Beta Coefficient

What is a Beta Coefficient?

Beta coefficient is a measure of sensitivity of a company’s stock price to movement in the broad market index. It is an indicator of a stock’s systematic risk which is the undiversifiable risk inherent in the whole financial system.

Beta coefficient is an important input in the capital asset pricing model (CAPM). CAPM estimates a stock’s required rate of return (cost of equity) as the sum of the risk free interest rate and the stock’s equity risk premium. A stock’s equity risk premium is the product of the stock’s beta coefficient and the market risk premium, the difference between equity market return and the risk free interest rate.

Markowitz (1952) began modern portfolio theory (MPT) which can be used to explain the relationship between risk and return for assets, particularly stocks. Stock of companies that have higher rates of return have higher levels of risk. In order to achieve a lower level of risk, an investor must accept a lower expected rate of return. This concept is called the dominance principle and allows for the creation of the efficient frontier. MPT partitions risk into non-systematic risk, which can be eliminated from a portfolio through diversification, and systematic risk that is market wide and cannot be diversified. Non-systematic risk is company specific and is reduced to zero in a large, well diversified portfolio. In order to determine systematic risk for a stock, analysts use the market model developed by Sharpe (1964). The returns for a stock are regressed as the dependent variable against a market index used as the independent variable. The slope coefficient of the regression is the measure of systematic risk for the stock. Systematic risk measures the degree to which a stock moves with the market. A higher beta coefficient implies that returns for the stock move more than the market and a lower beta coefficient implies that returns for the stock move less that the market. The former are aggressive stocks and the latter are defensive stocks.

The beta formula is used in the CAPM model to calculate the Cost of Equity as shown below

`Cost of Equity = Risk Free Rate + Beta x Risk Premium`

Systematic vs Unsystematic Risk

We can think about unsystematic risk as “stock-specific” risk and systematic risk as “general-market” risk. If we hold only one stock in a portfolio, the return of that stock may vary wildly compared to the average gain or loss of the overall market as reflected by a major stock index such as the S&P 500. However, as we continue adding more to the portfolio, the portfolio’s returns will gradually start more closely resembling the overall market’s returns. As we diversify our portfolio of stocks, the “stock-specific” unsystematic risk is reduced.

Systematic risk is the underlying risk that affects the entire market. Large changes in macroeconomic variables, such as interest rates, inflation, GDP, or foreign exchange, are changes that impact the broader market and that cannot be avoided through diversification. The Beta coefficient relates “general-market” systematic risk to “stock-specific” unsystematic risk by comparing the rate of change between “general-market” and “stock-specific” returns.

What Does Beta Coefficient Mean?

A beta coefficient can measure the volatility of an individual stock compared to the systematic risk of the entire market. In statistical terms, beta represents the slope of the line through a regression of data points. In finance, each of these data points represents an individual stock’s returns against those of the market as a whole.

In the Capital Asset Pricing Model, the beta coefficient is used to calculate the rate of return of a portfolio or stock.

The calculation of Beta is a form of regression analysis, as it typically represents the slope of the security’s characteristic line; a straight line which shows the relationship between the rate of return of a stock and the rate of return from the market. That is simply a representation of the likelihood of a change in the rate of return of a stock or security as a result of a change in the rate of market returns. This can be ascertained through dividing the covariance of market return with stock return by the variance of market return, as shown below:

`β = Covariance of Market Return with Stock Return / Variance of Market Return`

If the coefficient is 1, then the price of the stock or security moves with the market. If the coefficient is less that one, then the security’s returns are less likely to respond to movements in the market. If the β coefficient is greater than 1, then the security’s returns are more likely to respond to movements in the market; more volatile.

Covariance equals the product of standard deviation of the stock return, standard deviation of the market return and correlation coefficient. Using this relationship, we arrive at another formula for beta coefficient which shows that the beta coefficient equals correlation coefficient multiplied by standard deviation of stock returns divided by standard deviation of market returns.

`β = Correlation coefficient between market and stock * (Standard deviation of stock returns ÷ Standard deviation of market returns)`

Portfolio beta can be estimated as the weighted-average of beta coefficients of individual stocks.

Beta effectively describes the activity of a security’s returns as it responds to swings in the market. A security’s beta is calculated by dividing the product of the covariance of the security’s returns and the market’s returns by the variance of the market’s returns over a specified period.

`Beta coefficient (β) = Covariance(Re, Rm) ÷ Variance(Rm)`

where
Re​ – the return on an individual stock;
Rm​ – the return on the overall market;
Covariance – how changes in a stock’s returns are related to changes in the market’s returns;
Variance – how far the market’s data points spread out from their average value.

The beta calculation is used to help investors understand whether a stock moves in the same direction as the rest of the market. It also provides insights about how volatile – or how risky – a stock is relative to the rest of the market. For beta to provide any useful insight, the market that is used as a benchmark should be related to the stock. For example, calculating a bond ETF’s beta using the S&P 500 as the benchmark would not provide much helpful insight for an investor because bonds and stocks are too dissimilar.

Ultimately, an investor is using beta to try to gauge how much risk a stock is adding to a portfolio. While a stock that deviates very little from the market doesn’t add a lot of risk to a portfolio, it also doesn’t increase the potential for greater returns.

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

βP = Σ wi × βi

where wi is the proportion of a given security in a portfolio, βi is the beta coefficient of a given security, and i=1 to N. 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 = 0.85
• 35% of Stock Y with β = 1.1
• 25% of Stock Z with β = 1.35

The beta coefficient of Portfolio XYZ is 1.0625.

βXYZ = 0.4 × 0.85 + 0.35 × 1.1 + 0.25 × 1.35 = 1.0625

Types of Beta Values

Beta Coefficient = 1

If a stock has a beta of 1.0, it indicates that the security return and market return move in the same direction and have equal volatility. A stock with a beta of 1.0 has systematic risk. However, the beta calculation can’t detect any unsystematic risk. Adding a stock to a portfolio with a beta of 1.0 doesn’t add any risk to the portfolio, but it also doesn’t increase the likelihood that the portfolio will provide an excess return.

Beta Coefficient < 1

A beta value that is less than 1.0 means that the security is theoretically less volatile than the market. Including this stock in a portfolio makes it less risky than the same portfolio without the stock. For example, utility stocks often have low betas because they tend to move more slowly than market averages.

Beta Coefficient > 1

A beta that is greater than 1.0 indicates that the security’s price is theoretically more volatile than the market. For example, if a stock’s beta is 1.2, it is assumed to be 20% more volatile than the market. Technology stocks and small cap stocks tend to have higher betas than the market benchmark. This indicates that adding the stock to a portfolio will increase the portfolio’s risk, but may also increase its expected return.

Beta Coefficient < 0

Some stocks have negative betas. A beta of -1.0 means that the stock is inversely correlated to the market benchmark. This stock could be thought of as an opposite, mirror image of the benchmark’s trends. Put options and inverse ETFs are designed to have negative betas. There are also a few industry groups, like gold miners, where a negative beta is also common.

Importance

Beta is the hedge ratio of an investment with respect to the stock market. For example, to hedge out the market-risk of a stock with a market beta of 2.0, an investor would short \$2,000 in the stock market for every \$1,000 invested in the stock. Thus insured, movements of the overall stock market no longer influence the combined position on average.

Beta thus measures the contribution of an individual investment to the risk of the market portfolio that was not reduced by diversification. It does not measure the risk when an investment is held on a stand-alone basis.