Finance Questions

Please see the attachments.

1. Scheuer Enterprises has a beta of 1.10, the real risk-free rate is 2.00%, investors expect
a 3.00% future inflation rate, and the market risk premium is 4.70%. What is Scheuer’s
required rate of return?

a. 9.67%

b. 10.17%

c. 9.92%

d. 9.43%

e. 10.42%

 

2. Suppose rRF = 5%, rM = 8%, and rA = 14%.
a. Calculate Stock A’s beta. Round your answer to two decimal places.

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

Round your answer to two decimal places.
%

 

3. Suppose you manage a $5.08 million fund that consists of four stocks with the
following investments:
Stock Investment Beta

A $420,000 1.50

B 750,000 -0.50

C 1,460,000 1.25

D 2,450,000 0.75

If the market’s required rate of return is 8% and the risk-free rate is 4%, what is the fund’s
required rate of return? Round your answer to two decimal places.
%

 

4. Stock R has a beta of 1.2, Stock S has a beta of 0.85, the expected rate of return on an
average stock is 10%, and the risk-free rate is 6%. By how much does the required return
on the riskier stock exceed the required return on the riskier stock exceed that on the less
risky stock? Round your answer to two decimal places.
%

 

5. An individual has $45,000 invested in a stock with a beta of 0.4 and another $70,000
invested in a stock with a beta of 1.1. If these are the only two investments in her
portfolio, what is her portfolio’s beta? Round your answer to two decimal places.

6.
 Assume that you manage a $10.00 million mutual fund that has a beta of 1.05 and a
9.50% required return. The risk-free rate is 4.20%. You now receive another $5.00
million, which you invest in stocks with an average beta of 0.65. What is the required rate
of return on the new portfolio? (Hint: You must first find the market risk premium, then
find the new portfolio beta.)

a. 8.83%

b. 9.05%

c. 9.51%

d. 9.74%

e. 9.27%

 

7. Assume that the risk-free rate is 3% and that the market risk premium is 3%.
What is the required rate of return on a stock with a beta of 1.1? Round your answer to
two decimal places.
%
What is the required rate of return on a stock with a beta of 1.7? Round your answer to
two decimal places.
%

 

Chapte

r

4/

1

1/10

2

>Chapter

6. Tool Kit for Risk and

Return

RETURNS ON INVESTMENTS (Section

6.1

)

Amount invested $1,000 Amount received in one year $1,100 Dollar

return

(Profit) $100 Rate of return

=

Profit/

Investment

=

10%

STAND-ALONE RISK (Section

6.2

)

The relationship between risk and return is a fundamental axiom in finance. Generally speaking, it is

to

tally logical to assume that investors are only willing to assume additional risk if they are adequately compensated with additional return. This idea is rather fundamental, but the difficulty in finance arises from interpreting the exact nature of this relationship (accepting that risk aversion differs from investor to investor). Risk and return interact to determine security prices, hence it is of paramount importance in finance. PROBABILITY DISTRIBUTION A proba

bi

lity distribution is a listing of all possible outcomes and their corresponding probabilities. Figure 6-1.

Probability

Distributions for Sale.Com and

Basic Foods

Inc. Demand for the Probability of this Rate of Return on

Stock Company’s Products Demand Occurring if this Demand Occurs Sale.com

Basic Foods
Strong 0.

3

0 90

% 45

% Normal 0.4

0 15

% 15% Weak 0.30 −

60% −15% 1.0

0 EXPECTED RATE OF RETURN The expected rate of return is the rate of return that is expected to be realized from an investment. It is found as the weighted average of the probability distribution of returns. Figure 6-2. Calculation of Expected Rates of Return: Payoff Matrix Demand for the Probability of this Sale.com Basic Foods
Company’s Products
(1) Demand Occurring
(2) Rate of Return
(3) Product
(2) x (3) = (4) Rate of Return
(5) Product
(2) x (5) = (6) Strong 0.3

90% 27.0% 45% 1

3.5% Normal 0.4 15%

6.0%

15% 6.0%
Weak 0.3 −60%

−1

8.0%

−15%

−

4.5% 1.0

Alternative procudure: Expected Rate of Return =
Sum of Products = 1

5.0% 15.0% Excel’s SUMPRODUCT function can be used to find the expected values. This function takes the values in the first argument array, multiplies them by the arguments in the second argument array, and sums these product. If you put the probabilities in the first array and the outcomes in the second array, then you get the expected value. Expected return

=

15%

=SUMPRODUCT($B$44:$B$46,C44:C46) MEASURING STAND-ALONE RISK: THE STANDARD DEVIATION The standard deviation is a measure of a distribution’s dispersion. Figure 4. Probability Distributions of Sale.com’s and Basic Foods’ Rates of Return Prep work for chart
Mike Ehrhardt: This is so that percentages will be formatted “without” % sign. Sale.com Basic Foods
0.3 90 45
0.4 15 15
0.3

-60 -15 Calculating Standard Deviation Here are the steps used to calculate the standard deviation. First, find the differences of all the possible returns from the expected return. Second, square those differences. Third, multiply the squared numbers by the probability of their occurrence. Fourth, find the sum of all the weighted squares. Finally, take the square root of that number. Here are the calculations for Sale.com and Basic Foods. Figure 6-5. Calculating Sale.com’s and Basic Foods’ Standard Deviations Panel a.

Sale.com
Probability of Occurring
(1) Rate of Return on Stock
(2) Expected Return

(3) Deviation from Expected Return
(2) − (3) = (4) Squared Deviation
(4)2 = (5) Sq. Dev. × Prob.
(5) x (1) = (6) 0.3 90% 15%

75.0% 56.

25% 16.88% 0.4 15% 15%

0.0% 0.00

%

0.00%
0.3 −60% 15%

−75.0%

56.25% 16.88%

Alternative procedures: 1.0

Sum =

Variance = 33.75% Use SUMPRODUCT to find variance by putting probabilities in first argument array, the putting outcomes minus the expected value in the second array. Variance=

33.75%

=SUMPRODUCT($B$44:$B$46,C44:C46-D48,C44:C46-D48) Std. Dev.

= Square root of variance = 58.09% Use SD of a population (all possible outcomes). If there is a 30% probablity of Std dev =

58.09%

=STDEVP(C44,C44,C44,C45,C45,C45,C45,C46,C46,C46) = and outcome, enter the outcome 3 times, 4 times for

40%

probability, etc. Panel b.

Basic Foods

Probability of Occurring
(1) Rate of Return on Stock
(2) Expected Return
(3) Deviation from Expected Return
(2) − (3) = (4) Squared Deviation
(4)2 = (5) Sq. Dev. × Prob.
(5) x (1) = (6)

0.3 45% 15%

30.0% 9.00% 2.70%

0.4 15% 15% 0.0% 0.00% 0.00%

0.3 −15% 15%

−30.0%

9.00% 2.70%
1.0 Sum = Variance =

5.40%

Alternative procedures:
Std. Dev. = Square root of variance =

23.24% =STDEVP(E44,E44,E44,E45,E45,E45,E45,E46,E46,E46) =

23.24%
If Sales.com’s and Basic Foods’ stock return distributions are from normal distributions, then we can find confidence intervals. 0.6826 Expected Return

Std. Deviation 1-s range around expected return Sale.com 15% 58.09%

-43.09%

to

73.09% Basic Foods 15% 23.24%

-8.24%

to

38.24% USING HIST

OR

ICAL DATA TO MEASURE RISK Figure 6-7. Standard Deviation Based On a Sample of Historical Data Realized Year

return
2008

15.0%
2009 −5.0%

2010

20.0% Average =

AVERA

GE

(D122:D124) = 10.0% Standard deviation

=

STDEV(D122:D124) = 1

3.

2% Use STDEV, the SD for a sample, not STDEVP, the SD for a complete data set as used above. MEASURING STAND-ALONE RISK: THE COEFFICIENT OF VARIATION The coefficient of variation indicates the risk per unit of return, and it is calculated by dividing the standard deviation by the expected return. Std. Dev. Expected return

CV Sale.com 58.09% 15%

3.87 Basic Foods 23.24% 15%

1.5

5

RISK IN A PORTFOLIO CONTEXT (Section

6.3

)

Portfolio Expected Return The expected return on a portfolio is simply a weighted average of the expected returns of the individual assets in the portfolio. The weights are the percentage of total portfolio funds invested in each asset. Consider the following portfolio and the hypothetical illustrative returns data. Figure 6-8. Expected Returns on a Portfolio of Stocks Stock

Amount of Investment Portfolio

Weight Expected
Return Weighted Expected
Return Southwest Airlines $300,000

0.3 15.0% 4.5%
Starbucks $100,000 0.1 1

2.0

% 1.2

% FedEx $200,000 0.2

10.0% 2.0%
Dell $400,000

0.4

9.0% 3.

6% Total

investment = $1,000,000

1.0
Portfolio’s Expected Return = 1

1.3% Portfolio Standard Deviation Portfolios of stocks are created to diversify investors from unnecessary risk. The diversifiable, or idiosyncratic, risk is eliminated as more stocks are added. Diversification effects are strongest when combining uncorrelated assets. The following figures illustrate how creating two-stock portfolios with different correlations between the stocks affects the expected return and risk of various fictional portfolios. Figure 6-9. Portfolio Risk: Perfect Negative Correlation Stock W Stock M Weights 0.5

0.5

Year Stock W Stock M

Portfolio WM

2006

40%

-10%

15%
2007

-10% 40% 15%
2008

35% -5%

15%
2009 -5% 35% 15%
2010 15% 15% 15%
Average return = 15.00%

15.00% 15.00%
Standard deviation =

22.64%

22.64% 0.00%
Correlation coefficient = –

1.00 CONCLUSION: When two stocks are perfectly negatively correlated, diversification is its strongest, and in this case the portfolio return is a certain (no risk) 15%. Of course, this situation is very rare. Figure 6-10. Portfolio Risk: Perfect Positive Correlation

Stock W

Stock W’

Weights 0.5 0.5

Year Stock W Stock W’

Portfolio WW’

2006 40% 40% 40%
2007 -10% -10% -10%
2008 35% 35% 35%
2009 -5% -5% -5%

2010 15% 15% 15%
Average return = 15.00% 15.00% 15.00%

Standard deviation = 22.64% 22.64% 22.64%
Correlation coefficient = 1.00
CONCLUSION: When two stocks are perfectly positively correlated, diversification has no effect, and the portfolio’s risk is a weighted average of its individual stocks’ risks. Note that in this graph only the portfolio returns are visible, but realize that the stocks’ returns follow an identical path. Figure 6-11. Portfolio Risk: Imperfect (Partial) Correlation

Stock W

Stock Y

Weights 0.5 0.5

Year Stock W Stock Y

Portfolio WY

2006 40% 40%

40.00% 2007 -10% 15%

2.50% 2008 35% -5% 15.00%
2009 -5% -10%

-7.50% 2010 15% 35%

25.00%

Average return = 15.00% 15.00% 15.00%

Standard deviation = 22.64% 22.64%

18.62% Correlation coefficient =

0.35 CONCLUSION: In the case where two stocks are somewhat correlated, diversification is effective in lowering portfolio risk. Here, the portfolio return is an average of the stock returns and risk is reduced from 22.64% for the individual stocks to 18.62% for the portfolio. Notice that the portfolio’s return is always between that of the two stocks. If more similarly-correlated stocks were added, risk would continue to fall, but as we shall see, there is a limit to how low risk (the portfolio’s SD) can go.

Contribution to

Market

Risk:

Beta The beta coefficient measures the amount of risk that a stock contributes to a well-diversified portfolio. It also reflects the tendency of a stock to move up and down with the market. Shown below in the chart and in the table are the returns for three stocks and for the stock market. Figure 6-13. Relative Returns of Stocks H, A, and L Historical Returns

Year Market

Stock H Stock A Stock L 1

19.0% 26.0%

19.0%

12.0% 2

25.0% 35.0%

25.0% 15.0%
3

-15.0% -25.0%

-15.0%

-5.0% Average =

9.7%

12.0% 9.7%

7.3% Standard deviation =

2

1.6% 32.4%

21.6%

1

0.8

% Beta =

1.5 1.0 0.5
Note: These three stocks plot exactly on their regression lines. This indicates that they are exposed only to market risk. Portfolios that concentrate on stocks with betas of 1.5, 1.0, and 0.5 have patterns similar to those shown in the graph. Standard deviation is calculated with the Excel STDEV function because the data come from an historical sample. Probability Distributions for H, A, and L Notice that Stock L has the lowest average return, but it also has the tightest distribution. On the other hand, Stock H has the highest average return, but the widest distribution. Prep work for chart:
Mike Ehrhardt: This shows how “high” the pdf is at the mean for each stock. Stock A Stock H Stock L
1.2329251877 1.8493877816 3.6987755632 Calculating Beta for H, A, and L First, calculate correlation and covariance. Correlation of stock with Market, ri,M

1.00 1.00 1.00

We use Excel’s correlation function, CORREL:
=CORREL($B$310:$B$312,C310:C312) Covariance of stock with Market, COVi,M
Mike Ehrhardt: For the standard deviation and corrrelation functions, we use the Excel function for the sample, not assuming we have the entire population. But for the COV function, Excel only provides the formula for the entire population. Therefore, we must multiply the Excel result by n/(n-1), or 3/2 in our example, to get the sample covariance. 6.98%
Mike Ehrhardt: For the standard deviation and corrrelation functions, we use the Excel function for the sample, not assuming we have the entire population. But for the COV function, Excel only provides the formula for the entire population. Therefore, we must multiply the Excel result by n/(n-1), or 3/2 in our example, to get the sample covariance. 4.65%
Mike Ehrhardt: For the standard deviation and corrrelation functions, we use the Excel function for the sample, not assuming we have the entire population. But for the COV function, Excel only provides the formula for the entire population. Therefore, we must multiply the Excel result by n/(n-1), or 3/2 in our example, to get the sample covariance. 2.33%
Mike Ehrhardt: For the standard deviation and corrrelation functions, we use the Excel function for the sample, not assuming we have the entire population. But for the COV function, Excel only provides the formula for the entire population. Therefore, we must multiply the Excel result by n/(n-1), or 3/2 in our example, to get the sample covariance. Mike Ehrhardt: This shows how “high” the pdf is at the mean for each stock. We use Excel’s covariance function, COV, modified to reflect the fact that this is a sample and not a population: =COVAR(C310:C312,$B$310:$B$312)*(3/2) Method 1: bi = ri,M (si / sM)

1.5 1.0 0.5

=C355*(C314/$B$314) Method 2: bi = COVi,M / (sM)2

1.5 1.0 0.5

=C356/($B$314^2) Method 3: Slope of regression
Beta =

1.5 1.0 0.5

We use Excel’s SLOPE function: =SLOPE(C310:C312,$B$310:$B$312) CALCULATING BETA COEFFICIENTS (Section 6.4) Now we show how to calculate beta for an actual company, General Electric. Step 1. Retrive Data We downloaded stock prices and dividends from http://finance.yahoo.com for General Electric, using its ticker symbol GE, and for the S&P 500 Index ( symbol ^SPX), which contains 500 actively traded large stocks. For example, to download the GE data, enter its ticker symbol in the upper left section and click Go. Then select Historical Prices from the upper left side of the new page. After the daily prices come up, click monthly prices, enter a start and stop date, and click “Get Prices.” When presenting monthly data, the date shown is for the first date in the month, but the data are actually for the last day of trading in the month, so be alert for this. Note that these prices are “adjusted” to reflect any dividends or stock splits. Scroll to the bottom of the page and click “Download to Spreadsheet.” The downloaded data are in csv format. Convert to xls by opening a new Excel worksheet, copying the date and adjusted index price data to it, and saving as an xls file. Then repeat the process to get the S&P index data. At this point you have returns data for GE and the S&P Index, as we show below. Step 2. Calculate Returns Next, calculate the percentage change in adjusted prices (which already reflect dividends) for GE and the S&P to obtain returns, with the spreadsheet set up as shown below. At this point, we are ready to calculate some statistics and to find GE’s beta coefficient. This is shown below the data. Figure 6-14. Stock Return Data for GE and the S&P 500 Index Month Market Level
(S&P 500 Index) at Month End Market’s
Return GE Adjusted Stock Price at Month End
Michael C. Ehrhardt: Yahoo actually adjusts the stock prices to reflect any stock splits or dividend payments. For example, suppose the stock price is $100 in July, the company has a 2-for-1 split, and the actual price in August is $60. The reported adjusted price for August would be $60, but the reported price for July would be $50, which reflects the stock split. This gives an accurate stock return of

20%

: ($60-$50)/$50 = 20%, the same as if there had not been a split, in which case the return would have been ($120-$100)/$100 = 20%.
Or suppose the actual price in September is $50,the company pays a $10 dividend, and the actual price in October is $60. Shareholders have had a return of ($60+$10-$50)/$50 = 40%. Yahoo reports an adjusted price of $60 for October, and an adjusted price of $42.857 for September, which gives a return of ($60-$42.857)/$42.857 = 40%.
In other words, the percent change in the adjusted price accurately reflects the actual return. GE’s
Return March 2009 797.87 8.5% $10.11 18.8% February 2009 735.09 -1

1.0% $8.51 -27.8% January 2009 825.88 -8.6% $11.78 -25.2% December 2008 903.25

0.8%

$15.74 -3.8% November 2008 896.24 -7.5% $16.37 -12.0% October 2008 968.75 -16.8% $18.60 -23.5% September 2008 1,164.74 -9.2% $24.30 -8.1% August 2008 1,282.83

1.2%

$26.43 –

0.7% July 2008 1,267.38 -1.0% $26.61

6.0%
June 2008 1,280.00

-8.6%

$25.10 -1

2.1% May 2008 1,400.38 1.1% $28.57 -6.1% April 2008 1,385.59 4.8% $30.42 -11.6% March 2008 1,322.70 -0.6% $34.43 11.7% February 2008 1,330.63 -3.5% $30.83 -5.4% January 2008 1,378.55

-6.1%

$32.59 -4.6% December 2007 1,468.36 -0.9% $34.17 -2.4% November 2007 1,481.14 –

4.4% $35.00 -7.0% October 2007 1,549.38 1.5% $37.62

-0.6%
September 2007 1,526.75

3.6%

$37.84 7.2% August 2007 1,473.99

1.3%

$35.29 0.3% July 2007 1,455.27 -3.2% $35.19

1.3%
June 2007 1,503.35 -1.8% $34.75 2.6% May 2007 1,530.62 3.3% $33.87

2.0%
April 2007 1,482.37 4.3% $33.22 4.2% March 2007 1,420.86

1.0%

$31.87

1.3%
February 2007 1,406.82 -2.2% $3

1.4

7

-2.4%
January 2007 1,438.24 1.4% $32.24 -3.1% December 2006 1,418.30

1.3%

$33.28 6.3% November 2006 1,400.63

1.6%

$31.32 0.5% October 2006 1,377.94

3.2%

$31.17 -0.5% September 2006 1,335.85 2.5% $31.34

4.4%
August 2006 1,303.82

2.1%

$30.02

4.2%
July 2006 1,276.66

0.5%

$28.81 -0.8% June 2006 1,270.20

0.0%

$29.05

-3.1%
May 2006 1,270.09

-3.1%

$29.97

-1.0%
April 2006 1,310.61

1.2%

$30.26

-0.6%
March 2006 1,294.87

1.1%

$30.43 5.8% February 2006 1,280.66

0.0%

$28.76

1.1%
January 2006 1,280.08

2.5%

$28.44 -6.6% December 2005 1,248.29 -0.1% $30.44 -1.2% November 2005 1,249.48

3.5%

$30.80 5.3% October 2005 1,207.01

-1.8%

$29.24

0.7%
September 2005 1,228.81

0.7%

$29.03

0.8%
August 2005 1,220.33 -1.1% $28.79 -2.6% July 2005 1,234.18

3.6%

$29.55 –

0.4% June 2005 1,191.33 -0.0% $29.68 -4.4% May 2005 1,19

1.50 3.0% $31.05

0.7%
April 2005 1,156.85 -2.0% $30.82

0.4%
March 2005 1,180.59 NA $30.70

NA
Description of Data Average return (annual)

: -8.5% -22.9% Standard deviation (annual)

: 15.9% 28.9% Minimum monthly return:

-16.8% -27.8%
Maximum monthly return:

8.5% 18.8%
Correlation between GE and the market: 0.76 Beta: bGE = rGE,M (sGE / sM) 1.37 Beta (using the SLOPE function)

:

1.37
Intercept (using the INTERCEPT function): -0.01 R2 (using the RSQ function): 0.57 Step 3. Examine the Data Using the AVERAGE function and the STDEV function, we found the average historical return and standard deviation for GE and the market. (We converted these from monthly figures to annual figures. Notice that you must multiply the monthly standard deviation by the square root of 12, and not 12, to convert it to an annual basis.) These are shown in the rows above. We also used the CORREL function to find the correlation between GE and the market. We used the SLOPE, INTERCEPT, and RSQ functions to estimate the regression for beta. Step 4. Plot the Data and Calculate Beta Using the Chart Wizard, we plotted the GE returns on the y-axis and the market returns on the x-axis. We also used the menu Chart > Options to add a trend line, and to display the regression equation and R2 on the chart. The chart is shown below. Interesting note: We did the above analysis in July of 2008 and then updated it in March of 2009, after the market crash. Here are some resulting differences: As of March 2009 As of July 2008 Market GE Market GE
Average return (annual)

-8.45% -22.94% 3.83% -0.14% Standard deviation (annual)

15.92% 28.93% 9.60% 15.66% Beta (using the SLOPE function)

1.3744 0.5987 Correlation between GE and the market. 0.7562 0.3673 R2 (using the Excel function) 0.5719 0.1349 Average returns for GE and the market both fell. SDs rose, indicating higher stand-alone risk. GE’s beta rose, indicating greater sensitivity to changes in the market. GE’s correlation with the market rose, indicating that much of GE’s decline was due to the market drop. R2 rose, indicating that GE’s returns were more closely related to market changes than in the earlier period. These changes are all logical, but perhaps the most interesting one, for our purposes is the change in beta. We will use beta when we estimate a firm’s cost of capital, and the change in beta indicates a significant change in the cost of equity. Based on its low beta (well below 1.0) in July, GE appeared to have a low cost of equity. It’s sharper-than-average price drop indicated that it was, ex post, really more risky than average. That indicates that its “true risk” in July was risker than the average investor thought. People have tried to forecast beta, and if they can do so, they can earn abnormal returns in the market. At any rate, forecasting betas by adjusting historical betas for changes in leverage and other factors is widely practiced.

THE RELATIONSHIP BETWEEN RISK AND RATES OF RETURN (Section

6.5

)

The SML shows the relationship between the stock’s beta and its required return, as predicted by the CAPM. rRF

6%

<< Varies over time, but is constant for all firms at a given time. rM 11%

<< Varies over time, but is constant for all firms at a given time. bi 0.5

<< Varies over time, and varies from firm to firm. The SML predicts stock i’s required return to be: ri =

rRF + bi(RPM) ri = rRF + bi(rM – rRF) ri = 6% + 0.5(11% – 6%) ri = 8.5%

=B536+B538*(B537-B536) With the above data, we can generate a Security Market Line that is flexible enough to allow for changes in any of the input factors. We generate a table of values for beta and expected returns, and then plot the graph as a scatter diagram.

Beta

Required Return 0.00 6.0%
0.50

8.5%
1.00

11.0% 1.50

13.5% 2.00 16.0% The Security Market Line shows the projected changes in expected return, due to changes in the beta coefficient. However, we can also look at the potential changes in the required return due to variations in other factors, for example the market return and risk-free rate. In other words, we can see how required returns can be influenced by changing inflation and risk aversion. The level of investor risk aversion is measured by the market risk premium (rM – rRF), which is also the slope of the SML. Hence, an increase in the market return results in an increase in the maturity risk premium, other things held constant. We will look at two potential conditions as shown in the following columns: OR
Scenario 1. Interest rates increase: Scenario 2. Investors become more risk averse: Risk-free rate

6% Risk-free rate 6%
Beta 0.50 Beta 0.50
Old market return

11% Old market return 11%
Change in interest rates

2%

Increase in MRP

2.5%
New market return 13%

New market return 13.5%
Required return 10.5%

Required return

9.75% Now we can see how these two factors can affect a Security Market Line, using a data table for the required return with different beta coefficients. Required Return
Beta

Original Situation Interest Rate Increases Risk Aversion Increases 8.5% 10.5% 9.75%
0.00

6.00% 8.00%

6.00%
0.50

8.50% 10.50%

9.75%
1.00

11.00% 13.00% 13.50% 1.50 13.50%

15.50% 17.25% 2.00

16.00% 18.00% 21.00% 1. As beta, which measures risk, increases, the required return on securities increases, given the existence of risk aversion. 2. If iinterest rates increase, the required return on all securities, regardless of risk increases by the increase in the risk-free rate. 3. If risk aversion increases, the return on all securities except the riskless asset (beta = 0) increase. However, the higher the beta, the greater the increase in the required return.

Figure 6-11.
Security Market Line

Required Return 0.0 0.5 1.0 1.5 2.0 0.06 0.085 0.11 0.135 0.16 Beta
Required Return

Stock W

Stock W 2006.0 2007.0 2008.0 2009.0 2010.0 0.4 -0.1 0.35 -0.05 0.15 2010
Return

Stock W’
Stock W’ 2006.0 2007.0 2008.0 2009.0 2010.0 0.4 -0.1 0.35 -0.05 0.15 2010
Return

Portfolio WW’
Portfolio WW’ 2006.0 2007.0 2008.0 2009.0 2010.0 0.4 -0.1 0.35 -0.05 0.15 2010
Return

Stock W
Stock W 2006.0 2007.0 2008.0 2009.0 2010.0 0.4 -0.1 0.35 -0.05 0.15 2010
Return

Stock Y
Stock Y 2006.0 2007.0 2008.0 2009.0 2010.0 0.4 0.15 -0.05 -0.1 0.35 2010
Return

Portfolio WY
Portfolio WY 2006.0 2007.0 2008.0 2009.0 2010.0 0.4 0.025 0.15 -0.075

0.25

2010
Return

0.12 0.0966666666666667 0.0733333333333333 1.232925187733316 1.849387781599975 3.698775563199949

Stock L 0.19 0.25 -0.15 0.12 0.15 -0.05 Stock A 0.19 0.25 -0.15 0.19 0.25 -0.15 Stock H 0.19 0.25 -0.15 0.26 0.35 -0.25

Return on the Market

Returns on Stocks
H, A, and L

The SML Under Inflation and Risk Aversion Increases

Original 0.0 0.5 1.0 1.5 2.0 0.06 0.085 0.11 0.135 0.16 Inflation up 0.0 0.5 1.0 1.5 2.0 0.08 0.105 0.13 0.155 0.18 Risk aversion up 0.0 0.5 1.0 1.5 2.0 0.06 0.0975 0.135 0.1725 0.21

Risk, bi

Required Returns

GE Returns vs. S & P

0.0854045082915017 -0.109931224875284 -0.0856573484638804 0.00782156565205747 -0.0748490322580645 -0.168269313323145 -0.0920542862265459 0.0121905032429104 -0.00985937499999991 -0.0859623816392694 0.0106741532487967 0.0475466848113706 -0.00595958305464334 -0.0347611620906023 -0.0611634748971641 -0.0086284888666839 -0.0440434238211414 0.0148223350253808 0.0357940013161555 0.0128635923230741 -0.0319819070742009 -0.0178163097306973 0.0325492286001471 0.0432906831074138 0.00997995479165775 -0.0218461452886862 0.0140590848198548 0.012615751483261 0.0164666095766144 0.0315080285960251 0.0245662744857419 0.0212742625287859 0.00508581325775471 8.66080356511171E-5 -0.0309169012902389 0.0121556604137867 0.0110958412068776 0.00045309668145753 0.025466838635253 -0.00095239619681792 0.0351861210760474 -0.0177407410421464 0.00694894004080865 -0.0112220259605569 0.0359682036043751 -0.00014267729752419 0.0299520248951896 -0.0201085897729102 0.188014101057579 -0.277589134125637 -0.251588310038119 -0.0384850335980452 -0.11989247311828 -0.234567901234568 -0.0805902383654937 -0.00676437429537766 0.0601593625498007 -0.12145607280364 -0.0608152531229455 -0.116468196340401 0.116769380473565 -0.0540042957962567 -0.0462393912788996 -0.023714 2857142857 -0.0696438064859117 -0.00581395348837225 0.0722584301501843 0.00284171639670365 0.0126618705035971 0.0259816947150872 0.0195665261890427 0.042359585817383 0.0127105179536067 -0.0238833746898264 -0.03125 0.0625798212005109 0.00481231953801728 -0.0054243777919591 0.0439706862091939 0.0419993057965984 -0.00826161790017218 -0.0306973640306973 -0.00958360872438872 -0.00558659217877089 0.0580667593880389 0.0112517580872011 -0.0657030223390276 -0.0116883116883117 0.0533515731874146 0.00723389596968644 0.00833622785689482 -0.0257191201353638 -0.00438005390835576 -0.0441223832528181 0.00746268656716419 0.00390879478827365

x-axis: Historical
Market Returns

y-axis: Historical
GE Returns

Stock W
Stock W 2006.0 2007.0 2008.0 2009.0 2010.0 0.4 -0.1 0.35 -0.05 0.15 2010
Return

Panel a. Sale.com

90.0 15.0 -60.0 0.3 0.4 0.3

Rate of
Return
(%)

Probability of Occurrence

Panel b. Basic Foods

45.0 15.0 -15.0 0.3 0.4 0.3 Rate of
Return
(%)
Probability of Occurrence

Stock M
Stock M 2006.0 2007.0 2008.0 2009.0 2010.0 -0.1 0.4 -0.05 0.35 0.15 2010
Return

Portfolio WM
Portfolio WM 2006.0 2007.0 2008.0 2009.0 2010.0 0.15 0.15 0.15 0.15 0.15 2010
Return

Stock L
Stock H
Stock A

6.1

for an investment that returns

in one year. What is the annual rate of return?

Amount invested $500
Amount received in one year $600

$100

Rate of return 20%

SECTION 6.1
			SOLUTIONS TO SELF-TEST
Suppose you pay	$500	$600
Dollar return

6.2

SOLUTIONS TO SELF-TEST

return. What is its expected return? What is its standard deviation?

Probability Return

20% 25% 5.0%
60% 10% 6.0%
20% -15%

8.0%

Probability Return

20% 25%

60% 10% 2.0%

20% -15%

Variance =

Standard deviation =

Realized
Year return

1 10%
2 -15%
3 35%

10.0%

25.0%

Expected return 15.0%
Standard deviation 30.0%

2.0

SECTION 6.2
An investment has a 20% chance of producing a 25% return, a 60% chance of producing a 10% return, and a 20% chance of producing a			-15%
Prob x Ret.
-3.0%
Expected return =
Deviation from expected return	Deviation2	Prob x Dev.2
17.0%	2.890%	0.578%
0.040%	0.024%
-23.0%	5.290%	1.058%
1.660%
12.9%
A stock’s returns for the past three years are 10%, -15%, and 35%. What is the historical average return? What is the historical sample standard deviation?
Average =
Standard deviation =
An investment has an expected return of 15% and a standard deviation of 30%. What is its coefficient of variation?
Coefficient of variation

6.3

SOLUTIONS TO SELF-TEST

invested in Dell,

invested in

, and $25,000 invested in

. Dell’s beta is estimated to be 1.20, Ford’s beta is estimated to be 0.80, and Wal-Marts beta is estimated to be 1.0. What is the estimated beta of the investor’s portfolio?

Stock Investment Beta Weight

Dell $25,000 1.2 0.25 0.30
Ford $50,000 0.8 0.50

Wal-Mart $25,000 1.0 0.25 0.25
Total $100,000

SECTION 6.3
An investor has a 3-stock portfolio with		$25,000	$50,000	Ford	Wal-Mart
Beta x Weight
0.40
Portfolio beta =	0.95

6.5

SOLUTIONS TO SELF-TEST

and the market risk premium is 5%. What is the stock’s required rate of return?

Beta 1.4
Risk-free rate 5.5%

5.0%

SECTION 6.5
A stock has a beta of 1.4. Assume that the risk-free rate is	5.5%
Market risk premium
Required rate of return	12.50%

=
r

ˆ

r

ˆ
=
r
ˆ
r
ˆ

Ch06 P14 Build a Model

,

, and the

, and then calculate average returns over the five-year period. (Hint: Remember, returns are calculated by subtracting the beginning price from the ending price to get the capital gain or loss, adding the dividend to the capital gain or loss, and dividing the result by the beginning price. Assume that dividends are already included in the index. Also, you cannot calculate the rate of return for

because you do not have 2004 data.)

Market Index

Stock Price Dividend

48.750

2005

Bartman Reynolds Index
2010

2009

2008
2007
2006
Bartman Reynolds Index

No answer should be entered here.

Bartman Reynolds Index

No answer should be entered here.

Year Index Bartman Reynolds
2010

0.0% 0.0%

2009 0.0% 0.0% 0.0%
2008 0.0% 0.0% 0.0%
2007 0.0% 0.0% 0.0%
2006 0.0% 0.0% 0.0%

(RPM) =

=

Risk-free rate

Market risk premium

= 6.040% + 5.000%
=

=

Required return =

=

Required return =
=

,

, and

, respectively. Calculate the new portfolio’s required return if it consists of

of Bartman,

of

, 40% of

, and

of

.

Bartman 25%
Stock A 0.769 15%
Stock B 0.985 40%
Stock C 1.223 20%

=

=
=

																																																																												Spring 1, 2013
	7/22/12
Chapter 6. Ch 06 P14 Build a Model
Except for charts and answers that must be written, only Excel formulas that use cell references or functions will be accepted for credit.
Numeric answers in cells will not be accepted.
a. Use the data given to calculate annual returns for					Bartman				Reynolds	Market				Index	2005
Data as given in the problem are shown below:
Bartman Industries	Reynolds Incorporated
	Year		Stock Price		Dividend	Includes Divs.
		2010	$17.250	$1.150	$	48.750	$3.000	11,663.98
		2009	15.000	1.060	52.300	2.900	8,785.70
		2008	16.500	1.000	2.750	8,679.98
		2007	10.750	0.950	57.250	2.500	6,434.03
		2006	11.375	0.900	60.000	2.250	5,602.28
7.625	0.850	55.750	2.000	4,705.97
We now calculate the rates of return for the two companies and the index:
Kenneth D. Jackson: Change in stock price plus any dividends, divided by the previous stock price
Average
Note: To get the average, you could get the column sum and divide by 5, but you could also use the function wizard, fx. Click fx, then statistical, then Average, and then use the mouse to select the proper range. Do this for Bartman and then copy the cell for the other items.
b. Calculate the standard deviation of the returns for Bartman, Reynolds, and the Market Index. (Hint: Use the sample standard deviation formula given in the chapter, which corresponds to the STDEV function in Excel.)
Use the function wizard to calculate the standard deviations.
Standard deviation of returns
		No answer should be entered here.
c. Now calculate the coefficients of variation Bartman, Reynolds, and the Market Index.
Coefficient of Variation
d. Construct a scatter diagram graph that shows Bartman’s and Reynolds’ returns on the vertical axis and the Market Index’s returns on the horizontal axis.
It is easiest to make scatter diagrams with a data set that has the X-axis variable in the left column, so we reformat the returns data calculated above and show it just below.
														0.0%
To make the graph, we first selected the range with the returns and the column heads, then clicked the chart wizard, then choose the scatter diagram without connected lines. That gave us the data points. We then used the drawing toolbar to make free-hand (“by eye”) regression lines, and changed the lines color and weights to match the dots.
e. Estimate Bartman’s and Reynolds’s betas as the slopes of regression lines with stock returns on the vertical axis (y-axis) and market return on the horizontal axis (x-axis). (Hint: use Excel’s SLOPE function.) Are these betas consistent with your graph?
Bartman’s beta										=
Reynolds’ beta =
f. The risk-free rate on long-term Treasury bonds is 6.04%. Assume that the market risk premium is 5%. What is the expected return on the market? Now use the SML equation to calculate the two companies’ required returns.
	Market risk premium		5.000%
	Risk-free rate		6.0	40%
Expected return on market =		+
11.040%
		Required return
Bartman:
Reynolds:
g. If you formed a portfolio that consisted of 50% Bartman stock and 50% Reynolds stock, what would be its beta and its required return?
The beta of a portfolio is simply a weighted average of the betas of the stocks in the portfolio, so this portfolio’s beta
would be:
Portfolio beta =
h. Suppose an investor wants to include Bartman Industries’ stock in his or her portfolio. Stocks A, B, and C are currently in the portfolio, and their betas are	0.769	0.985	1.223	25%	15%	Stock A	Stock B	20%	Stock C
Beta	Portfolio Weight
100%
Portfolio Beta =
Required return on portfolio:	Risk-free rate +	Market Risk Premium *	* Beta

Sheet2

7/22/12