 # Beta and risk

Beta reflects the additional risk an investment adds to an investor’s diversified portfolio (the diversified portfolio bit is essential, see risk section). Beta measures the volatility of the investment relative to the market as a whole.  A beta of 1 indicates the investment moves directly with the market.  A beta of less than 1 indicates the investment has less volatility than the market. A negative beta indicates the investment goes up when the overall market goes down.

## Historical beta

Beta is calculated by regressing a stock’s returns against returns on a market index over the same period; the beta is the slope of the regression line.  In calculating a beta, the business valuation expert needs to decide upon the following inputs:

• The regression period.  A longer regression period provides more data; however, there is the risk the firm may have changed over that period.
• Return interval.  Daily returns provide more data but include non-trading periods where returns will be zero.  These non-trading periods are more applicable to smaller, thinly traded stocks and so will distort the overall returns.
• Market index. The standard practice is to use the relevant capitalised weighted index for the market in which the stock trades, so for an Australian stock, this will be against an ASX index, for example, the S&P/ASX 200 index.

The graph below plots the weekly return for Woolworths Group Ltd for the five year period ended in October 2018 against the S&P/ASX 200 weekly return over the same period. Each point represents one week’s return.

The slope of the regression line, the beta, is 0.91.  A beta of 0.91 implies that the Woolworths stock is closely aligned to the market (as per the S&P ASX 200).  If the market returns increase 10%, returns on Woolworth stock increase 9%. ## Calculating Beta

The slope of the regression is calculated as the covariance (COV) of a stock’s return and a market’s return, divided by the variation of the market’s return. The formula is as follows, where x is the independent variable (market return) and y the dependent variable (stock’s return): The covariance is the average value of the deviation of the market’s return and a stock’s return from each of their respective means. In other words, the covariance is a measure of the joint variability of the market’s return and the stock’s return. The covariance between the market’s weekly returns and Woolworths’ weekly returns, in the five-year period to October 2018, is 0.027%.

The variance is the average value of the square of the deviation of the market’s return from its mean, for the period.  Essentially the variance shows how far the returns are spread away from the average return. The variance for the market’s weekly returns, in the five-year period, is 0.03%.

An alternative way to calculate Beta in Excel is to use the SLOPE function or the Regression tool in the Data Analysis module.

The return is simply calculated by the movement in price in the period plus dividends, divided by the opening price.  In the Woolworths example, this is the movement between weeks.

Importantly the stock and market return has to include both price movement and dividends in the period. Dividends are reflected in the adjusted price of stocks and markets.  The videos below explain in more detail how Beta can be calculated using Excel.

## R-squared

The R-squared (the squared correlation coefficient) is a measure of how well the regression line fits the data, it measures the proportion of a stocks return that can be explained by the market return. Therefore, R-squared is a measure of the percentage of total risk that is related to market risk and 1-R-squared is the balance related to specific risk. A high R-square implies that much of a stock’s risk is related to market risk, with a much lower exposure to firm-specific risk.  A low R-squared implies a much higher exposure to firm-specific risk.

The correlation coefficient, R, is equal to the covariance of the stock return with the market return, divided by the product of the standard deviation of the stock return and market return. In the Woolworths example, the R-squared is 0.36. The implication being that 36% of Woolworth’s risk comes from market risk and 64% comes from firm-specific risk.

## Standard of error

The standard of error is a measure of the statistical accuracy of an estimate.  The standard of error is the standard deviation of the distribution, and it can be used to predict the beta with degrees of confidence. The true beta can be predicted with 67% confidence that it will be within one standard error of the beta estimate and with 95% confidence that it will be between two standard errors.

The standard error in the Woolworth’s regression analysis is .074.  Therefore, the Woolworths beta can be predicted to be between 0.76 and 1.05 with 95% confidence.

## Intercept of the regression

The intercept of the regression line on the y-axis is known as alpha.  Alpha is a measure of the risk-adjusted excess return earned by the stock, relative to the performance of the market.  The risk-adjusted return, is after adjusting the return for the beta risk of the stock.

A positive alpha indicates the stock outperformed the market and a negative alpha indicates the stock underperformed.

In the Woolworths example above, the intercept, alpha, is -0.017%, which is equal to an annualised difference of  -0.87% (1+weekly return)^52-1). The negative alpha implies that Woolworths over the period performed worse than expected in the period.

The alpha is by definition the unexpected element, and so cannot be extrapolated and does not assist in predicting whether the stock would be a good or bad investment in the future.

Simon is a CA Business Valuation specialist, Chartered Accountant and a Certified Fraud Examiner. Simon specialises in providing valuation services. Prior to founding Lotus Amity, he was a Corporate Finance and Forensic Accounting partner with BDO Australia. Simon provides valuation services in disputes, for raising finance, for restructuring, transactions and for tax purposes.