The Problem With Using Beta (β) As A Proxy For Risk

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The following is a brief list of definitions for beta from various textbooks and research papers.  Though brief, this is a representative sampling of the modern theory regarding beta.

Damodaran – “standardized measure of the risk that investment adds to the market portfolio.”

“Statistically, the beta of an investment measures how it covaries with the market portfolio and thus the risk added to that portfolio. Beta’s features: Betas are standardized around one; an investment with a beta above (below) one is an above average (below average) risk investment.“

Kpthari and Shanken – “primary measure of non-diversifiable risk.”

Reilly and Brown – “non-diversifiable portion of that stock’s risk relative to the market as a whole.  Because of this, beta can be thought as an indexing of the asset’s systemic risk to that of the market portfolio.  A stock with a beta of 1.20 has a level of systematic risk that is 20% greater than the average for the entire market, while a stock with a beta of 0.70 is 30% less risky than the market.  By definition, the market portfolio itself will always have a beta of 1.00.”

Brigham and Ehrhardt – “An average risk stock is defined as one with a beta equal to 1.”  “Beta measures a stock’s tendency to move up and down with the market.”  “A measure of a stock’s market risk, or the extent to which the returns on a given stock move with the stock market.”

CFA Institute – “beta is a measure of how sensitive an asset’s return is to the market as a whole.”  “beta captures and asset’s systematic risk, or the portion of an asset’s risk that cannot be eliminated by diversification.”  “A measure of systematic risk that is based on the covariance of an asset’s or portfolio’s return of the overall market.”

Some definitions correlate beta to risk, some attribute a beta greater than unity to more risk and less than unity to less risk, and others speak of the market portfolio.  Most mention that beta represents non-diversifiable risk (i.e., systematic risk).

Diagram borrowed from Investopia

Why is beta deemed synonymous with the risk of an individual asset as it is added to the market portfolio?  Because it is the slope of the individual asset’s returns plotted against the market portfolio’s returns (i.e., S&P 500 typically used as a proxy), and an asset with a return greater than the market (i.e., slope greater than unity) must be higher risk than the market since return is assumed to be positively correlated with risk (see figure above).  Simply, a stock with a beta greater than unity is higher risk than a stock with a beta less than unity.  This all makes theoretical sense, right?  But does this theory apply to reality? – that is the question.  To address this question, let’s look at the following hypothetical stocks:

What is striking about these stocks is that they all have betas of unity, so by definition are just as risky as the market, and should thus have the same returns as the market to compensate for the equivalent levels of risk.  Looking at the stand-alone risk of each stock (i.e., the standard deviation σ for each stock), things look good in terms of beta and risk being even.  But when we look at the returns, striking differences exist.  How can these three stocks have the same beta and the same standard deviation, yet such different returns, ranging between (51.5%) and 179.5% over a six-year period?  It is simple, because standard deviation is only the fluctuation of the stock over time, and the beta is merely the correlation of the stock’s return with the market, where neither address the absolute return of the stock.  To properly characterize an investment, risk and return must always be accounted for, one or the other in isolation is not a complete picture.

Ranking these stocks instead using the Coefficient of Variation (CV), which takes into account the risk and return, we quickly see that Stock #3 is the best investment since it has the lowest CV.  Stock #2 technically has the lowest CV, but a negative return and negative CV is a bad investment however you cut it!  What this simple exercise quickly shows is that both beta and standard deviation, in isolation of return, cannot provide sufficient information to choose an investment.  I would also consider Stock #2 clearly a riskier investment than Stock #3, but beta, which is supposed to indicate risk, holds these two stocks comparable.  So what is beta measuring then?  Brigham and Ehrhardt are consistently the most accurate in their definition, simply pointing out that beta is a correlation.

Let’s take a look at another selection of hypothetical stocks (see below).  Stock #4 has a negative beta, but an identical standard deviation to that of the market.  Once again, in isolation without considering return, standard deviation does not provide enough information.  Note that negative betas are not a bad thing, and when constructing portfolios can serve to dampen downward momentum during a recession since they act opposite to the market.  So similar to placing a put and call option to lock in a guaranteed return, utilizing both positive and negative betas can stabilize your return.  Stock #5 has a beta below unity, so it must be less risky than the market – this is at least what most theory states, but notice that it has a higher standard deviation and CV than the market, so it actually is riskier than the market even though it has a beta below unity.  Stock #6 is the equivalent of a 6-yr bond yielding 3.7% – note that it has a beta of zero, yet a return equal to the market, and zero risk.  So any asset that does not correlate with the “market” as defined by the S&P 500 can obviously taint whatever analysis and portfolio weight that is placed on beta.  Finally, Stock #7 has a similar standard deviation, return, and CV to that of the market, yet it has a beta below unity and is negative.

Based on these examples, the standard definition of assuming that beta is synonymous with risk is clearly flawed, and beta in isolation should not be used to select investments.  This analysis can be found on the “Model” worksheet in the attached workbook.

Two stocks have been randomly selecting from the S&P 100 that had a beta above and below market by 35% and (55%) (tickers AVP and AMGN respectively).  These two stocks were then combined into a portfolio, using the Markowitz Portfolio Theory to estimate the standard deviation of these two portfolios.  The summary presented below can be found in the “Summary” worksheet of the attached workbook.  The results of this analysis are pictured below:

Immediately you can see that the market has a higher standard deviation than AVP, yet AVP has a higher beta, which is in direct contradiction to beta being a measure of risk.  Someone may argue that beta is for the individual asset being added to the market portfolio, whereas standard deviation is for stand-alone risk, but even when adding high and low beta stocks to the market portfolio, shifts in the portfolio’s standard deviation did not correlate with the beta of the asset being added to the portfolio.

Portfolio 1 is a 24:76 mix of AVP:AMGN, and portfolio 2 is a 58%:42% mix.  Portfolio 2 has a beta of 1, yet is significantly riskier than market based upon its standard deviation, while portfolio 2 has a beta below unity so is “less risky” than market (by definition), yet has a standard deviation that is significantly greater than market.

In summary, simple standard deviations in combination with CV appear to be a safer and more practical method for selecting stocks, and reflecting the true risk inherent with the investment.  Plus these metrics work directly in the Markowitz Portfolio Theory.  Also, beta seems unreliable for telling much other than the correlation of the asset’s historical returns with that of the market’s historical returns, which is exactly what beta is.  The only caveat to using standard deviation, returns, and CV is that a black swan will not be accounted for since these theories depend solely on Gaussian distributions, but then one needs to ask how a black swan can truly be predicted?

The Workbook used with this post can be downloaded here.

A PDF of this post can be downloaded here.

©2018 Ben Etzkorn

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