# Semideviation instead of Standard Deviation

A lot of investing is based on using standard deviations as a measure of risk. It is used in things like the Sharpe Ratio and the Mean-Variance Optimization (the “efficient frontier” from Modern Portfolio Theory).

Standard deviation is just a measure of volatility. You start with a list of numbers and then you calculate how much they differ from one another.

`2, 4, 4, 4, 5, 5, 7, 9`

The average of those numbers is 5 and the standard deviation is 2.13.

• 68% of the values are between (5–2.13) and (5+2.13). That is, between 2.87 and 7.13. In our data, 2 and 9 fall outside of that range.
• 95% of the values are between (5-(2 x 2.13)) and (5 + (2 x 2.13)). That is, between 0.74 and 9.26. All of our sample data fits in that range, but we only have a few values.

There are some issues with using standard deviation alone. It doesn’t tell you much about skewness. That is, does the distribution of returns “lean” in one direction or another? Are you more likely to get a gain or a loss? Stocks are more likely to have a gain than a loss, so they have “negative skew”.

It also doesn’t tell you anything about kurtosis or spread. That is, how tall is the peak of the curve and how fat are the tails?

But an even bigger problem is that when we use standard deviation as a measure of risk, then we are saying that any variance is bad. Even when our returns are more than the average, that is still considered bad.

The standard deviation of equity returns in Japan between 1950 and 1959 was 43%. As a point of comparison, the standard deviation of US equity returns from 1871–2015 was 18% and the standard deviation of US government bond returns from 1900–2015 was 4.3%. So this is a lot higher.

But is it actually riskier? After all, the worst case scenario was that you lost 5%. Mostly the standard deviation is saying, “You’re going to have great returns but I don’t know how much…could be 10% or could be 130%”

The high standard deviation is caused (in part) by gaining 138.45% in a single year. Is that really a bad thing?

There’s another way we can measure things: use the semideviation. The semideviation starts with a simple observation: people don’t care about accidentally earning too much money, they really only care about when returns are less than expected.

You calculate the average. Then you look at any values that fall below the average and see how far below the average they were. And…that’s it really.

When we used standard deviation, Japanese stocks in 1950–1959 looked crazy risky (43%) compared to US stocks (20%). But when we look at they semideviation Japanese stocks in that period they look substantially safer: 27% for Japanese stocks and 18% for US stocks.

That feels closer to how I think most people would react as well. If you think the stocks are supposed to return 6% a year, then you don’t get upset when they return 9% in a year. But you do get upset when they return 3% for a year.

If semideviation is so great, why isn’t it used more often? Partly because standard deviation isn’t totally useless. It does tell us something about risk, even if we could argue there are better choices. Partly because of inertia. When Modern Portfolio Theory was introduced by Markowitz he used standard deviation. I can’t find a quote but apparently he claimed that he only used standard deviation because it was well-known, not because he thought it was the best choice.

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