January 27, 2009
Risk as Volatility
Economists and investment analysts therefore use alternative definitions for describing the risk of a particular investment, such as a stock, bond, or mutual fund. The most basic and common, and the one that is taught in introductory finance courses, is that risk equals volatility of returns. Volatility can be treated mathematically. If you can measure it, this mathematical tractability allows its use for prediction of future uncertainties. Moreover, this definition has an intuitive appeal, because it is undeniable that many investors alternate between elation and despair as the market (or particular stocks) go up and down.2
Once you define financial risk as volatility, well-understood tools of probability and statistics permit you to plug a few values (parameters) into established mathematical formulæ and to calculate the chances of a rise or fall in the price of a stock, or the market, or a bond, or your entire portfolio, and therefore your chances of not having enough money when you need it.
Here’s how it works. On the left, you see a plot of the distribution of annual returns to the U.S. stock market (as represented by the S&P 500 index) between 1926 and 2008. It’s a graph of the number of times that returns fell within defined intervals. You see, for example, that there were very few years with returns that were less than -30% and only a couple of years with returns that were more than +50%, but there were a lot of years with returns that were between 10% and 20%; the average is 11.67%.3 The shape of this graph looks vaguely like the graph on the right, the familiar “bell curve,” which is also known as the “normal distribution.” The bell curve is described by a mathematical formula. This formula requires that two, and only two, values be plugged in for the entire scale and shape to be defined. One of these values represents the middle or peak of the curve (the average) and the other represents how spread-out the curve is. For investment returns, the “spread-outness,” so to speak, is the volatility. (If volatility were low, most returns wouldn’t vary much, and they’d be tightly clustered around the average, and the curve would be very tall and narrow; if volatility were very high, the curve would be very spread out, with a low peak.)
So, for example, given the actual historical returns that underlie the graph on the left, I can calculate the average and the “spread-outness” and plug them into the formula, which then tells me that there is a 1-in-20 chance that the return on the U.S. stock market will be less than -22.2% in a single year. (I can calculate the return corresponding to any chance I choose: 1 in 10, 1 in 50, 1 in 1000, and so on.) With a just a little more mathematics, this can be pushed a bit further to show that if you invest $1,000,000 in the U.S. stock market today, your investment stands a 1-in-20 chance of being worth less than $1,101,873 in ten years. In a wry sort of way, this is comforting information; we can reword that last statement to say that we ought to expect that in only one decade during two centuries, your portfolio will grow by less than 10.2% in value. A total of 10.2% growth over ten years corresponds to an average rate of growth of 0.97% per year.4 That’s not enough to compensate for expected inflation, but at least it’s positive.
We’re getting a lot closer to being able to deal with the real risk to your financial wellbeing.
Equating investment risk with volatility has been extraordinarily fruitful. Much of the structure of modern finance rests upon this definition and its implications. And by “finance,” I mean not just the theory and practice of investing, but also corporate finance, the way companies finance themselves, evaluate decisions to pay for the creation of new products and factories, and buy and sell businesses. It also applies to the ways that financial institutions, primarily banks and insurance companies, estimate the business risks that they will face in the event of a credit crisis.
Recall Graham’s “margin of safety.” Modern financial practitioners, using the mathematical tools of probability and statistics, have developed a different form of margin of safety, called “Value at Risk,” or VaR (pronounced “varr”). Much as I just calculated that there was a 1-in-20 chance of a $1,000,000 portfolio being worth less than $1,101,873 in ten years, the risk managers—mathematicians who are employed by banks and other financial institutions for just this purpose—can calculate how much a bank’s capital or an insurance company’s reserves, say, might drop in a month’s time with a 1-in-1000 probability (meaning that the bank might experience such a loss in one month out of eighty-three years), or a 1-in-10,000 probability. The calculations are much more complex than the one underlying my example, but the basic concept is the same. Recognizing that the bears haven’t signed a treaty, and that there is always some chance, however small, of a catastrophic loss, they can now determine what that loss (or worse) might be for any probability they select. Whereas Graham and his followers recognize that some risk exists, and concentrate on ascertaining that there is a sufficiently large cushion to absorb the impact of a big loss, modern financial practice (for banks and insurance companies) explicitly calculates the size of a loss that can occur at any selected probability, and adjusts the portfolio to ensure that the worst losses are very unlikely.
2 Note that I am referring to the volatility of returns, not of prices. There are several reasons for preferring return volatility to price volatility. The most obvious is that returns are percentages, and therefore can be compared across different investments. The price volatility of a stock whose price averages $6 per share is not comparable to the price volatility of a stock whose price averages $100 per share.
3 Data, except for 2008, from Ibbotson SBBI 2008 Classic Yearbook: Market Results for Stocks, Bonds, Bills, and Inflation 1926-2007 (Chicago: Morningstar, Inc., 2008).
4This is the compound rate of growth, which is smaller than the arithmetic average.
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