THE VARIANCE AND THE STANDARD DEVIATION The variance and the standard deviation are both measures of disper-

THE VARIANCE AND THE STANDARD DEVIATION The variance and the standard deviation are both measures of disper-

sion. In fact, they are related: The standard deviation is the square root of the variance. So why do we go beyond the calculation of the variance to get the standard deviation? For two reasons.

First, the variance is in terms of squared units of measure (say, squared dollars or squared returns), whereas the standard deviation is in terms of the original unit of measure. It gets tough trying to interpret squared dollars or squared returns.

Second, if the probability distribution is approximately normally distributed (that is, bell-shaped, with certain other characteristics), we can use the standard deviation to compactly describe the probability distribution; not so with the variance. There are uses for the variance in statistical analysis, but for purposes of describing and comparing proba- bility distributions, we focus on the expected value and the standard deviation.

The calculation of the standard deviation can be made manageable with a worksheet such as Exhibit 10.3, used to calculate the standard deviations of possible outcomes for Products A and B.

THE FUNDAMENTALS OF VALUATION

EXHIBIT 10.3 Calculation of the Standard Deviation of Possible Outcomes for

Product A and Product B Product A

Outcome p x

px

x – E(x) (x – E(x)) 2 p(x – E(x)) 2

Product B Outcome p

px

x – E(x) (x – E(x)) 2 p(x – E(x)) 2

Summarizing, we have calculated the following:

Expected Return Standard Deviation of Possible Outcomes

Product A 10% 9.90% Product B

10% 18.57% While both products have the same expected value, they differ in the

distribution of possible outcomes. When we calculate the standard devi- ation around the expected value, we see that Product B has a larger standard deviation. The larger standard deviation for Product B tells us that Product B has more risk than Product A since its possible outcomes are more distant more from its expected value.

Return and the Tolerance for Bearing Risk Which product investment do you prefer, A or B? Most people would

choose A since it provides the same expected return with less risk. Most people do not like risk—they are risk averse. Does this mean a risk averse person will not take on risk? No—they will take on risk if they feel they are compensated for it.

Risk and Expected Return

A risk neutral person is indifferent toward risk. Risk neutral per- sons do not need compensation for bearing risk. A risk preference per- son likes risk—someone even willing to pay to take on risk. Are there such people? Yes. Consider people who play the state lotteries, where the expected value is always negative: The expected value of the win- nings is less than the cost of the lottery ticket.

When we consider financing and investment decisions, we assume that most people are risk averse. Managers, as agents for the owners, make decisions that consider risk “bad” and that if risk must be borne, they make sure there is sufficient compensation for bearing it. As agents for the owners, managers cannot have the “fun” of taking on risk for the pleasure of doing so.

Risk aversion is the link between return and risk. To evaluate a return you must consider its risk: Is there sufficient compensation (in the form of an expected return) for the investment’s risk?