Defining Risk: The Dispersion of Returns

Defining Risk: The Dispersion of Returns

Risk is a most slippery and elusive concept. It’s hard for investors – let alone economists – to agree on a precise definition. The American Heritage Dictionary defines risk as the possibility of suffering harm or loss. If I buy one-year Treasury bills to yield 8 percent and hold them until they mature, I am virtually certain of earning an 8 percent monetary return, before incoming taxes. The possibility of loss is so small as to be considered nonexistent. If I hold common stock in my local power and light company for one year on the basis of an anticipated 9 percent dividend form, the possibility of loss is greater. The dividend of the company may be cut, and, more important, the market price at the end of the year may be much lower, causing me to suffer a serious net loss. Risk is the chance that expected security returns will not materialize and, in particular, that the securities you hold will fall in price.

Once academics accepted the idea that risk for investors is related to the chance of disappointment in achieving expected security returns, a natural measure suggested itself – the probable variability or dispersion of future returns. Thus, financial risk has generally been defined as the variance or standard deviation of returns. Being long-winded, we use the accompanying exhibit to illustrate what we mean. A security whose returns are not likely to depart much, if at all, from its average (or expected) return is said to carry little or no risk. A security whose returns from year to year are likely to be quite volatile (and for which sharp losses are typical in some years) is said to be risky.

Bloor street, Yorkville. Photo by Elena

Exhibit

Expected Return and Variance: Measures of Reward and Risk

This simple example will illustrate the concept of expected return and variance and how they are measured. Suppose you buy a stock from which you expect the following overall returns (including both dividends and price changes) under different economic conditions:

Normal economic conditions - Probability of occurrence - 1 chance in 3 – Expected return – 10 percent

Rapid real growth – 1 chance in 3 – 30 percent

Recession with inflation (stagflation) – 1chance in 3 - -10 percent.

If, on average, a third of past year have been “normal,” another third characterized by rapid growth, and the remaining third characterized by “stagflation,” it might be reasonable to take these relative frequencies of past events and treat them as our best guesses (probabilities) of the likelihood of future business conditions. We could then say that an investor’s expected return is 10 percent. A third of the time the investor gets 30 percent, another third 10 percent, and the rest of time he suffers a 10 percent loss. This means that, on average, his yearly return will turn out to be 10 percent.

Expected Return = 1/3 (0.30) + 1/3 (0.10) + 1/3 (-0.10) = 0.10.

The yearly returns will be quite variable, however, ranging from a 30 percent gain to a 10 percent loss. The “variance” is a measure squared deviation of each possible return from its average (or expected) value, which we just saw was 10 percent.

Variance = 1/3 (.30 - 0.10)2 + 1/3 (0.10 - 0.10)2 + 1/3 (-0.10 – 0.10)2 = 1/3 (0.20)2 + 1/3 (0.00)2 + 1/3 (-0.20)2 = 0.0267.

The square root of the variance is called the standard deviation. In this example, the standard deviation equals 0.1634.

Dispersion measures of risk such as variance and standard deviation have failed to satisfy everyone. “Surely riskiness is not related to variance itself,” the critics say. “If the dispersion results from happy surprises – that is, from outcomes turning out better than expected – no investors in their right minds would call that risk.”

It is, of course, quite true that only the possibility of downward disappointments constitutes risk. Nevertheless, as a practical matter, as long as the distribution of returns is symmetric – that is, as long as the chances of extraordinary gain are roughly the same as the probabilities for disappointing return and losses – a dispersion or variance measure will suffice as a risk measure. The greater the dispersion or variance, the greater the possibilities for disappointment.

While the pattern of historical returns from individual securities has not usually been symmetric, the returns from well-diversified portfolios of stocks do seem to be distributed approximately symmetrically. The following chart shows a twenty-five-year distribution of monthly security returns for a portfolio consisting of equal dollar amounts invested in 100 stocks. It was constructed by dividing the range of returns into equal intervals (of approximately 1 ¼ percent) and then noting the frequency (the number of months or 10.7 percent per year. In periods when the market declined sharply, however, the portfolio also plunged, losing as much as 13 percent in a single month.

For symmetric distributions, such as this one, a helpful rule of thumb is that two-thirds of the monthly returns tend to fall within one standard deviation of the average return and 95 percent of the returns fall within two standard deviations. Recall that the average return for this distribution was just under 1 percent per month. The standard deviation (our measure of portfolio risk) turns out to be about 4 1/2 percent per month. Thus, in two-thirds of the months the returns from the portfolio were between 5 1/2 percent and -8 percent. Obviously, the higher the standard deviation (the more spread out are the returns), the more probable it is (the greater the risk) that at least in some periods you will take a real bath in the market. That's why a measure of variability such as standard deviation is so often used and justified as an indication of risk (standard deviation and its square, the variance, are used interchangeable as risk measures. They both do the same thing and it's purely a matter of convenience which one we use).

Burton G. Malkiel. A Random Walk Down Wall Street, including a life-cycle guide to personal investing. First edition, 1973, by W.W. Norton and company, Inc.