CH13
ou own a portfolio equally invested in a risk-free asset and two stocks. If one of the stocks has a beta of 1.34 and the total portfolio is equally as risky as the market, what must the beta be for the other stock in your portfolio?
12. The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. If the portfolio is as risky as the market, it must have the same beta as the market. Since the beta of the market is one, we know the beta of our portfolio is one. We also need to remember that the beta of the risk-free asset is zero. It has to be zero since the asset has no risk. Setting up the equation for the beta of our portfolio, we get: βP = 1.0 = 1/3(0) + 1/3(1.34) + 1/3(βX)
A stock has a beta of 1.19 and an expected return of 12.4 percent. A risk-free asset currently earns 2.7 percent. What is the expected return on a portfolio that is equally invested in the two assets? If a portfolio of the two assets has a beta of .92, what are the portfolio weights? If a portfolio of the two assets has an expected return of 10 percent, what is its beta? If a portfolio of the two assets has a beta of 2.38, what are the portfolio weights? How do you interpret the weights for the two assets in this case? Explain.
Again we have a special case where the portfolio is equally weighted, so we can sum the returns of each asset and divide by the number of assets. The expected return of the portfolio is: E(RP) = (.124 + .027)/2 E(RP) = .0755, or 7.55% b. We need to find the portfolio weights that result in a portfolio with a beta of .92. We know the beta of the risk-free asset is zero. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: βP = .92 = wS(1.19) + (1 - wS)(0) .92 = 1.19wS + 0 - 0wS wS = .92/1.19 wS = .7731 And, the weight of the risk-free asset is: wRf = 1 - .7731 wRf = .2269 c. We need to find the portfolio weights that result in a portfolio with an expected return of 10 percent. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: E(RP) = .10 = .124wS + .027(1 - wS) .10 = .124wS + .027 - .027wS .073 = .097wS wS = .7526 So, the beta of the portfolio will be: βP = .7526(1.19) + (1 - .7526)(0) βP = .896 d. Solving for the beta of the portfolio as we did in part a, we find: βP = 2.38 = wS(1.19) + (1 - wS)(0) wS = 2.38/1.19 = 2 wRf = 1 - 2 = -1 The portfolio is invested 200% in the stock and -100% in the risk-free asset. This represents borrowing at the risk-free rate to buy more of the stock.
Announcement and News
At the beginning of the year, market participants will have some idea or forecast of what theyearly GDP will be. To the extent that shareholders have predicted GDP, that prediction will already be factored into the expected part of the return on the stock, E(R). On the other hand, if the announced GDP is a surprise, the effect will be part of U, the unanticipated portion of the return. As an example, suppose shareholders in the market had forecast that the GDP increase this year would be .5 percent. If the actual announcement this year is exactly .5 percent, the same as the forecast, then the shareholders don't really learn anything, and the announcement isn't news. There will be no impact on the stock price as a result. This is like receiving confirmation of something you suspected all along; it doesn't reveal anything new. Going back to Flyers, suppose the government announces that the actual GDP increase during the year has been 1.5 percent. Now shareholders have learned something—namely, that the increase is one percentage point higher than they had forecast. This difference between the actual result and the forecast, one percentage point in this example, is sometimes called the innovation or the surprise. To summarize, an announcement can be broken into two parts: the anticipated, or expected, part and the surprise, or innovation: announcements and news contain both expected component and a surprise component it is the surprise component that affects a stock's price and therefore its return this is very obvious when we watch how stock prices move when an unexpected announcement is made or earnings are different than anticipated The expected part of any announcement is the part of the information that the market uses to form the expectation, E(R), of the return on the stock. The surprise is the news that influences the unanticipated return on the stock, U.
THE SYSTEMATIC RISK PRINCIPLE
Based on our study of capital market history, we know that there is a reward, on average, for bearing risk. We now need to be more precise about what we mean by risk. The systematic risk principle states that the reward for bearing risk depends only on the systematic risk of an investment. The underlying rationale for this principle is straightforward: Because unsystematic risk can be eliminated at virtually no cost (by diversifying), there is no reward for bearing it. Put another way, the market does not reward risks that are borne unnecessarily. Based on our study of capital market history, we know that there is a reward, on average, for bearing risk. We now need to be more precise about what we mean by risk. The systematic risk principle states that the reward for bearing risk depends only on the systematic risk of an investment. The underlying rationale for this principle is straightforward: Because unsystematic risk can be eliminated at virtually no cost (by diversifying), there is no reward for bearing it. Put another way, the market does not reward risks that are borne unnecessarily. There is an obvious corollary to this principle: No matter how much total risk an asset has, only the systematic portion is relevant in determining the expected return (and the risk premium) on that asset
If we rearrange this, then we can write the equation for the SML as: This result is the famous capital asset pricing model (CAPM).
By the way, the CAPM works for portfolios of assets just as it does for individual assets. In an earlier section, we saw how to calculate a portfolio's β. To find the expected return on a portfolio, we use this β in the CAPM equation. Figure 13.4 summarizes our discussion of the SML and the CAPM. As before, we plot expected return against beta. Now we recognize that, based on the CAPM, the slope of the SML is equal to the market risk premium, E(RM) − Rf.
A stock has a beta of 1.15, the expected return on the market is 11.3 percent, and the risk-free rate is 3.6 percent. What must the expected return on this stock be?
CAPM states the relationship between the risk of an asset and its expected return. CAPM is: E(Ri) = Rf + [E(RM) - Rf] × βi Substituting the values we are given, we find: E(Ri) = .036 + (.113 - .036)(1.15) E(Ri) = .1246, or 12.46%
Calculating Expected Return [ return
E(R) = .10(-.17) + .60(.08) + .30(.27) E(R) = .1120, or 11.20%
Calculating Expected Return [ return:
E(R) = .20(-.13) + .80(.19) E(R) = .1260, or 12.60%
You have $10,000 to invest in a stock portfolio. Your choices are Stock X with an expected return of 12.4 percent and Stock Y with an expected return of 10.1 percent. If your goal is to create a portfolio with an expected return of 10.85 percent, how much money will you invest in Stock X? In Stock Y?
E(RP) = .1085 = .124wX + .101(1 - wX) We can now solve this equation for the weight of Stock X as: .1085 = .124wX + .101 - .101wX .0075 = .023wX wX = .3261
8. Calculating Expected Returns [ LO1] A portfolio is invested 45 percent in Stock G, 40 percent in Stock J, and 15 percent in Stock K. The expected returns on these stocks are 11 percent, 9 percent, and 15 percent, respectively. What is the portfolio's expected return? How do you interpret your answer?
E(RP) = .45(.11) + .40(.09) + .15(.15) E(RP) = .1080, or 10.80%
Expected Return
E(RU) = .50 × 30% + .50 × 10% = 20% In other words, you should expect to earn 20 percent from this stock, on average. For Stock L, the probabilities are the same, but the possible returns are different. Here, we lose 20 percent half the Page 429 time, and we gain 70 percent the other half. The expected return on L, E(RL), is 25 percent: E(RL) = .50 × −20% + .50 × 70% = 25%
Using CAPM [ LO4] A stock has an expected return of 11.85 percent, its beta is 1.08, and the risk-free rate is 3.9 percent. What must the expected return on the market be?
E(Ri) = .1185 = .039 + [E(RM) - .039](1.08) E(RM) = .1126, or 11.26%
iversification [ LO2] True or false: The most important characteristic in determining the expected return of a well-diversified portfolio is the variance of the individual assets in the portfolio. Explain.
False. The variance of the individual assets is a measure of the total risk. The variance on a well-diversified portfolio is a function of systematic risk only.
sset W has an expected return of 8.8 percent and a beta of .90. If the risk-free rate is 2.6 percent, complete the following table for portfolios of Asset W and a risk-free asset. Illustrate the relationship between portfolio expected return and portfolio beta by plotting the expected returns against the betas. What is the slope of the line that results?
First, we need to find the beta of the portfolio. The beta of the risk-free asset is zero, and the weight of the risk-free asset is one minus the weight of the stock, so the beta of the portfolio is: ßP = wW(.90) + (1 - wW)(0) = .90wW So, to find the beta of the portfolio for any weight of the stock, we multiply the weight of the stock times its beta. Even though we are solving for the beta and expected return of a portfolio of one stock and the risk-free asset for different portfolio weights, we are really solving for the SML. Any combination of this stock, and the risk-free asset will fall on the SML. For that matter, a portfolio of any stock and the risk-free asset, or any portfolio of stocks, will fall on the SML. We know the slope of the SML line is the market risk premium, so using the CAPM and the information concerning this stock, the market risk premium is: E(RW) = .088 = .026 + MRP(.90) MRP = .062/.90 MRP = .0689, or 6.89% So, now we know the CAPM equation for any stock is: E(RP) = .026 + .0689βP The slope of the SML is equal to the market risk premium, which is .0689. Using these equations to fill in the table, we get the following results:
DIVERSIFICATION AND UNSYSTEMATIC RISK
From our discussion of portfolio risk, we know that some of the risk associated with individual assets can be diversified away and some cannot. We are left with an obvious question: Why is this so? It turns out that the answer hinges on the distinction we made earlier between systematic and unsystematic risk. Here is the important observation: If we held only a single stock, the value of our investment would fluctuate because of company-specific events. If we hold a large portfolio, on the other hand, some of the stocks in the portfolio will go up in value because of positive company-specific events and some will go down in value because of negative events. The net effect on the overall value of the portfolio will be relatively small because these effects will tend to cancel each other out. Now we see why some of the variability associated with individual assets is eliminated by diversification. When we combine assets into portfolios, the unique, or unsystematic, events—both positive and negative—tend to "wash out" once we have more than just a few assets. Unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk.
Assume these securities are correctly priced. Based on the CAPM, what is the expected return on the market? What is the risk-free rate?
Here we have the expected return and beta for two assets. We can express the returns of the two assets using CAPM. If the CAPM is true, then the security market line holds as well, which means all assets have the same risk premium. Setting the risk premiums of the assets equal to each other and solving for the risk-free rate, we find: (.108 - Rf)/1.25 = (.082 - Rf)/.87 .87(.108 - Rf) = 1.25(.082 - Rf) .09396 - .87Rf = .1025 - 1.25Rf .38Rf = .00854 Rf = .0225, or 2.25% Now using CAPM to find the expected return on the market with both stocks, we find: .108 = .0225 + 1.25(RM - .0225) .082 = .0225 + .87(RM - .0225) RM = .0909, or 9.09% RM = .0909, or 9.09%
suppose the government announces that, based on a just-completed survey, the growth rate in the economy is likely to be 2 percent in the coming year, as compared to 5 percent for the past year. Will security prices increase, decrease, or stay the same following this announcement? Does it make any difference whether the 2 percent figure was anticipated by the market? Explain.
If the market expected the growth rate in the coming year to be 2 percent, then there would be no change in security prices if this expectation had been fully anticipated and priced. However, if the market had been expecting a growth rate other than 2 percent and the expectation was incorporated into security prices, then the government's announcement would most likely cause security prices in general to change; prices would drop if the anticipated growth rate had been more than 2 percent, and prices would rise if the anticipated growth rate had been less than 2 percent.
Risk Premium
In our previous chapter, we defined the risk premium as the difference between the return on a risky investment and that on a risk-free investment, and we calculated the historical risk premiums on some different investments. Using our projected returns, we can calculate the projected, or expected, risk premium as the difference between the expected return on a risky investment and the certain return on a risk-free investment.
Market Portfolios
It will be very useful to know the equation of the SML. There are many different ways we could write it, but one way is particularly common. Suppose we consider a portfolio made up of all of the assets in the market. Such a portfolio is called a market portfolio, and we will express the expected return on this market portfolio as E(RM). Because all the assets in the market must plot on the SML, so must a market portfolio made up of those assets. To determine where it plots on the SML, we need to know the beta of the market portfolio, βM. Because this portfolio is representative of all of the assets in the market, it must have average systematic risk. In other words, it has a beta of 1. We could express the slope of the SML as: look at image The term E(RM) − Rf is often called the market risk premium because it is the risk premium on a market portfolio.
If a portfolio has a positive investment in every asset, Page 458 can the expected return on the portfolio be greater than that on every asset in theportfolio? Can it be less than that on every asset in the portfolio? If you answer yes to one or both of these questions, give an example to support your answer.
No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return.
ortfolio Expected Return [ LO1] You own a portfolio that has $4,450 invested in Stock A and $9,680 invested in Stock B. If the expected returns on these stocks are 8 percent and 11 percent, respectively, what is the expected return on the portfolio?
So, the expected return of this portfolio is: E(RP) = ($4,450/$14,130)(.08) + ($9,680/$14,130)(.11) E(RP) = .1006, or 10.06%
Asset A vs B
The Fundamental Result The situation we have described for Assets A and B could not persist in a well-organized, active market, because investors would be attracted to Asset A and away from Asset B. As a result, Asset A's price would rise and Asset B's price would fall. Because prices and returns move in opposite directions, A's expected return would decline and B's would rise. This buying and selling would continue until the two assets plotted on exactly the same line, which means they would offer the same reward for bearing risk.
THE COST OF CAPITAL
The appropriate discount rate on a new project is the minimum expected rate of return an investment must offer to be attractive. This minimum required return is often called the cost of capital associated with the investment. It is called this because the required return is what the firm must earn on its capital investment in a project just to break even. It can be interpreted as the opportunity cost associated with the firm's capital investment. Notice that when we say an investment is attractive if its expected return exceeds what is offered in financial markets for investments of the same risk, we are effectively using the internal rate of return (IRR) criterion that we developed and discussed in Chapter 9. The only difference is that now we have a much better idea of what determines the required return on an investment. This understanding will be critical when we discuss cost of capital and capital structure in Part 6 of our book.
You own a stock portfolio invested 15 percentin Stock Q, 20 percent in Stock R, 30 percent in Stock S, and 35 percent in Stock T. Thebetas for these four stocks are .79, 1.23, 1.13, and 1.36, respectively. What is the portfolio beta?
The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. So, the beta of the portfolio is: βP = .15(.79) + .20(1.23) + .30(1.13) + .35(1.36) βP = 1.18
Portfolio Expected Return [ LO1] You own a portfolio that is invested 15 percent in Stock X, 35 percent in Stock Y, and 50 percent in Stock Z. The expected returns on these three stocks are 9 percent, 15 percent, and 12 percent, respectively. What is the expected return on the portfolio?
The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is: E(RP) = .15(.09) + .35(.15) + .50(.12) E(RP) = .1260, or 12.60%
THE SECURITY MARKET LINE
The line that results when we plot expected returns and beta coefficients is obviously of some importance, so it's time we gave it a name. This line, which we use to describe the relationship between systematic risk and expected return in financial markets, is usually called the security market line (SM
MEASURING SYSTEMATIC RISK
The specific measure we will use is called the beta coefficient, for which we will use the Greek symbol β. A beta coefficient, or beta for short, tells us how much systematic risk a particular asset has relative to an average asset. By definition, an average asset has a beta of 1.0 relative to itself. An asset with a beta of .50, therefore, has half as much systematic risk as an average asset; an asset with a beta of 2.0 has twice as much The important thing to remember is that the expected return, and thus the risk premium, of an asset depends only on its systematic risk. Because assets with larger betas have greater systematic risks, they will have greater expected returns. From Table 13.8, an investor who buys stock in Johnson & Johnson, with a beta of .70, should expect to earn less, on average, than an investor who buys stock in Sherwin-Williams, with a beta of about 1.23..
Risk: Systematic and Unsystematic
The unanticipated part of the return, that portion resulting from surprises, is the true risk ofany investment. After all, if we always receive exactly what we expect, then the investment is perfectly predictable and, by definition, risk-free. In other words, the risk of owning an asset comes from surprises—unanticipated events. The first type of surprise—the one that affects many assets—we will label systematic risk. A systematic risk is one that influences a large number of assets, each to a greater or lesser extent. Because systematic risks have marketwide effects, they are sometimes called market risks. The second type of surprise we will call unsystematic risk. An unsystematic risk is one that affects a single asset or a small group of assets. Because these risks are unique to individual companies or assets, they are sometimes called unique or asset-specific risks. We will use these terms interchangeably As we have seen, uncertainties about general economic conditions (such as GDP, interest rates, or inflation) are examples of systematic risks. These conditions affect nearly all companies to some degree. An unanticipated increase, or surprise, in inflation, for example, affects wages and the costs of the supplies that companies buy; it affects the value of the assets that companies own; and it affects the prices at which companies sell their products. Forces such as these, to which all companies are susceptible, are the essence of systematic risk. In contrast, the announcement of an oil strike by a company will primarily affect that company and, perhaps, a few others (such as primary competitors and suppliers). It is unlikely to have much of an effect on the world oil market, or on the affairs of companies not in the oil business, so this is an unsystematic event The distinction between the types of risk allows us to break down the surprise portion, U, of the return on the Flyers stock into two parts. Earlier, we had the actual return broken down into its expected and surprise components: R = E(R) + U We now recognize that the total surprise component for Flyers, U, has a systematic and an unsystematic component, so: R = E(R) + Systematic portion + Unsystematic portion
portfolio weights.
There are many equivalent ways of describing a portfolio. The most convenient approach is to list the percentage of the total portfolio's value that is invested in each portfolio asset. We call these percentages the portfolio weights. For example, if we have $50 in one asset and $150 in another, our total portfolio is worth $200. The percentage of our portfolio in the first asset is $50/$200 = .25. The percentage of our portfolio in the second asset is $150/$200, or .75. Our portfolio weights are .25 and .75
Reward-to-Risk Ratios [ LO4] Stock Y has a beta of 1.2 and an expected return of 11.5 percent. Stock Z has a beta of .80 and an expected return of 8.5 percent. If the risk-free rate is 3.2 percent and the market risk premium is 6.8 percent, are these stocks correctly priced?
There are two ways to correctly answer this question so we will work through both. First, we can use the CAPM. Substituting in the value we are given for each stock, we find: E(RY) = .032 + .068(1.20) E(RY) = .1136, or 11.36% It is given in the problem that the expected return of Stock Y is 11.5 percent, but according to the CAPM the expected return of the stock should be 11.36 percent based on its level of risk. This means the stock return is too high, given its level of risk. Stock Y plots above the SML and is undervalued. In other words, its price must increase to reduce the expected return to 11.36 percent. For Stock Z, we find: E(RZ) = .032 + .068(.80) E(RZ) = .0864, or 8.64% The return given for Stock Z is 8.5 percent, but according to the CAPM the expected return of the stock should be 8.64 percent based on its level of risk. Stock Z plots below the SML and is overvalued. In other words, its price must decrease to increase the expected return to 8.64 percent. We can also answer this question using the reward-to-risk ratio. All assets must have the same reward-to-risk ratio. The reward-to-risk ratio is the risk premium of the asset divided by its beta. We are given the market risk premium, and we know the beta of the market is one, so the reward-to-risk ratio for the market is .068, or 6.8 percent. Calculating the reward-to-risk ratio for Stock Y, we find: Reward-to-risk ratio Y = (.1150 - .032)/1.20 Reward-to-risk ratio Y = .0692, or 6.92% The reward-to-risk ratio for Stock Y is too high, which means the stock plots above the SML, and the stock is undervalued. Its price must increase until its reward-to-risk ratio is equal to the market reward-to-risk ratio. For Stock Z, we find: Reward-to-risk ratio Z = (.0850 - .032)/.80 Reward-to-risk ratio Z = .0663, or 6.63% The reward-to-risk ratio for Stock Z is too low, which means the stock plots below the SML, and the stock is overvalued. Its price must decrease until its reward-to-risk ratio is equal to the market
Your portfolio is invested 30 percent each in A and C, and 40 percent in B. What is the expected return of the portfolio? b. What is the variance of this portfolio? The standard deviation?
This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: RP = .30(.35) + .40(.40) + .30(.28) = .3490, or 34.90% Good: RP = .30(.16) + .40(.17) + .30(.09) = .1430, or 14.30% Poor: RP = .30(-.01) + .40(-.03) + .30(.01) = -.0120, or -1.20% Bust: RP = .30(-.10) + .40(-.12) + .30(-.09) = -.1050, or -10.50% And the expected return of the portfolio is: E(RP) = .15(.3490) + .45(.1430) + .30(-.0120) + .10(-.1050) E(RP) = .1026, or 10.26% b. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, then add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio are: σP2 = .15(.3490 - .1026)2 + .45(.1430 - .1026)2 + .30(-.0120 - .1026)2 + .10(-.1050 - .1026)2 σP2 = .01809 σP = .018091/2 σP = .1345, or 13.45%
The reward-to-risk ratio must be the same for all the assets in the market.
This result is really not so surprising. What it says is that, for example, if one asset has twice as much systematic risk as another asset, its risk premium will be twice as large. Because all of the assets in the market must have the same reward-to-risk ratio, they all must plot on the same line. This argument is illustrated in Figure 13.3. As shown, Assets A and B plot directly on the line and have the same reward-to-risk ratio. If an asset plotted above the line, such as C inFigure 13.3, its price would rise and its expected return would fall until it plotted exactly on the line. Similarly, if an asset plotted below the line, such as D in Figure 13.3, its expected return would rise until it, too, plotted directly on the line.
What is the expected return on an equally weighted portfolio of these three stocks?b. What is the variance of a portfolio invested 20 percent each in A and B and 60 percent in C?
To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. This portfolio is a special case since all three assets have the same weight. To find the expected return of an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the return of the portfolio in each state of the economy is: Boom: RP = (.07 + .18 + .27)/3 = .1733, or 17.33% Bust: RP = (.12 - .08 − .21)/3 = -.0567, or -5.67% To find the expected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum the products. Doing so, we find: E(RP) = .75(.1733) + .25(-.0567) E(RP) = .1158, or 11.58% Part II Boom: RP = .20(.07) +.20(.18) + .60(.27) = .2120, or 21.20% Bust: RP = .20(.12) +.20(-.08) + .60(−.21) = -.1180, or -11.80% And the expected return of the portfolio is: E(RP) = .75(.2120) + .25(−.1180) E(RP) = .1295, or 12.95% To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance of the portfolio is: σP2 = .75(.2120 - .1295)2 + .25(−.1180 - .1295)2 σP2 = .020419
Determining Portfolio Weights [ LO1] What are the portfolio weights for a portfolio that has 145 shares of Stock A that sell for $47 per share and 200 shares of Stock B that sell for $21 per share?
Total portfolio value = 145($47) + 200($21) Total portfolio value = $11,015 The portfolio weight for each stock is: WeightA = 145($47)/$11,015 = .6187 WeightB = 200($21)/$11,015 = .3813
Analyzing a Portfolio [ LO2, 4] You have $100,000 to invest in a portfolio containing Stock X and Stock Y. Your goal is to create a portfolio that has an expected return of 12.1 percent. If Stock X has an expected return of 10.28 percent and a beta of 1.20, and Stock Y has an expected return of 7.52 percent and a beta of .80, how much money will you invest in Stock Y? How do you interpret your answer? What is the beta of your portfolio?
We are given the expected return of the assets in the portfolio. We also know the sum of the weights of each asset must be equal to one. Using this relationship, we can express the expected return of the portfolio as: E(RP) = .1210 = wX(.1028) + wY(.0752) .1210 = wX(.1028) + (1 - wX)(.0752) .1210 = .1028wX + .0752 - .0752wX .0458 = .0276wX wX = 1.6594 And the weight of Stock Y is: wY = 1 - 1.6594 wY = -.6594 The amount to invest in Stock Y is: Investment in Stock Y = -.6594($100,000) Investment in Stock Y = -$65,942.03 A negative portfolio weight means that you short sell the stock. If you are not familiar with short selling, it means you borrow a stock today and sell it. You must then purchase the stock at a later date to repay the borrowed stock. If you short sell a stock, you make a profit if the stock decreases in value. To find the beta of the portfolio, we can multiply the portfolio weight of each asset times its beta and sum. So, the beta of the portfolio is: βP = 1.6594(1.20) + (-.6594)(.80) βP = 1.464
A stock has an expected return of 11.4 percent, the risk-free rate is 3.9 percent, and the market risk premium is 6.8 percent. What must the beta of this stock be?
We are given the values for the CAPM except for the beta of the stock. We need to substitute these values into the CAPM, and solve for the beta of the stock. One important thing we need to realize is that we are given the market risk premium. The market risk premium is the expected return of the market minus the risk-free rate. We must be careful not to use this value as the expected return of the market. Using the CAPM, we find: E(Ri) = .114 = .039 + .068βi βi = 1.10
24. Analyzing a Portfolio [ LO2, 4] You want to create a portfolio equally as risky as the market, and you have $1,000,000 to invest. Given this information, fill in the rest of the following table:
We know the total portfolio value and the investment in two stocks in the portfolio, so we can find the weight of these two stocks. The weights of Stock A and Stock B are: wA = $195,000/$1,000,000 = .195 wB = $365,000/$1,000,000 = .365 Since the portfolio is as risky as the market, the beta of the portfolio must be equal to one. We also know the beta of the risk-free asset is zero. We can use the equation for the beta of a portfolio to find the weight of the third stock. Doing so, we find: βP = 1 = wA(.80) + wB(1.09) + wC(1.23) + wRf(0) 1 = .195(.80) + .365(1.09) + wC(1.23) Solving for the weight of Stock C, we find: wC = .36272358 So, the dollar investment in Stock C must be: Investment in Stock C = .36272358($1,000,000) Investment in C = $362,723.58 We also know the total portfolio weight must be one, so the weight of the risk-free asset must be one minus the asset weights we know, or: 1 = wA + wB + wC + wRf wRf = 1 - .195 - .365 - .36272358 wRf = .07727642 So, the dollar investment in the risk-free asset must be: Investment in risk-free asset = .07727642($1,000,000) Investment in risk-free asset = $77,276.42
Reward-to-Risk Ratios [ LO4] In the previous problem, what would the risk-free rate have to be for the two stocks to be correctly priced?
We need to set the reward-to-risk ratios of the two assets equal to each other, which is: (.1150 - Rf)/1.20 = (.0850 - Rf)/.80 We can cross multiply to get: .80(.1150 - Rf) = 1.20(.0850 - Rf) Solving for the risk-free rate, we find: .0920 - .80Rf = .1020 - 1.20Rf Rf = .0250, or 2.50%
DIVERSIFICATION AND SYSTEMATIC RISK
We've seen that unsystematic risk can be eliminated by diversifying. What about systematic risk? Can it also be eliminated by diversification? The answer is no because, by definition, a systematic risk affects almost all assets to some degree. As a result, no matter how many assets we put into a portfolio, the systematic risk doesn't go away. For obvious reasons, the terms systematic risk and nondiversifiable risk are used interchangeably. Total risk = Systematic risk + Unsystematic risk Systematic risk is also called nondiversifiable risk or market risk. Unsystematic risk is also called diversifiable risk, unique risk, or asset-specific risk. For a well-diversified portfolio, the unsystematic risk is negligible. For such a portfolio, essentially all of the risk is systematic.
Slope
What is the slope of the straight line in Figure 13.2A? As always, the slope of a straight line is equal to "the rise over the run." In this case, as we move out of the risk-free asset into Asset A, the beta increases from zero to 1.6 (a "run" of 1.6). At the same time, the expected return goes from 8 percent to 20 percent, a "rise" of 12 percent. The slope of the line is thus 12%/1.6 = 7.5%. What this tells us is that Asset A offers a reward-to-risk ratio of 7.5 percent. Asset A has a risk premium of 7.5 percent per "unit" of systematic risk.
THE PRINCIPLE OF DIVERSIFICATION
What we have plotted is the standard deviation of return versus the number of stocks in the portfolio at the benefit in terms of risk reduction from adding securities drops off as we add more and more. By the time we have 10 securities, most of the effect is already realized; and by the time we get to 30 or so, there is little remaining benefit. First, some of the riskiness associated with individual assets can be eliminated by forming portfolios. The process of spreading an investment across assets (and thereby forming a portfolio) is called diversification. The principle of diversification tells us that spreading an investment across many assets will eliminate some of the risk. The blue shaded area in Figure 13.1, labeled "diversifiable risk," is the part that can be eliminated by diversification. he second point is equally important. There is a minimum level of risk that cannot be eliminated by diversifying. This minimum level is labeled "nondiversifiable risk" in Figure 13.1. Taken together, these two points are another important lesson from capital market history: Diversification reduces risk, but only up to a point. Put another way, some risk is diversifiable and some is not.
Portfolio Risk [ LO2] If a portfolio has a positive investment in every asset, can the standard deviation on the portfolio be less than that on every asset in the portfolio? What about the portfolio beta?
Yes, the standard deviation can be less than that of every asset in the portfolio. However, βP cannot be less than the smallest beta because βP is a weighted average of the individual asset betas
s it possible that a risky asset could have a beta of zero? Explain. Based on the CAPM, what is the expected return on such an asset? Is it possible that a risky asset could have a negative beta? What does the CAPM predict about the expected return on such an asset? Can you give an explanation for your answer?
Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument.
The government announces that inflation unexpectedly jumped by 2 percent last month. Big Widget's quarterly earnings report, just issued, generally fell in line with analysts' expectations. The government reports that economic growth last year was at 3 percent, which generally agreed with most economists' forecasts. The directors of Big Widget die in a plane crash. Congress approves changes to the tax code that will increase the top marginal corporate tax rate. The legislation had been debated for the previous six months.
a change in systematic risk has occurred; market prices in general will most likely decline. b. no change in unsystematic risk; company price will most likely stay constant. c. no change in systematic risk; market prices in general will most likely stay constant. d. a change in unsystematic risk has occurred; company price will most likely decline. e. no change in systematic risk; market prices in general will most likely stay constant assuming the market believed the legislation would be passed.
Efficient Markets
efficient markets are a result of investors trading on the unexpected portion of announcements efficient markets involve random price changes because we cannot predict surprises Our discussion of market efficiency in the previous chapter bears on this discussion. We are assuming that relevant information known today is already reflected in the expected return. This is identical to saying that the current price reflects relevant publicly available information. We are implicitly assuming that markets are at least reasonably efficient in the semistrong form. Henceforth, when we speak of news, we will mean the surprise part of an announcement and not the portion that the market has expected and therefore already discounted. Announcement = Expected part + Surprise
Using CAPM [ LO4] A stock has an expected return of 10.45 percent, its beta is .85, and the expected return on the market is 11.8 percent. What must the risk-free rate be?
ere we need to find the risk-free rate using the CAPM. Substituting the values given, and solving for the risk-free rate, we find: E(Ri) = .1045 = Rf + (.118 - Rf)(.85) .1045 = Rf + .1003 - .85Rf Rf = .0280, or 2.80%
THE EFFECT OF DIVERSIFICATION: ANOTHER LESSON FROM MARKET HISTORY
o illustrate the relationship between portfolio size and portfolio risk, Table 13.7 illustrates typical average annual standard deviations for equally weighted portfolios that contain different numbers of randomly selected NYSE securities. In Column 2 of Table 13.7, we see that the standard deviation for a "portfolio" of one security is about 49 percent. What this means is that if you randomly selected a single NYSE stock and put all your money into it, your standard deviation of return would typically be a substantial 49 percent per year. If you were to randomly select two stocks and invest half your money in each, your standard deviation would be about 37 percent on average, and so on. The important thing to notice in Table 13.7 is that the standard deviation declines as the number of securities is increased. By the time we have 100 randomly chosen stocks, the portfolio's standard deviation has declined by about 60 percent, from 49 percent to about 20 percent
lassify the following events as mostly systematic or mostly unsystematic. Is the distinction clear in every case? Short-term interest rates increase unexpectedly. The interest rate a company pays on its short-term debt borrowing is increased by its bank. Oil prices unexpectedly decline. An oil tanker ruptures, creating a large oil spill. A manufacturer loses a multimillion-dollar product liability suit. A Supreme Court decision substantially broadens producer liability for injuries suffered by product users.
systematic b. unsystematic c. both; probably mostly systematic d. unsystematic e. unsystematic f. systematic
7. Calculating Returns and Standard Deviations [ LO1] Based on the following information, calculate the expected return and standard deviation for Stock A and Stock B:
σA2 = .15(.04 - .1065)2 + .55(.09 - .1065)2 + .30(.17 - .1065)2 σA2 = .00202 σA = .002021/2 σA = .0450, or 4.50% σB2 = .15(-.17 - .1215)2 + .55(.12 - .1215)2 + .30(.27 - .1215)2 σB2 = .01936 σB = .019361/2 σB = .1392, or 13.92%
