# 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 :-) – vonjd Nov 22 '14 at 12:34

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