### Beta Explained

Beta is a measure of systematic risk. Statistically, it depends upon 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:

$$R_i – R_f = β_i [R_i – R_f] + e_i$$

Which we can also formulate as:

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

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

$$Cov(R_i, R_m) = Cov(β_i × R_m + e_i , R_m)$$

$$Cov(R_i, R_m) = β_i Cov(R_m, R_m) + Cov(e_i , R_m)$$

$$Cov(R_i, R_m) = β_i σ_m^2 + 0$$

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

$$βi = \frac { Cov(R_i, R_m )}{ σ_m^2}$$

Where $$Cov(R_i, R_m) = ρ_{ i,m} σ_i σ_m$$ which when substituted into the equation simplifies to $$β_i= \frac{ρ_{ i,m} σ_i } {σ_m}$$.

Beta provides a measure of the sensitivity of the asset returns to the market as a whole and captures the portion of the asset risk that cannot be diversified away.

## Estimating Beta

The variances and correlations required to calculate beta are usually determined using the historic returns for the asset and the market. A regression analysis can be performed which 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 but may be less accurate than a beta measured over 3 to 5 years as a short-term event may impact the data.

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

## Interpreting Beta

A positive beta indicates the asset moves in the same direction as the market, whereas a negative beta would indicate 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 beta of the market 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 beta of the asset?

A. 0.77

B. 0.47

C. 0.99

Solution

$$β_i= \frac{ρ_{ i,m} σ_i } {σ_m}$$

$$β_i= \frac{0.6 × 0.18 } {0.14}$$

$$β_i= 0.77$$

Calculate and interpret beta

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