CHAPTER 3

THE BASICS OF RISK

Risk, in traditional terms, is viewed as a negative and something to be avoided.

Webster™s dictionary, for instance, defines risk as “exposing to danger or hazard”. The

Chinese symbols for risk, reproduced below, give a much better description of risk “

The first symbol is the symbol for “danger”, while the second is the symbol for

“opportunity”, making risk a mix of danger and opportunity. It illustrates very clearly the

tradeoff that every investor and business has to make “ between the “higher rewards” that

potentially come with the opportunity and the “higher risk” that has to be borne as a

consequence of the danger. The key test in finance is to ensure that when an investor is

exposed to risk that he or she is “appropriately” rewarded for taking this risk.

In this chapter, we will lay the foundations for analyzing risk in corporate finance

and present alternative models for measuring risk and converting these risk measures into

“acceptable” hurdle rates.

Motivation and Perspective in Analyzing Risk

Why do we need a model that measures risk and estimates expected return? A

good model for risk and return provides us with the tools to measure the risk in any

investment and uses that risk measure to come up with the appropriate expected return on

that investment; this expected return provides us with the hurdle rate in project analysis.

What makes the measurement of risk and expected return so challenging is that it

can vary depending upon whose perspective we adopt. When analyzing Disney™s risk, for

instance, we can measure it from the viewpoint of Disney™s managers. Alternatively, we

can argue that Disney™s equity is owned by its stockholders, and that it is their

perspective on risk that should matter. Disney™s stockholders, many of whom hold the

stock as one investment in a larger portfolio, might perceive the risk in Disney very

differently from Disney™s managers, who might have the bulk of their capital, human and

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financial, invested in the firm. In this chapter, we will argue that risk in an equity

investment has to be perceived through the eyes of investors in the firm. Since firms like

Disney often have thousands of investors, often with very different perspectives, we will

go further. We will assert that risk has to be measured from the perspective of not just

any investor in the stock, but of the marginal investor, defined to be the investor most

likely to be trading on the stock at any given point in time. The objective in corporate

finance is the maximization of firm value and stock price. If we want to stay true to this

objective, we have to consider the viewpoint of those who set the stock prices, and they

are the marginal investors.

Finally, the risk in a company can be viewed very differently by investors in its

stock (equity investors) and by lenders to the firm (bondholders and bankers). Equity

investors who benefit from upside as well as downside tend to take a much more

sanguine view of risk than lenders who have limited upside but potentially high

downside. We will consider how to measure equity risk in the first part of the chapter and

risk from the perspective of lenders in the latter half of the chapter.

We will be presenting a number of different risk and return models in this chapter.

In order to evaluate the relative strengths of these models, it is worth reviewing the

characteristics of a good risk and return model.

1. It should come up with a measure of risk that applies to all assets and not be

asset-specific.

2. It should clearly delineate what types of risk are rewarded and what are not,

and provide a rationale for the delineation.

3. It should come up with standardized risk measures, i.e., an investor presented

with a risk measure for an individual asset should be able to draw conclusions

about whether the asset is above-average or below-average risk.

4. It should translate the measure of risk into a rate of return that the investor

should demand as compensation for bearing the risk.

5. It should work well not only at explaining past returns, but also in predicting

future expected returns.

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Equity Risk and Expected Returns

To understand how risk is viewed in corporate finance, we will present the

analysis in three steps. First, we will define risk in terms of the distribution of actual

returns around an expected return. Second, we will differentiate between risk that is

specific to an investment or a few investments and risk that affects a much wider cross

section of investments. We will argue that when the marginal investor is well diversified,

it is only the latter risk, called market risk that will be rewarded. Third, we will look at

alternative models for measuring this market risk and the expected returns that go with

this risk.

I. Measuring Risk

Investors who buy an asset expect to make a return over the time horizon that they

will hold the asset. The actual return that they make over this holding period may by very

different from the expected return, and this is where the

Variance in Returns: This is a

risk comes in. Consider an investor with a 1-year time

measure of the squared difference

horizon buying a 1-year Treasury bill (or any other between the actual returns and the

default-free one-year bond) with a 5% expected return. expected returns on an investment.

At the end of the 1-year holding period, the actual

return that this investor would have on this investment will always be 5%, which is equal

to the expected return. The return distribution for this investment is shown in Figure 3.1.

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This is a riskless investment, at least in nominal terms.

To provide a contrast, consider an investor who invests in Disney. This investor,

having done her research, may conclude that she can make an expected return of 30% on

Disney over her 1-year holding period. The actual return over this period will almost

certainly not be equal to 30%; it might be much greater or much lower. The distribution

of returns on this investment is illustrated in Figure 3.2:

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In addition to the expected return, an investor now has to consider the following. First,

the spread of the actual returns around the expected return is captured by the variance or

standard deviation of the distribution; the greater the deviation of the actual returns from

expected returns, the greater the variance. Second, the bias towards positive or negative

returns is captured by the skewness of the distribution. The distribution above is

positively skewed, since there is a greater likelihood of large positive returns than large

negative returns. Third, the shape of the tails of the distribution is measured by the

kurtosis of the distribution; fatter tails lead to higher kurtosis. In investment terms, this

captures the tendency of the price of this investment to “jump” in either direction.

In the special case of the normal distribution, returns are symmetric and investors

do not have to worry about skewness and kurtosis, since there is no skewness and a

normal distribution is defined to have a kurtosis of zero. In that case, it can be argued that

investments can be measured on only two dimensions - (1) the 'expected return' on the

investment comprises the reward, and (2) the variance in anticipated returns comprises

the risk on the investment. Figure 3.3 illustrates the return distributions on two

investments with symmetric returns-

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Figure 3.3: Return Distribution Comparisons

Low Variance Investment

High Variance Investment

Expected Return

In this scenario, an investor faced with a choice between two investments with the same

standard deviation but different expected returns, will always pick the one with the higher

expected return.

In the more general case, where distributions are neither symmetric nor normal, it

is still conceivable, though unlikely, that investors still choose between investments on

the basis of only the expected return and the variance, if they possess utility functions1

that allow them to do so. It is far more likely, however, that they prefer positive skewed

distributions to negatively skewed ones, and distributions with a lower likelihood of

jumps (lower kurtosis) over those with a higher likelihood of jumps (higher kurtosis). In

this world, investors will trade off the good (higher expected returns and more positive

skewness) against the bad (higher variance and kurtosis) in making investments. Among

1 A utility function is a way of summarizing investor preferences into a generic term called ˜utility™ on the

basis of some choice variables. In this case, for instance, investor utility or satisfaction is stated as a

function of wealth. By doing so, we effectively can answer questions such as “ Will an investor be twice as

happy if he has twice as much wealth? Does each marginal increase in wealth lead to less additional utility

than the prior marginal increase? In one specific form of this function, the quadratic utility function, the

entire utility of an investor can be compressed into the expected wealth measure and the standard deviation

in that wealth, which provides a justification for the use of a framework where only the expected return

(mean) and its standard deviation (variance) matter.

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the risk and return models that we will be examining, one (the capital asset pricing model

or the CAPM) explicitly requires that choices be made only in terms of expected returns

and variances. While it does ignore the skewness and kurtosis, it is not clear how much of

a factor these additional moments of the distribution are in determining expected returns.

In closing, we should note that the return moments that we run into in practice are

almost always estimated using past returns rather than future returns. The assumption we

are making when we use historical variances is that past return distributions are good

indicators of future return distributions. When this assumption is violated, as is the case

when the asset™s characteristics have changed significantly over time, the historical

estimates may not be good measures of risk.

˜: 3.1: Do you live in a mean-variance world?

Assume that you had to pick between two investments. They have the same

expected return of 15% and the same standard deviation of 25%; however, investment A

offers a very small possibility that you could quadruple your money, while investment

B™s highest possible payoff is a 60% return. Would you

a. be indifferent between the two investments, since they have the same expected return

and standard deviation?

b. prefer investment A, because of the possibility of a high payoff?

c. prefer investment B, because it is safer?

Illustration 3.1: Calculation of standard deviation using historical returns: Disney

We collected the data on the returns we would have made on a monthly basis for

every month from January 1999 to December 2003 on an investment in Disney stock.

The monthly returns are graphed in figure 3.4:

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Figure 3.4: Returns on Disney: 1999- 2003

30.00%

20.00%

Return on Disney (including dividends)

10.00%

0.00%

-10.00%

-20.00%

-30.00%

Feb-99

Apr-99

Jun-99

Aug-99

Oct-99

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Feb-00

Apr-00

Jun-00

Aug-00

Oct-00

Dec-00

Feb-01

Apr-01

Jun-01

Aug-01

Oct-01

Dec-01

Feb-02

Apr-02

Jun-02

Aug-02

Oct-02

Dec-02

Feb-03

Apr-03

Jun-03

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Dec-03

Month