This month, we’re going to get a bit technical to illustrate a point about investment risk. In contrast to returns, risk is challenging to quantify. Most often, professional investors and academics use volatility of returns as a measure of risk. That is, if your return from an asset is 8 percent every year, there’s no volatility—and thus no risk.
On the other hand, if the returns are plus 17 percent in year one, minus 10 percent in year two and plus 17 percent in year three, you still have an arithmetic average of 8 percent per year. But you also now have a volatility of returns of about 15.6 percent using the common statistical metric called “standard deviation.” It just so happens that 15.6 percent is fairly close to the standard deviation of stock market returns over the long run.
Why be concerned about volatility risk if you end up at the same place between the unvarying 8 percent annual returns and a stock portfolio with the same average returns but greater volatility? Because you don’t end up at the same place.
First, mathematically, the compound annual return in the second scenario is about 7.2 percent, not 8 percent. The greater the volatility, the greater the difference between the actual compounded return and the mathematical average. Increasing the standard deviation of returns to 24 percent would reduce the compounded return to 6 percent, for example, while at a standard deviation of 45.2 percent, our compound return goes to zero.
More important, the order of those volatile returns can make a big difference in one’s plan. Assume for the moment you’re planning for retirement income. If you could find assured 8 percent annual returns, you could plan to withdraw 5 percent each year and reinvest the remainder to help grow your portfolio as protection against future inflation. Your $1 million retirement savings portfolio would throw off $50,000 in cash to meet expenses.
But suppose you invested everything in the stock market with a standard deviation of returns of about 16 percent and you had a bad first year, reducing your portfolio by both the investment loss and the withdrawals. In the previous example of a 10 percent decline in the market, your $1 million portfolio would have $850,000 in it at the beginning of the second year. Even with a 17 percent gain in year two, you’d have—at best—only $944,500 at the end of the second year. And even with another 17 percent return in year three, your account would be worth $1,055,000, less than the $1,097,400 for the steady 8 percent return account.
Now suppose there was substantial risk of a really bad year in the first year or two of your plan. How about down 37 percent, as in 2008? Had you started your plan January 1 that year, you’d have found your balance at $580,000 at year’s end. More than likely, you’d have bailed out, licked your wounds and put your money in something safe (like Treasury bills), where the minuscule returns would guarantee you’d run out of money in about 12 years, by withdrawing $50,000 a year.
It gets worse. The statistical measure called standard deviation is derived from normal or “Gaussian” distributions. When plotted on a graph, Gaussian distributions take the familiar bell curve shape. As most of us have heard, 67 percent of normally distributed events fall within one standard deviation from the mean, 95 percent within two standard deviations, and so on.
Unfortunately, unlike flips of a coin and other random events, investment returns aren’t random or normally distributed. In fact, they’re considerably riskier.
Groundbreaking work on this matter was done in the 1960s by mathematician Benoit Mandelbrot, creator of the advanced mathematical concept of fractal geometry. As it happened, in the 1950s, Mandelbrot was responsible for tracking cotton prices. In doing so, he noticed certain patterns, which he studied in depth. In 1963, Mandelbrot wrote a paper in which he postulated that something called a Lévy distribution better approximated the pattern of market returns than the Gaussian model.
In the February/March issue of Morningstar Advisor, James Xiong of Ibbotsen Associates wrote an article noting how Mandelbrot’s 1963 model, modified to something called “Truncated Lévy Flight,” captured the historical pattern of stock market returns much better than the widely used Gaussian model (it did better with bond market returns, too).
The problem with assuming a normal distribution is that extreme events—large negative and positive market returns—occur more frequently than suggested by the Gaussian model, a problem known as “fat tails.” Another way to put it is that returns will fall within two standard deviations of the mean less frequently, and outside of two standard deviations more frequently, than is assumed in a normal distribution.
The Truncated Lévy Flight approach addresses the fat-tail problem quite well, giving a much better handle on just how much risk an investor’s portfolio actually has. Xiong pointed out that the Gaussian model suggests a one-in-100-year risk that a “moderate” portfolio of 60 percent stocks and 40 percent bonds could lose as much as 20 percent in a given year. The Truncated Lévy Flight model, on the other hand, suggests such a loss is a one-in-40-year event. Losses of that degree have actually occurred three times in the past 70 years.
Advisers frequently use a tool known as Monte Carlo simulation to estimate the likelihood of a given level of loss over any particular time period. Using the Truncated Lévy Flight model inside a Monte Carlo simulation, Xiong estimates there’s a 1 percent chance that a 60 percent stock and 40 percent bond portfolio could lose one-third its value in a given six-year period. Now that’s what I call risk, particularly for investors approaching retirement.
