Calculation

  1. Top hedge fund manager Sally Buffit believes that a stock with the same market risk as the S&P 500 will sell at year-end at a price of $49. The stock will pay a divided at year-end of $300. Assume that risk-free Treasury securities currently offer an interest rate of 2.1%

 

Average rates of return on Treasury bills, government bonds, and common stocks, 1900-2017 (figures in percent per year) are as follows

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Portfolio Average Annual Rate of Return (%) Average Premium (Extra return versus Treasury bills) (%)
Treasury bills

 

 3.8  
Treasury bonds 5.3 1.5
Common stocks 11.5 7.7

 

  1. What is the discount rate on the Stock?
  2. What price should she be willing to pay for the stock today?

 

  1. Assume these are the stock market and treasury bill return for a 5-year period

 

Year Stock Market Return (%) T-Bill Return (%)
2013 35.90 0.21
2014 15.20 0.21
2015 -5.10 0.21
2016 16.90 0.08
2017 25.90 0.10

 

  1. What was the risk premium on common stock in each year?
  2. What was the average risk premium?
  3. What was the standard deviation of the risk premium?

 

  1. A stock is selling today for $40 per share. At the end of the year, it pays a dividend of $2 per share and sells for $46.
  2. What is the total rate of return on the stock?
  3. What are the divided yield and percentage capital gain?
  4. Now suppose the year-end stock price after the dividend is paid is $34. What are the divided yield and percentage capital gain in this case?

 

  1. You purchase 100 shares of stock for $50 a share. The stock pays a $3 per share dividend at year-end.
  2. What is the rate of return on your investment if the end-of -year stock price is

 

  1. 47
  2. 50
  • 55
  1. What is your real (inflation-adjusted) rate of return if the inflation is 5%?
  2. 47
  3. 50
  • 55

 

  1. Consider the following scenario analysis
Scenario Probability Stocks

(rate of return)

 Bonds

(rate of return)
Recession 0.20 -5% 19%
Normal economy 0.60 20% 10%
Boom 0.20 27% 4%

 

  1. Is it reasonable to assume that treasury bond will provide higher returns in recessions than in the booms? Yes or NO

 

  1. Calculate the expected rate of return and standard deviation for Stocks
  2. Calculate the expected rate of return and standard deviation for bonds
  3. Which investment do you prefer?

 

  1. In a recent 5-year period, mutual fund manager Diana Sauros produced the following percentage rates of return for the Mesozoic fund. Rates of return on the market index are given for comparison

 

  1 2 3 4 5
Fund -1.4 +24.0 +41.9 +10.9 +0.2
Market Index -0.5 +16.0 +31.4 +9.9 -0.3

 

Calculate

 

  1. The average return on Mesozoic Fund Return and Market Portfolio Return
  2. The standard deviation of the returns on Mesozoic fund return and Market portfolio return
  3. Did Ms. Sauros do better or worse than the market index on these measures?

 

  1. Consider the following scenario analysis

 

Recession – Probability 0.2, Stocks -7% and Bonds 15%

Normal economy – Probability 0.7, Stocks 16%, and Bonds 7%

Boom – Probability 0.1, Stocks 25%, and bonds 6%

 

Assume a portfolio with weights of 0.60 in stocks and 0.40 in bonds

 

  1. What is the rate of return on?
  2. Recession%
  3. Normal economy%
  4. Boom %

 

  1. What is the expected rate of return and standard deviation of the portfolio?

 

  1. Would you prefer to invest in the portfolio, in stocks only, or in bonds only?

 

 

  1. Here are the returns on two stocks
  Digital cheese Executive Fruit
Jan +16 +9
Feb -3 +2
Mar +5 +6
Apr +7 +14
May -4 +3
June +3 +7
July -2 -3

Aug                                                                 -8                                                                  -2

  1. Calculate the variance and standard deviation of each stock
  2. Which stock is riskier if held on its own?
  3. Now calculate the returns in each month of a portfolio that invest an equal amount each month in the two stocks
  4. Is the variance more or less than halfway between the variance of the two individual stocks

 

  1. A stock with a beta of 1.2 has an expected rate of return of 14%. If the market returns this year turns out to be 8% points below expectations, what is your best guess as to the rate of return on the stock.

Stock return%?

  1. The risk-free rate is 5% and the expected rate of return on the market portfolio is 10%
  2. Calculate the required rate of return on a security with a beta of 1.23.
  3. If the security is expected to return 12%, is it overpriced or underpriced?

 

  1. Consider the following two scenarios for the economy and the expected returns in each scenario for the market portfolio, an aggressive stock A, and a defensive stock D

 

  Market Stock A Defensive Stock
Bust -9% -10% -7%
Boom 25 28 20

 

  1. Find the beta of stock A and D
  2. If each scenario is equally likely, find the expected rate of return on the market portfolio, stock A, and stock D
  3. If the T-bill rate is 3%, what does the CAPM say about the fair expected rate of return on stock A and D?
  4. Which stock seems to be a better buy on the basis of your answers to (a) through (c)

 

  1. A share of stock with a beta of 0.65 now sells for $40. Investors expect the stock to pay a year-end divided of $2. The T-bill rate is 6% and the market risk premium is 9%. If the stock is perceived to be fairly priced today what must be investor’s expectation of the price of the stock at the end of the year?

 

  1. Suppose that the S&P 500, with a beta of 1.0, has an expected return of 14% and T-bills provide a risk-free return of 5%

 

  1. What would be the expected return and beta of portfolios constructed from these two assets with weights in the S&P 500 of

 

    Expected return Beta
1 0    
11 0.25    
111 0.50    
1V 0.75    
V 1.0    

 

How does expected return vary with beta?

The expected return (decrease or increase) by         % for a one unit increase in beta

 

  1. The treasury bill rate is 4% and the expected return on the market portfolio is 14%. According to the capital asset pricing model

 

  1. What is the risk premium on the market? %
  2. Return on investment? %
  3. NPV
  4. Beta

 

  1. A project under consideration has an internal rate of return of 13% and a beta of 0.8. the risk -free rate is 3% and the expected rate of return on the market portfolio is 13%
  2. What is the required rate on the project? %
  3. Should the project be accepted?
  4. What is the required rate of return on the project if its beta is 1.80? %
  5. Accept the project

 

  1. The treasury bill rate is 5% and the market risk premium is 8%

 

Project Beta Internal rate of return %
P 0.95 14
Q 0.00 12
R 2.00 21
S 0.35 13
T 1.60 23

 

  1. What is the project cost of capital for new ventures with betas of 0.70 and 1.59?
  2. Which of the capital investments shows above have positive (non-zero) NPV’s?

P, Q, S, T, R

 

  1. You are considering the purchase of real estate that will provide perpetual income that should average $65,000 per year. How much will you pay for the property if you believe its market risk is the same as the market portfolio’s? The T-bill rate is 6% and the expected market return is 10%

 

Property value?

 

ANSWER

  1. Top hedge fund manager Sally Buffit believes that a stock with the same market risk as the S&P 500 will sell at year-end at a price of $49. The stock will pay a divided at year-end of $3.00. Assume that risk-free Treasury securities currently offer an interest rate of 2.1%

Average rates of return on Treasury bills, government bonds, and common stocks, 1900-2017 (figures in percent per year) are as follows

Portfolio Average Annual Rate of Return (%) Average Premium (Extra return versus Treasury bills) (%)
Treasury bills

 

 3.8  
Treasury bonds 5.3 1.5
Common stocks 11.5 7.7

 

  1. What is the discount rate on the Stock?

7.7%+ 2.1% =9.8%

  1. What price should she be willing to pay for the stock today?

Price= ($3.00 + $49)/ (1+0.098) = $47.36

  1. Assume these are the stock market and treasury bill return for a 5-year period 3
Year Stock Market Return (%) T-Bill Return (%) Risk Premium Standard Deviation  (Mean) Deviation2
2013 35.90 0.21 35.69 18.09 327.25
2014 15.20 0.21 14.99 -2.61 6.81
2015 -5.10 0.21 -5.31 -22.91 524.87
2016 16.90 0.08 16.82 -0.78 0.61
2017 25.90 0.10 25.8 8.2 67.24
TOTAL 87.99   926.78
AVERAGE 17.598

17.6

 Variance 185.356

185.36
STANDARD DEVIATION 185.360.5

13.6%

 

  1. What was the risk premium on common stock in each year?

87.99

  1. What was the average risk premium?

17.6

  1. What was the standard deviation of the risk premium?

13.6%

  1. A stock is selling today for $40 per share. At the end of the year, it pays a dividend of $2 per share and sells for $46. 1
  2. What is the total rate of return on the stock?

Total Rate= (Capital gain+ dividend)/Initial SP

= [($46.1-$40) + $2/$40]

= 0.2025

= 20.25%

  1. What are the divided yield and percentage capital gain?

Dividend Yield = Dividend/Initial SP

= $2/$40

= 0.05

= 5%

Capital Gains Yield = Capital gain/Initial SP

= ($46.1-$40)/$40

= 0.1525

= 15.25%

  1. Now suppose the year-end stock price after the dividend is paid is $34. What are the divided yield and percentage capital gain in this case?

Dividend Yield = Dividend/Initial SP

= $2/$40

= 0.05

= 5%

Capital Gains Yield = Capital gain/Initial SP

= ($34-$40)/$40

= -0.15

= -15%

  1. You purchase 100 shares of stock for $50 a share. The stock pays a $3 per share dividend at year-end. 2
  2. What is the rate of return on your investment if the end-of -year stock price is
  3. 47

Rate of return = capital gain + dividend/ Initial SP

= [($47-$50) + $3]/$50

=0.0

= 0%

  1. 50

Rate of return = capital gain + dividend/ Initial SP

= [($50-$50) + $3]/$50

= 0.06

= 6%

  • 55

Rate of return = capital gain + dividend/ Initial SP

= [($55-$50) + $3]/$50

=0.16

= 16%

  1. What is your real (inflation-adjusted) rate of return if the inflation is 5%?
  2. 47

Rate of return= [(1+Nominal Rate of return)/ (1+Inflation rate)] -1

= [(1+0)/ (1+0.05)]-1

= -0.0476

= -4.76%

  1. 50

Rate of return= [(1+Nominal Rate of return)/ (1+Inflation rate)] -1

= [(1+0.06)/ (1+0.05)]-1

= 0.0095

= 0.95%

  • 55

Rate of return= [(1+Nominal Rate of return)/ (1+Inflation rate)] -1

= [(1+0.16)/ (1+0.05)]-1

= 0.105

= 10.5%

  1. Consider the following scenario analysis
Scenario Probability Stocks

(rate of return)

 Bonds

(rate of return)
Recession 0.20 -5% 19%
Normal economy 0.60 20% 10%
Boom 0.20 27% 4%

 

  1. Is it reasonable to assume that Treasury bond will provide higher returns in recessions than in the 5booms? Yes or NO

Yes. This is because interest rates may fall during an onset of recession but rise during a boom period. Hence, bond prices move inversely with interest rates, therefore, it is expected that the bonds will provide higher returns during recessions, when the interest rates fall.

  1. Calculate the expected rate of return and standard deviation for Stocks

= (0.2x (-5%) + (0.6×20%) + (0.2×27%)

=0.164

=16.4%

= {0.2 × (−5 − 16.4)2} + {0.6 × (20 − 16.4)2} + {0.2 × (27 − 16.4)2}

=121.84

=121.840.5

=11.038, =11.04%

  1. Calculate the expected rate of return and standard deviation for bonds

= (0.2x (19%) + (0.6×10%) + (0.2×4%)

=0.106

=10.6%

= {0.2 × (19 − 10.6)2} + {0.6 × (10 − 10.6)2} + {0.2 × (4 − 10.6)2}

=23.04

=23.040.5

=4.8, =4.8%

  1. Which investment do you prefer?

The most preferred investment are the stocks as they have higher expected returns and volatility, and align with my less risk averse. For investors with high risk averse, bonds could be the most preferred investments.

  1. In a recent 5-year period, mutual fund manager Diana Sauros produced the following percentage rates of return for the Mesozoic fund. Rates of return on the market index are given for comparison
Year Fund Return (%) Standard deviation mean Deviation 2 Market index return (%) Standard Deviation  (Mean) Deviation2
1 -1.4 -16.52 272.91 -0.5 -11.8 139.24
2 24.0 8.88 77.44 16.0 4.7 22.09
3 41.9 26.78 717.17 31.4 20.1 404.01
4 10.9 -4.22 17.81 9.9 -1.4 1.96
5 0.2 -14.92 222.61 -0.3 -11.6 134.56
TOTAL 75.6                     1,307.94         56.5   701.86
AVERAGE 15.12                     261.59              11.3   140.37
SD   16.17   11.85

Calculate

  1. The average return on Mesozoic Fund Return and Market Portfolio Return

Fund return = (-1.4+24+41.9+10.9+0.2)/5 = 15.12

Market index return = -0.5+16+31.4+9.9+ (-0.3)]/5 = 11.3

  1. The standard deviation of the returns on Mesozoic fund return and Market portfolio return

SD of the Fund = 16.17 and the market portfolio = 11.85

  1. Did Ms. Sauros do better or worse than the market index on these measures?

The fund provides a greater expected return of 16.17 compared to the market index 11.85 due to higher variance and standard deviation.

  1. Consider the following scenario analysis

Recession – Probability 0.2, Stocks -7% and Bonds 15%

Normal economy – Probability 0.7, Stocks 16%, and Bonds 7%

Boom – Probability 0.1, Stocks 25%, and bonds 6%

Assume a portfolio with weights of 0.60 in stocks and 0.40 in bonds

  1. What is the rate of return on?
  2. Recession%

(-7% x 0.60) + (15% x 0.40) = 0.018, =1.8%

  1. Normal economy%

(16% x 0.60) + (7% x 0.40) = 0.124, =12.4%

  1. Boom %

(25% x 0.60) + (6% x 0.40) = 0.174, =17.4%

  1. What is the expected rate of return and standard deviation of the portfolio?

(0.2×1.8%) + (0.7×12.4%) + (0.1×17.4%) = 0.1078, = 10.78%

Portfolio = {0.2 × (1.8 − 10.78)2} + {0.7 × (12.4 − 10.78)2} + {0.1 × (17.4 − 10.78)2}

=22.3476. = 22.35

=4.7%

Stock = [0.2x (-7%)] + (0.7×16%) + (0.1×25%) = 0.123. =12.3%

{0.2 × (-7 –12.3)2} + {0.7 × (16 – 12.3)2} + {0.1 × (25 – 12.3)2} = 100.21

= 10.01%

Bond = [0.2x 15%)] + (0.7×7%) + (0.1×6%) = 0.085. =8.5%

{0.2 × (15 – 8.5)2} + {0.7 × (7– 8.5)2} + {0.1 × (6 – 8.5)2} = 10.65

= 3.26%

  1. Would you prefer to invest in the portfolio, in stocks only, or in bonds only?
  Mean return Standard deviation
Portfolio 10.78% 4.7%
Stocks 12.3% 10.01%
Bond 8.5% 3.26%

The best investment would be a portfolio as it provides a high-expected return and a lower volatility.

  1. Here are the returns on two stocks
  Digital cheese Standard deviation Deviation2 Executive Fruit Standard deviation Deviation 2 Portfolio return

Assuming Equal Investment is $100
Jan +16 14.25 203.06 +9 4.5 20.25 12.5%
Feb -3 -4.75 22.56 +2 -2.5 6.25 -0.5%
Mar +5 3.25 10.56 +6 1.5 2.25 5.5%
Apr +7 5.25 27.56 +14 9.5 90.25 10.5%
May -4 -5.75 33.06 +3 -1.5 2.25 -0.5%
June +3 1.25 1.56 +7 2.5 6.25 5%
July -2 -3.75 14.06 -3 -7.5 56.25 -2.5%
Aug -8 -9.75 95.06 -2 -6.5 42.25 -5%
TOTAL 14   407.48 36   226  
AVERAGE 1.75 VAR 50.94 4.5 VAR 28.25 79.19
SD     7.14%     5.32%  

 

  1. Calculate the variance and standard deviation of each stock

Digital cheese = variance 50.94 and SD 7.14%

Executive fruit = variance 28.25 and SD 5.32%

  1. Which stock is riskier if held on its own? Digital Cheese
  2. Now calculate the returns in each month of a portfolio that invest an equal amount each month in the two stocks
  3. Is the variance more or less than halfway between the variance of the two individual stocks

The variance is less.

  1. A stock with a beta of 1.2 has an expected rate of return of 14%. If the market returns this year turns out to be 8% points below expectations, what is your best guess as to the rate of return on the stock.

Stock return%?

1.2×8% = 9.6%

14%-9.6% = 4.4%

  1. The risk-free rate is 5% and the expected rate of return on the market portfolio is 10%
  2. Calculate the required rate of return on a security with a beta of 1.23.
  3. Required Rate of Return = rf + β (rm − rf)

= 5% + [1.23 x (10%-5%)]

= 11.15%

  1. If the security is expected to return 12%, is it overpriced or underpriced?

It is underpriced considering the expected return is 12%.

  1. Consider the following two scenarios for the economy and the expected returns in each scenario for the market portfolio, an aggressive stock A, and a defensive stock D
  Market Stock A Defensive Stock
Bust -9% -10% -7%
Boom 25 28 20

 

  1. Find the beta of stock A and D

Stock A= 28%-(-10%)/ (32%-(-9%)

=0.93

Stock D = 20%-(-7%)/ (25%-(-9%)

= 0.79

  1. If each scenario is equally likely, find the expected rate of return on the market portfolio, stock A, and stock D

Stock A = (28%+ 10%)/2 = 19%

Stock D= (20%+7%)/2 = 13.5%

Market = (25%+9%)/2 = 17%

  1. If the T-bill rate is 3%, what does the CAPM say about the fair expected rate of return on stock A and D?

Stock A = 3% + 0.93(17%-3%) = 16.02%

Stock D = 3% + 0.79 (17%-3%) = 6.16%

  1. Which stock seems to be a better buy on the basis of your answers to (a) through (c)

Stock A as it provides a higher expected return rate.

  1. A share of stock with a beta of 0.65 now sells for $40. Investors expect the stock to pay a year-end divided of $2. The T-bill rate is 6% and the market risk premium is 9%. If the stock is perceived to be fairly priced today what must be investor’s expectation of the price of the stock at the end of the year?

r = rf + β (rm – rf)

6%+ (0.65×9%) = 11.85

r = 0.1185

Stock price end year = [2+ (p1-40)]/40 = $42.37

Fair stock price= ($2+42)/1.0925 =$40.27

  1. Suppose that the S&P 500, with a beta of 1.0, has an expected return of 14% and T-bills provide a risk-free return of 5%
  2. What would be the expected return and beta of portfolios constructed from these two assets with weights in the S&P 500 of
    Expected return Beta
1 0 (0x14%) + (1.0×5%)

= 0.05

=5%

 = (0x1) + (1×0)

= 0
11 0.25 (0.25×14%) + (0.75×5%)

= 0.0725

=7.2%

 = (0.25×1) + (0.75×0)

= 0.25
111 0.50 (0.50×14%) + (0.50×5%)

=0.095

=9.5%

 = (0.50×1) + (0.50×0)

= 0.50
1V 0.75 (0.75×14%) + (0.25×5%)

=0.1175

= 11.75%

 = (0.75×1) + (0.25×0)

= 0.75
V 1.0 (1×14%) + (0x5%)

=0.14

=14%

 = (1×1) + (0x0)

= 1

 

How does expected return vary with beta?

The expected return increases by 14% for a one-unit increase in beta.

  1. The Treasury bill rate is 4% and the expected return on the market portfolio is 14%. According to the capital asset pricing model
  2. What is the risk premium on the market? %
  3. Return on investment? %
  4. NPV
  5. Beta
  6. A project under consideration has an internal rate of return of 13% and a beta of 0.8. the risk -free rate is 3% and the expected rate of return on the market portfolio is 13%
  7. What is the required rate on the project? %

0.13-0.03= 0.1

=10%

  1. Should the project be accepted?

Expected return = risk free rate + (beta x risk premium)

= 3% + (0.8×10%)

=11%. Yes, it should be accepted

  1. What is the required rate of return on the project if its beta is 1.80? %

3% + (1.8×10%)

21%

  1. Accept the project

Yes

  1. The treasury bill rate is 5% and the market risk premium is 8%
Project Beta Internal rate of return % Expected return on capital asset
P 0.95 14 7.85
Q 0.00 12 5.00
R 2.00 21 11.00
S 0.35 13 6.05
T 1.60 23 9.80

 

  1. What is the project cost of capital for new ventures with betas of 0.70 and 1.59?

Cost of capital = risk free rate + beta x market risk premium

5% + (0.70x 8%) = 10.6%

5% + (1.59×8%) = 17.72%

  1. Which of the capital investments shows above have positive (non-zero) NPV’s?

P, Q, S, T, R

None.

  1. You are considering the purchase of real estate that will provide perpetual income that should average $65,000 per year. How much will you pay for the property if you believe its market risk is the same as the market portfolios? The T-bill rate is 6% and the expected market return is 10%

Property value? If the systematic risk were similar and comparable to the expected market return, the discount rate would be 10%. Hence, the property’s value and worth would be:

$65,000/0.1 = $650,000

                                                                                          

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