What is risk?
Volatility of returns
The standard deviation of returns
Interpretations of the standard deviation
Mean returns and the standard deviation
The big picture
Excel section
Challenge section
S
o much for the ‘good’ stuff. In the previous chapter we focused on two ways of summarizing the return performance of an asset. Here comes the ‘bad’ stuff: in this chapter we’ll focus on one way of summarizing an asset’s risk. Keep this in mind, though: the concept of risk is hard to pin down, so we’ll explore alternative definitions in forthcoming chapters.What is risk?
Silly question? Well, not really. The fact is that, simple as it may sound, academics and practitioners in finance have been wrestling with this definition for many years. And it gets worse. Nobody seems to have provided an answer that everybody else agrees with. As is often heard, it may well be the case that risk, like beauty, is in the eyes of the beholder.
But don’t throw up your arms in despair just yet. The fact that there is no universally accepted definition of risk doesn’t mean that risk cannot be quantified in a variety of ways. Before we get into definitions and formulas, take a look at Exhibit 3.1, which depicts the indices for Exxon and Intel that generated the returns we discussed in the previous chapter. Just to make the comparison easier, the indices are normalized so that they both start at 100.
EXHIBIT 3.1
Intel v. Exxon, indices
1,200
1,000
800
600
400
200
0
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Year
Index
Intel
Exxon
Although the concept of risk may be hard to pin down, your eyes probably won’t fool you: while Exxon seems to have had a rather steady climb, Intel seems to have been on a roller coaster, with pronounced rises and falls. Just by looking at the picture, most reasonable people would agree that Intel seems to be riskier (that is, more volatile or more unpredictable) than Exxon.
Think about it this way: the more a price fluctuates over time, the greater the uncertainty about where that price may be at some point in time in the future.
And the greater that uncertainty, the greater the risk. Does that make sense? If it does, then read on for a similar and complementary way of thinking about risk.
Volatility of returns
Instead of thinking about prices, as in Exhibit 3.1, we can think of risk in terms of returns. Exhibit 3.2 depicts the annual returns of Exxon and Intel during the 1994–2003 period. These are the same returns as those reported in Table 2.2.
EXHIBIT 3.2
Intel v. Exxon, returns
Again, without getting into definitions or formulas, your eyes won’t fool you.
At the same time that Exxon consistently delivered returns between, roughly, –10% and 40% (no small range, to be sure), Intel delivered far more volatile returns, with annual gains in excess of 130% and annual losses in excess of 50%.
160 140 120 100 80 60 40 20 0 –20 –40 –60
1994 1995 1996 1997 1998 1999 2000 2001 2002 2003
Intel
Exxon
Year
Return (%)
By looking at Exhibit 3.2, most reasonable people would again conclude that Intel is riskier than Exxon.
The standard deviation of returns
Now it is time to formalize the concept of risk. Let’s focus on Exhibit 3.2. As we just discussed, an obvious way of thinking about risk is in terms of volatility (or variability) in returns, which led us to conclude that Intel is riskier than Exxon.
One way to formally capture this volatility is to compute the standard deviation of returns (SD), which is (hold on to your seat) the square root of the average quadratic deviation from the arithmetic mean return. If after reading that again it still sounds like Sylvester Stallone speaking Chinese, stop reading this chapter and go to the stats review in Chapter 27. Otherwise, keep reading for a bit of extra insight on this measure of risk.
The standard deviation of a series of returns is formally given by the expression
(3.1)
where Rrepresents returns, AMrepresents the (arithmetic) mean return of the series of returns, tindexes time, and Tis the number of observations. (Note that sometimes the standard deviation is calculated by dividing the sum of quadratic deviations by T– 1 instead of by T. For practical purposes, you don’t really have to worry about this distinction.)
Let’s take a quick look at the calculation of the standard deviation of returns of Intel. Table 3.1 shows the returns in the second column, the deviations from the mean return in the third, and the square of those numbers in the fourth. The average of the numbers in the fourth column is the varianceof returns, but it is not widely used as a measure of risk. The number in the intersection between the last row and the last column, the square root of the average of quadratic deviations, is the standard deviation of returns, which in the case of Intel is 55.8%.
SD=
√
(1/T) ·Σ
Tt=1(Rt– AM)2TABLE 3.1
Year R R– AM (R– AM)2
(%) (%)
1994 3.4 –32.9 0.1080
1995 78.2 41.9 0.1754
1996 131.3 95.0 0.9029
1997 7.4 –28.8 0.0831
1998 69.0 32.8 0.1073
1999 39.1 2.8 0.0008
2000 –26.9 –63.1 0.3986
2001 4.9 –31.4 0.0985
2002 –50.3 –86.6 0.7497
2003 106.6 70.3 0.4948
Average 36.3% 0.3119
Square root 55.8%
Of course, you don’t have to go through all these calculations to estimate a standard deviation; Excel calculates this magnitude in the blink of an eye and in just one cell. But the table shows where the number that Excel calculates comes from.
Interpretations of the standard deviation
Let’s focus now on the interpretation of the standard deviation as a measure of risk. The easiest way to think about it is as follows: the larger this number, the riskier the asset. This way of thinking of the standard deviation confirms our previous argument that Intel appears to be riskier than Exxon; the standard deviation of the former is 55.8% and of the latter only 15.0%. (You will be asked to calculate this last number in the Challenge section at the end of the chapter.)
Basically, a small standard deviation indicates that returns fluctuate ‘closely’
around the mean return, and a large standard deviation indicates the opposite.
In other words, the larger the standard deviation, the more that returns tend to depart from the mean return.
Another way to think about the standard deviation is to recall that, under normality, 68.3% of the returns cluster one standard deviation around the mean.
It is also the case that 95.4% and 99.7% of the returns cluster two and three standard deviations around the mean, respectively. (See Chapter 28 if you need
to refresh your memory.) Keeping this in mind, think of two hypothetical stocks, both with a mean return of 20%, and standard deviations of 5% (stock 1) and 30% (stock 2).
Note that there is a 95% probability that the returns of stock 1 fluctuate between 10% and 30%, that is, two standard deviations around the mean.
However, in the case of stock 2, returns will fluctuate, with a 95% probability, in the interval [–40%, 80%], a range so large as to be basically useless. We could drive a train sideways between these two numbers.
This simple example illustrates another way of thinking about the standard deviation: we can use it to estimate the interval within which returns will fluctuate with any chosen probability. The larger the interval, the larger the uncertainty, and the riskier the stock.
Going back to Exxon and Intel, our previous line of reasoning would suggest that there is a 95% probability that the returns of Exxon fluctuate between –15.6% and 44.3%, and those of Intel between –75.4% and 148.0%. Because the range between the lower and the higher ends of the interval in the case of Intel (over 220%) is far larger than that for Exxon (under 60%), we confirm the fact that Intel is riskier than Exxon. (Note, however, that the intervals calculated are valid under normality, which may be a questionable assumption in this context.)
Mean returns and the standard deviation
We intuitively know that risk is ‘bad,’ and the discussion in the previous section attempts to explain why the standard deviation may be a good measure of how
‘bad’ an asset may be. Essentially, the standard deviation is a measure of volatility and uncertainty, both of which, most investors would agree, are ‘bad.’
Now we’ll take another (usually less explored) look at why volatility is bad for an investor. Consider the six hypothetical stocks in Table 3.2, all of which have an arithmetic mean return (AM) of 10% but different volatility (SD). Note that, as we move from stock A to stock F, volatility increases, so that, as we move from left to right, the stocks become riskier.
TABLE 3.2
Year A B C D E F
(%) (%) (%) (%) (%) (%)
1 10.0 12.0 15.0 20.0 25.0 40.0
2 10.0 8.0 5.0 0.0 –5.0 –20.0
3 10.0 12.0 15.0 20.0 25.0 40.0
4 10.0 8.0 5.0 0.0 –5.0 –20.0
5 10.0 12.0 15.0 20.0 25.0 40.0
6 10.0 8.0 5.0 0.0 –5.0 –20.0
7 10.0 12.0 15.0 20.0 25.0 40.0
8 10.0 8.0 5.0 0.0 –5.0 –20.0
9 10.0 12.0 15.0 20.0 25.0 40.0
10 10.0 8.0 5.0 0.0 –5.0 –20.0
AM 10.0% 10.0% 10.0% 10.0% 10.0% 10.0%
SD 0.0% 2.0% 5.0% 10.0% 15.0% 30.0%
GM 10.0% 10.0% 9.9% 9.5% 9.0% 5.8%
TW $25,937 $25,895 $25,671 $24,883 $23,614 $17,623
Now take a look at the geometric means (GM). As we move from left to right, the arithmetic mean return remains constant, volatility increases, and the geometric mean return decreases. This is sometimes referred to as the
‘variance drag,’ which is just a fancy way of saying that volatility has a negative impact on mean compound returns.
As you’ll remember from our discussion in the previous chapter, an investment does not compound over time at its arithmetic mean return but at its geometric mean return. So here we have another way to rationalize why volatility is bad: because it lowers the compound return of an investment, thus having a negative impact on its terminal value.
Table 3.2 illustrates this point. An initial investment of $10,000, compounded over 10 years at the geometric mean returns reported in the next-to-last row, yields the terminal wealth (TW) reported in the last row. To state the obvious:
the terminal value of an investment is negatively related to the volatility of the asset’s returns.
Formally, for any series of returns the relationship between the arithmetic mean, the geometric mean, and volatility is given by the expression
(3.2) GM≈ exp
{
ln(1 + AM) – (1/2) SD2(1 + AM)2
}
–1which holds well, as an approximation, for returns not much larger than ±30%.
This expression is, in fact, a better approximation to the geometric mean than the more widely used (and simpler) approximation given by GM≈ AM– (1/2) · (SD2).
The big picture
Risk is one of the most elusive concepts in finance. One of the most widely accepted ways to define it, however, is as volatility measured by the standard deviation of returns. This volatility can be thought of as uncertainty about the future price of an asset, or as dispersion around the asset’s mean return.
Volatility also causes a drag on mean compound return, which is one of the reasons we consider it detrimental. In other words, the higher the volatility of an asset, the lower the asset’s ability to compound wealth over time.
Excel section
Just as in the Excel sections of the previous two chapters, the stuff in this section is rather straightforward.
■ To calculate a square root in Excel you need to use the ‘sqrt’ function. With it, calculating the square root of any number x is as simple as typing
‘=sqrt(x)’ and hitting ‘Enter.’
Calculating a standard deviation in Excel is also very simple. Suppose you have a series of ten returns in cells A1 through A10. Then, you do the following:
■ To calculate a standard deviation that divides the average of squared deviations from the mean by T, simply type ‘=stdevp(A1:A10)’ in cell A11 and hit ‘Enter.’
■ To calculate a standard deviation that divides the average of squared deviations from the mean by T – 1, simply type ‘=stdev(A1:A10)’ in cell A11 and hit ‘Enter.’
Challenge section
1 Given the returns of Exxon shown in Table 2.2, confirm that (as suggested in the text) the standard deviation of those returns is 15.0%.
2 Given the returns of American Express (Amex) between 1994 and 2003 shown in Table 3.3, calculate the standard deviation of returns.
(Just for the sake of completeness, calculate this number with respect to both T and T – l.) Is Amex riskier than Exxon? Is it riskier than Intel?
TABLE 3.3
Year Return
(%)
1994 13.5
1995 42.8
1996 39.8
1997 59.9
1998 15.7
1999 63.4
2000 –0.3
2001 –34.5
2002 0.2
2003 37.7
3 Given the returns of Amex during the period 1994–2003, calculate the (arithmetic) mean return. Then, assuming normality, estimate the interval within which Amex returns should fluctuate with a probability of 95%. How does this interval compare with those discussed in the text for Exxon and Intel?
4 Calculate the geometric mean return of Amex in the way discussed in the previous chapter. Then, using equation (3.2) in this chapter, calculate the approximate geometric mean return for Exxon, Intel, and Amex. Does the approximation seem to work? Does it work in some cases better than in others? Why?