Relationship between Beta and Standard Deviation

  • I was doing some financial analysis on two firms in the coffee industry. After calculating Beta and Standard Deviation for both firms, I seem to have stumbled on some weird phenomenon.

    It appears that firm A has a higher standard deviation than firm B, while also possessing a lower beta coefficient.

    How is this possible? I had the impression that standard deviation and beta were both measures of risk / volatility, and a higher standard deviation would naturally lead to a higher beta.

    Your help would be greatly appreciated. Thanks and have a nice day!

    A warm welcome to Quant.SE and thank you for your question! If you find the answers helpful please upvote them and accept one of them. Thank you and looking forward to interacting with you more in the future :-)

  • beta_A = correlation_A_Index * (stdd_A / stdd_Index )

    The difference you see is due to correlation. The correlation between A and the index is lower than B and the index, and that's why you're seeing a lower beta.

    The moral of the story is that risk is subjective, and in fact you need to understand how your portfolio is correlated with these stocks in order to have an idea how buying the stock will impact your portfolio.

  • Intuitively put you can say that volatility is the within variation and beta is the between variation. Within means the variation that A has within its own time-series, whereas between means between A and the index.

  • The standard deviation (and variance) of the returns of an asset has two sources: the market beta times the market's standard deviation, and the asset's own idiosyncratic (market independent) standard deviation. Hence, an asset with high idiosyncratic standard deviation can have a high standard deviation despite a low beta.

    Definition of A:s beta to the Market: retA = beta * retMarket+ epsA

    Definition of A:s idiosyncratic return (epsA): Correlation(epsA, retMarket) = 0

    Hence: Variance(retA) = beta^2*Variance(retMarket) + variance(epsA).

    And, if Variance(epsA) (=idiosyncratic variance) is high enough, Variance(retA) can be high too regardless of beta and the same goes of course for standard deviation.

  • Let me give you an example to show how this can happen. Suppose you invest 0.50 in a coin flip that will pay 1 on heads and 0 on tails a month later. The monthly variance will be .5*(1-.5)^2+.5*(0-.5)^2=.5 so the standard deviation will be .25. This is significantly higher standard deviation than a market index or almost all stocks. So by one measure this is a very risky bet.

    But, if you owned a portfolio of a ton of these things it actually would be a very boring investment. Moreover, the market does not compensate you with positive returns for risk that can be diversified away. The coinflip has no priced risk, but it has a lot of non-priced risk.

    Put another way, the apparent risk of individual securities is not the same as their contribution to overall risk when held in a portfolio. Diversified portfolios that add a small amount of security A will have lower standard deviation than diversified portfolios that add a small amount of security B, even though A is the higher standard deviation stock.

    Higher standard deviation does naturally lead directly to higher beta, but for diversified portfolios only, not necessarily for individual securities.

    This concept is important when thinking about things like Venture Capital Investments where founders are forced to put almost all their wealth in one firm. If I had to choose to be the founder of firm B or firm A I would choose firm B, but I'd put A in my retirement portfolio all else equal.

  • TLDR:

    Beta = systematic risk

    Standard deviation = total risk

    Long Answer:

    There are two types of risk, systematic and unsystematic risk. Systematic risk affects the entire stock market. The recession of '08 is a good example of systematic risk. It affected all stocks. On the other hand, unsystematic risk is risk that only affects a particular security. For example, the risk of Tesla declaring bankruptcy is an unsystematic risk. It does not affect the entire market.

    Unsystematic risk can be eliminated with a well-diversified portfolio (see Modern Portfolio Theory for more information on that). But basically, by holding enough uncorrelated securities, unsystematic risk can be eliminated. However, if investors were compensated for taking risk that can be eliminated, the return of unsystematic risk would be arbitraged to zero. Therefore, investors are only compensated for systematic risk.

    This is where beta and standard deviation come in. Standard deviation represents total risk, the sum of systematic and unsystematic risk (i.e., the sum of variances). Beta measures systematic risk only, which is what return should be based on in an efficient market. Assuming you have a well-diversified portfolio, you are more focused on the systematic risk of a security because that is what returns are based on. Therefore, you look at beta to measure risk/return. However, if you have no portfolio to start with, unsystematic risk is more relevant to you. In this case, standard deviation is your friend because it accounts for both risk types.

  • Beta is volatility in relation to a benchmark whereas Standard Deviation is volatility in relation to actual returns vs expected returns

    beta is not volatility: it is the multiplier to apply to the benchmark returns to obtain the best estimate of the instrument returns: r = beta * b + TE, where TE is the tracking error. Hence you can have a very low beta if you are independent from the benchmark, and a large volatility.

License under CC-BY-SA with attribution

Content dated before 7/24/2021 11:53 AM