files.transtutors.com · web view6. portfolio theory. objectives: after reading this chapter, you...

110

6. PORTFOLIO THEORY

Objectives: After reading this chapter, you will be able to

1. Understand the basic reason for portfolio formation.2. Calculate the risk and return characteristics of a portfolio.

6.1 Video 0 6 A Portfolio Formation

A portfolio is a collection of projects, or securities, or investments, held together as a bundle. For example, you may buy 100 shares of Boeing, 200 shares of Microsoft, and 5 PP&L bonds in an account. This is your portfolio of investments. If you own a home, then you may include the value of the house in your portfolio. A portfolio may also include less tangible items as your professional education, or even a license to practice law. The total value of your portfolio may fluctuate with time.

Corporations also have a portfolio of different projects. They carefully select profitable projects and invest in them. The banks loan money to individuals and corporations. They have a loan portfolio, and they try to monitor the quality of their portfolios. The quality of a loan portfolio can deteriorate if too many loans are non-performing.

Why do people, or corporations, form a portfolio? The simple answer is diversification. You do not want to put all your eggs in one basket. It is a good idea to diversify your risk, and if some of the investments do not pan out, the others will keep the value of the portfolio intact.

6.2 Portfolio Theory

We can form a portfolio by carefully selecting a set of securities. The two main features of a portfolio are its risk and expected return. In 1952, H a r r y M a rko w itz first developed the ideas of portfolio theory based on statistical reasoning. He showed that one could reduce the risk for a given return by putting together unrelated or negatively correlated securities.

111

Harry Markowitz (1927- )

We may define the realized return of a single security as the sum of price appreciation and the dividends, divided by the acquisition cost of the security. For example, if we buy a stock at $50 a share, get a $2 dividend on it, and then sell it for $52, then the return, in dollars, is $4. The return, as a percentage, is 4/50 = 8%. In general, we may define the return as

P1 − P 0 + D 1

R = P0(6.1)

ij

Analytical Techniques 6. Portfolio Theory

112

where the purchase price of the stock is P0, its selling price is P1, and D1 is the dividend paid, if any, at the end. The quantity P1 − P0 is the price appreciation of the stock, and along with the dividend, is the total change in the value of the investment.

Modifying (6.1), we can represent the expected return of a security as

E(R) =E( P 1 ) − P 0 + E ( D 1 )

P0(6.2)

where E(•) is the expectations operator. Suppose we have a probability distribution pi, with i = 1 ... n, describing n states of the economy and we also have the returns Ri under each state, then

n –E(R) = piRi = R

i=1

(6.3)

The second component of any investment is the amount of risk inherent in that investment. We may use the standard deviation of return, σ(R), as a measure of risk. This is because the standard deviation of a random outcome represents the uncertainty, or spread, in that random variable. For a stock, it may represent the risk of that stock investment. Using the notation of (1.6 − 1.8), we write

n – 1/2

σ(R) = [pi(Ri −

R)2]i=1

(6.4)

To quantify the dependence of one stock on the other, recall equation (1.10) which defines the correlation coefficient, rij. Mathemathically,

r = c ov ( R i, R j)

(6.5)σ(Ri)σ(Rj)

For any two securities that are completely unrelated, the correlation coefficient between them is zero, rij = 0. For perfectly positively correlated securities, rij = 1, and


113

for those that are perfectly negatively correlated, rij = −1. In real life, most of the securities are partially positively correlated with one another.

6.3 Two-Security Portfolio

Let us first consider the simplest portfolio, the one that has only two securities in it, say the stock of two major corporations, GM and IBM. If we add up all the weights of securities in a given portfolio, the sum should be equal to one. For instance, if a portfolio has 75% assets invested in GM and 25% in IBM, then 0.75 + 0.25 = 1. In general,

w1 + w2 = 1 (6.6)


114

The return defined by (6.1) is the realized return. This is in the past. What is the future return of the stock? That is more difficult to predict. However, we can represent the expected return by the symbol E(R). Similarly, we can represent the expected return of a portfolio by E(Rp). For any two-security portfolio, the expected return of the portfolio, E(Rp), will depend upon the return of the two securities, and their weights. Indeed, the return is the weighted average of the returns of the individual securities. That is,

E(Rp) = w1 E(R1) + w2 E(R2) (6.7)

What is the risk of a two-security portfolio? The total risk will depend upon the weights of the two securities. If more money is invested in GM, then the portfolio risk will tend to be closer to the risk of GM.

Let us look at equation (6.4). This equation represents the risk of a portfolio. One way to measure risk is to use the standard deviation of returns. σ(R). For more risky securities, the standard deviation, or the spread of returns is higher. For less risky securities, the spread is less, meaning we are more confident what the return will be. For risk-free securities, the σ is zero.

For a two-security portfolio, the risk comes from both the securities. However, you cannot add risk linearly. In general, two units of risk of one security plus two units of risk from another security does not add up to four units of risk. In fact, the risk of one security may partially cancel the risk of another security when you hold them together as a portfolio. The greater is the diversification of the portfolio, the lesser is the risk of the portfolio.

One could measure diversification in terms of correlation coefficient, r12 between two stocks. You get greater diversification if you have stocks, which are unrelated to one another. In other words, their correlation coefficient is smaller.

To begin with, σ is a non-linear quantity. As seen in (6.4), you have to find it by taking the square root of a sum of squares. When you add the sigmas of two securities, you cannot say that the sigma of the portfolio is the sum of their individual sigmas. You have to add them by squaring them first, then adding them, and then you have to take the square root. It also depends on the weights of the securities. Finally, risk of the portfolio, σ(Rp) depends on their correlation coefficient, r12. It does become complicated. Using statistical theory, one can prove that (6.8) represents the risk of a portfolio correctly. Including the weights and risks of the two securities, and the correlation coefficient between them, we can write the total risk of a portfolio as follows:

1 σ

σ

1

2


115

σ(Rp) = w 22 +

w22

2 + 2w1w2 σ1σ2r12 (6.8)

Going from a two- to a three-security portfolio complicates the problem further. In this case, we have to consider the correlation coefficient between the first and the second stock, between the first and the third stock, and between the second and the third stock. That is, the formula should have r12, r13, and r23. This is what we find in equation (6.11).


116

It is better to use a computer program that can handle such equations to optimize the formation or a real portfolio with perhaps thirty stocks.

6.4 Portfolios with Three or More Securities

We can readily extend the results for the two-security portfolio to the portfolios with three or more securities. For a three-security portfolio, equations (6.6-8) become

w1 + w2 + w3 = 1 (6.9)

E(Rp) = w1 E(R1) + w2 E(R2) + w3 E(R3) (6.10)

2 2 2 2 2 2

σ(Rp) = w1 σ1 + w2 σ2

+ w3 σ3 + 2w1w2σ1σ2r12 + 2w1w3σ1σ3r13 + 2w2w3σ2σ3r23

(6.11)

We can extend our analysis for a portfolio with n securities. To summarize, let us define:

E(Rp) = expected return of the portfolio E(Ri) = expected return of the security iwi = weight of the security i, as a percentage of the total value of the portfolio σ(Rp) = standard deviation of the returns of the portfolioσi = standard deviation of the returns of the security i cov(i,j) = covariance between the returns of securities i and j rij = correlation coefficient between the securities i and j.

For a portfolio with n securities, (6.8) will become

We may write it as

w1 + w2 + w3 + ... + wn = 1

n

wi = 1 (6.12)

i=1


117

For an n-security portfolio, the expected return is

or,

E(Rp) = w1 E(R1) + w2 E(R2) + w3 E(R3) + ... + wn E(Rn)

n

E(Rp) = wi E(Ri) (6.13)

i=1

By definition, the covariance between the returns of the securities i and j is equal to the product of the correlation coefficient between these securities and the standard deviations of the returns of these two securities. Mathematically, we can write it as

For instance, we may write

cov(i,j) = σiσjrij (6.14)

w 3 3


118

σ12r11 = var(1) = cov(1,1) and σ1σ2r12 = cov(1,2)

Let us write (6.11) as follows:

2

2 1/2

w1 1

+ w1w212r12

2

2

+ w1w313r13

(Rp) = w2w121r21 + w2 2 + w2w323r23

w3w131r31 + w3w232r32 + 2 2

The terms along the principal diagonal are the variance terms, and those off the diagonal are the covariance ones. We can sum the terms in each row by using sigma notation with j as an index, and then sum the rows using i as the index. Finally, we get for n securities,

n n 1/2

σ(Rp) =

[wiwjcov(i,j)]i=1 j=1

(6.15)

If a portfolio is composed of projects whose expected returns Ri and their standard deviations i are all expressed in dollar amounts then we do not look at the weights of the projects. That is, we drop the w's from the previous equations. We may write (6.6) and (6.7) as

E(Rp) = E(R1) + E(R2) (6.16)

σ(Rp) = σ12 + σ2

2 + 2 σ1σ2r12 (6.17)

For an n-security portfolio, with dollar amounts of investments and returns, we rewrite (6.12) and (6.14) as

n

E(Rp) = E(Ri) (6.18)

i=1


119

n n 1/2

Examples

σ(Rp) =

[cov(i,j)]i=1 j=1

(6.19)

Video 06 . 01 6.1. Cooper Corporation has the opportunity to invest in two of the following three proposals. Which two projects should the company select, if the company wants to maximize the ratio between expected NPV and the standard deviation?

Project A Project B Project CExpected NPV $10,000 $11,000 $9,000

Standard deviation $2,000 $1,900 $1,500Corr. coeff. between (A,B) = .4 (A,C) = .5 (B,C) = .8

In this problem, we regard the expected NPV as the expected return. The project costs and returns are in dollars. For a two-security portfolio, with investments in projects 1 and 2, the sigma of the portfolio is given by


120

σ(Rp) = σ12 + σ2

2 + 2 σ1σ2r12 (6.17)

For the projects A + B, NPV = $21,000

σ = 20002 + 19002 + 2(0.4)(2000)(1900) = $3263

NPV/σ = 21,000/3263 = 6.4349

For A + C, NPV = $19,000

σ = 20002 + 15002 + 2(0.5)(2000)(1500) = $3041

NPV/σ = 19,000/3041 = 6.2472

For B + C, NPV = $20,000

σ = 19002 + 15002 + 2(0.8)(1900)(1500) = $3228

NPV/σ = 20,000/3228 = 6.1958

Cooper corporation should invest in projects A and B. ♥

6.2. Suppose you want to invest $40,000 in ExxonMobil, whose expected return is 15% with standard deviation 20%, and $10,000 in Boeing whose expected return is 18% with standard deviation 21%. The correlation coefficient between the securities is 0.75. Find the expected total value of your portfolio after one year, and its standard deviation in dollars.

The total value of the portfolio is 40,000 + 10,000 = $50,000. The weight of the first security is 40,000/50,000 = .8. The weight of the second security is thus .2. Therefore,

w1 = 0.8, w2 = 0.2.

The expected return of the portfolio comes out to


121

be

E(Rp) = 0.8(0.15) + 0.2(0.18) = 0.156

The standard deviation of the return of portfolio is

σ(Rp) = (.8)2(.2)2 + (.2)2(.21)2 + 2(.8)(.2)(.21)(.2)(.75) = .1935

To find the expected value of the portfolio after one year, and its sigma, in dollars, we calculate

E(V) = $50,000 (1.156) = $57,800 ♥and σ(V) = $50,000 (0.1935) = $9,675 ♥


122

6.3. Suppose you have $40,000 that you would like to invest equally in four securities A, B, C, and D. The expected returns from these securities are 10%, 11%, 12%, and 13%, respectively. The standard deviations of these returns are 12%, 14%, 16%, and 18%, respectively. The correlation coefficient between any two securities is 0.8. If the returns are normally distributed, what is the probability that the portfolio will be worth more than$50,000 after one year?

Since you want to invest your money equally among four securities, the weight of each security is 25%. This means w1 = w2 = w3 = w4 = .25. Because the correlation coefficient between any two securities is the same, 0.8, we have r12 = r13 = r14 = r23 = r24 = r34 = 0.8. Find E(Rp) by multiplying the weight of each security by its expected return, and then adding everything.

E(Rp) = 0.25(0.1) + 0.25(0.11) + 0.25(0.12) + 0.25(0.13) = 0.115 = 11.5%

Similarly, find the σ of the portfolio as follows,

2 2 2 2 2 2 2 2

(.25) (.12) + (.25) (.14) + (.25) (.16) + (.25) (.18) 1/2

σ(Rp) =

+ 2(.25)2(.12)(.14)(.8) + 2(.25)2(.12)(.16)(.8)+ 2(.25)2(.12)(.18)(.8) + 2(.25)2(.14)(.16)(.8)+ 2(.25)2(.14)(.18)(.8) + 2(.25)2(.16)(.18)(.8)

= 0.1384

If we require the portfolio to be worth more than $50,000, then the required return is (50,000 − 40,000)/40,000 = 0.25, or 25%. The expected return of the portfolio is 11.5%, it quite unlikely that the return will exceed 25%. To calculate the probability, find

z = (0.25 − 0.115)/0.1384 = 0.9755


123

Figure 6.1. The shaded area to the right of z = .9755 represents the probability that the return of the portfolio will be more than 25%.

Draw a normal probability distribution curve, with z = 0 in the center and z = .9755 to the right of center. Since the expected portfolio return is 11.5%, and we require it to have a return of more than 25%, it is unlikely that it will actually happen. The probability of it is the area under the tail of the curve, on the right side of z = .9755. Using the tables, calculate the probability as


124

P(R > 0.25) = .5 − [.3340 + .55(.3365 − .3340)] = 0.1643 = 16.43%

There is a 16.43% chance that the portfolio is worth more than $50,000 after one year. ♥

The following instructions will solve the problem in Excel.

A B C D E1 E(R) 0.1 .11 .12 .132 σ(R) 0.12 .14 .16 .183 Corr Coeff 0.84 Weights 0.255 E(Rp) =B4*(B1+C1+D1+E1)6 σ(Rp) =B4*SQRT(B2^2+C2^2+D2^2+E2^2+2*B3*(B2*C2+B2*D2+B2*E2+C2*D2+C2*E2+D2*E2))7 Req retn =(50000-40000)/400008 z =(B7-B5)/B69 Probability =1-NORMDIST(B8,0,1,true)

6.4. Capella Corporation has an expected return of 12% and sigma .25, the expected return of Rigel Corporation is 15% and its sigma 0.30. The coefficient of correlation between the two companies is 0.25. Make a portfolio of these stocks so that the expected return of the portfolio is 13%. What is the sigma of the portfolio?

We have to make a portfolio such that its expected return is 13%, but we do not know the weights of the two securities. To find the weights, w1 and w2, you will need two equations. The two equations are

and

w1 + w2 = 1 (6.6)

E(Rp) = w1 E(R1) + w2 E(R2) (6.7)

Substituting numbers in (6.6), we get

.13 = w1 (.12) + w2 (.15) (A)

To facilitate the calculation at W olf ra mAlph a , subsitute w1 = x and w2 = y, then use this expression,

WRA x+y=1,.13=.12*x+.15*y


125

The solution is x = .666667 and y = .333333, or w1 = 2/3, then w2 = 1/3. Using (6.8), we get

2 2 2 2

σ(Rp) = w1 σ1 + w2 σ2

+ 2w1w2σ1σ2r12

= (2/3)2(.25)2 + (1/3)2(.3)2 + 2(2/3)(1/3)(.25)(.3)(.25) = 0.2147 ♥

We notice that the σ(Rp) = 0.2147 is less than σ1 = .25 and σ2 = .3. In other words, it is quite possible to form a portfolio out of two securities so that the sigma of the portfolio is


126

less than the sigma of either of the two securities. This is because the risk of one security can possibly offset the risk of the other one.

Video 06 . 06 6.6. Elizabeth Corporation is starting two new projects. Project A requires an investment of $5,000, has expected return of 16% with standard deviation 14%. Project B has initial investment of $15,000, expected return of 15% with standard deviation 10%. The correlation coefficient between the projects is 0.75. Find the expected return, in dollars, of the portfolio of these two projects. What is the probability that this return is less than $4,000?

The total value of the portfolio is $20,000 and the weights are 0.25 and 0.75. We calculate the expected return of the portfolio as

E(Rp) = 0.25(0.16) + 0.75(0.15) = 0.1525 = 15.25%

and in dollars, E(Rp) = 0.1525(20,000) = $3,050

σ(Rp) = (.25)2(.14)2 + (.75)2(.1)2 + 2(.25)(.75)(.14)(.1)(.75) = 0.1038629

If the return is less than $4,000, it is less than 4,000/20,000 = 0.2 = 20%.

The expected return of the portfolio is 20%, but the required return is 15.25%, or less. The portfolio can have a return of 15.25% quite easily and thus the required probability is more than 50%. To find it, first calculate

z = (0.2 − 0.1525)/0.1038629 = 0.4573.

Draw a normal probability distribution curve, with z = 0 in the center. z = .4573 will be somewhat to the right of center. The area under the hump of the curve, to the left of z =.4573, will give the required probability. From the tables,

P(R < $4,000) = 0.5 + 0.1736 + .73(.1772 − .1736) = 67.62% ♥

You can check the naswer at Excel by copying the following


127

instruction.

EXCEL =NORMDIST(.2,.1525,.1038629,TRUE)

Key Terms

correlation coefficient, 108,

109, 110, 112, 113, 114,

116

covariance, 110, 111

diversification, 107

expected return, 107, 108,

110, 111, 112, 114, 115

loan portfolio, 107

portfolio, 107, 108, 109, 110,

111, 112, 113, 114, 115,

116

portfolio formation, 107

portfolio theory, 107

probability distribution, 108,

114

realized return, 107, 108

risk, 107, 108, 109

standard deviation of return, 108

weights, 108, 109, 111, 115


128

Problems

6.7. Bankhead Corporation is considering the following projects, which are all acceptable.

Project A Project B Project CExpected return $4000 $5000 $6000

Standard deviation $2100 $2500 $2800Correlation coefficients (A,B) = .8 (A,C) = .7 (B,C) = .6

Bankhead can take any one, any two, or all three projects. If the company wants to maximize the ratio E(R)/σ(R), what is the best course of action?

Invest in B and C. Ratio = 2.3195 for B + C ♥

6.8. Suppose you have $50,000 that you would like to invest in two companies, Bethlehem Books and Allentown Audio. Bethlehem has a return of 10% and standard deviation 15%, while Allentown has return of 15% with a standard deviation of 20%. The correlation coefficient between them is .5. Your portfolio should have a return of 12%. Find the standard deviation of this portfolio's returns. σ(Rp) = 14.73% ♥

6.9. Costello Corporation is undertaking these three projects:

Project A Project B Project CCost $235,000 $455,000 $310,000

Expected return 12% 11% 13%Standard deviation 8% 9% 10%

Correlation coefficients (A,B) = .4 (A,C) = .5 (B,C) = .6

Find the probability that the return on the portfolio is more than 15%. 33.87% ♥

6.10. The Lambda Corporation has the opportunity to invest in any of the following three proposals:

Project A Project B Project CExpected return $10,000 $11,000 $12,000

Standard deviation $2,000 $2,500 $2,800Correlation coefficients (A,B) = .4 (A,C) = .5 (B,C) = .8


129

If the company can invest in one, two, or three projects, what should it do to maximize the ratio between expected return and standard deviation?

E(R)/ratios: A = 5, B = 4.4, C = 4.28, (A + B) = 5.5630, (A + C) = 5.2680, (B + C) = 4.5735, (A + B + C) = 5.2916. Take (A + B) ♥

6.11. Kagera Company has made a portfolio of these three securities:


130

Treasury bond $100,000 6% 0Kasai Corporation $80,000 15% 25%Limpopo Company $70,000 16% 30%

The correlation coefficient between Kasai and Limpopo is 0.6. If the returns are normally distributed, find the probability that the return of the portfolio is more than 15%.

41.04% ♥

6.12. Excel. The expected return from two stocks, Apple and Google, under different states of the economy are as follows:

State of the economy Probability Apple GooglePoor 10% 0% −50%

Average 30% 20% 20%Good 60% 20% 30%

You have invested $40,000 in Apple and $60,000 in Google to form a portfolio. Find the following.

(A) Expected return of Apple and of Google. 18%, 19% ♥(B) The σ of Apple and of Google. 6%, 23.43% ♥(C) Coefficient of correlation between the two stocks. .9816 ♥(D) Expected return and σ of the portfolio. 18.6%, 16.42% ♥(E) Probability that the return of the portfolio will be more than 15%. 58.68% ♥

7. CAPITAL ASSET PRICING MODEL

Objectives: After reading this chapter, you should

1. Understand the concept of beta as a measure of systematic risk of a security.

2. Calculate the beta of a stock from its historical data.

3. Understand the Capital Asset Pricing Model.

4. Apply it to determine the risk, return, or the price of an investment opportunity.

7.1 Beta

In the section on capital budgeting, we saw the need for a risk-adjusted discount rate for risky projects. The risk of an investment or a project is difficult to measure and to quantify. This difficulty arises from the fact that different persons have different perceptions of risk. What may be quite a risky project to one investor may appear to be fairly safe to another person. After all, how can you quantify courage, or patience, or risk, or beauty?

In the section on portfolio theory, we used σ as a measure of risk, which is really the standard deviation of returns. Another useful measure of risk is the β of an investment. Like σ, β is also a statistical measure of risk. We infer it from the observations of the past performance of a stock. For example, we may want to find the risk of buying and holding the stock of a particular corporation, such as IBM, and we are interested in finding the β of IBM. We can start by looking at the historical value of three variables:

1. The returns of IBM stock, Rj. We define the return on a stock by the relation

Rj =P1 − P 0 + D 1

P0(7.1)

In the above equation, P0 is the purchase price of the stock, P1 its price at the end of the holding period, and D1 is the dividend paid, if any, at the end. The quantity P1 − P0 is the price appreciation of the stock, and along with the dividend, is the total change in the value of the investment. The return is equal to be the change in the value of the investment divided by the original investment. For example to find the monthly rate of

return on the IBM stock, we may want to know the price of the stock at the beginning of each month, the price at the end of the month, and the dividends paid during that particular month. We have to develop a series of numbers representing the return for each month for the last 24 months, say.

2. The returns of the market, Rm. A market index provides an overall measure of the performance of the market. The oldest and the most popular market index is the Dow Jones Industrial Average. The problem with this index is that it uses only 30 stocks in its valuation. For a broader market index, we may have to look at S&P100, or S&P500 index. There is even an index for over-the-counter stocks called the NASDAQ Composite Index. The value of these indexes is available daily.

121

Analytical Techniques 7. Capital Asset Pricing Model

122

Let us track the market for the last 24 months. If we know the value of the index at the start and finish of each month, we can find the return of the market for that month. The dividend yield for the market is around 1.71% annually at present. Therefore, we define the overall return on the market as

Rm =M1 − M 0

M0+ d1 (7.2)

where M0 is the beginning value and M1 the ending value of the market index, and d1 is the dividend yield as a percent for that period. With some effort, we may be able to develop a set of market returns for each of the last 24 months.

3. The riskless rate of interest, r. The securities issued by the Federal government, such as the Treasury bills, bonds, and notes, are, by definition, riskless. They are the safest investments available, backed by the full faith and taxing power of the government. Their rate of return depends on their time to maturity, and for longer maturity, the return is generally higher. The Treasury yield curve is available on the Internet.

After some research, we may also get a series of riskless rates for each of the past 24 months.

Then we define two variables x and y as:

y = Rj − r x = Rm −

r

where Rj = return on the stock j each month for the last 24 months,

Rm = corresponding monthly returns on the market for the same period,and r = riskless rate of interest per month, for the last 24 months.

By subtracting the riskless rate of interest, we are able to see the return due to the risk inherent in the given stock, and the return from the risk in the market. Thus, we are comparing the returns exclusively due to the risk in the investments.


123

A regression line drawn between the various observed values of x and y will show a certain linear relationship between x and y. The slope of the line will give the rate of change of y with respect to x. In other words, the slope will signify how much the return on the stock will change corresponding to a given change in the return on the market. In this diagram let us say that the slope of the line is β, and the y-intercept is α. The quantity α is practically zero, and it is statistically insignificant. The quantity β, on the other hand, represents an important concept.

This responsiveness of the stock return to the changing market conditions is called the "beta" of the stock. Stocks with low betas will show very little movement due to the fluctuations in the stock market. High beta stocks will tend to be jumpy showing a large variation in response to small changes in the market.

σ


124

Fig. 7.1: A regression line between y and x.

High β stocks, due to their large volatility, will be more unpredictable, and therefore, more risky. Low beta stocks show relatively small volatility, and they are more predictable and safe.

Beta is a statistical quantity, and it is a measure of the systematic risk, or the market related risk of a stock. These results can also be expressed as a statistical formula,

βj =c ov ( R j, R m)

var(Rm) =r j m σ m σ j

2

=m

r j m σ j

σm(7.3)

where cov(Rj,Rm) is the covariance between the returns on the stock j and the market, and var(Rm) is the variance of the returns on the market. If we have collected sufficient statistical data, we may find β by using

n ( xy ) − ( x ) ( y )

β =nx2

− (x)2(7.4)

α = where


125

n is the number of x and y values. y − βx

n (7.5)

One can apply the concept of beta to a portfolio. The beta of a portfolio is simply the weighted average of the betas of the securities in the portfolio,

n

Beta of a portfolio, βp = w1β1 + w2β2 + w3β3 + ... = wiβi (7.6)

i=1

Year

Price ofHauk stock Dividend

per share

S&P 500index

S&P 500 Risklessdividend

rate yieldbeginning end beginning end


126

The advantage of using β as a measure of risk is that it can combine linearly for different securities in a portfolio, but the disadvantage is that it can measure only the market related risk of a security. On the other hand, σ can measure the risk independent of the market conditions, but its disadvantage is that it is non-linear in character and difficult to apply in practice. Both β and σ are incomplete measures of risk; they change with time, and are difficult to measure accurately.

By definition, the beta of a riskless investment is zero. Further, the beta of the market is1. This is seen by setting j = m in (7.3) and noting that the covariance of a random variable with itself is just its variance.

A security that has a high beta should show a large rise in price when there is an upward movement in the market, and has a large drop in price in case of a downward movement. These large price fluctuations can cause a considerable amount of uncertainty about the return of this security, and greater risk associated with it. Therefore, a high beta security is also a high-risk security. Thus, beta is frequently used as a measure of the risk of a security. A low beta security is a defensive security and a high beta of a stock means a more aggressive management stance.

The numerical value of β for different stocks is available from sources on the Internet, such as www.etrade.com, and www.yahoo.com.

Examples

Video 07 . 01 7.1. Calculate the β of Hauck Corporation from the following data. The prices are at the beginning and end of each year:

2005 $25 $27 $1.00 1000 1050 3.05% 6.00%2006 $27 $29 $1.00 1050 1100 3.00% 6.00%2007 $29 $32 $1.50 1100 1200 2.95% 5.95%2008 $32 $33 $1.50 1200 1250 2.80% 5.90%

The return from the security in 2005 is capital gains ($2) plus dividends ($1) divided by the initial price ($25), that is, 3/25 = 0.12. The riskless rate during 2005 was 0.06, thus the excess return was 0.12 − 0.06 = 0.06. The return on the market for the same year


127

was 5/100 + 0.0305 = 0.0805. The excess return was 0.0805 − 0.06 = 0.0205. Designating the excess return for security as y and that for the market as x, we can tabulate the calculations as:

Year Rj − r = y Rm − r = x2005 3.00/25 − .06 = .06 5/100 + .0305 − .06 = 0.02052006 3.00/27 − .06 = .051111 5/105 + .03 − .06 = 0.0176192007 4.50/29 − .0595 = .095672 10/110 + .0295 − .0595 = 0.0609092008 2.50/32 − .059 = .019125 5/120 + .028 − .059 = 0.010667


128

Here n = 4, the number of periods, or x, y pairs

Σxy = (.0205)(.06) + (.017619)(.051111) + (.060909)(.095672) + (.010667)(.019125)= 0.0081618

Σx = 0.0205 + 0.017619 + 0.060909 + 0.010667 = 0.109695

Σy = .06 + .051111 + .095672 + .019125 = 0.225908

Σx2 = (0.0205)2 + (0.017619)2 + (0.060909)2 + (0.010667)2 = 0.00455437

Using equation (7.4)

4 ( 0.0081618) − ( 0.1096 9 5 )( 0.225908) β = 4(0.00455437) − (0.109695)2 = 1.271927967 1.27

One can do the above problem with the help of Maple as follows:

#n is the number of periods, or returns#n+1 is the number of price data points

n:=4;

#Price is an array to store price of stock Price:=array(1..n+1,[25,27,29,32,33]);

#Div is an array to store dividends Div:=array(1..n,[1,1,1.5,1.5]);

#Market is the array to store market index data

Market:=array(1..n+1,[100,105,110,120,125]);

#Markdiv is the array to store market dividends as percent Markdiv:=array(1..n,[.0305,.03,.0295,.028]);

#RF is the array to store riskfree rate

RF:=array(1..n,[.06,.06,.0595,.059]);

#x, y are the arrays to store x, y values x:=array(1..n); y:=array(1..n);


129

for i to n do

x[i]:=(Market[i+1]-Market[i])/Market[i]+Markdiv[i]-RF[i];

y[i]:=(Price[i+1]-Price[i]+Div[i])/Price[i]-RF[i] od; unassign('i');

n*sum(x[i]*y[i],i=1..n)-sum(x[i],i=1..n)*sum(y[i],i=1..n); n*sum(x[i]^2,i=1..n)-sum(x[i],i=1..n)^2;

beta=%%/%;

7.2. Calculate the β of Maine Corporation from the following data. The prices are at the beginning and at the end of each year:


130

Year Price of

Dividend of Maine

S&P 500index

S&P 500dividend

Riskless rate

2000 25-27 $2.00 100-105 3.05% 8.0%2001 27-29 $2.00 105-110 3.20% 8.5%2002 29-32 $2.50 110-120 3.50% 7.5%2003 32-33 $2.50 120-125 4.00% 7.0%

β = 0.89 ♥

7.2 Capital Asset Pricing Model

Beta is a measure of the market risk, or the systematic risk, of a security. A security with a large beta will have large swings in its price in relation to the changes in the market index. This will lead to a higher standard deviation in the returns of the security, which will indicate a greater uncertainty about the future performance of the security.

Draw a diagram with the β of various securities along the X-axis and their expected return along the Y-axis. We have already noticed that β is a linear measure of risk. If we assume that a linear relationship exists between the risk and return, then only two points are sufficient to draw a straight line in this diagram. The line, representing the relationship between risk and expected return, is called the security market line. Under equilibrium conditions, all other securities will also lie along this line. Higher β securities will have a correspondingly higher expected return. Figure 7.2 shows this graphically.


131

Fig. 7.2: Security market line.

By definition, beta of the market is equal to 1. The securities with more than average risk will have beta greater than 1, and less risky securities have beta less than 1. On this scale, the beta of a riskless security is zero. Such securities will provide riskless rate of return, r, to the investors. An example of such a security is the Treasury bill.


132

The security market line represents the risk-return characteristics of various securities, assuming that there is linear relationship between risk and return. Point A represents a riskless security with beta equal to zero and return r. Point B shows a market-indexed security which could be a very large mutual fund portfolio, which is invested in a large number of securities all weighted according to assets of the corporations whose securities make up the portfolio. Point C shows an individual security whose beta is βi and whose expected return is E(Ri). Since A, B, C all lie along the same straight line, then

Slope of segment AC = slope of segment AB

This gives,

E( R i) − r

βi=

E( R m) − r 1

Or, E(Ri) = r + βi [E(Rm) − r] (7.7)

Equation (7.7) gives the expected return of a security i in terms of its risk, expected return on the market, and the riskless rate. It is a forward-looking model, and thus gives the expected values of the returns. This equation represents what is known as the "C a pit a l Ass e t Pri c ing Mode l ", CAPM for short, and was developed in the 1960s by W illi a m S h a rp e , J a n M ossin , and J ohn L intn er . The use of this model is illustrated by the following examples.

William Sharpe (1934- )

Positive Alpha: Too Good to be True?New research from Robert Jarrow suggests that positive alpha is improbable.

During the past 25 years, an entire segment of the investment industry was constructed on the belief that positive alphas exist and can be exploited by portfolio managers to yield greater profit at less risk. New research by the Johnson School's Robert Jarrow strongly suggests that positive alphas are rare to nonexistent.

"Every hedge fund in the world claims to have positive alpha, but I say it can't be," says Jarrow, Ronald P. and Susan E. Lynch Professor of Investment Management at the Johnson School. "The claims for positive alpha are too strong—professional investment managers are taking risks that are hidden."


133

Alpha, an estimate of an asset's future performance, after adjusting for risk, is a measure routinely calculated by portfolio managers. Positive alpha suggests that an investor can realize higher returns at lower risk than by holding an index. In other words, by investing in assets with positive alpha, one can "beat the market," without exposure to the risk otherwise associated with the promised rate of return.

Jarrow used mathematical modeling to prove that positive alphas are equivalent to arbitrage opportunities. And arbitrage opportunities—risk-free trading of an asset between two markets to take advantage of a price differential—are rare in financial markets. According to Jarrow's research, positive alpha can exist only in the presence of a true arbitrage opportunity. For this to occur, two stringent conditions must be met. First, there must exist a market imperfection that enables the arbitrage opportunity to persist, even as arbitrageurs capitalize upon it; second, there must be a source of financial wealth, on which the arbitrageurs draw, either knowingly or unknowingly.


134

"Academics have looked for arbitrage opportunities in financial markets, and haven't found many. So it seems implausible to have so many positive alphas out there." Jarrow says. "To have positive alpha for any length of time means that someone is consistently losing money to someone else, and that's hard to believe."

In his paper "Active Portfolio Management and Positive Alphas: Fact or Fantasy?" forthcoming in the Journal of Portfolio Management, Jarrow outlines his model and offers examples of both true and false positive alphas, drawn from the pivotal events of the credit market crisis. His conclusions include a word of caution to investors.

"The moral of this paper is simple," Jarrow writes. "Before one invests in an investment fund that claims to have positive alphas, one should first understand the market imperfection that is causing the arbitrage opportunity and the source of the lost wealth. If the investment fund cannot answer those two questions, then the positive alpha is probably fantasy and not fact."

The Journal of Portfolio Management, Summer 2010, Vol. 36, No. 4: pp. 17-22

Examples

Video 07 . 03 7.3. Chicago Corp stock will pay a dividend of $1.32 next year. Its current price is $24.625 per share. The beta for the stock is 1.35 and the expected return on the market is 13.5%. If the riskless rate is 8.2%, what is the expected growth rate of Chicago?

Using the capital asset pricing model (CAPM),

E(Ri) = r + βi [E(Rm) − r] (7.7)

We first find the expected rate of return as

E(Ri) = 0.082 + 1.35 [0.135 − 0.082] = 0.15355 = R

The expected rate of return E(Ri), for a security is also its required rate of return R by the investors. Using the growth model for a stock, equation (3.6),

0


135

P = D1

R − g

we get, R − g = D1/P0, or g = R − D1/P0,

which gives g = 0.15355 − 1.32/24.625 = 0.1. Thus the growth rate is 10%. ♥

Video 07 . 04 7.4. Peggotty Services common stock has a β = 1.15 and it expects to pay a dividend of $1.00 after one year. Its expected dividend growth rate is 6%. The riskless rate is currently 12%, and the expected return on the market is 18%. What should be a fair price of this stock?

E(Ri) = r + βi [E(Rm) − r] (7.7)

we get E(Ri) = 0.12 + 1.15 [0.18 − 0.12] = 0.189


136

Thus, the expected return on the stock is 0.189, and the expected growth rate is 0.06. Using (3.1) once again,

1 P0 = 0.189 − 0.06 = $7.75 ♥

Video 0 7. 0 5 7.5. The beta of Vega Inc is 1.15, its rate of growth is 10%, it will give a dividend of $3.00 next year, and its common stock sells for $50 a share. The riskless rate is 8%. By careful planning and by selecting more secure projects, Vega has reduced its risk. Its new beta is estimated to be 1, while everything else (income, dividends, growth rate, capital structure, market return, etc.) is the same. What is its new share value?

The total return on a stock is the sum of its dividend return and the growth rate. If r is the required rate of return, E(Ri) is the expected rate of return, g is the growth rate, D1 is the dividend to be paid next year, and P0 is its price now, then

D1 3

R = P0 + g = 50 + 0.1 = 0.16 = E(Ri)

Use E(Ri) = r + βi [E(Rm) − r] (7.7)

Drop the subscript i, and solve for E(Rm), to get

Or, E(Rm) = r +

E( R ) − r β

Or, E(Rm) = 0.08 + (0.16 − 0.08)/1.15 = 0.1496

The new β is 1, and since the β of the market is also 1, this implies that

E(R) = E(Rm) = 0.1496

Thus P0 = 3/(0.1496 − 0.1) = $60.53 ♥

0 1 0


137

7.6. Eastern Oil stock currently sells at $120 a share. The stockholders expect to get a dividend of $6 next year, and they expect that the dividend will grow at the rate of 5% per annum. The expected return on the market is 12% and the riskless rate is 6%. This morning Eastern announced that it has won the multimillion dollar navy contract, and in response to the news, the stock jumped to $125 a share. Find the beta of the stock before and after the announcement.

Using Gordon's growth model, P = D1 , we get R = D /P + g, which is also the

R − g

expected return on the stock, E(R). But by CAPM,

E(Ri) = r + βi [E(Rm) − r]


138

we get β = E( R i) − r E(Rm) − r

Thus β =D1 / P 0 + g − r

E(Rm) − r

=

6/120+0.05−0.06

0.12 − 0.06 = 0.667, before. ♥

And β =

6/125+0.05−0.06

0.12 − 0.06 = 0.633, after. ♥

7.7. Jupiter Gas Company is planning to acquire Saturn Water Company. The additional pre-tax income from the acquisition will be $100,000 in the first year, but it will increase by 2% in future years. Because of diversification, the beta of Jupiter will decrease from1.00 to 0.9. Currently the return on the market is 12% and the riskless rate is 6%. What is the maximum price that Jupiter should pay for Saturn? The tax rate of Jupiter is 35%.The new beta for Jupiter is 0.9. Using CAPM, its expected return, and hence the cost of capital will be

E(R) = 0.06 + 0.9(0.12 − 0.06) = 0.114After tax income = 100,000 (1 − 0.35) = $65,000.

The total value of a firm is the present value of its future earnings, properly discounted. Thus, the value added to Jupiter due to the acquisition of Saturn is the present value of future after-tax earnings of Saturn, discounted at a rate equal to the cost of capital of Jupiter, and summed up to infinity. Thus

PV =65‚0 0 0 1.114 +

65‚0 0 0 ( 1.02) 1.1142 +

65‚000 (1.02)2

1.1143 ... ∞ = $691,489

WRA Sum[65000*1.02^(i-1)/1.114^i,{i,1,infinity}]

Jupiter should pay at most $691,489 for Saturn. ♥


139

7.8. Hamlin Dairies stock has a beta of 1.33. It has just paid its annual dividend of $1.20, and it sells for $30 a share. Shareholders believe that Hamlin is growing at the rate of 7% annually and will maintain a constant dividend payout ratio. Due to the unexpected death of the chairperson, Hannibal Hamlin, the company is facing an uncertain future, and the price per share dropped to $25. There is no other change in the company (dividends, growth, sales, etc.) or in the market. The riskless rate is 6%. In light of the greater risk of the company, find its new beta.

If the current dividend is $1.20, next year it will be 1.20(1.07) = $1.284. Apply Gordon’s growth model, (3.6), to find the required rate of return for the stockholders. Before Hamlin’s death, it is

R = D1/P0 + g = 1.20(1.07)/30 + 0.07 = 0.1128 (before)


140

After Hamlin’s death, the stock price drops suddenly, but the growth potential and the current dividend remains intact. Thus the required rate of return after the death is

R = 1.20(1.07)/25 + 0.07 = 0.12136 (after)

The expected and the required rate of return for the stock are the same, meaning R = E(R). We can use these numbers in CAPM, (7.7), to get two equations:

Before, 0.1128 = 0.06 + 1.33 [E(Rm) − 0.06]

After, 0.12136 = 0.06 + β[E(Rm) − 0.06]

Put β = x and E(Rm) = y temporarily. Copy and paste the following instruction atW olf ra mAlpha to solve the two equations simultaneously.

WRA .1128=.06+1.33*(y-.06),.12136=.06+x*(y-.06)

The approximate solution is x 1.54562 and y 0.0996992. Solving for beta, we get,

β = 1.55 ♥

The new β, 1.55, is higher than the previous β, 1.33, because of the uncertainty created by the death of the chairperson. Greater uncertainty also means greater risk.

7.9. Epperly Fund invests in S&P500 companies and thereby simulates a market portfolio. The expected return of Epperly is 13.5%, with a standard deviation of 10%. Suppose you are able to borrow $10,000 at the riskless rate of 9%, and you already have$10,000 of your own money. If you invest this $20,000 in Epperly Fund, what is the probability that you will have a return greater than 25% on your own money?

The β of the market is 1, by definition. Epperly Fund mimics the market and therefore, its β is also 1. When you borrow money to buy securities, the amount of borrowing is equivalent to a negative cash position in your account. The β of cash is zero, because the value of cash does not change due to fluctuations in the stock market. The total value of the portfolio you own is $10,000, which equals your investment. Its composition is as follows:


141

Value β WeightEpperly Fund $20,000 1 2

Cash −10,000 0 −1Portfolio 10,000 2 1

To find the β of the portfolio, use

βp = w1β1 + w2β2 = 2(1) + (−1)(0) = 2

This is highlighted in the previous table. With the help of CAPM, find


142

E(Rp) = .09 + 2(.135 − .09) = .18

The portfolio is consisting of two items with weights 2 and −1. The σ of the components is, 10% for Epperly Fund and zero for cash. Using equation (6.4) for the σ of a two- security portfolio, we have

σp = (2)2(.1)2 + (−1)2(0) + 2(2)(−1)(.1)(0)r12

Solving the above equation, we get σp = .2. We note that the expected return of the portfolio is 18% with a standard deviation of 20%. The required return is more than 25%.The probability of getting that return is less than 50%. To calculate it, first find

z = (25 − 18)/20 = 0.35

Draw a normal probability distribution curve, with z = 0 at the center and z = .35 to the right of center. The probability of getting a return of greater than 25% is equal to the shaded area to the right of z = .35. From the table, we get its value as,

P(R > 0.25) = 0.5 − .1368 = .3632 = 36.32%. ♥

EXCEL =1-NORMDIST(.25,.18,.2,TRUE)

7.10. Markham Co paid a dividend of $3.00 yesterday, but these dividends are expected to grow at the rate of 5% in the long run. The beta of Markham is 0.95, the expected return on the market is 15%, and the riskless rate is 10% at present. Find the price of one share of Markham stock.


143

Using the CAPM, we have, E(R) = 0.10 + 0.95(0.15 − 0.1) = 0.1475

Using Gordon's growth model, we get the price of a share as

3 ( 1.05) P0 = 0.1475 − 0.05 = $32.31♥

7.11. You have developed the following information about two mutual funds:

133


Name of fund Beta Expected returnDione Market Fund 1.00 14%Rhea Energy Fund 0.80 13%

You have $5,000 to invest and you put $3,000 in Dione and $1,000 each in Rhea and riskless bonds. Find the beta and expected return of your portfolio.

Let us first find the riskless rate. Dione has β of 1, the same as that of the market. Thus the expected return of the market is also 0.14. Using CAPM, and using the information about Rhea,

0.13 = r + 0.8(0.14 − r)

which gives the riskless rate, r = .09. The weights of securities are

w1 = 0.6, w2 = 0.2, and w3 = 0.2.

The beta of the portfolio is just the weighted average of the betas of the individual securities. That is,

βp = 0.6(1.00) + 0.2(0.8) + 0.2(0) = 0.76 ♥

Similarly, the expected return on the portfolio is given by

E(Rp) = 0.6(0.14) + 0.2(0.13) + 0.2(0.09) = 0.128 ♥

You can also calculate the expected return of the portfolio by substituting β = .76, r = .09 and E(Rm) = .14 in CAPM. This gives

E(Rp) = .09 + .76[.14 − .05] = .128

7.12. Pindar Corporation stock is selling for $80 a share and its dividend next year is expected to be $2. S&P500 index is 1437 at present, and it is expected to go up to 1550 after one year. The average dividend yield for the S&P500 is 1.52%, and the riskless rate is 5.14%. If the beta of Pindar is 1.14, find the expected price of one share of Pindar after one year.

Using the information about the market, find the expected percentage return on

134


the market as the sum of the dividend yield of the market (.0152) and its price appreciation (1550 − 1437)/1437. This gives

E(Rm) = 0.0152 + (1550 − 1437)/1437 = .09384

Next, find the expected return of Pindar using CAPM. Put r = .0514, β = 1.14, and E(Rm)= .09384.

E(Rj) = 0.0514 + 1.14(.09384 − 0.0514) = .09978

If x is the expected price of the stock next year, then the return of the stock is,

134


.09978 = (x − 80 + 2)/80

This gives x = 80(.09978) + 80 – 2 = $85.98 ♥

7.13. A portfolio is formed as follows:

Stock Amount invested β σ(R)Childs Corporation $12,000 1.25 25%Jermyn Company $13,000 1.20 22%

The riskless rate is 7%, and the expected return on the market is 14%. The covariance between the two stocks is 0.0385. Find the expected return, and the standard deviation of returns of the portfolio, in dollars, and as a percentage.

With the total value of the portfolio = 12,000 + 13,000 = $25,000, it is easy to see that the weights are w1 = 12/25 = 0.48, and w2 = 0.52

Using CAPM, the expected returns are

E(R1) = 0.07 + 1.25(0.14 − 0.07) = 0.1575

and E(R2) = 0.07 + 1.2(0.14 − 0.07) = 0.1540

The portfolio return is E(Rp) = 0.48(0.1575) + 0.52(0.154) = 15.568% ♥

= 0.15568(25,000) = $3892 ♥

c o v ( 1,2) 0.0385 The correlation coefficient, r12 =

σ1σ2= (0.25)(0.22) = 0.7

The portfolio sigma, σ(Rp) = 0.482 0.252 + 0.522 0.222 + 2(0.48)(0.52)(0.25)(0.22)(0.7)

135


= 21.61% ♥

And in dollars, σ(Rp) = 0.2161(25,000) = $5403 ♥

7.14. The β of Blakely Company is 1.2. Blakely is planning to acquire Waymart Corporation which will result in the combined company to have a β of 1.3. The riskless rate is 6%, and the expected return on the market is 12%. Waymart Corporation is expected to have first year earnings after taxes of $40,000, and these earnings are expected to increase by 3% per annum in future. How much should Blakely pay for Waymart?

With β = 1.3, use CAPM to find the risk adjusted discount rate = 0.06 + 1.3(0.12 − 0.06)= 0.138

1


136

40‚000 40‚00 0( 1.03)

40‚000(1.03)2

40‚000

1 . 1 3 8

PV of earnings = 1.138 + 1.1382

+

1.1383 + ... ∞ = 1.03 = $370,3701.138

The value of Waymart is thus $370,370 ♥

7.15. McNamara Fund has expected return of 10.5%, standard deviation of 17.5%, and beta of 0.8. Schlesinger Fund has expected return of 12.5%, standard deviation of 21% and beta of 1.1. The two mutual funds have correlation coefficient of 0.7. Find the expected return and standard deviation of the market. What is the riskless rate of return?

First we use the CAPM, and put the numbers for the two funds, which gives

us McNamara: 0.105 = r + 0.8 [E(Rm) − r]Schlesinger: 0.125 = r + 1.1 [E(Rm) − r]

To solve the equations, put r = x and E(Rm) = y temporarily. Then copy and paste the following instruction at W olf ra mAlpha .

WRA .105=x+.8*(y-x), .125=x+1.1*(y-x)

Solving the two equations, we get E(Rm) = 0.1183, and r = 0.05167 ♥

Next we construct the market portfolio out of these two funds. The β of the market, by definition, is 1. The weights are w1 and w2, and they are combined to get β = 1 for the market portfolio. Thus, we have

w1 + w2 = 1

0.8 w1 + 1.1 w2 = 1


137

To solve the equations, put w1 = x and w2 = y temporarily. Then copy and paste the following instruction at W olf ra mAlpha .

WRA x+y=1,8/10*x+11/10*y=1

Solving these two equations we get, w1 = 1/3 and w2 = 2/3. The same result can be obtained by combining the expected returns of the two funds to get the expected return of the market. Now we can find the sigma of the market as follows:

σ(Rm) = (1/3)2 0.1752 + (2/3)2 0.22 + 2(1/3)(2/3)(0.175)(0.21)(0.7) = 0.1856 ♥

7.16. Armstrong Corporation $6 preferred stock sells for $50 a share. The beta of this stock is 1.25. The current riskless rate is 8%. Just yesterday, Louis Armstrong, the founder and CEO, died and the stock dropped to $47 a share in response to the news. Find the new beta of Armstrong preferred.


138

A preferred stock has fixed dividends, that is, there is no expectation of growth. This means g = 0 in Gordon’s growth model, P0 = D1/(R – g), which becomes P0 = D1/R. Rewrite it as R = D1/P0. This implies that the current return of the stock is 6/50 = .12.

This is quite reasonable. If you buy a stock for $50 a share and it pays a dividend of $6 annually, without any growth opportunity, the return is indeed 12%.

Using CAPM,

.12 = .08 + 1.25[E(Rm) − .08]

Solve this equation to get E(Rm) = (.12 − .08)/1.25 + .08 = .112

This is quite reasonable, because the stock, with its β1 = 1.25, has a return of .12; and the market with its β = 1, should have a lower expected return, perhaps around 11%.

Let β1 = 1.25, beta of the stock before Armstrong’s death and β2 = beta of the stock after his death. Now assume that Armstrong is just an insignificant player in the stock market, and the market will ignore his demise. The expected return on the market will remain at.112 and the risk-free rate at .08. The new return on the stock is 6/47. The CAPM gives us

6/47 = .08 + β2[.112 − .08]

This gives β2 = (6/47 − .08)/(.112 − .08) = 1.49 ♥

The answer is quite reasonable, because Louis Armstrong was a very important individual at Armstrong Company. His departure has introduced a substantial measure of uncertainty, or risk, in the company, thereby increasing its β from 1.25 to 1.49.

Problems

7.17. The Washington Corp stock has a β of 1.15 and it will pay a dividend of $2.50 next year. The expected rate of return of the market is 17% and the current riskless rate is 9%. The expected rate of growth of Washington is 4%. Find the value of its common stock.

$17.61 per share ♥


139

7.18. Molopo Company has β = 1.2, whereas the return on the market is expected to be 12%, with a standard deviation of 8%. The riskless rate is 6% at present. The stock of Molopo is selling at $100 a share, but it does not pay any dividends. Find the probability that it will be selling for more than $120 by next year. Assume that the entire change in the stock price is due to the change in the market. 23.94% ♥

7.19. Cheever Corp stock is selling at $40 a share. Its dividend next year will be $2 a share and its beta is 1.25. Crane Company has the same growth rate as Cheever. The current stock price of Crane is $55 a share, and its dividend this year is $3. The riskless rate is 8% and the expected return on the market is 16%. Find the beta of Crane stock.

1.3955 ♥


140

7.20. Kingston Corporation has β = 1.2. It is interested in buying Plains Corporation which also has β = 1.2. Kingston believes that after the acquisition, its β will be 1.1. The expected after-tax earnings from Plains will be $50,000 for the first year, but this figure is expected to increase by 3% per year in future. The expected return on the market is 12%, and the riskless rate is 6%. Find the amount that Kingston should spend on this acquisition. $520,833 ♥

7.21. Toledo Corporation estimates its β as 1.3, whereas the risk-free rate is 5% at present. The expected return on the market is 11%, with a standard deviation of 7%. Assume that the variation in the Toledo stock price is entirely due to the fluctuations of the market. If you invest $10,000 in Toledo stock now, what is the probability that the value of your investment will be more than $12,000 by next year? 21.45% ♥

7.22. Palmer Company stock has paid a dividend of $1.25 this year, which is in line with its long-term growth rate of 5%. Its current β is 1.2 and the expected return of the market is 12%. Today, after the company won the multimillion-dollar contract from the navy, the stock jumped 3%, to $15.45 a share, in response to the good news. Find the risk-free rate and the new β of the stock. r = 3.25%, new β = 1.171 ♥

7.23. Johnson Corporation preferred stock sells for $37 a share and pays an annual dividend of $4. The β of this stock is 1.3. The current riskless rate is 3%. The common stock of Johnson was upgraded by the analysts from ‘hold’ to ‘buy’ today. In response to the news, the preferred stock jumped in price by $1. Find the new β of Johnson preferred.

1.253 ♥

Key Terms

beta, 116, 117, 118, 119, 120,

121, 123, 124, 125, 126,

127, 128, 129, 130, 131

Capital Asset Pricing Model, 116, 121, 122

capital budgeting, 116

CAPM, 122, 123, 124, 125,

127, 128, 129, 130

discount rate, 116, 129 dividend, 116, 117, 119, 121,

123, 124, 125, 127, 128,

130, 131

Dow Jones Industrial Average, 116

linear relationship, 117, 121,

122

market index, 116, 117, 120,


141

121

regression line, 117, 118

return, 116, 117, 119, 121,

122, 123, 124, 125, 126,

127, 128, 129, 130, 131

risk, 116, 117, 118, 119, 121,

122, 124, 125, 130, 131

S&P500, 116, 126, 128

security market line, 121 standard deviation of returns,

116, 128

systematic risk, 116, 121


142

files.transtutors.com · web view6. portfolio theory. objectives: after reading this chapter, you...

Documents