BUS 650 Week 4 Assignment

Chapter 7

Finding the Required Rate of Return for an Investment

Associated Press

Learning Objectives

A�er studying this chapter, you should be able to:

Explain the significance of required return and its components. Describe the rela�onship between risk and return and how to measure for both. Iden�fy how to use required return to determine valua�on.

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Ch. 7 Introduction

Investors come in many forms. They may be individuals who invest in corporate stocks, re�rement accounts that invest in bonds, partnerships that invest in apartment buildings, or corpora�ons that invest in produc�ve projects. One thing all these investors have in common is their desire to increase their wealth, which is done by iden�fying projects whose value is expected to exceed their cost. If we invest $100 today in a project that produces cash flows worth $125 in today's terms, then we increase our wealth by $25. Equa�on (7.1) is the basic formula for es�ma�ng the value of an investment, which is found by discoun�ng the expected future cash flows back to today's equivalent value at a rate of return that is appropriate given the investment's risk. This fundamental formula for assessing value was first introduced in Chapter 2 and further developed in Chapters 4 and 5, while Chapter 3 explored cash flows in some detail.

One part of the formula that hasn't been covered is how to es�mate the required return that is appropriate to use as the discount rate in the valua�on calcula�on. Finding the required rate of return is the topic of this chapter (and is expanded upon in Chapter 8).

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Like children, who need to be bribed with the promise of a reward for their good behavior, investors require a worthwhile incen�ve before they will commit to an investment.

Beyond/SuperStock

7.1 The Building Blocks of the Required Return

In Chapter 2, we introduced the idea that investors are assumed to be ra�onal and risk averse. Because they are (mostly!) ra�onal, investors will give up control of their money for a period of �me by inves�ng only if they expect to increase their wealth. Therefore, investors have an almost ins�nctual return requirement as they invest. For example, a ra�onal investor would always want to earn at least the risk-free rate of return when inves�ng in some security or project. Otherwise, they would be se�ling for a return lower than what they could be assured of by simply deposi�ng the funds in a savings account that is guaranteed by both the bank and the government through the Federal Deposit Insurance Corpora�on (FDIC). The FDIC guarantees the first $250,000 of funds deposited to an individual's bank account. So we establish that the first building block for assessing a required return is the risk-free interest rate.

For most investments, however, the risk-free rate is only the first component of the required return. Virtually all investments have some risk associated with them, so investors also require what is known as a risk premium to compensate them for this risk exposure. Recall that we assume that investors are risk averse, which implies that to bear risk they require compensa�on in order to subject themselves to distasteful uncertainty. This is a li�le like one of the authors of this text who used to pay his children five cents if they would eat all of their broccoli because his kids were "broccoli averse."

Now we have the two fundamental building blocks of the required return for an investment: the risk-free return and a risk premium.

R(r) = Risk-free rate of return + Risk premium

Given these intui�ve building blocks, we will now take a closer look at returns, risk, and their rela�onship to one another in order to fully develop the methods for more precisely es�ma�ng the required rate of return for an investment.

Different Types of Returns

It's useful to do some thinking about different kinds of returns that investors might discuss when they are considering investment performance. One type of return is the historical return, also known as an actual or realized return. If you buy a share of stock for $20 and a year later you sell it for $22, you have earned a historical or realized return equal to 10% per year ($2 gain on a $20 investment). The actual return you earned is 10%. This may be the same or it may be quite different than the expected return that you were hoping for when you bought the stock. Perhaps your friend who is a stock broker told you that she had calculated a target selling price of $30 for the stock. If you believed her forecast, then you were expec�ng a 50% return when you decided to buy the stock. Clearly, if you were expec�ng a 50% return but the actual return was only 10%, then it's likely that you were disappointed in result. But were you sa�sfied with the 10% that you earned? To answer that, we need to know your required return for the stock. The es�ma�on of the required return for an investment is the subject of this chapter, but it is generally acknowledged that risk contributes to one's required return. So if this was a super risky stock, you may have had a required return equal to 25%. In this case, you would have been pre�y unhappy with the result. On the other hand, if the stock was considered a low risk investment, then you might have had a return requirement of only 8%, and you were probably very sa�sfied with the 10% actual return, given the stock's low risk.

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Changes in interest rates and other factors can affect rates of return. Investors must be aware of the risks they are taking and understand probabili�es and expected rate of return in their business. At what point would you consider a poten�al investment to be too risky? Why must financial managers consider the risk-return tradeoff when making financial decisions?

7.2 Risk and Return

The trade-off between risk and return is second nature to us: We understand that if we are going to invest in a bond issued by United Airlines, we would do so only if we expected to receive a rate of return greater than we would receive if we invested in a bond issued by the U.S. Treasury. Why? United Airlines is generally considered riskier than the US government. One of the great intellectual challenges of finance over the past fi�y years was to find a method for measuring risk, and then find a formula that quan�fies the rela�onship between risk and return. Using our example, we need to find a method for quan�fying how much risk United Airlines has and then discover a method for es�ma�ng the return that investors should require given that level of risk.

Risk-Return Tradeoff

Measuring Return

But before delving into how to measure risk, let's look at how to measure returns. For simplicity's sake, we will use stocks to illustrate returns and risk. A single period's historical return is given by the formula

(7.2) Return over a period = Rt = (Pricet – Pricet – 1 + Dividendt)/Pricet – 1

Example: If you buy a share of stock for $40, hold it for one year during which you collect a dividend of $2 a share, and then sell the stock for $40.50, what was your return?

The answer is ($40.50 – $40 + $2)/$40 = 2.50/40 = 0.0625 = 6.25%.

This stock formula can be generalized for any investment's return:

(7.3) Return over a period = Rt = (Valuet – Valuet – 1 + Cash flowt)/Valuet – 1

In words, the return for the period is equal to the change in value of the asset during the period, plus any cash flows paid by the asset during the period, divided by the value of the asset at the beginning of the period.

It would be really useful to predict returns (for one thing, you would get rich if you could consistently forecast returns!). Unfortunately, in order to predict returns, you would like to know what price changes will be in the future so these future prices can be plugged into the return formula. But, as you learned in Chapter 2, market efficiency implies that compe��ve market prices reflect all available informa�on. Therefore, we cannot say what future price changes will be and therefore what returns will be. This is because price changes will only reflect new informa�on, and it's anybody's guess whether that informa�on will be good news or bad news for the company or for the economy. Because it is nearly impossible to predict returns, we o�en use the historical average return as our best es�mate of the expected future return. Take cau�on with this approach. When using an average to predict the future, one should use a rela�vely long run average since almost anything can happen in the short term. For example, between 1950 and 2010, the average annual return for the stock market (as proxied by the S&P 500 index) has been about 11% per year. This is considered a be�er es�mate than, say, the five-year average stock market return between 2007 and 2012, which averaged about zero!

Measuring Risk

We begin our discussion of risk by considering uncertainty. We are not certain what return we will receive in the future when we invest. With some investments, we feel a greater level of uncertainty than we do with others. For example, if an investor chooses to buy a five-year cer�ficate of deposit at an FDIC-insured bank, most investorsProcessing math: 0%

Returns are based on returns from 3/2010 to 3/2012.

would feel there is very li�le uncertainty about how much their deposit will be worth a�er the five-year period. However, if the funds were invested in Facebook common stock, there is a wide range of poten�al values that the stock could have five years a�er the investment is made. One might wonder, "How much riskier is Facebook stock than a cer�ficate of deposit? Is it twice as risky? Ten �mes as risky? Twenty �mes as risky?"

In order to answer that ques�on, we need a metric for measuring risk. We begin by introducing the concept of an investment's total risk. We will define total risk as the variability of returns, measured by their standard devia�on. For simplicity's sake we will be using historical returns to measure risk because, as previously discussed, future returns are difficult to predict. Note that we are assuming in this case that past risk is a good predictor of future risk, which may be OK, but as you become a more sophis�cated analyst, this es�mate may be adjusted up or down depending on what you know about the prospects of the firm or the investment that you're analyzing.

(7.4) Total risk = Standard devia�on =

Standard devia�on measures the typical distance (or devia�on) of a return from the average (or expected) return. So a stock that has a standard devia�on of 15% has more uncertainty regarding its returns than a stock with a standard devia�on of only 10%. To see this, look at Figure 7.1, which illustrates the distribu�on of returns for two stocks, Peabody Coal and Pacific Gas and Electric (PG&E). These histograms show the frequency of weekly returns from March 2010 un�l March 2012. No�ce that PG&E's returns are much more �ghtly clustered, whereas Peabody's have long tails, par�cularly a long tail to the le� of its center. Both of these distribu�ons have about the same average return (0.00), but there is much more uncertainty about the return of Peabody because the standard devia�on of its returns is 0.068 per week while Pacific Gas and Electric's standard devia�on is only 0.023. Therefore, judging by this historical data, risk-averse investors would be much more concerned about owning Peabody because of the uncertainty surrounding its returns.

Figure 7.1: Historical distribu�on of returns for Peabody Coal and PG&E

It is important to keep in mind where the uncertainty illustrated in Figure 7.1 actually comes from: Risk is measured by the variability of returns, and returns are generated by price changes as we saw in Equa�on (7.4). Recall that price changes are caused by the arrival to the marketplace of new informa�on, which investors and analysts anxiously await in order to adjust their view of the company's or investment's worth. Risk, therefore, has at its founda�on informa�on and the investment's sensi�vity to that informa�on. As an example, consider what might happen to a company's stock value, and therefore its returns, if the United States announced that it will impose a significant tax on carbon emissions. The prices of oil companies would likely fall drama�cally as one would imagine gasoline costs increasing and demand decreasing, lowering oil company profits. However, a hydroelectric-based u�lity company might see li�le change in its value since it does not produce carbon so its cost and pricing structure would remain unchanged—its value might actually increase as demand for clean energy would likely rise.

The risk of adverse price movements can be decreased by diversifica�on. For example, in the previous example, consider what would happen to an investor's por�olio (collec�on of investment assets) if the investor held both oil company stocks and hydroelectric u�lity stocks. The oil stock values would fall because of the carbon tax, but this risk would be mi�gated by the posi�ve response to the tax by hydroelectric firms. In this case, the fall in gas stock prices is offset by the posi�ve response of the u�lity stocks. Risk is decreased in this case because of the different reac�ons by the two industries to the same informa�on. When one has investments in a variety of companies, there is a good chance that what affects one company nega�vely may actually have li�le impact, or perhaps a posi�ve impact, on the value of another stock in the por�olio.

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Por�olio diversifica�on refers to the reduced risk to individuals who invest in various assets with different characteris�cs. How has financial globaliza�on led to more opportuni�es for por�olio diversifica�on?

The Benefits of Financial Globaliza�on: Risk Diversifica�on

Total risk can, therefore, be broken down into risk that may be diversified away (called diversifiable risk) and risk that cannot be avoided or mi�gated (called systema�c or nondiversifiable risk). Diversifiable risk is o�en characterized as "firm-specific" risk and "industry-specific" risk. Nondiversifiable risk is o�en referred to as market risk. Just so you know all the terms you might run into, diversifiable risk is also referred to as unsystema�c risk, whereas market risk is also called systema�c risk. Investors are primarily concerned with nondiversifiable risk because they can eliminate a great deal of the diversifiable risk by simply holding a large number of different stocks. Table 7.1 breaks down diversifiable and nondiversifiable risk.

(7.5) Total risk = Diversifiable risk + Nondiversifiable risk

(7.6) Diversifiable risk = Firm- and industry-specific risk = Unsystema�c risk

(7.7) Nondiversifiable risk = Market risk = Systema�c risk

Table 7.1: Classifying risk

Names for risk that can be diversified away Names for risk that cannot be diversified away

Industry- and firm-specific risk Market risk

Diversifiable risk Nondiversifiable risk

Idiosyncra�c risk Systema�c risk

Unsystema�c risk Economy-wide risk

Example: CEO quits Example: Recession

Firm-specific risk is associated with events such as a company making a poor product decision, being sued, having a CEO get indicted or die, having a big fire at a factory, or having a compe�tor develop a new product. All of these would adversely affect the value of the company, but if an investor is well-diversified, the impact will be minimal to the overall por�olio because such an event impacts only a single firm. Also, with enough firms in a por�olio, there is a good chance that when a firm-specific bad event happens to company A, you may have another company that experiences firm-specific good news. For example, suppose that on the same day that Firm A losses a lawsuit, Firm B discovers oil, so these events would tend to offset one another in your por�olio. Some�mes, a news event for one company ripples through the industry. If Apple announces a new, more powerful but less expensive iPad, that will almost certainly affect the prospects of other companies making tablet computers. If one airline company has several planes grounded for safety inspec�ons, other airlines might benefit as passengers switch their flight plans.

Industry-specific risks are also largely avoidable via diversifica�on because events that harm a par�cular type of industry will not necessarily have a nega�ve effect on other stocks in a por�olio that represent firms in other industries. For example, low interest rates may hurt the profits in the banking industry, yet they actually help the housing- building industry. Therefore, holding a por�olio of stocks (in other words, being diversified) enables the nega�ve impact of low interest rates on one industry to be offset by the posi�ve effect these rates have on other industries within the investor's por�olio.

Nondiversifiable risks are difficult to avoid regardless of how many stocks you own, or how diversified your investment por�olio becomes. Some events have nega�ve effects that pervade the en�re economy. For example, unemployment hurts almost all companies as consumer demand falls lowering sales and profits, and savings fall making capital scarce. These kinds of far-reaching events are referred to as nondiversifiable or market risks. High infla�on, war, economic recessions, and oil embargos all have a nega�ve impact on almost all of the firms in one's por�olio, regardless of how many stocks you own!

Because much of the risk of inves�ng may be avoided simply by diversifying one's por�olio, it is argued that we need not concern ourselves with these diversifiable risks. It is, for example, just as easy to buy a mutual fund that holds shares of 500 different companies as it is to load up on a single firm's stock. Clearly, the mutual fund strategy avoids much of the risk that the investor in a single security faces. In fact, the standard devia�on (the variability) of a diversified por�olio's returns can easily be reduced by

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The por�olio standard devia�on of returns decreases as the number of randomly selected stocks in the por�olio increases.

Luxury markets like Porsche dealerships are highly suscep�ble to the rise and fall of the economy. Do you tend to feel more or less inclined to invest in these companies?

Associated Press

about half compared to the average standard devia�on of the individual stocks in the por�olio. The diversifica�on effect of lowering risk is shown in Figure 7.2 where we can see that increasing the number of even randomly selected stocks in a por�olio can drama�cally reduce the por�olio's standard devia�on of returns.

Figure 7.2: The diversifica�on effect

Since investors can easily and inexpensively eliminate most diversifiable risk from their por�olios, we assume that everyone does so. Thus, the risk that is relevant to the investor is the nondiversifiable risk of an investment. The ques�on becomes, how can we measure this risk? To measure market risk, we u�lize a metric called beta. Beta measures a firm's typical responsiveness to informa�on that impacts the en�re market—like informa�on about economic growth, poli�cal news, infla�onary expecta�ons, the balance of trade, natural disasters, and so on. By defini�on, the average sensi�vity to this kind of informa�on would be measured by the responsiveness of the market por�olio. The market por�olio theore�cally would be totally diversified and would include virtually all investment instruments including all of the stocks and bonds that are traded. In prac�ce, there is no such thing as a true market por�olio so a proxy is used as an approxima�on. Typically, the S&P 500 Index is used as that proxy. The beta of the market por�olio is defined as being equal to posi�ve one.

A firm may be more sensi�ve than the average to economic informa�on, in which case the firm's beta would be greater than one. A company that is twice as sensi�ve as the average firm to economic events will have a beta of 2.00, whereas a firm that is less sensi�ve than average will have a beta below one. Here is an example. Take a firm that sells luxury goods, like a Porsche automobile dealership. We might assume that when the economy is booming, this business does really well, but when the economy is doing poorly, luxury sports car sales suffer dras�cally. Let's suppose that Porsche dealership's beta is 1.70, meaning that it is 1.7 �mes as sensi�ve as the overall market to "macroeconomic" type events. So, if the government announces that economic growth is very strong, we might hear that the S&P 500 por�olio had a return of 5% that day in response to this good economic news. But because a Porsche dealership is more sensi�ve than average to such informa�on, its stock would likely return around 8.5% on the same day (found by taking the product of 1.7 × 5% = 8.5%). If, on the other hand, the market por�olio declines by 10% one month because of bad economic news, then the dealership's stock would be expected to fall by around 17%.

Of course, these are the expected returns for the Porsche dealership and may not be equal to the firm's actual returns on those days because there are always firm-specific factors that may affect a single stock's return. For example, on the day that the government announces strong economic growth (good news and we expect the

8.5% return), it may be the dealership also learns that the company is being sued, so the firm's stock could actually fall in value on the date because of this nega�ve firm- specific announcement.

Some businesses are less sensi�ve to market-level informa�on than the average firm is. An example might be an electric u�lity company, say Pacific Gas and Electric (PG&E). When the economy is doing well, Pacific Gas and Electric does well because there is more demand for electricity. When �mes are bad, demand for power falls, but it doesn't fall too far because, unlike Porsche sports cars, electricity is close to being a necessity. So with this rela�vely low sensi�vity, Pacific Gas and Electric has lower than average market risk, and its beta is less than one. In June 2012, Yahoo! Finance reported PG&E's beta as 0.29. If the country goes into a recession and the market as proxied by the S&P 500 declines by 10%, PG&E, with its beta of 0.29, would see its stock price drop by only about 3% on average.

Betas are typically es�mated using historical returns and linear regression es�ma�on. Linear regression is a sta�s�cal technique for es�ma�ng a best-fit line through points plo�ed in an x-y coordinate system like the graphs typically used in algebra. The idea is that the slope of this line will capture the average rela�onship between the x- and the y-variables. So if the slope is 1.5 for a regression line, then for each unit increase in the x-value, the y-value will (on average) increase by one and a half units. Regression is used to es�mate a variety of rela�onships, like the effect that the �me spent studying has on the grade point average of students. For our purposes, we use returns for the S&P 500 as our x-values, and corresponding returns for the stock that we are interested in as our y-values. The regression's slope, therefore, is an es�mate of the stock's average responsiveness to market-wide returns, or its beta.Processing math: 0%

A stock's beta is the slope of the line-of-best fit through the sca�erplot of market returns (S&P500) and the company's returns.

Since betas are sta�s�cal es�mates, they can vary depending on the sample of data being used for the es�mate. Figure 7.3 es�mates the beta for PG&E by plo�ng the corpora�on's returns against those of the S&P 500. As you can see, the beta we es�mate (0.43) differs from the beta reported by Yahoo! in June 2012 (0.29). The difference can be a�ributed to different data sets; Yahoo! based their report on 36 monthly returns, whereas we have based our es�mate on 24 weekly returns.

Figure 7.3: Es�ma�ng beta for PG&E

The Capital Asset Pricing Model

Now we know how to measure returns and how to measure nondiversifiable risk (by using beta). Next we need to learn how to u�lize these metrics to es�mate the required rate of return for an investment. Recall from the beginning of the chapter that the "building blocks" of a required return include the risk-free rate and a risk premium. These two elements are present in an equa�on called the capital asset pricing model (CAPM). We need this model to quan�fy the rela�onship between investor's required rate of return and the risk of an investment. Here is the CAPM as it was originally developed:

(7.8) Required return for an investment = Rf + Beta[E(Rmkt) – Rf]

where

Rf = the risk-free rate of return

E(Rmkt) = the expected return on the market por�olio

Beta = the stock's beta

This theore�cal rela�onship between risk and return was one of the path breaking achievements in economics in the 1960s, for which several academics were awarded the Nobel Prize. Like many theories, however, there are challenges when using the CAPM in prac�ce. For example, no one knows what the expected return on the market, E(Rmkt), is equal to. The model also assumes there is a single, observable risk-free rate, when in reality there is no investment free of risk and there is more than one

possible rate that can be used as a close proxy for risk-free. Because of these problems, most prac��oners use a different form of the model which is given here:

(7.9) Required return for an investment = Rf + Beta(Market risk premium)

Rf can be thought of as the rate that links the CAPM to current market condi�ons. This is important because interest rates are constantly changing due to changes in infla�on, economic ac�vity, or government policies. We use yields on outstanding debt issued by the U.S. government as a proxy for the risk-free rate, choosing the Treasury bill or bond that best matches the life of the asset we are evalua�ng. So, for stocks that have an almost perpetual life, long-term U.S. Treasury bond yields are o�en used for the risk-free rate. The market risk premium (MRP) is the amount of return yielded by the market por�olio over and above the treasury yield. It can be thought of as the return required for each addi�onal unit of risk as measured by beta. O�en, the MRP is assumed to equal its historical average, which is about 5% to 7%, depending on whose data you use.

Here is an example of using the CAPM. Let's es�mate the required return for Nordstrom's (Ticker: JWN) stock given that Nordstrom's beta is 1.58, as reported on Yahoo! Finance in June 2012 (h�p://finance.yahoo.com/q/ks?s=JWN+Key+Sta�s�cs (h�p://finance.yahoo.com/q/ks?s=JWN+Key+Sta�s�cs) ). Nordstrom's has a fairly high beta because it is considered a high-end or almost luxury retailer, not dealing in necessity goods. Consequently, when �mes are tough, some people may discon�nue shopping at Nordstrom's and may buy their shoes and clothing at a more moderately priced retailer. Let's also assume that T-bonds are yielding 4.5% per year, and that the historical average market risk premium (the average return of the market por�olio over and above the risk-free return) is about 6%. Using this informa�on, we may es�mate the required return for Nordstrom's stock using the CAPM.

RNordstroms = Rf + BNordstroms(Market risk premium) = 0.045 + 1.58(0.06) = 0.1398 = 13.98% Processing math: 0%

A second example would be required rate of return for PG&E's stock. Using the published beta of 0.29 and the risk-free rate and market risk premium above we would compute PG&E's required rate of return as:

RPG&E = Rf + BPG&E(Market risk premium) = 0.045 + 0.29(0.06) = 0.0624 = 6.24%

No�ce that the much lower beta of PG&E results in a much lower required rate of return compared to Nordstrom.

Applying Finance: Finding Beta Using Excel

Finding an asset's beta using Excel is a two-step process: first, compute returns from prices; second, use the LINEST func�on to compute the beta (slope of a regression line).

We downloaded stock prices for Dow Chemical (Ticker: DOW) from Yahoo! Finance using its historical price feature. We then downloaded the index data for the S&P 500 (Yahoo! Ticker: ^GSPC). We use the adjusted close price to make sure dividends are included in the price. Our data are weekly and run from Friday, February 3, 2012, through Friday, June 22, 2012. We compute returns using the formula from the text:

(Change in price)/Beginning price

Since we are using adjusted prices we don't need to explicitly include dividends in the returns equa�on. Figure 7.4 shows the spreadsheet of prices showing the price series, the returns formulas and the LINEST formula.

Figure 7.4: DOW Chemical spreadsheet, LINEST

Figure 7.5 shows the same spreadsheet with the numerical results shown. Using this small sample of data, Dow's beta es�mate is 1.63.

Figure 7.5: DOW Chemical spreadsheet, numerical

Determining an Asset's Beta Processing math: 0%

During the discussions of the Porsche dealership, Nordstrom's, and PG&E, we said that it is the nature of the products and services that are sold by a company that determine the company's risk. Luxury goods, like sports cars, tend to have more market risk than do necessi�es like electricity. Demand for durable goods, items that are long-lived and can be repaired like cars and appliances, will fluctuate more as the economy rises and falls than demand for food or medicine. If a company has invested in produc�ve assets to make luxury or durable goods, then it is likely to have a high beta (higher than 1 or the market average beta). Similarly, if the factories and equipment make food or electricity or things that are necessary (or that have steady demand), the company will have a lower beta (less than one). Thus, it is the assets of a company and the products those assets make that determine the company's beta. Companies that produce similar goods that are sold in similar markets will have similar betas because those companies will be impacted similarly by the kind of macroeconomic news that creates market risk. Assess your expecta�ons of beta by comple�ng the exercise in the Field Trip: Expecta�ons of Beta feature.

Field Trip: Expecta�ons of Beta

Pick three firms that you think may have low betas and three that you think may have high betas. Visit Yahoo! Finance or Google Finance to look up their betas.

Visit Yahoo finance: h�p://finance.yahoo.com/ (h�p://finance.yahoo.com/)

Google finance: h�p://www.google.com/finance (h�p://www.google.com/finance)

Reflec�on Ques�on

Are there any surprises? Can you think of some reasons that these betas are different from what you expected?

In theory, the asset beta of a Porsche dealership should be very nearly the same as the asset beta of, say, a BMW dealership. This is because they are similar businesses offering similar products. If both the Porsche and the BMW dealerships had no debt financing, then both of their asset betas would be iden�cal to the betas of their stock. This is because the stock would represent the only claim against the assets, so the risk of the assets would translate directly to the risk of the stock. In this case, both dealerships' stock, being in the same business, would probably have almost iden�cal betas and both would also have almost iden�cal required rates of return.

However, most companies use debt financing in addi�on to equity financing. This use of debt is also known as leverage. The use of leverage increases the risk of equity because debt, with its priority claim, forces equity holders to bear the risk that there will be lower cash flows available for them a�er debt payments are made. For this reason, the betas of stock differ even among firms in the same industry because of the varying amount of debt that the companies borrow. Asset betas, therefore, depend primarily on the nature of a company's business, whereas equity betas—the betas of inves�ng in just a company's stock—depend on both a firm's asset beta and on its use of leverage. A specific technique used for es�ma�ng an asset beta, called the pure-play approach, is covered in the next chapter.

Portfolio Betas

There are �mes when investors may want to es�mate the required return for a por�olio or stocks or other investment assets. For example, we may want to compare the actual, realized return on a por�olio to the required return on that por�olio in order to assess the performance of the manager who is in charge of the por�olio's investments. If the realized return exceeds the required return given the por�olio's risk, then the manager is performing at or above expecta�ons. If, on the other hand, the realized return is below the required return for the por�olio, then the manager is performing below expecta�ons, and may find his or her posi�on in jeopardy.

Por�olio betas are found by taking the weighted average of the betas of the assets held in the por�olio, where the weights are determined by the amount invested in each asset. For example, consider the por�olio in Table 7.2.

Table 7.2: Sample por�olio

Stock Stock's beta Amount invested Weight

Acme, Inc. 1.20 $100,000 .10

XYZ Corp. 1.50 $150,000 .15

ABC Corp. 0.70 $500,000 .50

Delphi, Inc. 1.00 $250,000 .25

The weights represent the propor�on of total investment that is invested in each asset. For example, the total investment in this por�olio is $1,000,000, so the investment of $150,000 in XYZ's stock represents 15% of the total, making the weight for XYZ .15. In this case, the por�olio's beta is the sum:

Beta por�olio = (1.20)(.10) + (1.50)(.15) + (.70)(.50) + (1.00)(.25) = .945

To illustrate the usefulness of this number, it could be used to es�mate the risk of this por�olio versus another por�olio or a mutual fund. It could also be plugged into the CAPM to give an es�mate of the required return for this por�olio.

One warning: beta is not a good measure of risk unless the investor is what is known as "well diversified." Usually, if you own more than 20 stocks, you are considered well- diversified. However, if these stocks are concentrated in only a couple of industries, then you are probably not effec�vely diversified. Diversifica�on is a subjec�ve and rela�ve term, so, as a rule, it's be�er to be more diversified than to be less diversified. Next, we will look at why it is prudent to invest in several stocks and other assets (like bonds and real estate) that are not concentrated in only a few industries (or geographic loca�ons).

Correlation and Diversification

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This graph shows what perfect nega�ve correla�on between two stocks (A and B) might look like.

In a more advanced mathema�cal presenta�on of diversifica�on's effect, you would learn that diversifica�on depends on asset returns having imperfect correla�on with one another. In other words, if all stocks in a por�olio went up and down together, then they would be perfectly correlated, and there would be no point in diversifying. For example, if a disaster happened to one stock, it would also happen to all the other stocks because they are perfectly posi�vely correlated to one another. As a result, because they would likely have high posi�ve correla�ons with one another, it is not a good idea to concentrate your por�olio in only one industry or even just a few industries.

On the other hand, stocks with lower correla�ons make great choices for forming a well-diversified por�olio. In fact, the most efficient diversifica�on happens when we mix nega�vely correlated assets in our por�olios because their risks will offset one another. To illustrate, consider this example. Suppose you owned stock A whose returns were perfectly nega�vely correlated with another, stock B. Whenever A's return went up, B's went down and vice versa. Imagine that they both varied around an average return of 10%. If you were lucky enough to locate these two stocks, and you put your money in each one to form a special two-stock por�olio, your por�olio would earn exactly 10% but you would experience zero variability. That is because every �me A had a return below 10%, B would have a return above 10% because of its perfect nega�ve correla�on. In other words, you could have a risk-free por�olio with a 10% return. Of course, it is not easy (perhaps impossible) to find two stocks with perfect nega�ve correla�on. Figure 7.6 shows a graphic representa�on of two stocks with perfect nega�ve correla�on.

Figure 7.6: Returns with perfect nega�ve correla�on

The good news is that you can always reduce risk by mixing assets whose returns are imperfectly correlated, and you do this without lowering their average return. Furthermore, since almost all investment assets are imperfectly correlated, you can get the risk-reducing benefits of diversifica�on by simply mixing together even a randomly selected bunch of stocks (as was shown in Figure 7.2). Of course, with a li�le insight, you can improve the benefits of diversifica�on by being sure to not focus on one industry group and by mixing in, for example, a few interna�onal stocks (can you figure out why, in terms of correla�on?).

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Knowing the market price of a stock is beneficial, but it can o�en be more advantageous to know how to make your own es�mate of a required return.

Stockbyte/Ge�y Images

7.3 Required Returns and Valuation

Now that we know how to es�mate required returns, we want to consider once again the �me value of money problems and security valua�on. This is because required returns are used as the discount rates in these valua�on formulas. To illustrate, suppose that PG&E just paid an annual dividend of $1.82 per share and that we believe dividends will grow at a 2% annual rate in the future. In this case, we can use the constant growth stock valua�on formula to es�mate the value of PG&E's stock (and we will use the form of the model with D0 in the numerator because we were given the last dividend paid). Recalling that PG&E's required return was 6.24%,

Value of PG&E stock = ($1.82)(1.02)/(0.0624 – 0.02) = $1.8564/.0424 = $43.78

Suppose we did this es�ma�on of value and looked up an actual quote for PG&E's stock on the Internet. If the price is currently $43.00, this would mean that the stock appears to be underpriced on the market, which would indicate that it is a bargain according to our es�mates. However, before we run out to buy PG&E stock, which we think may be worth $1.64 more per share than its price, we need to consider market efficiency. Remember that market efficiency says that the market price (the $43.00) is the best available es�mate of value. Now, we must decide whose es�mate we put more faith in: our es�mate ($44.64) or the market's ($43.00)? If we believe in market efficiency, we would probably not make the investment.

There are �mes, however, when we need to value an asset or a closely held stock for which no market price exists. In this case, we have li�le choice but to rely on our own es�mates. In these cases, the ability to es�mate the required return is essen�al. For example, before Facebook went public in 2012, there was no exis�ng market price for the stock, yet the firm's ownership had to place an ini�al price on the shares. If the price they chose was too high, no one would buy the stock, and the offer would be unsuccessful. If the offer price was too low, then the original owners would be selling a stake in their company for too li�le. To get the price correct, Facebook's management, ownership, and financial advisors had to es�mate the firm's value, which depended on an accurate es�mate of its risk and the required return of investors given that risk level.

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Ch. 7 Conclusion

In this chapter, the building blocks of required returns were introduced. Most of the chapter used stock to illustrate and explore the rela�onships between risk and investors' return requirements. The calcula�on of returns, of total risk (standard devia�on of returns), and of market risk (beta) were covered. The benefit of diversifica�on by elimina�ng certain types of risk was discussed as was the effec�veness of how diversifica�on is linked to the correla�on between investments' returns. The capital asset pricing model, used to es�mate the required return for an investment, was also covered and illustrated. The concept of an asset beta was also introduced. Chapter 8 extensively uses the CAPM along with the asset beta as a means to discover the overall required return for the en�re firm rather than just the return requirement for its equity, which we focused on in this chapter. The theories and techniques explored in Chapter 7 will be vitally important for those of you who one day will enter a career in the investments field, but the insights will also be important for everyone who becomes an investor whether they are inves�ng for re�rement or for their child's college fund.

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Ch. 7 Learning Resources

Key Ideas

A ra�onal investor will always want to earn at least the risk-free rate of return when inves�ng in some security or project. Otherwise, they would be se�ling for a return lower than what they could be assured of by simply deposi�ng the funds in a savings account. Virtually all investments have some risk associated with them, so investors also require what is known as a risk premium to compensate them for this risk exposure. The two fundamental building blocks of the required return for an investment are the risk-free return and a risk premium. The required return for an investor = (Risk-free rate of return) + (Risk premium). An investment's total risk is the variability of returns and is measured by their standard devia�on. For simplicity's sake, we will be using historical returns to measure risk because expected returns are difficult to predict. Returns are generated by price changes caused by the arrival to the marketplace of new informa�on, which investors and analysts anxiously await in order to adjust their view of the company's or investment's worth. Total risk can be broken down into risk that may be diversified away and risk that cannot be avoided or mi�gated. Firm-specific risk is associated with events specific to the firm that adversely affect the value of the company, but if an investor is well-diversified, the impact will be minimal to the overall por�olio because such an event impacts only a single firm. Industry-specific risks are largely avoidable via diversifica�on because events that harm a par�cular type of industry will not necessarily have a nega�ve effect on other stocks in a por�olio that represent firms in other industries. Nondiversifiable or market risks are difficult to avoid regardless of how many stocks you own or how diversified your investment por�olio becomes. Some events have nega�ve effects that pervade the en�re economy. To measure market risk, we u�lize a metric called beta. Beta measures a firm's typical responsiveness to informa�on that impacts the en�re market. The market risk premium (MRP) is the amount of return yielded by the market por�olio over and above the Treasury yield. It can be thought of as the return required for each addi�onal unit of risk as measured by beta. Companies that produce similar goods, sold in similar markets, will have similar betas because those companies will be affected similarly by the kind of macroeconomic news that creates market risk. Por�olio betas are found by taking the weighted average of the betas of the assets held in the por�olio, where the weights are determined by the amount invested in each asset. Diversifica�on depends on asset returns having imperfect correla�on with one another. In other words, if all stocks in a por�olio went up and down together, then they would be perfectly correlated, and there would be no point in diversifying. Stocks with lower correla�ons make great choices for forming a well-diversified por�olio. In fact, the most efficient diversifica�on happens when we mix nega�vely correlated assets in our por�olios because then their risks will offset one another. Required returns are used as the discount rates in valua�on formulas such as �me value of money problems and security valua�on.

Key Equa�ons

Cri�cal Thinking Ques�ons

1. Total risk is measured by the standard devia�on of returns. Jot down the formula for the standard devia�on and then comment on what part of the formula has to do with devia�ons and what part is related to calcula�ng the standard of these devia�ons. (Hint: devia�on may be thought of as departure from what is typical and standard may be thought of as the average or what is expected).

2. The theore�cal development of the CAPM calls for use of the risk-free rate. Swiss government bonds have typically had a lower risk ra�ng than any other government bond. Why isn't the Swiss bond, therefore, the standard for use in the CAPM worldwide?

3. The average return on the S&P 500 has been in the neighborhood of 11%, and in 1980 U.S. Treasury bonds were yielding about 15%. Imagine you were an investor in 1980. How do the circumstances at that �me help explain why it is be�er to use the market risk premium (of around 6%) rather than [Rmkt - Rf] when es�ma�ng the required return

with the CAPM? 4. Beta is the measure of market risk. Look at the businesses listed below and see if you can iden�fy one that could very likely have a rela�vely high total risk but a low beta.

Explain your reasoning.

a. The manufacturer of diamond-encrusted dog collars.

b. A company that specializes in finding and salvaging old shipwrecks from the Age of Discovery (the 1500s and 1600s).

SLIDE 1 OF 4

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c. A casino

5. When you es�mate betas using historical data and linear regression, you must make some of the following choices: what index to use as a proxy for the market por�olio (the S&P 500, the Wilshire 5000, and the NYSE Composite Index are a few of the possibili�es); the length of the return period (daily returns, weekly returns, and monthly returns are all used); the length of the historical record (two, three, or five years are candidates). Each combina�on of these choices will yield a slight (or maybe even a major) difference in the es�mated beta. How many different betas for a single firm could you and your classmates get on the same day by making different choices among these op�ons?

6. Security A has a standard devia�on of returns equal to 20% and a beta of 1.50. Security B has a standard devia�on of 16% and a beta of 1.80. Which security probably has the higher required return? Explain.

Key Terms

Click on each key term to see the defini�on.

beta (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A measure of an asset's systema�c risk, also known as its market risk. The average stock has a beta of one, while stocks with greater than average market risk will have betas greater than one and those with less risk will have betas less than one. Beta is used to find the required return for an asset using the CAPM.

capital asset pricing model (CAPM) (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A formula that quan�fies the connec�on between an investment's market risk and its required rate of return, specifically the required return on an asset equals the riskfree rate plus a risk premium. The risk premium is the asset's beta �mes the market risk premium. The CAPM is the equa�on of the security market line.

correla�on (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A sta�s�cal measure of the co-movement of asset returns. Correla�on varies between the nega�ve return and the posi�ve one, while a correla�on of zero means that the two assets are "uncorrelated" and move independently. Perfect posi�ve correla�on means that the two assets' values move together in the same direc�on. Nega�vely correlated assets tend to move in opposite direc�ons.

diversifiable risk (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Risk that can be avoided through diversifica�on.

diversifica�on (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The mixing of investments in a single por�olio that can reduce risk exposure. Diversifica�on's benefits are most drama�c when the correla�ons between assets in the por�olio are low or even nega�ve.

diversifica�on effect (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The reduc�on in risk (standard devia�on) that occurs through the blending of stocks into a por�olio.

historical return (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The past performance of a security or index.

leverage (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A descrip�on of the propor�on of debt used in a firm's capital structure.

market por�olio (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A theore�cal bundle of investments that includes every kind of asset available in the financial market. Because a market por�olio is completely diversified, it is subject only to systema�c risk.

market risk premium (MRP) (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The difference between the rate of return on the market (e.g., S&P 500) and the risk-free return (e.g., Treasury bonds).

por�olio (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A collec�on of assets or investments. Inves�ng in a por�olio can reduce risk exposure compared to inves�ng in a single asset.

por�olio betas (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/froProcessing math: 0%

The weighted average of the betas of the assets held in the por�olio, where the weights are determined by the amount invested in each asset.

required rate of return (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The minimum return investors must expect in order to be interested in inves�ng in an asset.

risk averse (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Characteris�c in which people focus more on losses than on equivalent gains. Risk aversion implies that investors must be paid to bear risk.

risk-free rate of return (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The return an investor earns on a risk-free asset.

risk premium (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The added return necessary to compensate investors for taking added risk.

S&P 500 Index (Standard & Poor's 500) (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A price index of 500 stocks represen�ng a broad cross sec�on of industries o�en used to represent the en�re stock market's ac�vity.

systema�c or nondiversifiable risk (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Risk that cannot be avoided through diversifica�on.

total risk (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The overall poten�al for financial loss. The variability of returns, measured by their standard devia�on.

unsystema�c risk (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Risk that affects primarily one company or industry. Unique risk may be mi�gated by diversifying one's por�olio.

Web Resources

This chapter has introduced some measures of risk. Follow this link to take a quiz assessing your own risk tolerance: h�p://njaes.rutgers.edu/money/riskquiz/ (h�p://njaes.rutgers.edu/money/riskquiz/ )

Professor Aswath Damodaran of NYU maintains a list of betas for different industries. These may be viewed and the sectors compared at the following website. It might be interes�ng to compare the average betas in two sectors, like gambling versus natural gas u�li�es to see if the betas conform with your intui�on. h�p://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/Betas.html (h�p://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/Betas.html)

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