ABOUT numericals on investment

1. Chapter 3 – Individual security risk and return
1. An investor buys a stock for $47 and gets an annual cash dividend of $2.25. Will he
experience a capital gain or a capital loss if he sells the stock for $51.75 one year later?
What is the investor’s dollar and percentage return?
2. Another investor purchases a stock for $29.80 and gets an annual cash dividend of $1.60.
This investor sells the stock three years later for $26.40 Will he experience a capital gain
or a capital loss? What is this investor’s dollar and percentage return?
3. Suppose you buy 400 shares of stock at the beginning of the year for $23/share and sell it
in 6 months for $26/share. While you held the stock you received quarterly dividends of
$0.25/share.
1. Calculate your income and capital gains over the six months you held the stock.
What is your total (dollar) return?
2. Calculate your holding period return.
3. What is your dividend yield and your capital gains yield over the six month
period?
4. Annualized your HPR using APR and EAR.
4. Find the EAR for the following periodic returns.
1. HPR =1% over 1 month.
2. HPR=2.5% over 3 months.
3. HPR=3% over 6 months.
4. HPR=0.04% over one day.
5. Imagine you buy a bond for $17,000, which pays $200 interest every 3 months. What are
the total income, the amount of capital gain/loss, and the total dollar return you
experience if you sell the bond for $15,500 nine months later? What is the HPR?
6. For the following problems, determine (1) the current income, (2) the capital gain or loss,
and (3) the total return in both dollars and as a percentage of the initial investment.
1. An investor purchased a stock for $75 one year ago and received four dividend
payments totaling $3.80. Today, the investor sells the share for $90.
2. An investor sells his stock for $121. He bought the stock 18 months ago for $137
and received dividend payments of $8.10 every 6 months.
3. An investor purchased a bond for $9,100 15 months ago. The bond paid $90
interest every 3 months and is now sold for $8,700.
7. Bob is interested in a stock that is currently trading at $32 per share and that pays
monthly dividends of $0.80 per share. His expectation is to sell the stock in four months
for $35. Steve is interested in a different stock that is trading at $34 per share and that
pays quarterly dividends of $0.95 per share. He expects to sell the stock in six months for
$38. Which stock is more lucrative in terms of the annualized holding period return?
8. You are buying a stock for $81 per share and are expecting to sell it in one year for $89.
The Stock pays $1.27 per share in quarterly dividends. At the same time, your friend
purchases a share of stock for $69. This stock is paying quarterly dividends of $1.09 per
share and your friend is expecting to sell it in 3 months for $71. Will you or your friend
have a better annualized holding period return?

2. 9. Imagine that an investment of $3,500 today is likely to return $6,000 in 8 years. Calculate
the IRR on this investment. Decide whether it is worth doing this investment if a
minimum rate of return of 6% is required. What is your decision if the required minimum
rate of return is 8%?
10. Another investment of $7,500 promises a return of $9,600 in 5 years. Estimate the IRR
on this investment. Can you recommend this investment if a minimum rate of return of
6% is required? What if the required minimum rate of return is 7%?
11. Imagine you purchased 250 shares of the investment described in the table at the
beginning of 2004.
1. Calculate the yearly total return in dollars.
2. What is the aggregate (7-year) return in dollars and as a percentage of the initial
investment if you sell the shares at the end of 2010?
3. Calculate the IRR if income is received at the end of the year.
Market Value
Year Income Market Value Beginning
Ending
2004 $2.00 $45.00 $48.00
2005 2.15 48.00 51.50
2006 2.30 51.50 58.00
2007 2.50 58.00 55.50
2008 2.75 55.50 54.00
2009 2.95 54.00 60.00
2010 3.20 60.00 65.00
12. Assume an investor purchases shares of stock for $4.500 today and will sell them for
$5,300 in 5 years. During those 5 years he receives $38, $40, $43, $39, and $41 in
dividends. Calculate the internal rate of return on this investment.
13. You invest $11,000 in a stock that pays $98, $105, $102, and $105 in dividends over the
next 4 years. After the 4 years, you decide to sell the stock for $ $12,300. Determine the
internal rate of return on your investment.
14. Determine the IRR for the investments described in the following situations.
Future
Situation Initial investment End of year
value
1 $3,000 $7,000 8
2 900 4,000 25
3 6,000 6,500 4
15. The returns for two different investments are shown in the following table.
1. Calculate the arithmetic and geometric average returns for each investment.
2. Which of the two investments seems to be more risky? Explain why.
3. What are the standard deviation of returns of each investment?

3. Year Return on A Return on B
2002 5% 10%
2003 16 14
2004 11 8
2005 1 13
2006 9 9
2007 24 12
16. The table below shows the returns for three alternative investments.
1. Calculate the arithmetic and geometric average returns for each investment.
2. What is the (average) risk premium of each investment?
3. Determine which of the three investments seems to be most risky. Which appears
to be the safest investment?
4. What are the standard deviations of returns of each investment?
Year Return on A Return on B Return on C
2003 8 16 24
2004 8 5 16
2005 8 9 10
2006 8 14 3
2007 8 1 7
Answers
1. gain, total dollar return=$7, or 14.89%
2. loss, +1.40 per share, or 4.70%
3.
1. Income=$200, cg=$1200, total=$1400
2. HPR=15.22%
3. Div yield=2.17%, cg yield=13.04% (NOTE: when looking at quotes in the
financial press, dividend yields are usually annualized).
4. APR=30.44%, EAR=32.76%
4.
1. 12.68%
2. 10.38%
3. 6.09%
4. 15.72%
5. Income=$600, cap loss=$1500, total dollar return= -$900, HPR= -5.29%
6.

4. 1. (1) $3.80 (2) $15 (3) $18.80; 25.07%
2. (1) $24.30 (2) -$16 (3) $8.30; 6.06%
3. (1) $450 (2) -$400 (3) $50; 0.55%
7. Bob APR=58.13%; Steve APR=34.71%
8. Your APR=16.15%; Your friend APR=17.91%
9. 6.97%, invest if we only require 6%, but if we require 8%, we don't invest
10. 5.06%, don't invest
11.
1. dollar return by year: 1,250.00; 1,412.50; 2,200.00; 0; 312.50; 2,237.50; 2,050.00,
as a percent: 11.11%; 11.77%; 17.09%; 0%; 2.25%; 16.57%; 13.67%
2. income=$4,462.50; capital gains=$5,000; total dollar return=$9,462.50;
HPR=84.11%
3. 10.16%
12. 4.16%
13. 3.73%
14. (1) 11.17% (2) 6.15% (3) 2.02%
15.
1. Arithmetic: A=11%, B=11%; geometric: A=10.75%, B=10.98%
2. A's returns seem more volatile, higher hi's and lower lo's
3. A=8.17%, B=2.37%
16.
1. Arithmetic: A=8%, B=9%, C=12%; geometric: A=8%, B=8.86%, C=11.76%
2. Arithmetic average risk premium: A=0%, B=1%, C=4% (note that security A is
the risk free asset)
3. A=0%, B=6.20%, C=8.22%
Chapter 4 - Investing in stocks: the basics
1. An investor purchases 150 shares of a certain stock at the following prices and margins.
Calculate the loan for each transaction as well as the equity necessary to make these
margin transactions.
Transaction Price/Share Margin Loan Equity
A 45 60%
b 128 45%
c 22 73%
d 10 66%
e 84 57%
f 38 62%
2. Use the information from the previous question. Ignore any interest paid on loans.
1. What is the investor’s new margin position if the stock price increases by $10 per
share?
2. What is the margin position if the share price decreases by $10?

5. 3. Bob purchased 150 shares of stock for $120 per share. The initial margin was 70% and
the maintenance margin was 40%. At what price will Bob face a margin call?
4. Kelly decides to buy 100 shares of a stock at a price of $62/share, using an initial margin
of 60% and the maintenance margin being 30%. How far does the stock have to drop
before Kelly faces a margin call?
5. McDonald’s stock is currently selling at $48 per share. An investor purchases 100 shares
of this stock using a margin of 70%. The annual dividends are $2 per share and the
investor can obtain a margin loan at an annual interest cost of 6%. What is the return on
invested capital that the investor can get if the stock price increases to $55 in 12 months?
6. Steve bought 200 shares of UFO stock 12 months ago. The share price was $38 per share
and the initial margin requirement used was 60%. Today, Steve decides to sell the shares.
During the last 12 months, the stock paid $2 per share in cash dividends and the annual
interest on the margin loan charged was 7%. There was a minimum maintenance margin
of 35%.
1. What is the initial value of the transaction? Determine the loan amount and the
equity position on Steve’s transaction.
2. Imagine the share price is i) $50, ii) $28, iii) $16, and iv) $45. What is the actual
margin percentage for each situation? When would Steve be subject to a margin
call?
3. Imagine that after the 12-month holding period the sales prices are the following:
i) $30, ii) $35, iii) $40, iv) $45, and v) $50. What is the rate of return for each
situation?
7. An investor has borrowed 300 shares of a stock from a broker. He decides to short sell
them for $25 a share with the initial margin being 60%.
1. Calculate the amount of money that will be in the investor’s account after the
transaction.
2. What is the margin if the stock falls to 22/share?
3. What is the holding period return if the stock falls to $22 per share?
4. If the maintenance margin is 35%, when will the investor receive a margin call?
8. An investor has borrowed 150 shares of stock from a broker. He decides to short sell
them for $30 per share with an initial margin of 40% and the maintenance margin being
20%. Calculate the margin and indicate if there will be a margin call for each of the
following situations.
1. The stock price falls to $18 per share.
2. The stock price falls to $25 per share.
3. The stock price rises to $37 per share.
4. The stock price rises to $42 per share.
9. Jimmy makes a cash purchase of 100 shares of a stock for $40 per share. Louie buys the
same stock but uses 50% margin and the loan has an 8% interest rate. Each investor
holds the stock for a year, receiving a $2 dividend. Suppose they close their positions at
the end of the year and all trades are charged a commission of $20.
1. What is the return for each investor if the stock price is $50 per share at the end of
the year?
2. What is the return for each investor if the stock price is $30 per share at the end of
the year?

6. Answers
Transaction Loan Equity Margin Margin
(+10) (-10)
a 2700 4050 67.27% 48.57%
b 10560 8640 48.99% 40.34%
c 891 2409 81.44% 50.50%
d 510 990 83.00% N/A
e 5418 7182 61.57% 51.19%
f 2166 3534 69.92% 48.43%
60
35.42
New equity = 4173.60, return = 24.21%
1. Stock=7600, loan=3040, equity=4560
2. if you assume the stock price changes instantly so you can ignore interest and dividends
for this question: i) 69.6%, ii) 45.71%, iii) 5.00%, iv) 66.22%. Only in (iii) will there be a
margin call
3. i) ending equity = 3147.20, return =-30.98%, ii)-9.05%, iii) +12.88%, iv) +34.81%,
v)+56.74%
1. Loan=7500, equity =4500, assets=cash=12,000
2. 81.82%
3. 20%
4. 29.63
1. 133.33%
2. 68%
3. 13.5%
4. wipes out equity completely
1. +50%
2. -50%

7. Chapter 6 - Portfolio
1. Calculate the average return of the following portfolio
Investment Avg. Return Amount invested
Stocks 15% $100,000
Bonds 5% 50,000
Real estate 10% 250,000
2. Assume you are considering a portfolio containing two assets, A and B. Asset A will
represent 45% of the dollar value of the portfolio, and asset B will account for the other
55%. The expected returns over next 6 years, 2009-2014, for each of these assets are
summarized in the following table.
Year Return on A Return on B
2009 13% 20%
2010 13 19
2011 15 15
2012 16 13
2013 16 12
2014 20 11
3.
1. Calculate the average return and standard deviation of returns for Assets A and B.
2. Find the portfolio’s expected return for EACH of the 6 years.
3. Calculate the (arithmetic) average expected portfolio return, over the 6-year period.
4. Calculate the standard deviation of expected portfolio returns, over the 6-year
period.
5. How would you characterize the correlation of the returns of the two assets A and B
(no calculations are necessary)?
6. Discuss any benefits of diversification achieved through creation of the portfolio.
4. You have been asked for your advice in selecting a portfolio of assets and have been
supplied with the following future expected return data.

8. Year Asset Asset Asset
P Q R
2009 10% 14% 9%
2010 12 12 12
2011 14 10 15
5. You have been told that you can create 2 portfolios—one consisting of assets P and Q
and the other consisting of assets P and R—by investing equal proportions (50%) in each
of the 2 component assets.
1. What is the (arithmetic) average expected return for each asset over the 3-year
period?
2. What is the standard deviation of returns for each asset?
3. What is the average for each of the two portfolios?
4. What is the standard deviation of returns for each portfolio?
5. How would you characterize the correlations of the 2 assets making up each of the
2 portfolios (no calculations are necessary)?
6. Which portfolio do you recommend? Why?
6. Referring to the above problem, what would happen of you constructed a portfolio
consisting of assets P, Q, and R, equally weighted? Would this reduce risk or enhance
return?
7. Assume you wish to evaluate the risk and return behaviors associated with the various
combinations of assets A and B under three assumed degrees of correlation: perfect
positive, uncorrelated, and perfect negative. The following average return and risk values
were calculated for these assets.
Asset Avg Return Risk (Std
Deviation)
A 8 6
B 12 10
1. If the returns of assets A and B are perfectly positively correlated (correlation
coefficient = +1), what is the range of return and risk for all possible portfolio
combinations.
2. If the returns of assets A and B are uncorrelated (correlation coefficient =
0),describe the approximate range of return and risk for all possible portfolio
combinations.
3. If the returns of assets A and B are perfectly negatively correlated (correlation
coefficient = -1), describe the range of return and risk for all possible portfolio
combinations.

9. 8. You are evaluating 2 possible stock investments, Sprinkles Co. and Jimmies Corp.
Sprinkles Co. has an average return of 15%, and a beta of 1. Jimmies Corp. has an
average return of 15%, and a beta of 1.3. Based only on this data, which stock should you
buy and why?
9. Referring to above problem, if you expected a significant market rally, would your
decision be altered? Explain.
10. Assume you have a portfolio of $25,000 invested in each of Investment P, Q, and R.
What is your portfolio beta? Betas for securities P, Q, and R are as shown below:
Security Beta
P 1.30
Q 0.50
R -0.80
11. Calculate the beta of the following portfolio.
Stock Amt invested Beta
ABC $1,000 1.1
JKL 3,000 0.6
PQR 7,000 1.4
XYZ 9,000 0.9
12. Use the capital asset pricing model (CAPM) to find the required return for each of the
following securities in light of the data given.
Security Risk-free Return on Beta
rate market
P 6% 9% 1.20
Q 9 12 0.85
R 10 14 -0.30
S 11 16 1.00
T 8 13 0.70
13. Mark is reviewing his portfolio of investments, which include certain stocks and bonds.
He has large amount tied up in the risk free rate of 3%. He is considering moving some of
his funds from the risk free rate into a stock. The stock has a beta of 1.20. If Mark expects
a return of 12% from the stock (a little better than the current market return of 10%),
should he buy the stocks or leave his funds in the risk free asset?

10. 14. Portfolios A through J, which are listed in the following table along with their returns
[r(p)] and risk (measured by the standard deviation), represent all currently available
portfolios in the feasible or attainable set.
Portfolio Avg Risk (Std Dev)
Return
A 9% 7%
B 4 9
C 12 10
D 15 14
E 8 10
F 12 8
G 10 16
H 7 12
I 9 11
J 10 12
1. Plot the feasible or attainable set represented by these data on a set of portfolio risk
(x-axis) and portfolio return (y-axis).
2. Approximate the efficient frontier on the graph in part a.
3. Which portfolios lie on the efficient frontier? Why do these portfolios dominate all
others in the feasible or attainable set?
4. How would an investor’s utility function or risk-indifference curves be used with
the efficient frontier to find the optional portfolio?
15. The following tables present returns on a pair of stocks for five periods. Without doing
any calculations, can you characterize the correlation that would best describe the
relationship between these pairs of stocks? Your choices are: (a) correlation is -1, (b)
correlation is negative, (c) correlation is zero, (d) correlation is positive, and (e)
correlation is +1

11. (i) (ii)
Period Stock A Stock B Period Stock C Stock D
1 2% 4% 1 10% 2
2 6 12 2 2 2
3 -2 -4 3 8 6
4 10 20 4 4 6
5 5 10 5 6 10
(iii) (iv)
Period Stock E Stock F Period Stock G Stock H
1 10% -10% 1 10% 2%
2 6 -6 2 0 1
3 -5 5 3 -4 2
4 0 0 4 12 4
5 -2 2 5 -8 -1
16. A professional money manager is considering two investments. The first is a stock and
the second is a bond. The average return and risk of the securities is shown below:
Asset Avg return Std Dev
Stock 15% 35%
Bond 8 20
1. Suppose the correlation between the securities is 0.7. Draw the investment
opportunity set of the two securities. To do this, very the weights applied to the
stock in the bond from zero to 100% in increments of 25%. Determine the average
return and risk of each combination and plot the results on a graph.
2. Repeat question a. assuming the correlation between the securities is -0.3.
3. What is the impact of the difference in the correlation coefficient?
17. Calculate the beta of a security if investors demand and return of 11%, the risk free rate is
5%, and the market risk premium is 3%.
18. You are given that the beta of a stock is 0.6, the risk free rate is 7%, and the market risk
premium is 5%.
1. Draw the security market line for this situation.
2. A stock analyst estimates that, based on the future prospects of the company, the
stock is expected to earn 9%. Plot this stock on the graph and evaluate the
attractiveness of this stock.

12. Answers
1. 10.625%
2. parts 1-4 (see below), parts 5-6. Note that as A goes up, B goes down – correlation is
negative. Therefore, there appear to be some diversification benefits. Risk goes down
quite a bit with little change to average return.
Year Return on A Return on B Portfolio
2009 13% 20% 16.85
2010 13 19 16.3
2011 15 15 15.00
2012 16 13 14.35
2013 16 12 13.80
2014 20 11 15.05
Avg 15.50 15.00 15.23
Std Dev 2.59 3.74 1.16
3. Parts 1-4 see below, parts 5-6 The correlation between PQ is -1, while the correlation
between PR =+1. Therefore, there are no diversification benefits for PR – note that the
risk is the average risk of the individual securities (this only works when correlation =
+1). There is perfect diversification for PQ since correlation = -1.
Year Asset P Asset Asset R PQ PR
Q
2009 10% 14% 9% 12 9.5
2010 12 12 12 12 12
2011 14 10 15 12 14.5
Avg 12 12 12 12 12
Std DEv 2 2 3 0 2.5
4. Doesn’t enhance return and risk is not zero
Year Asset P Asset Asset R PQR
Q
2009 10% 14% 9% 11
2010 12 12 12 12
2011 14 10 15 13
Avg 12
Std DEv 1
5. average returns are always between 8 and 12% regardless of correlation. The more
invested in A the more the return is toward 8%.

13. 1. when correlation = +1, there are no diversification benefits. The risk will be
between the risk of the individual securities – between 6-10%
2. if correlation = 0, the risk may fall below the 6% (the less risky security) but the
risk will never be zero. Also, the risk will never be above 10% (the most risky
security).
3. if correlation = -1, there is one portfolio with A and B that actually eliminates risk
completely (std dev = 0). So risk is between zero and 10%.
6. Jimmies should offer a higher return since it is riskier (its beta = 1.3). Since Sprinkles
offers higher return for lower risk, choose Sprinkles.
7. If you were completely clairvoyant and you could perfectly predict when a rally was
coming, you might pick the stocks that are likely to rally the most. Beta is a measure of
the sensitivity of a stock’s return to changes in the market. If the beta of Jimmies is 1.3,
then it is expected to move more than the market.
8. 0.3333
9. 1.04
10. r(P)=9.6%, r(Q)=11.55%, r(R)=8.8%, r(S)=16%, r(T)=11.5%
11. He should invest. CAPM say return should be 3 +1.2*(10-3)=11.4. If he expects 12%
return, it more than compensates for the risk in the security.
12. Efficient portfolios are the ones that offer the highest return for each level of risk. A, D,
and F offer the highest returns for their level of risk. Note that there are some portfolios
which can be formed by combining D & F. Combining D & F would dominate all other
portfolios given. Note that portfolio G will not be chosen since F offers higher return for
less risk.

14. 13. (i) correlation = +1, (ii) correlation is negative, (iii) correlation is -1, (iv) correlation
is positive
14. If correlation =0.7
Stock Bond Return Risk
0% 100% 8.00% 20.00%
25 75 9.75 22.03
50 50 11.50 25.52
75 25 13.25 29.96
100 0 15.00 35.00
If Correlation = -0.3
Stoc Bon Retur Risk
k d n
0% 100 8.00 20.00%
% %
25 75 9.75 14.93
50 50 11.5 17.36
0
75 25 13.2 25.21
5
100 0 15.0 35.00
0

15. 15. 2.00
16. stock is expected to earn less than required (9%<10%). It is a bad investment