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Copyright Information (bibliographic)

Document Type: Book Chapter

Title of Book: Financial Management Theory and Practice (16th Edition)

Author(s) of Book: Eugene F. Brigham, Michael C. Ehrhardt

Chapter Title: Chapter 6 Risk and Return

Author(s) of Chapter: Eugene F. Brigham, Michael C. Ehrhardt

Year: 2020

Publisher: Cengage Learning

Place of Publishing: the United States of America

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Risk and Return

What a difference a year makes! At the beginning of 2016, many investors purchased

shares of stock in the NASDAQ companies Madrigal Pharmaceuticals and Frontier

Communications. By year end, Madrigal had gone up by 500% while Frontier had fallen

by 87% (yet still remained listed on NASDAQ). Big gains and losses weren't limited to small

companies. Investors were feeling good with Vertex Pharmaceuticals, gaining 100%

during the year. At the other extreme, Under Armour Inc. went down by 50%.

Did investors in Frontier and Under Armour make bad decisions? Before you answer,

suppose you were making the decision back in January 2016, with the information

available then. You now know the decision's outcome was poor, but that doesn't mean

the decision itself was badly made. Investors must have known these stocks were risky,

with a chance of a gain or a loss. But given the information available to them, they

certainly invested with the expectation of a gain. What about the investors in Madrigal

and Vertex? They also realized the stock prices could go down or up but were probably

pleasantly surprised that the stocks went up so much.

These examples show that what you expect to happen and what actually happens

are often very different-the world is risky! Therefore, it is vital that you understand

risk and the ways to manage it. As you read this chapter and think about risk, keep

these companies in mind.

243

244 Part 3 Stocks and Options

Intrinsic Value, Risk, and Return

The intrinsic value of a company is the present value of its ex­

pected future free cash flows (FCFs) discounted at the weighted

average cost of capital (WACC). This chapter shows you how to

measure a firm's risk and the rate of return expected by share­

holders, which affects the WACC. All else held equal, higher risk

increases the WACC, which reduces the firm's value.

uu:..ce

The textbook's Web site

contains an Excel file that

w/11 guide you through the

chapter's calculations.

The file for this chapter is

Ch06 Tool Kit.xlsx, and we

encourage you to open the

file and follow along as

you read the chapter.

Net operating profit after taxes

Required investments in operating capital

Free cash flow (FCF)

K� K� K� Value=-----+-----+···+-----

(1 + WACC)l (1 + WACC)2 (1 + WACC) '"

Market interest rates

Weighted average cost of capital (WACC)

Cost of debt Cost of equity

Firm's debt/equity mix

Firm's business risk

In this chapter, we start from the basic premise that investors like returns and dislike risk; this is called risk aversion. Therefore, people will invest in relatively risky assets only if they expect to receive relatively high returns: the higher the perceived risk, the higher the expected rate of return an investor will demand. In this chapter, we define exactly what the term "risk" means as it relates to investments, we examine procedures used to measure risk, and we discuss more precisely the relationship between risk and required returns. In later chapters, we extend these relationships to show how risk and return interact to determine security prices. Managers must understand and apply these concepts as they plan the actions that will shape their firms' futures, and investors must understand them in order to make appropriate investment decisions.

6-1 Investment Returns and Risk

With most investments, an individual or business spends money today with the expecta­ tion of earning even more money in the future. However, most investments are risky. Fol­ lowing are brief definitions of return and risk.

6-la Returns on Investments

The concept of return provides investors with a convenient way to express the financial performance of an investment. To illustrate, suppose you buy 10 shares of a stock for $1,000. The stock pays no dividends, but at the end of 1 year, you sell the stock for $1,100. What is the return on your $1,000 investment?

0

0

Chapter 6 Risk and Return

One way to express an investment's return is in dollar terms:

Dollar return = Amount to be received - Amount invested = $1,100 - $1,000 = $100

245

If instead, at the end of the year, you sell the stock for only $900, your dollar return will be -$100.

Although expressing returns in dollars is easy, two problems arise: (1) To make a meaningful judgment about the return, you need to know the scale (size) of the invest­ ment; a $100 return on a $100 investment is a great return (assuming the investment is held for 1 year), but a $100 return on a $10,000 investment would be a poor return. (2) You also need to know the timing of the return; a $100 return on a $100 investment is a great return if it occurs after 1 year, but the same dollar return after 100 years is not very good.

The solution to these scale and timing problems is to express investment results as rates of return, or percentage returns. For example, the rate of return on the I-year stock investment, when $1,100 is received after 1 year, is 10%:

f Amount received - Amount invested

Rate o return=-------------­ Amount invested

Dollar return $ 100

Amount invested $1,000

= 0.10 = 10%

The rate of return calculation "standardizes" the dollar return by considering the annual return per unit of investment. Although this example has only one outflow and one inflow, the annualized rate of return can easily be calculated in situations where multiple cash flows occur over time by using the time value of money concepts discussed in Chapter 4.

6-lb Stand-Alone Risk versus Portfolio Risk

Risk is defined in Webster's as "a hazard; a peril; exposure to loss or injury." Thus, risk refers to the chance that some unfavorable event will occur. For an investment in financial assets or in new projects, the unfavorable event is ending up with a lower return than you expected. An asset's risk can be analyzed in two ways: (1) on a stand-alone basis, where the asset is considered in isolation; and (2) as part of a portfolio, which is a collection of assets. Thus, an asset's stand-alone risk is the risk an investor would face if she held only this one asset. Most assets are held in portfolios, but it is necessary to understand stand-alone risk in order to understand risk in a portfolio context.

SELF-TEST

Compare and contrast dollar returns and rates of return.

Why are rates of return superior to dollar returns when com poring different potential invest­ ments? (Hint: Think about size and timing.)

If you pay $500 for on investment that returns $600 in 1 year, what is your annual rate of return? (20%)

6-2 Measuring Risk for Discrete Distributions Political and economic uncertainties affect stock market risk. For example, the market went up by over 3% in November 2017. At that time, one scenario might have been contin­ ued economic growth and a successful congressional effort to cut corporate taxes. On the

246 Part 3 Stocks and Options

other hand, another possible scenario was military conflict with North Korea. At the risk of oversimplification, these outcomes represented distinct (or discrete) scenarios for the market, with each scenario having a very different market return.

Risk can be a complicated topic, so we begin with a simple example that has discrete possible outcomes.1

6-2a Probability Distributions for Discrete Outcomes

An event's probability is defined as the chance that the event will occur. For example, a weather forecaster might state: "There is a 40% chance of rain today and a 60% chance that it will not rain." If all possible events, or outcomes, are listed, and if a probability is assigned to each event, then the listing is called a discrete probability distribution. (Keep in mind that the probabilities must sum to 1.0, or 100%.)

Suppose an investor is facing a situation similar to the debt ceiling crisis and believes there are three possible outcomes for the market as a whole: (1) Best case, with a 30% probability; (2) Most Likely case, with a 40% probability; and (3) Worst case, with a 30% probability. The investor also believes the market would go up by 37% in the Best scenario, go up by 11% in the Most Likely scenario, and go down by 15% in the Worst scenario.

Figure 6-1 shows the probability distribution for these three scenarios. Notice that the probabilities sum to 1.0 and that the possible returns are dispersed around the Most Likely scenario's return.

We can calculate expected return and risk using the probability distribution, as we il­ lustrate in the next sections.

6-2b Expected Rate of Return for Discrete Distributions

The rate-of-return probability distribution is shown in the "Inputs" section of Figure 6-2; see Columns (1) and (2). This portion of the figure is called a payoff matrix when the out­ comes are cash flows or returns.

If we multiply each possible outcome by its probability of occurrence and then sum these products, as in Column (3) of Figure 6-2, the result is a weighted average of outcomes.

FIGURE6-1

Discrete Probability Distribution for Three Scenarios

Probability of Scenario

0.5

0.4

0.3

0.2

0.1

0

Worst Case

-15%

Most Likely

11%

Best Case

37%

Outcomes: Market Returns for 3 Scenarios

'The following discussion of risk applies to all random variables, not just stock returns.

FIGURE6-2

Chapter 6 Risk and Return 247

Calculating Expected Returns and Standard Deviations: Discrete Probabilities

A I B I C D E I F G 61 INPUTS: Exuected Return Standard Deviation

1-- Product of Deviation from

Probability Market Rate Probability and Expected Squared Prob. x Sq.

of Scenario of Return Return Return Deviation Dev.

62 Scenario (1) (2) (3) = (1) X (2) (4) = (2) - D66 (5) = (4) 2 (6): (1) X (5)

63 Best Case 0.30 37% 11.1% 0.2600 0.0676 0.0203

64 Most Likely 0.40 11% 4.4% 0.0000 0.0000 0.0000 -

65 Worst Case 0.30 -15% -4.5% -0.2600 0.0676 0.0203

66 -

1.00 I Exp. ret. = Sum = 11.0% Sum = Variance = 0.0406 Std. Dev. = Square

67 root of variance= 20.1%

Source: See the file Ch06 Tool Kft.xlsx. Calculations are not rounded in intermediate steps.

r..esource

See Ch06 Toof Kit.xfsx on

the textbook's Web site.

The weights are the probabilities, and the weighted average is the expected rate of return (r), called "r-hat," which is the mean of the probability distribution. 2 As shown in cell D66 in Figure 6-2, the expected rate ofreturn is 11%.3

The calculation for expected rate of return can also be expressed as an equation that does the same thing as the payoff matrix table:

Expected rate ofreturn = r = Pl1 + Pl2 + · · · + pnrn

Here r i is the return if outcome i occurs, p

i is the probability that outcome i occurs,

and n is the number of possible outcomes. Thus, r is a weighted average of the possible outcomes (the r

i values), with each outcome's weight being its probability of occurrence.

Using the data from Figure 6-2, we obtain the expected rate of return as follows:

r = P1(r1) + P2(r) + p3(r3)

= 0.3(37%) + 0.4(11%) + 0.3(-15%)

= 11%

6-2c Measuring Stand-Alone Risk: The Standard Deviation of a Discrete Distribution

For simple distributions, it is easy to assess risk by looking at the dispersion of possible outcomes: a distribution with widely dispersed possible outcomes is riskier than one with

2Jn other chapters, we will use r d

and r, to signify expected returns on bonds and stocks, respectively. However, this distinction is unnecessary in this chapter, so we just use the general term r to signify the expected return on an investment. 1Don't worry about why there is an 11 % expected return for the market. We discuss the market return in more detail later in the chapter.

248

resource

See Ch06 Tool Kit.xlsx an

the textbook's Web site for

oil calculations.

Part 3 Stocks and Options

narrowly dispersed outcomes. For example, we can look at Figure 6-1 and see that the possible returns are widely dispersed. But when there are many possible outcomes and we are comparing many different investments, it isn't possible to assess risk simply by look­ ing at the probability distribution-we need a quantitative measure of the tightness of the probability distribution. One such measure is the standard deviation (er), the symbol for which is er, pronounced "sigma." A large standard deviation means that possible outcomes are widely dispersed, whereas a small standard deviation means that outcomes are more tightly clustered around the expected value.

To calculate the standard deviation, we proceed as shown in Figure 6-2, taking the following steps:

1. Calculate the expected value for the rate of return using Equation 6-1. 2. Subtract the expected rate of return (r) from each possible outcome (r) to obtain a

set of deviations about r, as shown in Column (4) of Figure 6-2:

Deviation 1 = r

i - r

3. Square each deviation as shown in Column (5). 4. Multiply the probability of the occurrence (shown in Column 1) by the squared de­

viations in Column (5); these products are shown in Column (6). 5. Sum these products to obtain the variance of the probability distribution:

n

Variance = CT2 = L pi(ri - r) 2

i=l

Thus, the variance is essentially a weighted average of the squared deviations from the expected value.

6. Finally, take the square root of the variance to obtain the standard deviation:

n

Standard deviation = er = L P1<ri - r) 2

i•l

The standard deviation provides an idea of how far above or below the expected value the actual value is likely to be. Using this procedure in Figure 6-2, our hy­ pothetical investor believes that the market return has a standard deviation of about 20%.

SELF -TEST

What does "investment risk" mean?

Set up an illustrative probability distribution for an investment.

What is a payoff matrix?

How does one calculate the standard deviation?

An investment has a 20% chance of producing a 25% return, a 60% chance of producing a 10% return, and a 20% chance of producing a -15% return. What is its expected return? (8%) What is its standard deviation? (12.9%)

FIGURE 6-3

Chapter 6 Risk and Return 249

6-3 Risk in a Continuous Distribution

Investors usually don't estimate discrete outcomes in normal economic times but instead use the scenario approach during special situations, such as the debt ceiling crisis, the European bond crisis, oil supply threats, bank stress tests, and so on. Even in these situ­ ations, they would estimate more than three outcomes. For example, an investor might add more scenarios to our example of possible stock market returns; Panel A in Figure 6-3 shows 15 scenarios for the market and has a standard deviation of20.2%.

Recall that the standard deviation provides a measure of dispersion that provides in­ formation about the range of possible outcomes. Panel B of Figure 6-3 shows 15 possible

Discrete Probability Distributions for 15 Scenarios

Panel A: Market Return for 15 Scenarios: Standard Deviation = 20.2%

Probability

0.25

0.20

0.15

0.10

0.05

-66%-55%-44%-33%-22%-11% 0% 11% 22% 33% 44% 55% 66% 77% 88%

Outcomes: Market Return

Panel B: Single Company's Stock Return for 15 Scenarios: Standard Deviation = 36.2%

Probability 0.25

0.20

0.15

0.10

0.05

-66%-55%-44%-33%-22%-11% 0% 11% 22% 33% 44% 55% 66% 77% 88%

Outcomes: Stock Return

250

e .. s...cv.u c e

For more discussion of

probability distributions,

see Web Extension

6A, ovoi/oble on the

textbook's Web site.

FIGURE6-4

Part 3 Stocks and Options

Probability Ranges for a Normal Distribution

-3rr

Notes:

I

I

I

I

JI'".-------- 95.46%---1-----.... ""' ..... 99.74%----i----+----"-...�--i

-2rr -Irr +Irr +2rr +3rr

1. The area under the normal curve always equals 1.0, or 100%. Thus, the areas under any pair of normal curves drawn on the same scale, whether they are peaked or flat, must be equal.

2. Half of the area under a normal curve is to the left of the mean, r, indicating that there is a 50% probability that the actual outcome will be less than the mean, and half is to the right, indicating a 50% probability that it will be greater than the mean.

3. Of the area under the curve, 68.26% is within ± lCT of the mean, indicating that the probability is 68.26% that the actual outcome will be within the range r - CT to r + CT.

outcomes for a single company's stock for the same scenarios in Panel A. The single stock has a standard deviation of 36.2%. Notice how much more widely dispersed the single stock's outcomes are than those of the stock market. We will have much more to say about this phenomenon when we discuss portfolios.

We live in a complex world, and sometimes this complexity can't be described ad­ equately with several or even many scenarios. In such cases, instead of adding more and more scenarios, most analysts turn to continuous probability distributions, which have an infinite number of possible outcomes. The normal distribution, with its familiar bell-shaped curve, is widely used in many areas, including finance. One feature of a normal distribution is that the actual return will be within ± 1 standard deviation of the expected return 68.26% of the time. Figure 6-4 illustrates this point, and it also shows the situation for ±2a and ±3a. For our 3-scenario example, r = 11% and CT = 20%. If returns come from a normal distribution with the same ex­ pected value and standard deviation as the discrete distribution, there would be a 68.26% probability that the actual return would be in the range of 11 % ± 20%, or from -9% to 31%.

When using a continuous distribution, it is common to use historical data to estimate the standard deviation, as we explain in the next section.

SELF-TEST

For a normal distribution, what is the probability of being within 1 standard deviation of the expected value? {68.26%)

'

Chapter 6 Risk and Return

What Does Risk Really Mean?

As explained in the text, the probability of being within 1

standard deviation of the expected return is 68.26%, so the

probability of being further than 1 standard deviation from

the mean is 100% - 68.26% = 31.74%. There is an equal

probability of being above or below the range, so there is a

15.87% chance of being more than 1 standard deviation be­

low the mean, which is roughly equal to a 1 in 6 chance (1 in

6 is 16.67 %).

For the average firm listed on the New York Stock

Exchange (NYSE), rr has been in the range of 35% to 40% in

recent years, with an expected return of around 8% to 12%.

One standard deviation below this expected return is about

10% - 35% = -25%. This means that, for a typical stock in

a typical year, there is about a 1 in 6 chance of having a 25%

loss. You might be thinking that 1 in 6 is a pretty low probabil­

ity, but what if your chance of getting hit by a car when you

crossed a street were 1 in 6? When put that way, 1 in 6 sounds

pretty scary.

You might also correctly be thinking that there would be

a 1 in 6 chance of getting a return higher than 1 standard de­

viation above the mean, which would be about45% for a typi­

cal stock. A 45% return is great, but human nature is such that

most investors would dislike a 25% loss a whole lot more than

they would enjoy a 45% gain.

251

You might also be thinking that you'll be OK if you hold

the stock long enough. But even if you buy and hold a diversi­

fied portfolio for 10 years, there is still roughly a 10% chance

that you will lose money. If you hold it for 20 years, there is

about a 4% chance of losing. Such odds wouldn't be worri­

some if you were engaged in a game of chance that could be

played multiple times, but you have only one life to live and

just a few rolls of the dice.

We aren't suggesting that investors shouldn't buy

stocks; indeed, we own stock ourselves. But we do believe

investors should understand more clearly how much risk in­

vesting entails.

6-4 Using Historical Data to Estimate Risk Suppose that a sample of returns over some past period is available. These past realized rates of return (i\) are denoted as i\ ("r bar t"), where t designates the time period. The average return (r Avg) over the last T periods is defined as:

T

�r, 1-1

rAvg = T

The standard deviation of a sample of returns can then be estimated using this formula:4

Estimated rr = S =

T

""'Cr - r )2 .£.J I Avg t=l

T - I

When estimated from past data, the standard deviation is often denoted by S.

'Because we are estimating the standard deviation from a sample of observations, the denominator in Equation 6-5 is "T - 1" and not just "T." Equations 6-4 and 6-5 are built into all financial calculators. For example, to find the sample standard deviation, enter the rates of return into the calculator and press the key marked S (or S,) to get the standard deviation. See your calculator's manual for details.

252

6-4a Calculating the Historical Average and Standard Deviation

Part 3 Stocks and Options

To illustrate these calculations, consider the following historical returns for a company:

Year

2017

2018

2019

Return

15%

-5%

20%

Using Equations 6-4 and 6-5, the estimated average and standard deviation, respec­ tively, are:

Estimated a (or S) =

15% - 5% + 20% ------- = 10.0%

3

(15% - 10%)2 + (-5% - 10%)2 + (20% - 10%)2

3 - 1

= 13.2%

The average and standard deviation can also be calculated using Excel's built-in func­ tions, shown here using numerical data rather than cell ranges as inputs:

=AVERAGE(0.15,-0.05,0.20) = 10.0% =STDEV(0.15,-0.05,0.20) = 13.2%

The historical standard deviation is often used as an estimate of future variabil­ ity. Because past variability is often repeated, past variability may be a reasonably good estimate of future risk. However, it is usually incorrect to use rAvg based on a past period as an estimate of r, the expected future return. For example, just because a stock had a 75% return in the past year, there is no reason to expect a 75% return this year.

6-4b Calculating MicroDrive's Historical Average and Standard Deviation

Figure 6-5 shows 48 months of recent stock returns for two companies, MicroDrive and SnailDrive; the actual data are in the Excel file Ch06 Tool Kit.xlsx. A quick glance is enough to determine that MicroDrive's returns are more volatile.

We could use Equations 6-4 and 6-5 to calculate the average return and standard de­ viation, but that would be quite tedious. Instead, we use Excel's AVERAGE and STDEV functions and find that MicroDrive's monthly average return was 1.00% and its monthly standard deviation was 14.94%. SnailDrive had an average monthly return of 0.77% and a standard deviation of 9.87%. These calculations confirm the visual evidence in Figure 6-5: MicroDrive had greater stand-alone risk than SnailDrive.

We often use monthly data to estimate averages and standard deviations, but we nor­ mally present data in an annualized format. Multiply the monthly average return by 12 to get MicroDrive's annualized average return of 1.00%(12) = 12.0%. As noted earlier, the past average return isn't a good indicator of the future return.

0

Chapter 6 Risk and Return 253

FIGURE&·S

Historical Monthly Stock Returns for Micro Drive and SnailDrive

A B

201

202 Monthly Rate of

203 Return

204 40%

205

206 30%

207

208 20%

209

210 10%

211

212 0%

213

214 -10%

215

216 -20%

217

218 -30%

219

220 -40%

221 0 6

222

12 18

C D

MicroDrive

24 Month of Return

30

E F

36 42 48

Average Return (annualized)

Standard Deviation (annualized)

MicroDrive

12.0%

51.8%

SnailDrive

9.3%

34.2%

To annualize the standard deviation, multiply the monthly standard deviation by the square root of 12. MicroDrive's annualized standard deviation was 14.94%(V12) = 51.8%.5 SnailDrive's average annual return was 9.3%, and its annualized standard devia­ tion was 34.2%.

Notice that MicroDrive had higher risk than SnailDrive (a standard deviation of 51.8% versus 34.2%) and a higher average return (12.0% versus 9.3%) during the past 48 months. However, a higher return for undertaking more risk isn't guaranteed-if it were, then a riskier investment wouldn't really be risky!

The file Ch06 Tool Kit.xlsx calculates the annualized average return and standard deviation using just the most recent 12 months. Here are the results:

Results for Most Recent 12 Months

Average return (annual)

Standard deviation (annual)

Micro Drive

-28.9%

52.4%

SnailDrive

11.6%

31.4%

5If we had calculated the monthly variance, we would annualize it by multiplying it by 12, as intuition (and mathematics) suggests. Because standard deviation is the square root of variance, we annualize the monthly standard deviation by multiplying it by the square root of 12.

254 Part 3 Stocks and Options

The Historic Trade-Off between Risk and Return

The table accompanying this box summarizes the historical

trade-off between risk and return for different classes of invest­

ments. The assets that produced the highest average returns also

had the highest standard deviations and the widest ranges of re­

turns. For example, small-company stocks had the highest aver­

age annual return, but their standard deviation of returns also

was the highest. In contrast, U.S. Treasury bills had the lowest

standard deviation, but they also had the lowest average return.

Note that a T-bill is riskless if you hold it until maturity,

but if you invest in a rolling portfolio of T-bills and hold the

portfolio for a number of years, then your investment in­

come will vary depending on what happens to the level of

interest rates in each year. You can be sure of the return you

will earn on an individual T-bill, but you cannot be sure of

the return you will earn on a portfolio of T-bills held over a

number of years.

Realized Returns, 1926-2016

Small- Large- Long-Term Long-Term U.S.

Company Company Corporate Government Treasury

Stocks Stocks Bonds Bonds Bills Inflation

Average return 16.6% 12.0% 6.3% 6.0% 3.4% 3.0%

Standard deviation 31.9 19.9 8.4 9.9 3.1 4.1

Excess return over T-bonds• 10.6 6.0 0.3

'The excess return overT-bonds is called the "historical risk premium." This excess return will also be the current risk premium that is reflected in security prices if and only if investors expect returns in the future to be similar to returns earned in the past.

Source: Data from the 2016 SBBI Yearbook: Stocks, Bonds, Bills and Inflation, Roger Ibbotson, Roger J. Grabowski, James P. Harrington and Carla Nunes (Hoboken, NJ: John Wiley & Sons, 2016).

Even though MicroDrive's standard deviation was well above that of SnailDrive over the last 12 months of the sample period, MicroDrive experienced an annualized aver­ age loss of almost 29%, while SnailDrive gained almost 12%.6 MicroDrive's stockholders certainly learned that higher risk doesn't always lead to higher actual returns.

SELF-TEST

A stock's returns for the past 3 years were 10%, -15%, and 35%. What is the historical average return? (10%) What is the historical sample standard deviation? (25%)

6-5 Risk in a Portfolio Context

Most financial assets are actually held as parts of portfolios. Banks, pension funds, insur­ ance companies, mutual funds, and other financial institutions are required by law to hold diversified portfolios. Even individual investors-at least those whose security holdings constitute a significant part of their total wealth-generally hold portfolios, not the stock of only one firm, because diversification can reduce risk exposure.

6-Sa Creating a Portfolio

A portfolio is a collection of assets. The weight of an asset in a portfolio is the percentage of the portfolio's total value that is invested in the asset. For example, if you invest $1,000

•During the last 12 months, MicroDrive had an average monthly loss of2.41%, but it had a compound loss for the year of over 34%. We discuss the difference between arithmetic averages and geometric averages (based on compound returns) in Chapter 9.

Chapter 6 Risk and Return 255

in each of 10 stocks, your portfolio has a value of $10,000, and each stock has a weight of $1,000/$10,000 = 10%. If instead you invest $5,000 in 1 stock and $1,000 apiece in 5 stocks, the first stock has a weight of $5,000/$10,000 = 50%, and each of the other 5 stocks has a weight of 10%. Usually it is more convenient to talk about an asset's weight in a portfolio rather than the dollars invested in the asset. Therefore, when we create a portfolio, we choose a weight (or a percentage) for each asset, with the weights summing to 1.0 (or the percentages summing to 100%).

Suppose we have a portfolio of n stocks. The actual return on a portfolio in a particular period is the weighted average of the actual returns of the stocks in the portfolio, with w

1

denoting the weight invested in Stock i:

r =wr +wr + .. ·+wr p II 22 n n

n

=�wl1 i=l

The average portfolio return over a number of periods is also equal to the weighted aver­ age of the stock's average returns:

n

r = w.r . -

� -Avg,p 1 Avg., i=I

Recall from the previous section that SnailDrive had an average annualized return of 9.3% during the past 48 months and MicroDrive had a 12.0% return. A portfolio with 75% invested in SnailDrive and 25% in MicroDrive would have had the following return:

fA = 0.75(9.3%) + 0.25(12%) = 10.0%vg,p

Notice that the portfolio return of 10.0% is between the returns of SnailDrive {9.3%) and MicroDrive (12.0%), as you would expect.

Suppose an investor with stock only in SnailDrive came to you for advice, saying, "I would like more return, but I hate risk!" How do you think the investor would react if you suggested taking 25% of the investment out of the low-risk SnailDrive {with a standard deviation of 34.2%) and putting it into the high-risk MicroDrive {with a standard deviation of 51.8%)? As just shown, the return during the 48-month period would have been 10.0%, well above the return on SnailDrive. But what would have happened to risk?

The file Ch06 Tool Kit.xlsx calculates the portfolio return for each month (us­ ing Equation 6-6) and calculates the portfolio's standard deviation by applying Excel's STDEV function to the portfolio's monthly returns. Imagine the investor's surprise in learning that the portfolio's standard deviation is 27.1 %, which is less than that of SnailDrive's 34.2% standard deviation. In other words, adding a risky asset to a safer asset can reduce risk!

How can this happen? MicroDrive sells high-end memory storage, whereas SnailDrive sells low-end memory, including reconditioned hard drives. When the economy is doing well, MicroDrive has high sales and profits, but SnailDrive's sales lag because customers prefer faster memory. But when times are tough, customers resort to SnailDrive for low­ cost memory storage. Take a look at Figure 6-5. Notice that SnailDrive's returns don't move in perfect lockstep with MicroDrive: Sometimes MicroDrive goes up and SnailDrive goes down, and vice versa.

256 Part 3 Stocks and Options

6-Sb Correlation and Risk for a Two-Stock Portfolio

The tendency of two variables to move together is called correlation, and the correlation coefficient (p) measures this tendency. The symbol for the correlation coefficient is the Greek letter rho, p (pronounced "roe"). The correlation coefficient can range from + LO, denoting that the two variables move up and down in perfect synchronization, to -LO, denoting that the variables always move in exactly opposite directions. A correlation coefficient of zero indicates that the two variables are not related to each other at all-that is, changes in one variable are independent of changes in the other.

The estimate of correlation from a sample of historical data is often called "R." Here is the formula to estimate the correlation between stocks i and j Ci\

1 is the actual return

for Stock i in period t, and r. A

is the average return during the T-period sample; similar I, vg notation is used for Stock j):

T

� (r.t - rl.A )(r.t - r.A > ,"-,/ 1, vg J, J,Vg t = l

Estimated p = R = --;;::=====;;;:;::=====� T T

� (ri,t - rl.Av/ � (rj,t - rj.Avl t=l t=l

Fortunately, it is easy to estimate the correlation coefficient with a financial calcula­ tor or Excel. With a calculator, simply enter the returns of the two stocks and then press a key labeled "r."7 In Excel, use the CORREL function. See Ch06 Tool Kit.xlsx, where we calculate the correlation between the returns of MicroDrive and SnailDrive to be -0.13. The negative correlation means that when SnailDrive is having a poor return, MicroDrive tends to have a good return; when SnailDrive is having a good return, MicroDrive tends to have a poor return. This means that adding some of MicroDrive's stock to a portfolio that only had SnailDrive's stock tends to reduce the volatility of the portfolio.

Here is a way to think about the possible benefit of diversification: If a portfolio's stan­ dard deviation is less than the weighted average of the individual stocks' standard devia­ tions, then diversification provides a benefit. Does diversification always reduce risk? If so, by how much? And how does correlation affect diversification? Let's consider the full range of correlation coefficients, from -1 to + L

If two stocks have a correlation of -1 (the lowest possible correlation), when one stock has a higher than expected return then the other stock has a lower than expected return, and vice versa. In fact, it would be possible to choose weights such that one stock's de­ viations from its mean return completely cancel out the other stock's deviations from its mean return. 8 Such a portfolio would have a zero standard deviation but would have an expected return equal to the weighted average of the stock's expected returns. In this situ­ ation, diversification can eliminate all risk: For correlation of -1, the portfolio's standard deviation can be as low as zero if the portfolio weights are chosen appropriately.

If the correlation were + 1 (the highest possible correlation), the portfolio's standard deviation would be the weighted average of the stock's standard deviations. In this case, diversification doesn't help: For correlation of+ 1, the portfolio's standard deviation is the weighted average of the stocks' standard deviations.

'See your calculator manual for the exact steps. Also, note that the correlation coefficient is often denoted by the term "r." We use p to avoid confusion with r, which we use to denote the rate of return. 1If the correlation between Stocks I and 2 is equal to -!, then the weights for a zero-risk portfolio are w

1 = rr/(rr, + rr,) and w2 = rr/(rr, + rr,).

Chapter 6 Risk and Return 257

For any other correlation, diversification reduces, but cannot eliminate, risk: For cor­ relation between -1 and + 1, the portfolio's standard deviation is less than the weighted average of the stocks' standard deviations.

The correlation between most pairs of companies is in the range of 0.2 to 0.3, so diver­ sification reduces risk, but it doesn't completely eliminate risk.9

6-Sc Diversification and Multi-Stock Portfolios

Figure 6-6 shows how portfolio risk is affected by forming larger and larger portfolios of randomly selected New York Stock Exchange (NYSE) stocks. Standard deviations are plotted for an average one-stock portfolio, an average two-stock portfolio, and so on, up

FIGURE6-6

Effects of Portfolio Size on Portfolio Risk for Average Stocks

Portfolio Risk, crp

(%)

35

30

25

15 Portfolio's Total Risk:

Declines 10 as Stocks

Are Added

5

10

Diversifiable Risk

Portfolio's Market Risk: Remains Constant

20 30 40

Minimum Attainable Risk in a Portfolio of Average Stocks

2,000+

Number of Stocks in the Portfolio

•During the period 1968-1998, the average correlation coefficient between two randomly selected stocks was 0.28, while the average correlation coefficient between two large-company stocks was 0.33; see Louis K. C. Chan, Jason Karceski, and JosefLakonishok, "On Portfolio Optimization: Forecasting Covariance and Choos­ ing the Risk Model," The Review of Financial Studies, Vol. 12, No. 5, Winter 1999, pp. 937-974. The average correlation fell from around 0.35 in the late 1970s to less than 0.10 by the late 1990s; see John Y. Campbell, Martin Lettau, Burton G. Malkiel, and Yexiao Xu, "Have Individual Stocks Become More Volatile? An Empiri­ cal Exploration of Idiosyncratic Risk," Journal of Finance, February 2001, pp. 1-43.

258 Part 3 Stocks and Options

to a portfolio consisting of all 2,000-plus common stocks that were listed on the NYSE at the time the data were plotted. The graph illustrates that, in general, the risk of a portfolio consisting of stocks tends to decline and to approach some limit as the number of stocks in the portfolio increases. According to data from recent years, a

1 , the standard deviation

of a one-stock portfolio (or an average stock), is approximately 35%. However, a portfolio consisting of all shares of all stocks, which is called the market portfolio, would have a standard deviation, a

M ' of only about 20%, which is shown as the horizontal dashed line

in Figure 6-6. Thus, almost half of the risk inherent in an average individual stock can be eliminated

if the stock is held in a reasonably well-diversified portfolio, which is one containing 40 or more stocks in a number of different industries. The part of a stock's risk that cannot be eliminated is called market risk, while the part that can be eliminated is called diversifi­ able risk. 10 The fact that a large part of the risk of any individual stock can be eliminated is vitally important because rational investors will eliminate it simply by holding many stocks in their portfolios, thus rendering it irrelevant. This simple observation forms the basis of modern portfolio theory!

Market risk stems from factors that affect most firms: war, inflation, recessions, and high interest rates. Because most stocks are affected by these factors, market risk cannot be eliminated by diversification. Diversifiable risk is caused by such random events as lawsuits, strikes, successful and unsuccessful marketing programs, winning or losing a major contract, and other events that are unique to a particular firm. Because these events are random, their effects on a portfolio can be eliminated by diversification-bad events in one firm will be offset by good events in another.

SELF-TEST

Explain the following statement: •�n asset held as part of a portfolio is generally less risky than

the same asset held in isolation."

What is meant by perfect "positive correlation," "perfect negative correlation," and "zero correlation"?

In general, can the risk of a portfolio be reduced to zero by increasing the number of stocks in the

portfolio? Explain.

Stock A's returns in the pasts years have been 10%, -15%, 35%, 10%, and -20%. Stock B's

returns have been -5%, 1%, -4%, 40%, and 30%. What is the correlation coefficient for returns

between Stock A and Stock B? (-0.35)

6-6 The Relevant Risk of a Stock: The Capital Asset Pricing Model (CAPM)

We assume that investors are risk averse and demand a premium for bearing risk; that is, the higher the risk of a security, the higher its expected return must be to induce investors to buy it or to hold it. All risk except that related to broad market movements can, and pre­ sumably will, be diversified away. After all, why accept risk that can be eliminated easily? This implies that investors are primarily concerned with the risk of their portfolios rather than the risk of the individual securities in the portfolio. How, then, should the risk of an individual stock be measured?

The Capital Asset Pricing Model (CAPM) provides one answer to that question. A stock might be quite risky if held by itself, but-because diversification eliminates about

10Diversifiable risk is also known as company-specific risk or unsystematic risk. Market risk is also known as nondiversifiable risk or systematic risk; it is the risk that remains after diversification.

Chapter 6 Risk and Return 259

half of its risk-the stock's relevant risk is its contribution to a well-diversified portfolio's risk, which is much smaller than the stock's stand-alone risk.11

6-6a Contribution to Market Risk: Beta

A well-diversified portfolio has only market risk. Therefore, the CAPM defines the rel­ evant risk of an individual stock as the amount of risk that the stock contributes to the market portfolio, which is a portfolio containing all stocks.12 In CAPM terminology, P;M is the correlation between Stock i's return and the market return, <T

i is the standard deviation

of Stock i's return, and a M is the standard deviation of the market's return. The relevant measure of risk is its beta coefficient (b), which is often called just beta; the beta of Stock i, denoted by b

i , is calculated as:13

I This formula shows that a stock with a high standard deviation, <T

i , will tend to have a high

beta, which means that, other things held constant, the stock contributes a lot of risk to a well­ diversified portfolio. This makes sense because a stock with high stand-alone risk will tend to destabilize a portfolio. Note too that a stock with a high correlation with the market, p

i M' will

also tend to have a large beta and hence be risky. This also makes sense because a high correla­ tion means that diversification does not help much; the stock performs well when the portfolio also performs well, and the stock performs poorly when the portfolio also performs poorly.

Suppose a stock has a beta of 1.4. What does that mean? Specifically, how much risk will this stock add to a well-diversified portfolio? To answer that question, we need two important facts. First, the beta of a portfolio, b , is the weighted average of the betas of the

p

stocks in the portfolio, with the weights equal to the same weights used to create the port- folio. This can be written as:

b =w b +w b +···+w b p I I 2 2 n n

For example, suppose an investor owns a $100,000 portfolio consisting of$25,000 in­ vested in each of four stocks; the stocks have betas of 0.6, 1.2, 1.2, and 1.4. The weight of each stock in the portfolio is $25,000/$100,000 = 25%. The portfolio's beta will be b = 1.1:

p

b = 25%(0.6) + 25%(1.2) + 25%(1.2) + 25%(1.4) = 1.1 p

"Nobel Prizes were awarded to the developers of the CAPM, Professors Harry Markowitz and William F. Sharpe.

"In theory, the market portfolio should contain all assets. In practice, it usually contains only stocks. Many analysts use returns on the S&P 500 Index to estimate the market return.

"If you express the change in the market portfolio's standard deviation relative to a slight change in the amount of Stock i in the market portfolio (w.), this differential change is:

cJCJM -=b.cr., aw, .

If you compare two stocks with different betas, the stock with the larger beta contributes more risk to the market portfolio, as the preceding equation shows.

260 Part 3 Stocks and Options

The second important fact is that the variance of a well-diversified portfolio is ap­ proximately equal to the product of its squared beta and the market portfolio's variance:14

0-2 = b2 0-2 p p M ■

Now take the square root of each side of Equation 6-10. If we only consider portfolios with positive portfolio betas, which are typical, then the standard deviation of a well­ diversified portfolio, a- , is approximately equal to the product of the portfolio's beta and

p

the market standard deviation:15

o- =b o- p p M ■

Equation 6-11 shows three things: (1) A portfolio with a beta greater than 1 will have a bigger standard deviation than the market portfolio. (2) A portfolio with a beta equal to 1 will have the same standard deviation as the market. (3) A portfolio with a beta less than 1 will have a smaller standard deviation than the market. For example, suppose the mar­ ket standard deviation is 20%. Using Equation 6-11, a well-diversified portfolio with a beta of 1.1 will have a standard deviation of 22%:

a- = 1.1(20%) = 22% p

By substituting Equation 6-9 into Equation 6-11, we can see the impact that each indi­ vidual stock beta has on the risk of a well-diversified portfolio:

n

= �w1 b1 0-M i=l

This means Stock S with beta bs and weight w 5

in a well-diversified portfolio con­ tributes a total of w sb

5 o-

M to the portfolio's standard deviation. For a numerical example,

consider the four-stock portfolio in our example. Although a well-diversified portfolio would have more than four stocks, let's suppose our portfolio is well diversified. If that is the case, then Figure 6-7 shows how much risk each stock contributes to the portfolio.16

"This relationship is only valid for a very well-diversified portfolio having a large number of stocks with weights that are similar in size. If this is not true, then the portfolio will contain a significant amount of diver­ sifiable risk, and Equation 6-10 will not be a good approximation.

"If a portfolio's beta is negative, then the portfolio's standard deviation would depend upon the absolute value of its beta: <TP"' lbPI er.,. However, a portfolio's beta can be negative in only two situations, neither of which occurs in practice. First, the portfolio could be invested heavily in stocks with negative betas. This is not practical because there aren't enough negative beta stocks to create a well-diversified portfolio, which means Equations 6-10 and 6-11 don't apply. Second, it is possible to have a negative portfolio beta if the portfolio has negative weights (which means the stock is sold short) on stocks with large betas and positive weights on stocks with small betas. However, magnitudes of weights necessary for this strategy will result in a portfolio that is not diversified well enough for Equations 6-10 and 6-11 to be good approximations.

"If the portfolio isn't well diversified, then bPcr M (or lbPl(cr Ml ifbP < O) measures the part of the portfolio's stan­ dard deviation due to market risk. For undiversified portfolios, b,w,cr

M measures the amount of the portfolio's

standard deviation that is due to the market risk of Stock i.

Chapter 6 Risk and Return 261

FIGURE6-7

The Contribution of Individual Stocks to Portfolio Risk: The Effect of Beta

Portfolio Standard Deviation = 22%

Market Standard Deviation = rrM = 20%

Stock 1

Stock2

Stock 3

Stock4

Weight in Contribution to Contribution to Stock Beta: Portfolio: Portfolio Beta: Portfolio Risk:

b1 W1 b;X W1 b1 X W1 X CTM

0.6 25.0% 0.150 3.0%

1.2 25.0% 0.300 6.0%

1.2 25.0% 0.300 6.0%

1.4 25.0% 0.350 7.0% b

p = 1.100 CT

p = 22.0%

Out of the total 22% standard deviation of the portfolio, Stock 1 contributes w 1 b

1 cr

M =

(25%)(0.6)(20%) = 3%. Stocks 2 and 3 have betas that are twice as big as Stock l's beta, so Stocks 2 and 3 contribute twice as much risk as Stock 1. Stock 4 has the largest beta, and it contributes the most risk.

We demonstrate how to estimate beta in the next section, but here are some key points about beta: (1) Beta determines how much risk a stock contributes to a well-diversified portfolio. If all the stocks' weights in a portfolio are equal, then a stock with a beta that is twice as big as another stock's beta contributes twice as much risk. (2) The average of all stocks' betas is equal to l; the beta of the market also is equal to 1. Intuitively, this is because the market return is the average of all the stocks' returns. (3) A stock with a beta greater than 1 contributes more risk to a portfolio than does the average stock, and a stock with a beta less than 1 contributes less risk to a portfolio than does the average stock. (4) Most stocks have betas that are between about 0.4 and 1.6.

6-6b Estimating Beta

The CAPM is an ex ante model, which means that all of the variables represent before­ the-fact, expected values. In particular, the beta coefficient used by investors should reflect the relationship between a stock's expected return and the market's expected return during some future period. However, people generally calculate betas using data from some past period and then assume that the stock's risk will be the same in the future as it was in the past.

262 Part 3 Stocks and Options

Many analysts use 4 to 5 years of monthly data, although some use 52 weeks of weekly data. To illustrate, we use the 4 years of monthly returns from Ch06 Tool Kit.xlsx and cal­ culate the betas of MicroDrive and SnailDrive using Equation 6-8:17

Market Micro Drive SnailDrive

Standard deviation (annual): 19.89% 51.75% 34.17% Correlation with the market: 0.511 0.264 b

i = piM(rr/rr M) 1.33 0.45

Table 6-1 shows the betas for some well-known companies as provided by two different financial organizations, Yahoo!Finance and Value Line. Notice that their estimates of beta usually differ because they calculate it in slightly different ways. Given these differences, many analysts choose to calculate their own betas or else average the published betas.

Calculators and spreadsheets can calculate the components of Equation 6-8 (p1M, rri, and CT M), which can then be used to calculate beta, but there is another way.

18 The covari­ ance between Stock i and the market, COV

1M , is defined as:19

TABLE6-1

Beta Coefficients for Some Actual Companies

Stock (Ticker Symbol)

Amazon.com (AMZN)

Apple (AAPL)

Coca-Cola (KO)

Duke Energy Corp. (DUK)

Energen Corp. (EGN)

General Electric (GE)

Microsoft Corp. (MSFT)

Procter & Gamble (PG)

Tesla (TSLA)

Value Line

1.2 0.95 0.7 0.60 1.70 1.05 1.10 0.70 1.30

Yahoo! Finance

1.39 1.34 0.63 0.01 2.10 0.99 1.50 0.47 0.73

■

Sources: www.valueline.com and finance.yahoo.com, December 2017. For Value Line, enter the ticker symbol. For Yahoo!Finance, enter the ticker symbol. When the results page comes up, select Statistics to find beta.

17 As with any estimation, more observations usually lead to tighter confidence intervals. However, a longer esti­ mation period means that beta may change during the period. In our consulting, we use 252 to 504 days of daily data when calculating beta. We have found this to be the best trade-off between tighter confidence intervals due to more observations and the risk due to a changing beta. We use monthly data in this example to reduce the number of observations on the graph and the number of rows in Ch06 Tool Kit.xlsx. "For an explanation of computing beta with a financial calculator, see Web Extension 6B on the textbook's Web site. 19Using historical data, the sample covariance can be calculated as:

� Cr,., - r,,..")CrM., - rM.Mgl Sample covariance from historical data = COV,

M = -'-1--'1--------

T - l

Calculating the covariance is somewhat easier than calculating the correlation. So if you have already cal­ culated the standard deviations, it is easier to calculate the covariance and then calculate the correlation as p

1M = COV,./(a,a").

Chapter 6 Risk and Return 263

Substituting Equation 6-13 into Equation 6-8 provides another frequently used ex­ pression for calculating beta:

cov�M b=--1 CT�

Suppose you plotted the stock's returns on the y-axis of a graph and the market portfo­ lio's returns on the x-axis. The formula for the slope of a regression line is the same as the formula for beta in Equation 6-14. Therefore, to estimate beta for a security, you can estimate a regression with the stock's returns on the y-axis and the market's returns on the x-axis:

r. 1 = a. + b. r

M 1 + e.

1 1, 1 1 , I,

where r 1 ., and r

M ., are the actual returns for the stock and the market for date t; a

1 and b

1 are

the estimated regression coefficients; and e 1 •1 is the estimated error at date t.

20

Figure 6-8 illustrates this approach. The blue dots represent each of the 48 data points, with the stock's returns on the y-axis and the market's returns on the x-axis. For refer­ ence purposes, the thick black line shows the plot of market versus market. Notice that MicroDrive's returns are generally above the market's returns (the black line) when the market is doing well but below the market when the market is doing poorly, suggesting that MicroDrive is risky.

We used the Trendline feature in Excel to show the regression equation and R2 on the chart (these are colored red): MicroDrive has an estimated beta of 1.33, the same as we calculated earlier using Equation 6-8. It is also possible to use Excel's SLOPE function to estimate the slope from a regression: =SLOPE(known_y's,known_x's). The SLOPE function is more convenient if you are going to calculate betas for many different compa­ nies; see Ch06 Tool Kit.xlsx for more details.

6-Gc Interpreting the Estimated Beta

First, always keep in mind that beta cannot be observed; it can only be estimated. The R2 value shown in the chart measures the degree of dispersion about the regression line. Statistically speaking, it measures the proportion of the variation that is explained by the regression equation. An R2 of 1.0 indicates that all points lie exactly on the regression line and hence that all of the variations in the y-variable are explained by the x-variable. Micro­ Drive's R2 is about 0.26, which is a little less than a typical stock's R2 of 0.32. This indicates that about 26% of the variation in MicroDrive's returns is explained by the market return; in other words, much of MicroDrive's volatility is due to factors other than market gyra­ tions. If we had done a similar analysis for a portfolio of 40 randomly selected stocks, then the points would probably have been clustered tightly around the regression line, and the R2 probably would have exceeded 0.90. Almost 100% of a well-diversified portfolio's vola­ tility is explained by the market.

Ch06 Tool Kit.xlsx demonstrates how to use the Excel function LINEST to calculate the confidence interval for MicroDrive's estimated beta and shows that the 95% confi­ dence interval around MicroDrive's estimated beta ranges from about 0.7 to 2.0. This means that we can be 95% confident that MicroDrive's true beta is between 0.7 and 2.0.

20Theory suggests that the risk-free rate (r•F.,) should be subtracted from the stock return and the market return for each observation. However, the estimated coefficients are virtually identical to those estimated without subtracting the risk-free rate, so it is common practice to ignore the risk-free rate when estimating beta with this model, which is called the market model.

264 Part 3 Stocks and Options

FIGUREG-8

Stock Returns of Micro Drive and the Market: Estimating Beta

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

A B C

y-axis: Historical

Micro Drive

Returns

35% •

• •

•

•

•

•

-35%

•

•

- •

D

•

•

•

•

E

y = 1.33x - 0.002 R2 = 0.2612

35%

x-axis: Historical

Market Returns

F

Market vs. ►

Market

•

•

•

-35%

Notice that this is a fairly big range, which is also typical for most stocks. In other words, the estimated beta truly is an estimate!

MicroDrive's estimated beta is about 1.33. What does that mean? By definition, the average beta for all stocks is equal to l, so MicroDrive contributes 33% more risk to a well-diversified portfolio than does a typical stock (assuming they have the same portfo­ lio weight). Notice also from Figure 6-8 that the slope of the estimated line is about 1.33, which is steeper than a slope of 1. When the market is doing well, a high-beta stock like MicroDrive tends to do better than an average stock, and when the market does poorly, a high-beta stock also does worse than an average stock. The opposite is true for a low-beta stock: When the market soars, the low-beta stock tends to go up by a smaller amount; when the market falls, the low-beta stock tends to fall less than the market.

Finally, observe that the intercept shown in the regression equation on the chart is -0.002. This is a monthly return; the annualized value is 12(-0.2%) = -2.4%. This in­ dicates that MicroDrive lost about 2.4% per year as a result of factors other than general market movements.

Chapter 6 Risk and Return 265

The Benefits of Diversifying Overseas

Figure 6-6 shows that an investor can significantly re­

duce portfolio risk by holding a large number of stocks.

The figure accompanying this box suggests that inves­

tors may be able to reduce risk even further by holding

stocks from all around the world, because the returns

Portfolio Risk, a P (%)

on domestic and international stocks are not perfectly

correlated.

Source: See Kenneth Kasa, "Measuring the Gains from International Portfo­

lio Diversification," Federal Reserve Bonk of Son Francisco Weekly Letter, No.

94-14 (April 8, 1994).

U.S. Stocks

U.S. and International Stocks

Number of Stocks in Portfolio

For more on calculating beta, take a look at Ch06 Tool Kit.xlsx, which shows how to download data for an actual company and calculate its beta.

SELF-TEST

What is the average beta? If a stock has a beta of 0.8, what does that imply about its risk relative to the market?

Why is beta the theoretically correct measure of a stock's risk?

If you plotted the returns of a particular stock versus those on the S&P 500 Index over the past 5 years, what would the slope of the regression line tell you about the stock's market risk?

What types of data are needed to calculate a beta coefficient for an actual company?

What does the R2 measure? What is the R2 for a typical company?

An investor has a three-stock portfolio with $25,000 invested in Apple, $50,000 invested in Ford, and $25,000 invested in Walmart. Apple's beta is estimated to be 1.20, Ford's beta is estimated to be 0.80, and Walmart's beta is estimated to be 1.0. What is the estimated beta of the investor's portfolio? (0.95)

6-7 The Relationship between Risk and Return in the Capital Asset Pricing Model

In the preceding section, we saw that beta measures a stock's contribution to the risk of a well-diversified portfolio. The CAPM assumes that the marginal investors (i.e., the inves­ tors with enough cash to move market prices) hold well-diversified portfolios. Therefore,

266 Part 3 Stocks and Options

beta is the proper measure of a stock's relevant risk. However, we need to quantify howrisk affects a stock's required rate of return: For a given level of risk as measured by beta,what rate of expected return do investors require to compensate them for bearing thatrisk? To begin, we define the following terms: ;; = Expected rate of return on Stock i. r

1 = Required rate of return on Stock i. This is the minimum expected returnthat is required to induce an average investor to purchase the stock.

r = Realized, after-the-fact return. r

RF = Risk-free rate of return. In this context, r

RF is generally measured by theexpected return on long-term U.S. Treasury bonds.

b 1

= Beta coefficient of Stock i. r

M = Required rate of return on a portfolio consisting of all stocks, which iscalled the market portfolio.

RP M

= Risk premium on "the market." RP M

= (r M

- r RF

) is the additional return over the risk-free rate required to induce an average investor to invest in themarket portfolio. RP

1 = Risk premium on Stock i: RP

1 = b

1 (RP

M ).

6-7a The Security Market Line (SML)

In general, we can conceptualize the required return on Stock i (r 1 ) as the risk-free rateplus the extra return (i.e., the risk premium) needed to induce the investor to hold thestock. The CAPM's Security Market Line (SML) formalizes this general concept by show­ing that a stock's risk premium is equal to the product of the stock's beta and the marketrisk premium:

R . d S k . Ri k fr +(Risk premium)eqmre return on toe 1 = s - ee rate for Stock i . . . (Beta of) (Market risk)Reqmred return on Stock 1 = Risk-free rate + St k . . oc 1 premium

r 1 = r

RF + b1(RPM) = r

RF + b;(rM - rRF)

Let's take a look at the three components of required return (the risk-free rate, themarket risk premium, and beta) to see how they interact in determining a stock's requiredreturn. THE RISK-FREE RATE

Notice that a stock's required return begins with the risk-free rate. To induce an investorto take on a risky investment, the investor will need a return that is at least as big as therisk-free rate. The yield on long-term Treasury bonds is often used to measure the risk-freerate when estimating the required return with the CAPM.

Chapter 6 Risk and Return

THE MARKET RISK PREMIUM

267

The market risk premium (RP M

) is the extra rate of return that investors require to invest in the stock market rather than purchase risk-free securities. It is also called the equity premium or the equity risk premium.

The size of the market risk premium depends on the degree of risk aversion that in­ vestors have on average. When investors as a whole are very risk averse, the market risk premium is high; when investors are less concerned about risk, the market risk premium is low. For example, suppose that investors (on average) need an extra return of 5% before they will take on the stock market's risk. If Treasury bonds yield r

RF = 6%, then the re­

quired return on the market, r M

, is 11%:

r M

= r RF +RPM=

6% + 5% = 11%

If we had instead begun with an estimate of the required market return (perhaps through scenario analysis similar to the example in Section 6-2), then we can find the implied market risk premium. For example, if the required market return is estimated as 11 %, then the market risk premium is:

RP M

= r M

- r RF

= 11% - 6% = 5%

We discuss the market risk premium in detail in Chapter 9, but for now you should know that most analysts use a market risk premium in the range of 4% to 7%.

THE RISK PREMIUM FOR AN INDIVIDUAL STOCK

The CAPM shows that the risk premium for an individual stock (RP) is equal to the product of the stock's beta and the market risk premium:

Risk premium for Stock i = RP; = b 1 (RP

M ) ■

For example, consider a low-risk stock with b l o w

= 0.5. If the market risk premium is 5%, then the risk premium for the stock (RP

L o) is 2.5%:

RP Low = (5%)(0.5) = 2.5%

Using the SML in Equation 6-15, the required return for our illustrative low-risk stock is then found as follows:

r Low

= 6% + 5%(0.5) = 8.5%

If a high-risk stock has b High = 2.0, then its required rate of return is 16%:

r High = 6% + (5%)2.0 = 16%

An average stock, with b Avg = 1.0, has a required return of 11 %, the same as the market return:

r = 6% + (5%)1.0 = 11% = r � M

Figure 6-9 shows the SML when r RF

= 6% and RP M

= 5%. Note the following points:

1. Required rates of return are shown on the y-axis, while risk as measured by beta is shown on the x-axis. This graph is quite different from the regression line shown in Figure 6-8, where the historical returns of the same individual stock were plotted on the y-axis, and historical returns of the market index were plotted on the x-axis.

268

FIGURE6-9

Part 3 Stocks and Options

The Security Market Line (SML)

Required Rate of �eturn {%)

rl = 8.5

SML: r; = rRF + (RPM) b; = 6%+ (5%)b;

Aver-age-Risk Stock's Risk Premium: 5%. This is also the ma,ket risk premium.

High-Risk Stock's Risk Premium: 10%

Risk-Free Rate, rRF

0 0.5 1.0 1.5 2.0 Risk, b;

For the SML in Figure 6-9, the required (or expected) returns of different stocks (or portfolios) are plotted on the y-axis, and the betas of different stocks (or portfolios) are plotted on the x-axis. In other words, the betas of different stocks are estimated in Figure 6-8 but are used in Figure 6-9 to determine the required returns.

2. Riskless securities have b i = O; therefore, r

RF is the y-axis intercept in Figure 6-9. If

we could construct a portfolio that had a beta of zero, then it would have a required return equal to the risk-free rate.

3. The slope of the SML (5% in Figure 6-9) reflects the degree of risk aversion in the economy: The greater the average investor's aversion to risk, then (a) the steeper the slope of the line, (b) the greater the risk premium for all stocks, and (c) the higher the required rate of return on all stocks. 21

6-7b The Impact on Required Return due to Changes in the Risk-Free Rate, Risk Aversion, and Beta

The required return depends on the risk-free rate, the market risk premium, and the stock's beta. The following sections illustrate the impact of changes in these inputs.

21S tudents sometimes confuse beta with the slope of the SML. The slope of any straight line is equal to the "rise" divided by the "run," or (Y, - Y

0 )/(X

1 - X.). Consider Figure 6-9. If we Jet Y = r and X = beta, and if we

go from the origin to b = 1.0, then we see that the slope is (r .,

- r RF)/(b., - bRF) = (11% - 6%)/(1 - O) = 5%.

Thus, the slope of the SML is equal to (r .,

- r RF

), the market risk premium. In Figure 6-9, r 1 = 6% + 5%(b

1 ), so

an increase of beta from 1.0 to 2.0 would produce a 5-percentage-point increase in r,.

Chapter 6 Risk and Return 269

THE IMPACT OF CHANGES IN THE RISK-FREE RATE

Suppose that some combination of an increase in real interest rates and in anticipated inflation causes the risk-free interest rate to increase from 6% to 8%. However, real interest rates and inflation don't necessarily affect investors' risk aversion, which is the primary determinant of the market risk premium. Therefore, a change in r

RF will not necessarily cause a change in the market risk premium. If there is no change in the market risk pre­ mium, then the CAPM shows that an increase in r

RF leads to an identical increase in the

required rate of return on an asset. This applies to all assets, so it also applies to the mar­ ket return. For this example, the risk-free rate increases by 2 percentage points. Therefore, the required rate of return on the market portfolio, r

M ' increases from 11 % to 13%. 22 Other

risky securities' required returns also rise by 2 percentage points. Figure 6-10 illustrates these points.

CHANGES IN RISK AVERSION

The slope of the Security Market Line reflects the extent to which investors are averse to risk: The greater the average investor's aversion to risk, the steeper the slope

FIGURE 6-10

Shift in the SML Caused by an Increase in the Risk-Free Rate

Required Rate of Return {%)

I

f-�,i,=;..--- Increase in Risk-Free Interest Rate

0 0.5

I

I

I

I

I

I

I

I

1.0 1.5 2.0 Risk, b1

22Think of a sailboat floating in a harbor. The distance from the ocean floor to the ocean surface is like the risk­ free rate, and it moves up and down with the tides. The distance from the top of the ship's mast to the ocean floor is like the required market return: It too moves up and down with the tides. The distance from the mast­ top to the ocean surface is like the market risk premium: it stays the same, even though tides move the ship up and down. Thus, other things held constant, a change in the risk-free rate also causes an identical change in the required market return, r.., resulting in a relatively stable market risk premium, r

M - rRF'

270

FIGURE&-11

Shift in the SML Caused by Increased Risk Aversion

Required Rate ofReturn (%) IT25 ------------------------

rMl = 11 9.75

Part 3 Stocks and Options

SML 2 = 6% + 7.5%(b;)

SML1 "" 6% + 5%(b;)

, .. �: -------+-r= Premium, rM2 - rRF = 7.5%

I

0

Original Market Risk Premium, rM1 - rRF = 5%

0.5 1.0

I I

--------1

I

I

I

I

I

I

I

I

I

I

I

I

1.5 2.0 Risk, b;

of SML. To see this, suppose all investors were indifferent to risk-that is, suppose they were not risk averse. In this case, they would require no risk premium to in­ duce them to invest in risky assets and the SML would be plotted as a horizontal line. If investors become more averse to risk, the becomes steeper. For example, Figure 6-11 shows an increase in slope due to an increase in the market risk premium from 5% to 7.5%.

The increase in the market risk premium causes r M

to rise from r M 1

= 11% to r M2

= 13.5%. The returns on other risky assets also rise, and the effect of the increased risk aversion is greater for riskier securities. For example, the required return on a stock with b

i = 0.5 increases by only 1.25 percentage points, from 8.5% to 9.75%; that on a

stock with b i = 1.0 increases by 2 .5 percentage points, from 11.0% to 13.5%; and that on

a stock with b i = 1.5 increases by 3.75 percentage points, from 13.5% to 17.25%.

CHANGES IN A STOCK'S BETA COEFFICIENT

Given risk aversion and a positively sloped SML as in Figure 6-9, the higher a stock's beta, the higher its required rate of return. As we shall see later in the book, a firm can influence its beta through changes in the composition of its assets and also through its use of debt: Acquiring riskier assets will increase beta, as will a change in capital struc­ ture that calls for a higher debt ratio. A company's beta can also change as a result of external factors such as increased competition in its industry, the expiration of basic patents, and the like. When such changes lead to a higher or lower beta, the required rate of return will also change.

Chapter 6 Risk and Return

6-7c Portfolio Returns and Portfolio Performance Evaluation

271

The expected rate of return on a portfolio ( r P) is the weighted average of the expected re­ turns on the individual assets in the portfolio. Suppose there are n stocks in the portfolio and the expected return on Stock i is r

r The expected return on the portfolio is:

I The required rate of return on a portfolio (r ) is the weighted average of the required

p

returns on the individual assets in the portfolio:

n r

P =Lwi ri i=I I

We can also express the required return on a portfolio in terms of the portfolio's beta:

■ Equation 6-19 means that we do not have to estimate the beta for a portfolio if we have

already estimated the betas for the individual stocks. All we have to do is calculate the portfolio beta as the weighted average of the stock's betas (see Equation 6-9) and then ap­ ply Equation 6-19.

This is particularly helpful when evaluating portfolio managers. For example, suppose the stock market has a return for the year of 9% and a particular mutual fund has a 10% return. Did the portfolio manager do a good job or not? The answer depends on how much risk the fund has. If the fund's beta is 2, then the fund should have had a much higher re­ turn than the market, which means the manager did not do well. The key is to evaluate the portfolio manager's return against the return the manager should have made given the risk of the investments.

6-7d Required Returns versus Expected Returns: Market Equilibrium

We explained in Chapter 1 that managers should seek to maximize the value of their firms' stocks. We also emphasized the difference between the market price and intrinsic value. Intrinsic value incorporates all relevant available information about expected cash flows and risk. This includes information about the company, the economic environment, and the political environment. In contrast to intrinsic value, market prices are based on investors' selection and interpretation of information. To the extent that investors don't select all relevant information and don't interpret it correctly, market prices can deviate from intrinsic values. Figure 6-12 illustrates this relationship between market prices and intrinsic value.

When market prices deviate from their intrinsic values, astute investors have profit­ able opportunities. For example, we saw in Chapter 5 that the value of a bond is the present

272

FIGURE&-12

Part 3 Stocks and Options

Determinants of Intrinsic Values and Market Prices

The company, the economic

environment, and the political climate

All relevant available

infofmation

Intrinsic value

Selected information and its interpretation

Market price

value of its cash flows when discounted at the bond's required return, which reflects the bond's risk. This is the intrinsic value of the bond because it incorporates all relevant available information. Notice that the intrinsic value is "fair" in the sense that it incorpo­ rates the bond's risk and investors' required returns for bearing the risk.

What would happen if a bond's market price were lower than its intrinsic value? In this situation, an investor could purchase the bond and receive a rate of return in excess of the required return. In other words, the investor would get more compensation than justified by the bond's risk. If all investors felt this way, then demand for the bond would soar as investors tried to purchase it, driving the bond's price up. But as the price of a bond goes up, its yield goes down. This means that an increase in price would reduce the subse­ quent return for an investor purchasing (or holding) the bond at the new price.23 It seems reasonable to expect that investors' actions would continue to drive the price up until the expected return on the bond equaled its required return. After that point, the bond would provide just enough return to compensate its owner for the bond's risk.

If the bond's price were too high compared to its intrinsic value, then investors would sell the bond, causing its price to fall and its yield to increase until its expected return equaled its required return.

A stock's future cash flows aren't as predictable as a bond's, but we show in Chapter 7 that a stock's intrinsic value is the present value of its expected future cash flows, just as a bond's intrinsic value is the present value of its cash flows. If the price of a stock is lower than its intrinsic value, then an investor would receive an expected return greater than the return required as compensation for risk. The same market forces we described for a mis­ priced bond would drive the mispriced stock's price up. If this process continues until its

23The original owner of the bond when it was priced too low would reap a nice benefit as the price climbs, but

the subsequent purchasers would only receive the now-lower yield.

C

Chapter 6 Risk and Return 273

Another Kind of Risk: The Bernie Madoff Story

In the fall of 2008, Bernard Madoff's massive Ponzi scheme

was exposed, revealing an important type of risk that's not

dealt with in this chapter. Madoff was a money manager in

the 1960s, and apparently through good luck he produced

above-average results for several years. Madoff's clients

then told their friends about his success, and those friends

sent in money for him to invest. Madoff's actual returns then

dropped, but he didn't tell his clients that they were losing

money. Rather, he told them that returns were holding up

well, and he used new incoming money to pay dividends and

meet withdrawal requests. The idea of using new money to

pay off old investors is called a Ponzi scheme, named after

Charles Ponzi, a Bostonian who set up the first widely publi­

cized such scheme in the early 1900s.

Madoff perfected the system, ran his scheme for about

40 years, and attracted about $50 billion of investors' funds.

His investors ranged from well-known billionaires to retir­

ees who invested their entire life savings. His advertising

was strictly by word of mouth, and clients telling potential

clients about the many wealthy and highly regarded people

who invested with him certainly helped. All of his investors

assumed that someone else had done the "due diligence"

and found the operation to be clean. A few investors who

actually did some due diligence were suspicious and didn't

invest with him, but for the most part, people just blindly

followed the others.

All Ponzi schemes crash when something occurs that

causes some investors to seek to withdraw funds in amounts

greater than the incoming funds from new investors. Some­

one tries to get out, can't do it, tells others who worry and

try to get out too, and almost overnight the scam unravels.

That happened to Madoff in 2008, when the stock market

crash caused some of his investors to seek withdrawals and

few new dollars were coming in. In the end, his investors lost

billions; some lost their entire life savings, and several have

committed suicide.

expected return equals its required return, then we say that there is market equilibrium (which is also called just equilibrium):

Market equilibrium: Expected return = Required return r=r

We can also express market equilibrium in terms of prices:

Market equilibrium: Market price = Intrinsic value

New information about the risk-free rate, the market's degree of risk aversion, or a stock's expected cash flows (size, timing, or risk) will cause a stock's price to change. But other than periods in which prices are adjusting to new information, is the market usually in equilibrium? We address that question in the next section.

SELF-TEST

Differentiate among the expected rate of return (P), the required rate of return (r), and the real­ ized, after-the-fact return (f) on a stock. Which must be larger to get you to buy the stock, P or r? Would P, r, and f typically be the same or different for a given company at a given point in time?

What are the differences between the relative returns graph (the regression line in Figure 6-8), where "betas are made," and the SML graph (Figure 6-9), where "betas are used"? Discuss how the graphs are constructed and the information they convey.

What happens to the SML graph in Figure 6-9 when inflation increases or decreases?

What happens to the SML graph when risk aversion increases or decreases? What would the SML look like if investors were completely indifferent to risk-that is, had zero risk aversion?

How can a firm's managers influence market risk as reflected in beta?

A stock has a beta of 0.8. Assume that the risk-free rate is 5.5% and that the market risk premium is 6%. What is the stock's required rate of return? {10.3%)

274

6-8 The Efficient Markets Hypothesis

Part 3 Stocks and Options

The Efficient Markets Hypothesis (EMH) asserts: (1) Stocks are always in equilibrium. (2) It is impossible for an investor to "beat the market" and consistently earn a higher rate of return than is justified by the stock's risk. In other words, a stock's market price is al­ ways equal to its intrinsic value. To put it a little more precisely, suppose a stock's market price is equal to the stock's intrinsic value, but new information that changes the stock's intrinsic value arrives. The EMH asserts that the market price will adjust to the new in­ trinsic value so quickly that there isn't time for an investor to receive the new information, evaluate the information, take a position in the stock before the market price changes, and then profit from the subsequent change in price.

Here are three points to consider. First, almost every stock is under considerable scru­ tiny. With 100,000 or so full-time, highly trained, professional analysts and traders, each following about 30 of the roughly 3,000 actively traded stocks (analysts tend to specialize in a specific industry), there are an average of about 1,000 analysts following each stock. Second, financial institutions, pension funds, money management firms, and hedge funds have billions of dollars available for portfolio managers to use in taking advantage of mis­ priced stocks. Third, SEC disclosure requirements and electronic information networks cause new information about a stock to become available to all analysts virtually simul­ taneously and almost immediately. With so many analysts trying to take advantage of temporary mispricing due to new information, with so much money chasing the profits due to temporary mispricing, and with such widespread dispersal of information, a stock's market price should adjust quickly from its pre-news intrinsic value to its post-news in­ trinsic value, leaving only a very short amount of time that the stock is "mispriced" as it moves from one equilibrium price to another. That, in a nutshell, is the logic behind the Efficient Markets Hypothesis.

The following sections discuss forms of the Efficient Markets Hypothesis and empiri­ cal tests of the hypothesis.

6-Sa Forms of the Efficient Markets Hypothesis

There are three forms of the Efficient Markets Hypothesis, and each focuses on a different type of information availability.

WEAK-FORM EFFICIENCY

The weak form of the EMH asserts that all information contained in past price move­ ments is fully reflected in current market prices. If this were true, then information about recent trends in stock prices would be of no use in selecting stocks; the fact that a stock has risen for the past three days, for example, would give us no useful clues as to what it will do today or tomorrow. In contrast, technical analysts, also called "chartists," believe that past trends or patterns in stock prices can be used to predict future stock prices.

To illustrate the arguments supporting weak-form efficiency, suppose that after study­ ing the past history of the stock market, a technical analyst identifies the following his­ torical pattern: If a stock has fallen for three consecutive days, its price rose by 10% (on average) the following day. The technician would then conclude that investors could make money by purchasing a stock whose price has fallen three consecutive days.

Weak-form advocates argue that if this pattern truly existed, then surely other inves­ tors would soon discover it, and if so, why would anyone be willing to sell a stock after it had fallen for three consecutive days? In other words, why sell if you know that the price is going to increase by 10% the next day? For example, suppose a stock had fallen three con­ secutive days to $40. If the stock were really likely to rise by 10% to $44 tomorrow, then its

0

Chapter 6 Risk and Return 275

price today, right now, would actually rise to somewhere close to $44, thereby eliminating the trading opportunity. Consequently, weak-form efficiency implies that any informa­ tion that comes from past stock prices is too rapidly incorporated into the current stock price for a profit opportunity to exist.

SEMISTRONG-FORM EFFICIENCY

The semistrong form of the EMH states that current market prices reflect all publicly available information. Therefore, if semistrong-form efficiency exists, it would do no good to pore over annual reports or other published data because market prices would have ad­ justed to any good or bad news contained in such reports back when the news came out. With semistrong-form efficiency, investors should expect to earn returns commensurate with risk, but they should not expect to do any better or worse other than by chance.

Another implication of semistrong-form efficiency is that whenever information is released to the public, stock prices will respond only if the information is different from what had been expected. For example, if a company announces a 30% increase in earnings and if that increase is about what analysts had been expecting, then the announcement should have little or no effect on the company's stock price. On the other hand, the stock price would probably fall if analysts had expected earnings to increase by more than 30%, but it probably would rise if they had expected a smaller increase.

STRONG-FORM EFFICIENCY

The strong form of the EMH states that current market prices reflect all pertinent in­ formation, whether publicly available or privately held. If this form holds, even insiders would find it impossible to earn consistently abnormal returns in the stock market.

6-Sb Is the Stock Market Efficient? The Empirical Evidence

Empirical studies are joint tests of the EMH and an asset pricing model, such as the CAPM. They are joint tests in the sense that they examine whether a particular strategy can beat the market, where "beating the market" means earning a return higher than that predicted by the particular asset pricing model. Before addressing tests of the particular forms of the EMH, let's take a look at market bubbles.

MARKET BUBBLES

The history of finance is marked by numerous instances in which the following string of events occurs: (1) Prices climb rapidly to heights that would have been considered ex­ tremely unlikely before the run-up. (2) The volume of trading is much higher than past volume. (3) Many new investors (or speculators?) eagerly enter the market. (4) Prices suddenly fall precipitously, leaving many of the new investors with huge losses. These instances are called market bubbles.

The stock market bubbles that burst in 2000 and 2008 suggest that, at the height of these booms, the stocks of many companies-especially in the technology sector in 2000 and the financial sector in 2008-vastly exceeded their intrinsic values, which should not happen if markets are always efficient. Two questions arise. First, how are bubbles formed? Behavioral finance, which we discuss in Section 6-10, provides some possible answers. Second, why do bubbles persist when it is possible to make a fortune when they burst? For example, hedge fund manager Mark Spitznagel reputedly made billions for his Universa funds by betting against the market in 2008. The logic underlying market equilibrium suggests that everyone would bet against an overvalued market and that their actions

276 Part 3 Stocks and Options

would cause market prices to fall back to intrinsic values fairly quickly. To understand why this doesn't happen, let's examine the strategies for profiting from a falling market: (1) Sell stocks (or the market index itself) short. (2) Purchase a put option or write a call option. (3) Take a short position in a futures contract on the market index. Following is an explanation for how these strategies work (or fail).

Loosely speaking, selling a stock short means that you borrow a share from a broker and sell it. You get the cash (subject to collateral requirements required by the broker), but you owe a share of stock. For example, suppose you sell a share of Google short at a current price of $500. If the price falls to $400, you can buy a share of the stock at the now-lower $400 market price and return the share to the broker, pocketing the $100 difference be­ tween the higher price ($500) when you went short and the lmyer price ($400) when you closed the position. Of course, if the price goes up, say to $550, you lose $50 because you must replace the share you borrowed (at $500} with one that is now more costly ($550). Even if your broker doesn't require you to close out your position when the price goes up, your broker certainly will require that you put in more collateral.

A put option gives you the option to sell a share at a fixed strike price. For example, suppose you buy a put on Google for $60 with a strike price of $500. If the stock price falls below the strike price, say to $400, you can buy a share at the low price ($400} and sell it at the higher strike price ($500), making a net $40 profit from the decline in the stock price: $40 = -$60 - $400 + $500. However, if the put expires before the stock price falls below the strike price, you lose the $60 you spent buying the put. You can also use call options to bet on a decline. For example, if you write a call option, you receive cash in return for an obligation to sell a share at the strike price. Suppose you write a call option on Google with a strike price of $500 and receive $70. If Google's price stays below the $500 strike price, you keep the $70 cash you received from writing the call. But if Google goes up to $600 and the call you wrote is exercised, you must buy a share at the new high price ($600} and sell it at the lower strike price ($500), for a net loss of $30: $70 - $600 + $500 = -$30. 24

With a short position in a futures contract on the market index (or a particular stock), you are obligated to sell a share at a fixed price. If the market price falls below the specified price in the futures contract, you make money because you can buy a share in the market and sell it at the higher price specified in the futures contract. But if the market price in­ creases, you lose money because you must buy a share at the now higher price and sell it at the price fixed in the futures contract. 25

Each of these strategies allows an investor to make a lot of money. And if all investors tried to capitalize on an overvalued market, their actions would soon drive the market back to equilibrium, preventing a bubble from forming. But here is the problem with these strategies. Even if the market is overvalued, it might take months (or even years) before the market falls to its intrinsic value. During this period, an investor would have to spend a lot of cash maintaining the strategies described earlier, including margin calls, settling options, and daily marking to market for futures contracts. These negative cash flows could easily drive an investor into bankruptcy before the investor was eventually proven correct. Unfortunately, there aren't any low-risk strategies for puncturing a market bubble.

Notice that the problem of negative cash flows doesn't exist for the opposite situation of an undervalued market in which the intrinsic value is greater than the market price. Investors can simply buy stock at the too-low market price and hold it until the market price eventually increases to the intrinsic value. Even if the market price continues to go

"Options are usually settled by cash rather than by actually buying and selling shares of stock.

25Futures contracts are actually settled daily and that they are usually settled for cash rather than the actual shares.

C

Chapter 6 Risk and Return 277

down before eventually rising, the investor experiences only paper losses and not actual negative cash flows. Thus, we would not expect "negative" bubbles to persist very long.

TESTS OF WEAK-FORM EFFICIENCY

Most studies suggest that the stock market is highly efficient in the weak form, with two exceptions. The first exception is for long-term reversals, with studies show­ ing that portfolios of stocks with poor past long-term performance (over the past 5 years, for example) tend to do slightly better in the long-term future than the CAPM predicts, and vice versa. The second is momentum, with studies showing that stocks with strong performance in the short-term past (over the past 6 to 9 months, for example) tend to do slightly better in the short-term future than the CAPM predicts, and likewise for weak performance. 26 Strategies based on taking advantage of long-term reversals or short­ term momentum produce returns that are in excess of those predicted by the CAPM. However, the excess returns are small, especially when transaction costs are considered.

TESTS OF SEMISTRONG-FORM EFFICIENCY

Most studies show that markets are reasonably efficient in the semistrong form: It is diffi­ cult to use publicly available information to create a trading strategy that consistently has returns greater than those predicted by the CAPM. In fact, the professionals who manage mutual fund portfolios, on average, do not outperform the overall stock market as mea­ sured by an index like the S&P 500 and tend to have returns lower than predicted by the CAPM, possibly because many mutual funds have high fees. 27

However, there are two well-known exceptions to semistrong-form efficiency. The first is for small companies, which have had historical returns greater than predicted by the CAPM. The second is related to book-to-market (B/M) ratios, defined as the book value of equity divided by the market value of equity (this is the inverse of the market/book ratio defined in Chapter 3). Companies with high B/M ratios have had higher returns than predicted by the CAPM. We discuss these exceptions in more detail in Section 6-9.

TESTS OF STRONG-FORM EFFICIENCY

The evidence suggests that the strong-form EMH does not hold because those who pos­ sessed inside information could make and have (illegally) made abnormal profits. On the other hand, many insiders have gone to jail, so perhaps there is indeed a trade-off between risk and return!

SELF-TEST

What is the Efficient Markets Hypothesis (EMH)?

What are the differences among the three forms af the EMH?

Why is it difficult ta puncture a market bubble?

What violations af the EMH have been demonstrated?

What is short-term momentum? What are long-term reversals?

'6For example, see N. Jegadeesh and S. Titman, "Returns to Buying W inners and Selling Losers: Implications for Stock Market Efficiency," Journal of Finance, March 1993, pp. 69-91; and W. F. M. DeBondt and R. H. Thaler, "Does the Stock Market Overreact?" Journal of Finance, July 1985, pp. 793-808.

''For a discussion of the performance of actively managed funds, see Jonathan Clements, "Resisting the Lure of Managed Funds," The Wall Street Journal, February 27, 2001, p. Cl.

278

TABLE 6-2

Part 3 Stocks and Options

6-9 The Fama-French Three-Factor Model28

Take a look at Table 6-2, which reports the returns for 25 portfolios formed by Profes­ sors Eugene Fama and Kenneth French. The Fama-French portfolios are based on the company's size as measured by the market value of its equity (MVE) and the company's book-to -market (B/M) ratio, defined as the book value of equity divided by the market value of equity. Each row shows portfolios with similarly sized companies; each column shows portfolios whose companies have similar B/M ratios. Notice that if you look across each row, the average return tends to increase as the B/M ratio increases. In other words, stocks with high B/M ratios have higher returns. If you look up each column (except for the column with the lowest B/M ratios), stock returns tend to increase: Small companies have higher returns.

This pattern alone would not be a challenge to the CAPM if small firms and high-B/M firms had large betas (and thus higher returns). However, even after adjusting for their betas, the small-stock portfolios and the high B/M portfolios earned returns higher than predicted by the CAPM. This indicates that either markets are inefficient or the CAPM isn't the correct model to describe required returns.

In 1992, Fama and French published a study hypothesizing that the SML should have three factors rather than just beta as in the CAPM. 29 The first factor is the stock's CAPM beta, which measures the market risk of the stock. The second is the size of the company, measured by the market value of its equity (MVE). The third factor is the book-to-market (B/M) ratio.

When Fama and French tested their hypotheses, they found that small companies and companies with high B/M ratios had higher rates of return than the average stock, just as they hypothesized. Somewhat surprisingly, however, they found that beta was not useful in explaining returns. After taking into account the returns due to the company's size and B/M ratio, high-beta stocks did not have higher than average returns and low-beta stocks did not have lower than average returns.

Average Annual Returns for the Fama-French Portfolios Based on Size and Book Equity to Market Equity, 1927-2016

Book Equity to Market Equity

Size Low 2 3 4 High

Small 12.1% 18.6% 19.7% 23.4% 29.2%

2 12.5 16.8 17.8 18.8 19.8

3 11.9 15.7 16.1 17.3 19.2

4 12.5 13.4 15.0 16.6 17.6

Big 10.9 12.2 13.5 13.5 16.4

Source: Raw data from Professor Kenneth French, http:f/mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. Thes e are equal­

weighted annual returns. Following is a d escription from Prof essor French's Web site d escribing the construction of the portfolios: "The portfolios,

which are constructed at the end of each June, ar e the int ers ections of 5 portfolios form ed on size (market equity, ME) and 5 portfolios formed on the

ratio of book equity to market equity (BE/ME). The size breakpoints foryeart are the NYSE market equity quintiles at the end of June oft. BE/ME for

June of yeart is the book equity for the last fiscal year end in t - 1 divided by ME for December of t - 1. The BE/ME breakpoints are NYSE quintil es.

The portfolios for July of year t to June oft+ 1 includ e all NYSE, AMEX, and NASDAQ stocks for which we have market equity data for December of

t - 1 and Jun e oft, and (positive) book equity data fort - 1."

"Some instructors may choose to omit this section with no loss in continuity.

"See Eugene F. Fama and Kenneth R. French, "The Cross-Section of Expected Stock Returns," Journal of Finance, Vol. 47, 1992, pp. 427-465. In 2013, Eugene Fama was awarded the Nobel Prize in Economics for this and his other work in asset pricing.

Chapter 6 Risk and Return 279

In 1993, Fama and French developed a three-factor model based on their previous results.Jo The first factor in the Fama-French three-factor model is the market risk pre­ mium, which is the market return, r

M , minus the risk-free rate, r

RF " Thus, their model

begins like the CAPM, but they go on to add a second and third factor.J1 To form the sec­ ond factor, they ranked all actively traded stocks by size and then divided them into two portfolios, one consisting of small stocks and one consisting of big stocks. They calculated the return on each of these two portfolios and created a third portfolio by subtracting the return on the big portfolio from that of the small one. They called this the SMB (small minus big) portfolio. This portfolio is designed to measure the variation in stock returns that is caused by the size effect.

To form the third factor, they ranked all stocks according to their book-to-market (B/M) ratios. They placed the 30% of stocks with the highest ratios into a portfolio they called the H portfolio (for high B/M ratios) and placed the 30% of stocks with the lowest ratios into a portfolio called the L portfolio (for low B/M ratios). Then they subtracted the return of the L portfolio from that of the H portfolio to derive the HML (high minus low) portfolio. Before showing their model, here are the definitions of the variables.

r i,t = Historical (realized) rate of return on Stock i in period t.

r RF.t = Historical (realized) rate ofreturn on the risk-free rate in period t.

r M,t = Historical (realized) rate ofreturn on the market in period t.

r 5MB,t = Historical (realized) rate of return on the small-size portfolio minus

the big-size portfolio in period t.

r HML.t

= Historical (realized) rate of return on the high-B/M portfolio minus the low-B/M portfolio in period t.

a 1

= Vertical axis intercept term for Stock i.

b;, c 1 , and d

1 = Slope coefficients for Stock i.

e 1 ., = Random error, reflecting the difference between the actual return on

Stock i in period t and the return as predicted by the regression line.

Their resulting model is shown here:

■ When this model is applied to actual stock returns, the "extra" return disappears for

portfolios based on a company's size or B/M ratio. In fact, the extra returns for the long­ term stock reversals that we discussed in Section 6-8 also disappear. Thus, the Fama­ French model accounts for many of the major violations of the EMH that we described earlier.

Because the Fama-French model explains so well a stock's actual return given the re­ turn on the market, the SMB portfolio, and the HML portfolio, the model is very useful

'0See Eugene F. Fama and Kenneth R. French, "Common Risk Factors in the Returns on Stocks and Bonds," Journal of Financial Economics, Vol. 33, 1993, pp. 3-56.

11 Although our description captures the essence of their process for forming factors, the actual procedure is a little more complicated. The interested reader should see their 1993 paper, cited in footnote 29.

280 Part 3 Stocks and Options

in identifying the market's reaction to news about a company. 32 For example, suppose a company announces that it is going to include more outsiders on its board of directors. If the company's stock falls by 2% on the day of the announcement, does that mean investors don't want outsiders on the board? We can answer that question by using the Fama-French model to decompose the actual return of the company on the announcement day into the portion that is explained by the environment (i.e., the market and the SMB and HML portfolios) and the portion due to the company's announcement.

To do this, we gather a sample of data (i\.,, rRF.t• rM,t• rSMB,t> and rHML.,) for T periods prior to the announcement date and then run a regression using a variation of Equation 2-20:

ri,I = ai + blM,I + clsMB,I + dlHML,t + ei,I This is similar to the regression approach for estimating beta, described in Section 6-6,

except there are three slope coefficients in this multiple regression. 33 Suppose the estimated coefficients are a

i = 0.0, b

i = 0.9, c

i = 0.2, and d

1 = 0.3. On the

day of the announcement, the stock market had a return of -3%, the SMB portfolio had a return of -1%, and the HML portfolio had a return of -2%. The predicted value of the error term in the Fama-French model, e. , is by definition equal to zero. Based on these as-

,., sumptions, the predicted return on the announcement day using the Fama-French three- factor model is:

Predicted return = a l + b.(rM ) + c.(rSMB ) + d.(r )I ,I I ,I I HML,t

= 0.0 + 0.9(-3%) + 0.2(-1%) + 0.3(-2%) = -3.5%

The unexplained return is equal to the actual return less the predicted return:

Unexplained return = -2.0% - (-3.5%) = 1.5%

Although the stock price went down by 2% on the announcement day, the Fama­ French model predicted that the price should have gone down by 3.5% due to market movements. Thus, the stock had a positive 1.5% reaction on the announcement day. This is just one company, but if we repeated this process for many companies that made similar announcements and calculated the average unexplained reaction, we could draw a conclusion regarding the market's reaction to adding more outside di­ rectors. As this example shows, the model is very useful in identifying actions that affect a company's value.

There is no question that the Fama-French three-factor model does a good job in ex­ plaining actual returns, but how well does it perform in explaining required returns? In other words, does the model define a relationship between risk and compensation for bearing risk?

Advocates of the model suggest that size and B/M are related to risk. Small companies have less access to capital markets than do large companies, which exposes small compa­ nies to greater risk in the event of a credit crunch-such as the one that occurred during

"Because the Fama-French model doesn't seem to explain short-term momentum, many researchers also use the four-factor model, which includes a factor for momentum; see Mark Carhart, "On Persistence in Mutual Fund Performance," Journal of Finance, Vol. 52, No. I., March 1997, pp. 57-82.

"Notice that the risk-free rate is not used in the regression for the same reasons it is not used in the market model to estimate beta; see footnote 20. Also, the annual risk-free rate is 6% in this example, so the daily rate is 6%/365 = 0.01%, which is so small that it can be ignored.

Chapter 6 Risk and Return 281

the global economic crisis that began in 2007. With greater risk , investors would require a higher expected return to induce them to invest in small companies.

Similar arguments apply for companies with high B/M ratios. If a company's prospects are poor, then the company will have a low market value, which causes a high B/M ratio. Lenders usually are reluctant to extend credit to a company with poor prospects, so an economic downturn can cause such a company to experience financial distress. In other words, a stock with a high B/M ratio might be exposed to the risk of financial distress, in which case investors would require a higher expected return to induce them to invest in such a stock.

If a company's sensitivity to the size factor and the B/M factor are related to financial distress risk, then the Fama-French model would be an improvement on the CAPM regarding the relationship between risk and required return. However, the evidence is mixed as to whether financially distressed firms do indeed have higher expected returns as compensation for their risk. In fact, some studies show financially distressed firms actually have lower expected returns instead of higher returns.34

A number of other studies suggest that the size effect no longer influences stock re­ turns, that there never was a size effect (the previous results were caused by peculiarities in the data sources), that the size effect doesn't apply to most companies, and that the book­ to-market effect is not as significant as first supposed.35

In summary, the Fama-French model is very useful in identifying the unexplained component of a stock's return. However, the model is less useful when it comes to estimat­ ing the required return on a stock because the model does not provide a well-accepted link between risk and required return.

SELF-TEST

What are the factors in the Fama-French model?

How can the model be used to estimate the predicted return on a stock?

Why isn't the model widely used by managers at actual companies?

An analyst has modeled the stock of a company using a Fama-French three-factor model and has estimated that a

i = 0, b

i = 0. 7, c

i = 1.2, and d

i = 0. 7. Suppose that the daily risk-free rate is

approximately equal to zero, the market return is 11%, the return on the SMB portfolio is 3.2%, and the return on the HML portfolio is 4.8% on a particular day. The stock had an actual return of

16.9% on that day. What is the stock's predicted return for that day? (14.9%) What is the stock's unexplained return for the day? (2%)

"For studies supporting the relationship between risk and return as related to size and the B/M ratio, see Nishad Kapadia, "Tracking Down Distress Risk," Journal of Financial Economics, Vol. 102, 2011, pp. 167-182;

Thomas J. George, "A Resolution of the Distress Risk and Leverage Puzzles in the Cross Section of Stock Returns," Journal of Financial Economics, Vol. 96, 2010, pp. 56-79; and Lorenzo Garlappi and Hong Yan, "Financial Distress and the Cross-Section of Equity Returns," Journal of Finance, June, 2011, pp. 789-822. For studies rejecting the relationship, see John Y. Campbell, Jens Hilscher, and Jan Szilagyi, "In Search of Distress Risk," Journal of Finance, December 2008, pp. 2899-2940; and Ilia D. Dichev, "Is the Risk of Bankruptcy a Systematic Risk?" Journal of Finance, June 1998, pp. 1131-1147.

35See Peter J. Knez and Mark J . Ready, "On the Robustness of Size and Book-to-Market in the Cross­ Sectional Regressions," Journal of Finance, September 1997, pp. 1355-1382; Dongcheol Kim, "A Reex­ amination of Firm Size, Book-to-Market, and Earnings Price in the Cross-Section of Expected Stock Returns," Journal of Financial and Quantitative Analysis, December 1997, pp. 463-489; Tyler Shumway and Vincent A. Warther, "The Delisting Bias in CRSP's Nasdaq Data and Its Implications for the Size Effect," Journal of Finance, December 1999, pp. 2361-2379; and Tim Loughran, "Book-to-Market across Firm Size, Exchange, and Seasonality: Is There an Effect?" Journal of Financial and Quantitative Analysis, September 1997, pp. 249-268.

282 Part 3 Stocks and Options

6-10 Behavioral Finance36

A large body of evidence in the field of psychology shows that people often behave irra­ tionally, but in predictable ways. The field of behavioral finance focuses on irrational but predictable financial decisions. The following sections examine applications of behavioral finance to market bubbles and to other financial decisions.

6-10a Market Bubbles and Behavioral Finance

We showed in Section 6-8 that strategies for profiting from a punctured bubble expose an investor to possible large negative cash flows if it takes a long time for the bubble to burst. That explains why a bubble can persist, but it doesn't explain how a bubble is created. There are no definitive explanations, but the field of behavioral finance offers some pos­ sible reasons, including overconfidence, anchoring bias, and herding.

Many psychological tests show that people are overconfident with respect to their own abilities relative to the abilities of others, which is the basis of Garrison Keillor's joke about a town where all the children are above average. Professor Richard Thaler and his col­ league Nicholas Barberis address this phenomenon as it applies to finance:

Overconfidence may in part stem from two other biases, self-attribution bias and hind- sight bias. Self-attribution bias refers to people's tendency to ascribe any success they have in some activity to their own talents, while blaming failure on bad luck, rather than on their ineptitude. Doing this repeatedly will lead people to the pleasing but erroneous conclusion that they are very talented. For example, investors might become overconfi- dent after several quarters of investing success [Gervais and Odean (2001)37]. Hindsight bias is the tendency of people to believe, after an event has occurred, that they predicted it before it happened. If people think they predicted the past better than they actually did, they may also believe that they can predict the future better than they actually can.38

Psychologists have learned that many people focus too closely on recent events when predicting future events, a phenomenon called anchoring bias. Therefore, when the mar­ ket is performing better than average, people tend to think it will continue to perform better than average. When anchoring bias is coupled with overconfidence, investors can become convinced that their prediction of an increasing market is correct, thus creating even more demand for stocks. This demand drives stock prices up, which serves to rein­ force the overconfidence and move the anchor even higher.

There is another way that an increasing market can reinforce itself. Studies have shown that gamblers who are ahead tend to take on more risks (i.e., they are playing with the house's money), whereas those who are behind tend to become more conservative. If this is true for investors, we can get a feedback loop: When the market goes up, investors have gains, which can make them less risk averse, which increases their demand for stock, which leads to higher prices, which starts the cycle again.

Herding behavior occurs when groups ofinvestors emulate other successful investors and chase asset classes that are doing well. For example, high returns in mortgage-backed securi­ ties during 2004 and 2005 enticed other investors to move into that asset class. Herding be­ havior can create excess demand for asset classes that have done well, causing price increases that induce additional herding behavior. Thus, herding behavior can inflate rising markets.

36Some instructors may choose to omit this section with no loss of continuity.

"See Simon Gervais and Terrance Odean, "Learning to Be Overconfident," Review of Financial Studies, Spring

2001, pp. 1-27.

"See page 1066 in an excellent review of behavioral finance by Nicholas Barberis and Richard Thaler, "A Survey of Behavioral Finance," in Handbook of the Economics of Finance, George Constantinides, Milt

Harris, and Rene Stulz, eds. (Amsterdam: Elsevier/North-Holland, 2003), Chapter 18.

Chapter 6 Risk and Return 283

Sometimes herding behavior occurs when a group of investors assumes that other investors are better informed-the herd chases the "smart" money. But in other cases, herding can occur even when those in the herd suspect that prices are overinflated. For example, consider the situation of a portfolio manager who believes that bank stocks are overvalued even though many other portfolios are heavily invested in such stocks. If the manager moves out of bank stocks and they subsequently fall in price, then the manager will be rewarded for her judgment. But if the stocks continue to do well, the manager may well lose her job for missing out on the gains. If instead the manager follows the herd and invests in bank stocks, then the manager will do no better or worse than her peers. Thus, if the penalty for being wrong is bigger than the reward for being correct, it is rational for portfolio managers to herd even if they suspect the herd is wrong.

Researchers have shown that the combination of overconfidence and biased self­ attribution can lead to overly volatile stock markets, short-term momentum, and long­ term reversals.39 We suspect that overconfidence, anchoring bias, and herding can contribute to market bubbles.

6-10b Other Applications of Behavioral Finance

Psychologists Daniel Kahneman and Amos Tversky show that individuals view potential losses and potential gains very differently. 40 If you ask an average person whether he or she would rather have $500 with certainty or flip a fair coin and receive $1,000 if it comes up heads and nothing if it comes up tails, most would prefer the certain $500 gain, which sug­ gests an aversion to risk-a sure $500 gain is better than a risky expected $500 gain. How­ ever, if you ask the same person whether he or she would rather pay $500 with certainty or flip a coin and pay $1,000 if it's heads and nothing if it's tails, most would indicate that they prefer to flip the coin, which suggests a preference for risk-a risky expected $500 loss is better than a sure $500 loss. In other words, losses are so painful that people will make irrational choices to avoid sure losses. This phenomenon is called loss aversion.

One way that people avoid a loss is by not admitting that they have actually had a loss. For example, in many people's mental bookkeeping, a loss isn't really a loss until the losing investment is actually sold. Therefore, they tend to hold risky losers instead of accepting a certain loss, which is a display of loss aversion. Of course, this leads investors to sell losers much less frequently than winners even though this is suboptimal for tax purposes.41

Many corporate projects and mergers fail to live up to their expectations. In fact, most mergers end up destroying value in the acquiring company. Because this is well known, why haven't companies responded by being more selective in their investments? There are many possible reasons, but research by Ulrike Malmendier and Geoffrey Tate suggests that overconfidence leads managers to overestimate their abilities and the quality of their projects.42 In other words, managers might know that the average decision to merge de­ stroys value, but they are certain that their decision is above average.

39See Terrance Odean, "Volume, Volatility, Price, and Profit When All Traders Are above Average," Journal of Finance, December 1998, pp. 1887-1934; and Kent Daniel, David Hirshleifer, and Avanidhar Subrahmanyam, "Investor Psychology and Security Market Under- and Overreactions," Journal of Finance, December 1998, pp. 1839-1885.

'°Daniel Kahneman and Amos Tversky, "Prospect Theory: An Analysis of Decision under Risk," Econometrica, March 1979, pp. 263-292.

"See Terrance Odean, "Are Investors Reluctant to Realize Their Losses?" Journal of Finance, October 1998, pp. 1775-1798.

42See Ulrike Malmendier and Geoffrey Tate, "CEO Overconfidence and Corporate Investment," Journal of Finance, December 2005, pp. 2661-2700.

284 Part 3 Stocks and Options

Finance is a quantitative field, but good managers in all disciplines must also under­ stand human behavior.43

SELF-TEST

What is behavioral finance?

What is anchoring bias? What is herding behavior? How can these can tribute to market bubbles?

6-11 The CAPM and Market Efficiency: Implications for Corporate Managers and Investors

A company is like a portfolio of projects: factories, retail outlets, R&D ventures, new prod­ uct lines, and the like. Each project contributes to the size, timing, and risk of the com­ pany's cash flows, which directly affect the company's intrinsic value. This means that the relevant risk and expected return of any project must be measured in terms of its effect on the stock's risk and return. Therefore, all managers must understand how stockholders view risk and required return in order to evaluate potential projects.

Stockholders should not expect to be compensated for the risk they can eliminate through diversification, only for the remaining market risk. The CAPM provides an im­ portant tool for measuring the remaining market risk and goes on to show how a stock's required return is related to the stock's market risk. It is for this reason that the CAPM is widely used to estimate the required return on a company's stock and, hence, the required returns that projects must generate to provide the stock's required return. We describe this process in more detail in Chapters 7 and 9, which cover stock valuation and the cost of capital. We apply these concepts to project analysis in Chapters 10 and 11.

Is the CAPM perfect? No. First, we cannot observe beta but must instead estimate beta. As we saw in Section 6-6, estimates of beta are not precise. Second, we saw that small stocks and stocks with high B/M ratios have returns higher than the CAPM predicts. This could mean that the CAPM is the wrong model, but there is another possible explanation. If the composition of a company's assets were changing over time with respect to the mix of physical assets and growth opportunities (involving, e.g., R&D or patents), then this would be enough to make it appear as though there were size and B/M effects. In other words, even if the returns on the individual assets conform to the CAPM, changes in the mix of assets would cause the firm's beta to change over time in such a way that the firm would appear to have size and book-to-market effects.44 Recent research supports this hypothesis.

Recall that the CAPM asserts that a stock's required return is related to its expo­ sure to systematic risk, not to its diversifiable risk. Therefore, you might expect the CAPM to do a very good job of explaining stock returns when news is announced that affects almost all companies, such as government reports regarding interest rate policy, inflation, and unemployment. Professors Savor and Wilson tested this hypoth­ esis and found an extremely strong relationship between betas and stock returns on announcement days.45

"Excellent reviews of behavioral finance are found in Advances in Behavioral Finance, Richard H. Thaler, ed. (New York: Russell Sage Foundation, 1993); and Andrei Shleifer, Inefficient Markets: An Introduction to Behavioral Finance (New York: Oxford University Press, 2000).

"See Jonathan B. Berk, Richard C. Green, and Vasant Naik, "Optimal Investment, Growth Options, and Security Returns," Journal of Finance, October 1999, pp. 1553-1608.

"See Pavel Savor and Mungo Wilson, "Asset Pricing: A Tale of Two Days," Journal of Financial Economics, Vol. 113, 2014, pp. 171-201.

Chapter 6 Risk and Return 285

Based on these results and its widespread use in practice, we will use the CAPM to estimate required returns in subsequent chapters.46

Regarding market efficiency, our understanding of the empirical evidence suggests it is very difficult, if not impossible, to beat the market by earning a return that is higher than justified by the investment's risk. This suggests that markets are reasonably efficient for most assets for most of the time. However, we believe that market bubbles do occur and that it is very difficult to implement a low-risk strategy for profiting when they burst.

SELF-TEST

Explain the following statement: "The stand-a/one risk of an individual corporate project may be

quite high, but viewed in the context of its effect on stockholders' risk, the project's true risk may be much lower."

This chapter focuses on the trade-off between risk and return. We began by discussing how to estimate risk and return for both individual assets and portfolios. In particu- lar, we differentiated between stand-alone risk and risk in a portfolio context, and we explained the benefits of diversification. We introduced the CAPM, which describes how risk affects rates of return.

• Risk can be defined as exposure to the chance of an unfavorable event. • The risk of an asset's cash flows can be considered on a stand-alone basis (each asset

all by itself) or in a portfolio context, in which the investment is combined with other assets and its risk is reduced through diversification.

• Most rational investors hold portfolios of assets, and they are more concerned with the risk of their portfolios than with the risk of individual assets.

• The expected rate of return on an investment is the mean value of its probability dis­ tribution of returns.

• The required return on a stock is the minimum expected return that is required to induce an average investor to purchase the stock.

• The greater the probability that the actual return will be far below the expected return, the greater the asset's stand-alone risk.

• The average investor is risk averse, which means that he or she must be compensated for holding risky assets. Therefore, riskier assets have higher required returns than less risky assets.

• An asset's risk has two components: (1) diversifiable risk, which can be eliminated by diversification, and (2) market risk, which cannot be eliminated by diversification.

• Market risk is measured by the standard deviation of returns on a well-diversified portfolio, one that consists of all stocks traded in the market. Such a portfolio is called the market portfolio.

• The CAPM defines the relevant risk of an individual asset as its contribution to the risk of a well-diversified portfolio. Because market risk cannot be eliminated by diver­ sification, investors must be compensated for bearing it.

• A stock's beta coefficient (b) measures how much risk a stock contributes to a well­ diversified portfolio.

"See Zhi Da, Re-Jin Guo, and Ravi Jagannathan, "CAPM for Estimating the Cost of Equity Capital: Interpreting the Empirical Evidence," Journal of Financial Economics, Vol. 103, 2012, pp. 204-220.

286 Part 3 Stocks and Options

• A stock with a beta greater than 1 has stock returns that tend to be higher than the market when the market is up but tend to be below the market when the market is down. The opposite is true for a stock with a beta less than 1.

• The beta of a portfolio is a weighted average of the betas of the individual securities in the portfolio.

• The CAPM's Security Market Line (SML) equation shows the relationship between a security's market risk and its required rate of return. The return required for any security i is equal to the risk-free rate plus the market risk premium multiplied by the security's beta: r; = r

RF + (RP

M )br

• In equilibrium, the expected rate of return on a stock must equal its required re- turn. However, a number of things can happen to cause the required rate of return to change: (1) The risk-free rate can change because of changes in either real rates or ex­ pected inflation. (2) A stock's beta can change. (3) Investors' aversion to risk can change.

• Because returns on assets in different countries are not perfectly correlated, global diversification may result in lower risk for multinational companies and globally diver­ sified portfolios.

• Market equilibrium is the condition under which the expected return on a security as seen by the marginal investor is just equal to its required return, r = r. Also, the stock's intrinsic value must be equal to its market price.

• The Efficient Markets Hypothesis (EMH) asserts: (1) Stocks are always in equilib­ rium. (2) It is impossible for an investor who does not have inside information to con­ sistently "beat the market." Therefore, according to the EMH, stocks are always fairly valued and have a required return equal to their expected return.

• The Fama-French three-factor model has one factor for the market return, a second factor for the size effect, and a third factor for the book-to-market effect.

• Behavioral finance assumes that investors can behave in predictable but irrational ways. Anchoring bias is the human tendency to "anchor" too closely on recent events when predicting future events. Herding is the tendency of investors to follow the crowd. When combined with overconfidence, anchoring and herding can contribute to market bubbles.

• Two Web extensions accompany this chapter: Web Extension 6A provides a discussion of continuous probability distributions, and Web Extension 6B shows how to calcu­ late beta with a financial calculator.

(6·1) Define the following terms, using graphs or equations to illustrate your answers where feasible.

a. Risk in general; stand-alone risk; probability distribution and its relation to risk b. Expected rate of return, r c. Continuous probability distribution d. Standard deviation, cr; variance, cr2

e. Risk aversion; realized rate of return, r f. Risk premium for Stock i, RP

1 ; market risk premium, RP

M

g. Capital Asset Pricing Model (CAPM) h. Expected return on a portfolio, r P; market portfolio

Chapter 6 Risk and Return

i. Correlation as a concept; correlation coefficient, p

j. Market risk; diversifiable risk; relevant risk k. Beta coefficient, b; average stock's beta 1. Security Market Line (SML); SML equation m. Slope of SML and its relationship to risk aversion n. Equilibrium; Efficient Markets Hypothesis (EMH); three forms of EMH o. Fama-French three-factor model p. Behavioral finance; herding; anchoring

287

(6-2) The probability distribution ofa less risky return is more peaked than that ofa riskier return. What shape would the probability distribution have for (a) completely certain returns and (b) completely uncertain returns?

(6-3) Security A has an expected return of 7%, a standard deviation of returns of 35%, a correla­ tion coefficient with the market of -0.3, and a beta coefficient of -1.5. Security B has an expected return of 12%, a standard deviation of returns of 10%, a correlation with the market of 0. 7, and a beta coefficient of 1.0. Which security is riskier? Why?

(6-4) If investors' aversion to risk increased, would the risk premium on a high-beta stock increase by more or less than that on a low-beta stock? Explain.

(6-5) If a company's beta were to double, would its expected return double?

S E L F - T E S T P R O B L E M S S O I_ U T i O f,! S S H O \IV i,! I f,! !\ P fJ E 1\I D I X A

(ST-1) Realized Rates

of Return

Stocks A and B have the following historical returns:

Vear rA fB

2015 -18% -24% 2016 44 24 2017 -22 -4 2018 22 8 2019 34 56

a. Calculate the average rate of return for each stock during the 5-year period. Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B. What would have been the realized rate of return on the portfolio in each year? What would have been the average return on the portfolio for the 5-year period?

b. Now calculate the standard deviation of returns for each stock and for the portfolio. Use Equation 6-5.

c. Looking at the annual returns data on the two stocks, would you guess that the correlation coefficient between returns on the two stocks is closer to 0.8 or to -0.8?

d. If you added more stocks at random to the portfolio, which of the following is the most accurate statement of what would happen to CJ' ?

p

1. CJ' P

would remain constant. 2. CJ'

P would decline to somewhere in the vicinity of 20%.

3. CJ' P

would decline to zero if enough stocks were included.

288

(ST-2) Beta and Required

Rate of Return

(6-1) Portfolio Beta

(6-2) Required Rate

of Return

(6-3) Required Rates

of Return (6-4)

Fama-French Three­ Factor Model

(6-5) Expected

Return: Discrete Distribution

Part 3 Stocks and Options

ECRI Corporation is a holding company with four main subsidiaries. The percentage of its business coming from each of the subsidiaries, and their respective betas, are as follows:

Subsidiary Percentage of Business Beta

Electric utility 60% 0.70 Cable company 25 0.90

Real estate 10 1.30

International/special 5 1.50 projects

a. What is the holding company's beta? b. Assume that the risk-free rate is 6% and that the market risk premium is 5%. What

is the holding company's required rate of return? c. ECRI is considering a change in its strategic focus: It will reduce its reliance on the

electric utility subsidiary so that the percentage of its business from this subsidiary will be 50%. At the same time, ECRI will increase its reliance on the international/ special projects division, and the percentage of its business from that subsidiary will rise to 15%. What will be the shareholders' required rate of return if management adopts these changes?

EASY PROBLEMS 1-4

Your investment club has only two stocks in its portfolio. $20,000 is invested in a stock with a beta of 0.7, and $35,000 is invested in a stock with a beta of 1.3. What is the portfolio's beta?

AA Corporation's stock has a beta of0.8. The risk-free rate is 4%, and the expected return on the market is 12%. What is the required rate of return on A.Ns stock?

Suppose that the risk-free rate is 5% and that the market risk premium is 7%. What is the required return on (1) the market, (2) a stock with a beta of 1.0, and (3) a stock with a beta of 1.7?

An analyst gathered daily stock returns for Feburary 1 through March 31, calculated the Fama-French factors for each day in the sample (SMB

1 and HML1), and estimated the

Fama-French regression model shown in Equation 6 -21. The estimated coefficients were a

i = 0, b

i = 1.2, c

i = -0.4, and d

i = 1.3. On April 1, the market return was 10%, the re­

turn on the SMB portfolio (r5 M

8) was 3.2%, and the return on the HML portfolio (r HML

) was 4.8%. Using the estimated model, what was the stock's predicted return for April l?

INTERMEDIATE PROBLEMS 5-10

A stock's return has the following distribution:

Demand for the Probability of This Rate of Return ifThis

Company's Products Demand Occurring Demand Occurs (%}

Weak 0.1 -50% Below average 0.2 -5 Average 0.4 16 Above average 0.2 25 Strong _Q,_l_ 60

.!.:Q.

Calculate the stock's expected return and standard deviation.

(6-6) Expected

Returns: Discrete Distribution

(6-7) Required Rate

of Return

(6-8) Required Rate

of Return

(6-9) Portfolio Beta

(6-10) Portfolio Required

Return

(6-11) Portfolio Beta

(6-12) Required Rate

of Return

Chapter 6 Risk and Return

The market and Stock J have the following probability distributions:

Probability

0.3 0.4 0.3

15% 9

18

20% 5

12

a. Calculate the expected rates of return for the market and Stock J. b. Calculate the standard deviations for the market and Stock J.

Suppose r RF

= 5%, r M

= 10%, and r A

= 12%.

a. Calculate Stock A's beta. b. If Stock A's beta were 2.0, then what would be A's new required rate of return?

289

As an equity analyst you are concerned with what will happen to the required return for Universal Toddler's stock as market conditions change. Suppose r

RF = 5%, r

M = 12%, and

bur = 1.4.

a. Under current conditions, what is rur• the required rate of return on UT stock? b. Now suppose r

RF (1) increases to 6% or (2) decreases to 4%. The market risk premium,

RP M' (i.e., the slope of the SML) remains constant. How would this affect rM and ru/

c. Now assume r RF

remains at 5% but r M

(1) increases to 14% or (2) falls to 11%. The market risk premium, RP

M ' {i.e., the slope of the SML) does not remain constant.

How would these changes affect ru/

Your retirement fund consists of a $5,000 investment in each of 15 different common stocks. The portfolio's beta is 1.20. Suppose you sell one of the stocks with a beta of 0.8 for $5,000 and use the proceeds to buy another stock whose beta is 1.6. Calculate your port­ folio's new beta.

Suppose you manage a $4 million fund that consists of four stocks with the following investments:

Stock Investment Beta

A $ 400,000 1.50 B 600,000 -0.50 C 1,000,000 1.25 D 2,000,000 0.75

If the market's required rate ofreturn is 14% and the risk-free rate is 6%, what is the fund's required rate ofreturn?

CHALLENGING PROBLEMS 11-14

You have a $2 million portfolio consisting of a $100,000 investment in each of 20 different stocks. The portfolio has a beta of 1.1. You are considering selling $100,000 worth of one stock with a beta of 0.9 and using the proceeds to purchase another stock with a beta of 1.4. What will the portfolio's new beta be after these transactions?

Stock R has a beta of 1.5, Stock S has a beta of0.75, the expected rate of return on an average stock is 13%, and the risk-free rate is 7%. By how much does the required return on the riskier stock exceed that on the less risky stock?

290

(6-13) Historical Realized

Rates of Return

(6-14) Historical Returns:

Expected and Required Rates

of Return

(6-15) Evaluating Risk

and Return

resource

Part 3 Stocks and Options

You are considering an investment in either individual stocks or a portfolio of stocks. The two stocks you are researching, Stock A and Stock B, have the following historical returns:

Vear j'A j'B

2014 -20.00% -5.00% 2016 42.00 15.00 2017 20.00 -13.00 2018 -8.00 50.00 2019 25.00 12.00

a. Calculate the average rate of return for each stock during the 5-year period. b. Suppose you had held a portfolio consisting of 50% of Stock A and 50% of Stock

B. What would have been the realized rate of return on the portfolio in each year? What would have been the average return on the portfolio during this period?

c. Calculate the standard deviation of returns for each stock and for the portfolio. d. Suppose you are a risk-averse investor. Assuming Stocks A and B are your only

choices, would you prefer to hold Stock A, Stock B, or the portfolio? Why?

You have observed the following returns over time:

Year StockX StockY Market

2015 14% 13% 12% 2016 19 7 10 2017 -16 -5 -12 2018 3 1 1 2019 20 11 15

Assume that the risk-free rate is 6% and the market risk premium is 5%.

a. What are the betas of Stocks X and Y? b. What are the required rates of return on Stocks X and Y? c. What is the required rate of return on a portfolio consisting of 80% of Stock X and

20% of Stock Y?

Start with the partial model in the file Ch06 Pl5 Build a Model.xlsx on the textbook's Web site. The file contains data for this problem. Goodman Corporation's and Landry Incor­ porated's stock prices and dividends, along with the Market Index, are shown here. Stock prices are reported for December 31 of each year, and dividends reflect those paid during the year. The market data are adjusted to include dividends.

Goodman Corporation Landry Incorporated Market Index

Year Stock Price Dividend Stock Price Dividend Includes Dividends

2019 $25.88 $1.73 $73.13 $4.50 17,495.97 2018 22.13 1.59 78.45 4.35 13,178.55 2017 24.75 1.50 73.13 4.13 13,019.97 2016 16.13 1.43 85.88 3.75 9,651.05 2015 17.06 1.35 90.00 3.38 8,403.42 2014 11.44 1.28 83.63 3.00 7,058.96

Chapter 6 Risk and Return 291

a. Use the data given to calculate annual returns for Goodman, Landry, and the Market Index, and then calculate average annual returns for the two stocks and the index. (Hint: Remember, returns are calculated by subtracting the beginning price from the ending price to get the capital gain or loss, adding the dividend to the capital gain or loss, and then dividing the result by the beginning price. Assume that dividends are already included in the index. Also, you cannot calculate the rate of return for 2014 because you do not have 2013 data.)

b. Calculate the standard deviations of the returns for Goodman, Landry, and the Market Index. (Hint: Use the sample standard deviation formula given in the chapter, which corresponds to the STDEV function in Excel.)

c. Construct a scatter diagram graph that shows Goodman's returns on the verti- cal axis and the Market Index's returns on the horizontal axis. Construct a similar graph showing Landry's stock returns on the vertical axis.

d. Estimate Goodman's and Landry's betas as the slopes of regression lines with stock return on the vertical axis (y-axis) and market return on the horizontal axis (x-axis). (Hint: Use Excel's SLOPE function.) Are these betas consistent with your graph?

e. The risk-free rate on long-term Treasury bonds is 6.04%. Assume that the market risk premium is 5%. What is the required return on the market? Now use the SML equation to calculate the two companies' required returns.

f. If you formed a portfolio that consisted of 50% Goodman stock and 50% Landry stock, what would be its beta and its required return?

g. Suppose an investor wants to include some Goodman Industries stock in his portfolio. Stocks A, B, and C are currently in the portfolio, and their betas are 0.769, 0.985, and 1.423, respectively. Calculate the new portfolio's required return if it consists of 25% Goodman, 15% Stock A, 40% Stock B, and 20% Stock C.

Assume that you recently graduated and landed a job as a financial planner with Cicero Services, an investment advisory company. Your first client recently inherited some assets and has asked you to evaluate them. The client owns a bond portfolio with $1 million invested in zero coupon Treasury bonds that mature in 10 years.47

The client also has $2 million invested in the stock of Blandy, Inc., a company that produces meat-and-potatoes frozen dinners. Blandy's slogan is, "Solid food for shaky times."

Unfortunately, Congress and the president are engaged in an acrimonious dispute over the budget and the debt ceiling. The outcome of the dispute, which will not be resolved until the end of the year, will have a big impact on interest rates one year from now. Your first task is to determine the risk of the client's bond portfolio. After consulting with the economists at your firm, you have specified five possible scenarios for the resolution of the dispute at the end of the year. For each scenario, you have estimated the probability of the scenario occurring and the impact on in­ terest rates and bond prices if the scenario occurs. Given this information, you have

"The total par value at maturity is $1.79 million and yield to maturity is about 6%, but that information is not necessary for this mini case.

292 Part 3 Stocks and Options

calculated the rate of return on 10-year zero coupon Treasury bonds for each scenario. The probabilities and returns are shown here:

Return on a 10-Year Zero

Probability of Coupon Treasury Bond

Scenario Scenario During the Next Year

Worst Case 0.10 -14% Poor Case 0.20 -4%

Most Likely 0.40 6%

Good Case 0.20 16%

Best Case 0.10 26%

1QQ

You have also gathered historical returns for the past 10 years for Blandy and Gour­ mange Corporation (a producer of gourmet specialty foods), and the stock market.

Historical Stock Returns

Year Market Blandy Gourmange

1 30% 26% 47% 2 7 15 -54

3 18 -14 15 4 -22 -15 7

5 -14 2 -28 6 10 -18 40 7 26 42 17

8 -10 30 -23

9 -3 -32 -4 10 38 28 75 Average return: 8.0% ? 9.2%

Standard deviation: 20.1% ? 38.6%

Correlation with the market: 1.00 ? 0.678 Beta: 1.00 ? 1.30

The risk-free rate is 4%, and the market risk premium is 5%.

a. What are investment returns? What is the return on an investment that costs $1,000 and is sold after 1 year for $1,060?

b. Graph the probability distribution for the bond returns based on the 5 scenarios. What might the graph of the probability distribution look like if there were an infi­ nite number of scenarios (i.e., if it were a continuous distribution and not a discrete distribution)?

c. Use the scenario data to calculate the expected rate of return for the 10-year zero coupon Treasury bonds during the next year.

d. What is the stand-alone risk? Use the scenario data to calculate the standard devia­ tion of the bond's return for the next year.

e. Your client has decided that the risk of the bond portfolio is acceptable and wishes to leave it as it is. Now your client has asked you to use historical returns to estimate

Chapter 6 Risk and Return

the standard deviation of Blandy's stock returns. (Note: Many analysts use

293

4 to 5 years of monthly returns to estimate risk, and many use 52 weeks of weekly returns; some even use a year or less of daily returns. For the sake of simplicity, use Blandy's 10 annual returns.)

f. Your client is shocked at how much risk Blandy stock has and would like to reduce the level of risk. You suggest that the client sell 25% of the Blandy stock and create a portfolio with 75% Blandy stock and 25% in the high-risk Gourmange stock. How do you suppose the client will react to replacing some of the Blandy stock with high­ risk stock? Show the client what the proposed portfolio return would have been in each year of the sample. Then calculate the average return and standard deviation using the portfolio's annual returns. How does the risk of this two-stock portfolio compare with the risk of the individual stocks if they were held in isolation?

g. Explain correlation to your client. Calculate the estimated correlation between Blandy and Gourmange. Does this explain why the portfolio standard deviation was less than Blandy's standard deviation?

h. Suppose an investor starts with a portfolio consisting of one randomly selected stock. As more and more randomly selected stocks are added to the portfolio, what happens to the portfolio's risk?

i. (1) Should portfolio effects influence how investors think about the risk of indi­ vidual stocks? (2) If you decided to hold a one-stock portfolio and consequently were exposed to more risk than diversified investors, could you expect to be compensated for all of your risk; that is, could you earn a risk premium on that part of your risk that you could have eliminated by diversifying?

j. According to the Capital Asset Pricing Model, what measures the amount of risk that an individual stock contributes to a well-diversified portfolio? Define this measurement.

k. What is the Security Market Line (SML)? How is beta related to a stock's required rate of return?

I. Calculate the correlation coefficient between Blandy and the market. Use this and the previously calculated (or given) standard deviations of Blandy and the market to estimate Blandy's beta. Does Blandy contribute more or less risk to a well-diversified portfolio than does the average stock? Use the SML to estimate Blandy's required return.

m. Show how to estimate beta using regression analysis. n. (1) Suppose the risk-free rate goes up to 7%. What effect would higher interest rates

have on the SML and on the returns required on high-risk and low-risk securi- ties? (2) Suppose instead that investors' risk aversion increased enough to cause the market risk premium to increase to 8%. (Assume the risk-free rate remains con­ stant.) What effect would this have on the SML and on returns of high- and low-risk securities?

o. Your client decides to invest $1.4 million in Blandy stock and $0.6 million in Gour­ mange stock. What are the weights for this portfolio? What is the portfolio's beta? What is the required return for this portfolio?

p. Jordan Jones (JJ) and Casey Carter (CC) are portfolio managers at your firm. Each manages a well-diversified portfolio. Your boss has asked for your opinion regarding their performance in the past year. JJ's portfolio has a beta of 0.6 and had a return of 8.5%; CC's portfolio has a beta of 1.4 and had a return of9.5%. Which manager had better performance? Why?

294 Part 3 Stocks and Options

q. What does market equilibrium mean? If equilibrium does not exist, how will it be established?

r. What is the Efficient Markets Hypothesis (EMH), and what are its three forms? What evidence supports the EMH? What evidence casts doubt on the EMH?

The following cases from CengageCompose cover many of the concepts discussed in this chapter and are available at compose.cengage.com.

Klein-Brigham Series:

Case 2, "Peachtree Securities, Inc. (A)."

Brigham-Buzzard Series:

Case 2, "Powerline Network Corporation (Risk and Return)."