NOTES
3.1 RECONSIDERING RISK
The common dictionary definition ofrisksays it is the chance of injury, damage, or loss. Although this definition is correct, it is not highly suitable for scientific analysis.
Analysis cannot proceed very far using verbal definitions, for several reasons.
(1) Verbal definitions are not exact; different people interpret them in different ways. (2) Verbal definitions do not yield to analysis; they can only be broken down into more verbose verbal definitions and examples. (3) Verbal definitions do not facilitate ranking or comparison because they are usually not explicit enough to allow measurement. A quantitative risk surrogate is needed to replace the verbal definition of risk if risk analysis and portfolio analysis are to proceed very far. Most sciences are moving to refine and quantify their studies. For example, biometrics, econometrics, and psychometrics are focusing on quantification of the studies of biology, economics, and psychology, respectively.
The model used here for analyzing risk focuses on probability distributions of quantifiable outcomes. Because the rate of return from an investment is the most relevant outcome from an investment, risk analysis focuses on probability distributions of returns. A probability distribution of holding period returns is illustrated in Figure 3.1.
The arithmetic mean or expected value of the probability distribution of returns, denotedE(r), represents the mathematical expectation of the possible rates of return.
Theexpected returnis
E(r)= N i=1piri
= p1r1+p2r2+ · · · +pNrN (3.1) wherei=1, 2, 3,. . .,Nare a series of integers that count the possible outcomes,Nis the total number of outcomes, andpiis the probability that rate of returnioccurs.
29
Prob (rt)
rt
0 E(r)
FIGURE 3.1 Probability Distribution of Rates of Return
Continuous probability distributions are used rarely in this book. The estimated returns assume finite values and have finite variances.
Rates of return belowE(r)represent disappointing outcomes to someone who has invested his or her funds in the asset’s probability distribution of risky returns.
The area within the probability distribution that lies to the left ofE(r) graphically represents the investor’s chance of injury, loss, or damage. These unfortunate outcomes align with the dictionary definition of risk. The semivariance (SV) of returns(s), defined in equation (3.2), is a quantitative risk surrogate that measures the area belowE(r)in the probability distribution of returns:
SV(r)= N
i=1
pi min
ri−E(r), 02
= L
i=1
pi
ri−E(r)2
= p1
r1−E(r)2
+p2
r2−E(r)2
+ · · · +pL
rL−E(r)2
(3.2) where ri represents below-average returns. These rates of return are less than E(r) (that is, ri<E(r)),N represents the total number of returns, andLrepresents the number of returns less than E(r). The square root of equation (3.2) is called thesemideviationof returns and is an equivalent financial risk surrogate that some people may find more intuitively pleasing. The semivariance is described in more detail in Chapter 10.
The semivariance and semideviation of returns are special cases of the variance and standard deviation of returns. The variance of returns, equation (2.2), measures the dispersion or width of the entire probability distribution of returns, rather than merely the portion of the distribution lying belowE(r).
σ2= N
i=1
pi
ri−E(r)2
= p1[r1−E(r)]2+p2[r2−E(r)]2+ · · · +pN[rN−E(r)]2 (2.2) The standard deviation of returns,σ, is the square root of equation (2.2).
3.1.1 Symmetric Probability Distributions
Figures 3.2, 3.3, and 3.4 contrast three different types of skewness in probability distributions of returns. If an asset’s probability distribution of rates of return are symmetric, as shown in Figure 3.3, rather than skewed to the left or right, then
‘‘an analysis based on expected return and standard deviation would consider these assets as equally desirable’’ relative to an analysis based on expected return and semideviation. Because most studies published thus far indicate the distributions of returns are approximately symmetric,1 the semideviation is abandoned in favor of the standard deviation of returns. As Markowitz points out, the standard deviation
pi
ri E(r)
FIGURE 3.2 Probability Distribution Skewed Left
pi
ri
E(r)
FIGURE 3.3 Symmetric Probability Distribution
pi
ri
E(r)
FIGURE 3.4 Probability Distribution Skewed Right
(or variance) ‘‘is superior with respect to cost, convenience, and familiarity’’ and ‘‘will produce the same set of efficient portfolios’’ as the semideviation (or semivariance) if the probability distributions are symmetric.2
The variance or standard deviation of returns is the surrogate for total risk that will be employed throughout this book. This is equivalent to defining risk as variability of return.
3.1.2 Fundamental Security Analysis
Analyzing only a firm’s rate of return may seem oversimplified when compared with fundamental security analysis techniques that use many financial ratios to analyze financial statements, management interviews, industry forecasts, and the economic outlook.3 However, there is no contradiction in these two approaches. After the fundamental security analyst completes her task, she need only convert the estimates into several possible rates of return and attach probability estimates to each. The security analyst’s consideration of such matters as how highly the firm is levered (that is, how much debt is used relative to the equity), its ability to meet fixed obligations, instability within the industry, the possibility of product obsolescence, the aggressiveness of competitors, the productivity of research and development, management depth and ability, and macroeconomic conditions are all dully reflected in the forecasted rates of return and their probabilities. Thus, variability of returns is a measure of risk grounded in fundamental analysis of the firm, its industry, and the economic outlook.4