Jeremy Siegel Misunderstands Standard Deviation

Jeremy Siegel is a distinguished professor of finance at the Wharton Business School. He’s written many important articles and books on investing and finance (here’s my review of his last book, “The Future for Investors”). However, I’m afraid Dr. Siegel misunderstands a basis mathematical concept.
This is from his most-recent “The Future For Investors” column:

Stock returns are composed of the sum of the average real return on safe bonds, such as U.S. government bonds, which has been about 3 percent, plus an extra risk premium that has averaged between 3 percent and 4 percent per year. This risk premium has propelled stocks above other asset classes in investor returns.
But this premium may be overly generous. Although stocks are indeed much riskier than bonds in the short run, in the long run they are safer. In fact my studies have shown that over periods 20 years or longer, a portfolio of diversified stocks has been more stable in purchasing power than a portfolio of long-term government bonds. As a result, a long-term stock investor gets rewarded for risk that basically only a short-term stock investor endures.

He says that over periods of 20 years or longer, a diversified portfolio of stocks has been more stable than a portfolio of long-term bonds. The problem is, that’s not what his research indicates.
In Chapter 2 of his book, “Stocks for the long Run,” Siegel looks at how stocks and bonds have performed over long periods. His point is that stocks are more volatile than bonds in the short-term, but over time, stocks have a very good track record of beating bonds. That’s a very important point for all investors.
But then, he makes a critical error. On page 33, he writes:

As the holding period increases, the dispersion of the average annual return on both stocks and bonds falls, but it falls faster for stocks than bonds. In fact, for a 20-year holding period, the dispersion of stock returns is less than for bonds and bills, and becomes even smaller as the holding period increases.

His numbers are right, but his conclusion is wrong. By dispersion, he’s referring to standard deviation. If you recall from your high school math, standard deviation measures variation against the mean. What his data shows is that the variation of returns decreases as you have progressively longer holding periods. That’s a tautology. It must happen, but it doesn’t say anything about inherent risk.
What Siegel says is that the variation of stock returns against the mean of stock returns decreases faster than the variation of bonds returns against the mean of bond returns. (That’s a mouthful!) But at no point are we comparing stocks against bonds. It’s merely stocks against themselves and bonds against themselves. Siegel is using the wrong instrument to make his point.
Let’s say we have an asset class that has returned, on average, 10% a year for 100 years. The one-year holding periods might be very volatile. However, the two-year and three-year holding periods will become progressively less volatile. But there’s no insight here, the holding periods have to. They’ll eventually zero in on 10% a year.
Siegel compares the rate at which stocks “zero in” to the rate at which bonds “zero in.” He says that since stocks zero in on their long-term average faster than bonds zero in on theirs, stocks are safer. They may indeed be safer, but his point doesn’t support that conclusion.

Posted by on March 14th, 2006 at 11:26 am

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