Measuring Stand Alone Risk The Standard Deviation
Risk is a difficult concept to grasp, and a great deal of controversy has surrounded attempts to define and measure it. However, a common definition, and one that is satisfactory for many purposes, is stated in terms of probability distributions such as those presented in Figure 32: The tighter the probability distribution of expected future returns, the smaller the risk of a given investment. According to this definition, U.S. Water is less risky than Martin Products because there is a smaller chance that its actual return will end up far below its expected return.
To be most useful, any measure of risk should have a definite value—we need a measure of the tightness of the probability distribution. One such measure is the standard deviation, the symbol for which is a, pronounced "sigma." The smaller the standard deviation, the tighter the probability distribution, and, accordingly, the lower
FIGURE 32 Continuous Probability Distributions of Martin Products' and U.S. Water's Rates of Return
Probability Density
FIGURE 32 Continuous Probability Distributions of Martin Products' and U.S. Water's Rates of Return
Probability Density
 Rate of Return
Expected Rate of Return
 Rate of Return
Expected Rate of Return
Note: The assumptions regarding the probabilities of various outcomes have been changed from those in Figure 31. There the probability of obtaining exactly 15 percent was 40 percent; here it is much smaller because there are many possible outcomes instead of just three. With continuous distributions, it is more appropriate to ask what the probability is of obtaining at least some specified rate of return than to ask what the probability is of obtaining exactly that rate. This topic is covered in detail in statistics courses.
the riskiness of the stock. To calculate the standard deviation, we proceed as shown in Table 33, taking the following steps:
1. Calculate the expected rate of return:
Expected rate of return = r = a Piri.
For Martin, we previously found r = 15%.
2. Subtract the expected rate of return (r) from each possible outcome (ri) to obtain a set of deviations about r as shown in Column 1 of Table 33:
(1) 
(ri — F)2 (2) 
(Fi — r)2Pi (3)  
100 
— 15 = 
85 
7,225 
(7,225)(0.3) = 2,167.5 
15 
— 15 = 
0 
0 
(0)(0.4) = 0.0 
70 
— 15 = 
85 
7,225 
(7,225)(0.3) = 2,167.5 
Variance = ct2 = 4,335.0  
Standard deviation = ct = 
VCT = 24,335 = 65.84%. 
3. Square each deviation, then multiply the result by the probability of occurrence for its related outcome, and then sum these products to obtain the variance of the probability distribution as shown in Columns 2 and 3 of the table:
Variance = ct2 = 2 ( i= 1 
ri  r )2pi. (32) 
4. Finally, find the square root of the variance to obtain the standard deviation:  
Standard deviation = ct = b 
/ (ri  r)2pi. (33) 
Thus, the standard deviation is essentially a weighted average of the deviations from the expected value, and it provides an idea of how far above or below the expected value the actual value is likely to be. Martin's standard deviation is seen in Table 33 to be ct = 65.84%. Using these same procedures, we find U.S. Water's standard deviation to be 3.87 percent. Martin Products has the larger standard deviation, which indicates a greater variation of returns and thus a greater chance that the expected return will not be realized. Therefore, Martin Products is a riskier investment than U.S. Water when held alone.
If a probability distribution is normal, the actual return will be within ± 1 standard deviation of the expected return 68.26 percent of the time. Figure 33 illustrates this point, and it also shows the situation for ±2ct and ±3ct. For Martin Products, r = 15% and ct = 65.84%, whereas r = 15% and ct = 3.87% for U.S. Water. Thus, if
FIGURE 33 Probability Ranges for a Normal Distribution
For more discussion of probability distributions, see the Chapter 3 Web Extension on the textbook's web site at http://ehrhardt. swcollege.com.
FIGURE 33 Probability Ranges for a Normal Distribution
Notes:
a. The area under the normal curve always equals 1.0, or 100 percent. Thus, the areas under any pair of normal curves drawn on the same scale, whether they are peaked or flat, must be equal.
b. Half of the area under a normal curve is to the left of the mean, indicating that there is a 50 percent probability that the actual outcome will be less than the mean, and half is to the right of r, indicating a 50 percent probability that it will be greater than the mean.
c. Of the area under the curve, 68.26 percent is within ±1 ct of the mean, indicating that the probability is 68.26 percent that the actual outcome will be within the range r — 1ct to r + 1ct.
d. Procedures exist for finding the probability of other ranges. These procedures are covered in statistics courses.
e. For a normal distribution, the larger the value of ct, the greater the probability that the actual outcome will vary widely from, and hence perhaps be far below, the expected, or most likely, outcome. Since the probability of having the actual result turn out to be far below the expected result is one definition of risk, and since ct measures this probability, we can use ct as a measure of risk. This definition may not be a good one, however, if we are dealing with an asset held in a diversified portfolio. This point is covered later in the chapter.
the two distributions were normal, there would be a 68.26 percent probability that Martin's actual return would be in the range of 15 ± 65.84 percent, or from —50.84 to 80.84 percent. For U.S. Water, the 68.26 percent range is 15 ± 3.87 percent, or from 11.13 to 18.87 percent. With such a small ct, there is only a small probability that U.S. Water's return would be significantly less than expected, so the stock is not very risky. For the average firm listed on the New York Stock Exchange, ct has generally been in the range of 35 to 40 percent in recent years.
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