Solution sketches
Test 2, Fall 2011
Level of difficulty
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On multiple choice questions…
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“Easy” denotes that about 80-100% of
students get this question correct
“Medium” denotes that about 60-80% of
students get this question correct
“Hard” denotes that about 40-60% of
students get this question correct
Level of difficulty
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On Problems…
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I take a 10% sample
“Easy” denotes that the average score is
80-100% of the points possible
“Medium” denotes that the average score
is 60-80% of the points possible
“Hard” denotes that that average score is
40-60% of the points possible
What is the beta of a stock?
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Risk-free rate is 6%
A stock has an expected return of 14%
Expected return on the market is 11%
What is beta?
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14% = 6% + ß (11% – 6%)
ß = 1.6
Easy question
What is the standard deviation?
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Sample of a stock’s returns
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14%, 5%, 6%
Average is 8.33%
Squared deviation of means
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(.14 - .0833)2 = 0.00321111
(.05 - .0833)2 = 0.00111111
(.06 - .0833)2 = 0.00054444
Sum of the three numbers
above, divided by N – 1
(remember that you lose a
degree of freedom with a
sample)
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Standard deviation is the
square root of the variance
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Variance is 0.00243333
0.04933
The closest answer is 5%
Easy problem
Geometric average
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What is the geometric average of annual
returns of 5%, 8%, 10%, –4%, and 15%?
Steps
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5th root of (1.05)(1.08)(1.1)(0.96)(1.15)
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Subtract 1
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1.066
6.6%
Easy problem
Carina Corona, real payment
of $50K 10 years from now
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Inflation will be 8% per year for the
first 4 years
Inflation will be 12% per year after the
first 4 years
Nominal payment?
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$50,000(1.08)4(1.12)6 = $134,268
Easy problem
Perpetuity
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A perpetuity makes payments of $1,000
every two years, starting one year
from today. If the effective annual
interest rate is 9%, what is the present
value of this perpetuity?
What is the effective rate every 2
years?
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1.092 – 1 = 18.81%
Perpetuity
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What is a perpetuity worth if $1,000 is
paid every two years, starting two
years from today?
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$1,000/.1881 = $5,316.32
The problem tells us that the perpetuity
starts in one year, not two
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We need to increase the value of each
payment
Perpetuity
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We need to find out how much this
perpetuity is worth if each payment is
made one year sooner
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$5,316.32(1.09) = $5,794.79
Very hard problem
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Only about 25% of students got this
correct
IRR problem
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If I invest $1,000 today in a project and
receive $1,800 five years from today in
return, my annual internal rate of return is
_____.
For what annual discount rate will the PV of
the costs and benefits be equal?
$1,000 (1 + IRR)5 = $1,800
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1 + IRR = 1.1247
IRR = 12.47%
Easy problem
You loan a friend $500
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Suppose that you loan your friend $500
and the effective annual interest rate
that is agreed on is 12%. Your friend
will pay you back $60 per year for the
first nine years (starting one year from
today). If the final payment will be
made 10 years from today, how much
will this payment have to be to
completely pay back the loan?
You loan a friend $500
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Each of the first nine years, you only pay
back the interest (12% of $500), or $60.
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Final payment: $560 total
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The balance does not change from year to year
Interest = 12% of $500, or $60
Pay off the balance: $500
Medium problem
Payback period method
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In your first job, suppose that you boss tells you to use
the payback period method (undiscounted), with the
cutoff date 3 years, 3 months from now. In other words,
the payback period is 3 years, 3 months. The effective
annual discount rate is 6%. Which of the following offers
should you accept if you use this method?
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$1,400 every two years forever, with the first payment made
two years from today
$40,000 every year forever, with the first payment made four
years from today
$450 per year forever, with the first payment made today
$500 per year forever, with the first payment made one year
from now
A one-time payment of $1,000,000 five years from today
Payback period method
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If we only look at the first 3 years, 3
months, how much is each offer worth?
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No discounting used in the calculations
Hard problem
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Not meant to be a hard problem
Make sure that you understand the
terminology used in the book and in
lectures
Payback period method:
How much after 3¼ years?
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$500 per year forever, starting in 1 year
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A one-time payment of $1M in 5 years
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1 payment: $1,400
$40K per year forever, starting in 4 years
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0 payments: $0
$1,400 every 2 years, starting in 2 years
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3 payments: $1,500
0 payments: $0
$450 per year forever, starting today
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4 payments: $1,800
Summary of MC questions
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5
1
1
1
easy questions
medium question
hard question
very hard question
Problem: Jackie Jackson
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Jackie Jackson has won the final round of a new fictional game
show, “Pay What You Can Afford.” Jackie gets to take out a 30year, fixed-rate mortgage for $500,000. The stated annual
interest rate of the mortgage is 6%, and is compounded
monthly. Jackie makes 360 monthly payments of $1,600 per
month, starting one month from today. The producers of the
show make an arrangement with the mortgage company so that
an additional payment is made 60 months (or 5 years) after
Jackie takes out the mortgage. How much will this payment be
if we assume that Jackie makes all 360 monthly payments and
that the balance of the loan will be zero after 30 years? Round
your answer to the nearest dollar.
Problem: Jackie Jackson
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Monthly r = 0.005 = 0.5%
PV of 360 monthly payments
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($1600/.005) * [1 – 1/(1.005)360]
$320,000 (0.833958)
$266,866.58
Problem: Jackie Jackson
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PV of additional payment
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FV of additional payment (in 5 years)
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$500,000 – $266,866.58
$233,133.42
$233,133.42(1.005)60
$314,462
Easy-medium problem
Problem: Prince Pitcher
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Prince Pitcher is interested in buying a stock. In order
to determine if this stock is worth buying, he
assumes an annual effective discount rate of 10%.
After a thorough analysis of the stock, he assumes
that the company will pay its next dividend of $1 per
share one year from today. For two years after the
next dividend payment, the dividend will increase by
15.5% per year. After that, the dividend will increase
by 8% per year. Based on these assumptions, what
is the present value of each share of this stock?
Round your answer to the nearest cent.
Problem: Prince Pitcher
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PV of payment in year 1
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PV of payment in year 2
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1.155 / 1.12 = $0.955
Easy-medium
problem
PV of payment in year 3
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1 / 1.1 = $0.909
Total of all
payments in PV
terms: $56.99
1.1552 / 1.13 = $1.002
PV of payment in years 4 and beyond
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(1.1)-3 * [(1.155)2(1.08)/(0.10 – 0.08) = $54.123
Discount by 3 years
Dividend 4 years
from today
Problem: 92 days of interest
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Today is November 17, 2011. If you
deposit $10,000 in the bank today and
the stated annual interest rate is 7.3%,
how much interest will you earn after
92 days? (Assume 365 days per year in
your calculation.) Round your answer to
the nearest cent.
Problem: 92 days of interest
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Daily interest rate
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7.3% / 365 = 0.0002 = 0.02%
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Total balance after 92 days
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$10,000(1.0002)92 = $10,185.68
Interest earned
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Not 0.2% (common mistake)
$10,185.68 - $10,000 = $185.68
Easy-medium problem
Problem: Stock A
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Stock A has an expected annual return of 5% and a standard deviation of 10%.
Stock B has an expected annual return of 15% and a standard deviation of 20%.
Draw a graph with “expected return on portfolio” on the vertical axis and
“standard deviation of portfolio’s return” on the horizontal axis.
(a) (3 points) Make sure that the following are well labeled, including indicating
numerical values on each axis for each of the following points: Put an “A” by the
point on the graph for Stock A and a “B” by the point on the graph for Stock B.
(Hint: You may want to see the graph on the cover sheet to see that it is well
labeled.)
(b) (3 points) Draw a line or curve to denote possible expected return-standard
deviation combinations of all of the possible portfolios with positive amounts of
both Stock A and Stock B. Assume that the correlation between Stock A and
Stock B is –0.5. (Note: Be careful how you draw this. However, I am only
looking for the shape of the line or curve, so it does not need to be exact.)
(c) (1 point) Label the minimum variance point MV. (Note that you do not
necessarily need to calculate the exact point of minimum variance.)
(d) (3 points) Clearly show or explain what the efficient set is. (Note: The
efficient set is also referred to as the efficient frontier.)
Problem: Stock A
expected return
on portfolio
(d) The points
between MV
and B are the
efficient set
(a): Easy
problem
(b): Medium
problem
(c): Medium
problem
15%
B
(d): Hard
problem
MV
5%
A
10%
20%
std. dev. of
portfolio’s return
Summary of Problems
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3 points of Easy problems
21 points of Easy-Medium problems
4 points of Medium problems
3 points of Hard problems