-free asset and the market is zero. The market’s beta is, by definition, 1, and most developed market stocks tend to exhibit high, positive betas.

## Question

If the correlation between an asset and the market is 0.6, the standard deviation of the asset is 18%, and the standard deviation of the market is 14%, what is the asset beta?

A. 0.77.

B. 0.47.

C. 0.99.

Solution

$$\beta_i= \frac{\rho_{ i,m} \sigma_i } {\sigma_m}$$

$$\beta_i= \frac{0.6 × 0.18 } {0.14}$$

$$\beta_i= 0.77$$

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