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Name_______________________________________
Econ 134A (common final for both lectures)
Test 3, Form A
Perm #_______________________________
John Hartman
March 18, 2017
Instructions:
YOU WILL TURN IN THE ENTIRE TEST, INCLUDING THE MULTIPLE-CHOICE
QUESTIONS.
You have 160 minutes to complete this test, unless you arrive late. Late arrival will lower the time available
to you, and you must finish at the same time as all other students.
Cheating will not be tolerated during any test. Any suspected cheating will be reported to the relevant
authorities on this issue.
You are allowed to use a nonprogrammable four-function or scientific calculator that is NOT a
communication device. You are NOT allowed to have a calculator that stores formulas, buttons that
automatically calculate IRR, NPV, or any other concept covered in this class. You are NOT allowed to have
a calculator that has the ability to produce graphs. If you use a calculator that does not meet these
requirements, you will be assumed to be cheating.
Unless otherwise specified, you can assume the following:
 Negative internal rates of return are not possible.
 Equivalent annual cost problems are in real dollars.
You are allowed to turn in your test early if there are at least 10 minutes remaining. As a courtesy to your
classmates, you will not be allowed to leave during the final 10 minutes of the test.
Your test should have 10 multiple choice questions (20 points) and 8 problems (49 points). The maximum
possible point total is 70 points. If your test is incomplete, it is your responsibility to notify a proctor to get a
new test.
Grading:
For your reference, an example of a well-labeled graph is below:
Filling in scantron correctly, putting name and perm #
on this page, & having photo ID
___/1
(automatic unless something is incorrect)
Multiple choice portion
_____/20
Problems
_____/49
Total score
_____/70
MULTIPLE CHOICE: Answer the following questions on your scantron. Each correct answer is worth 2
points. All incorrect or blank answers are worth 0 points. If there is an answer that does not exactly match
the correct answer, choose the closest answer.
1. At the beginning of the February 8 lecture, I read the cover story of the January 30 issue of Barron’s
magazine titled Next Stop: Dow __________. (Hint: This number is a projection of the Dow Jones Industrial
Average by the year 2025.)
A. 10,000
B. 15,000
C. 20,000
D. 25,000
E. 30,000
Answer: E. Discussed in lecture.
2. A European call option for a stock has an exercise price of $60, and an expiration date of one year from
today. This stock has a 10% chance of having each of the following values: $48, $49, $50, $51, $52, $53,
$54, $55, $56, and $57. What is the present value of the call option? The effective annual interest rate for
this stock is 50%.
A. $0
B. $1
C. $2
D. $3
E. $4
Answer: A. This call option is never in the money, so it has no value.
3. A firm currently has a capital structure of one part debt to two parts equity. The beta of this firm would
be 1.25 if it would be an all-equity firm with no debt. (In other words, the beta of the firm if unlevered is
1.25.) The beta value of any debt issued is 0.9. How much would the equity beta decrease by if the firm
decides to change to having one part debt to three parts equity?
A. 0.01
B. 0.05
C. 0.1
D. 0.25
E. 0.5
Answer: B. The 𝛽𝑤𝑎𝑐𝑐 should remain constant, so we use the equation once for before the change and once
afterward.
1
𝑏𝑒𝑓𝑜𝑟𝑒 2
𝑏𝑒𝑓𝑜𝑟𝑒
Before, 𝛽𝑤𝑎𝑐𝑐 = 1.25 = 𝛽𝑒
( ) + (0.9) ( ) ⇒ 𝛽𝑒
= 1.425
3
3
1
𝑎𝑓𝑡𝑒𝑟 3
After, 𝛽𝑤𝑎𝑐𝑐 = 1.25 = 𝛽𝑒
(4) + (0.9) (4) ⇒
𝑎𝑓𝑡𝑒𝑟
𝑏𝑒𝑓𝑜𝑟𝑒
Change: 𝛽𝑒
− 𝛽𝑒
= 0.05833
𝑎𝑓𝑡𝑒𝑟
𝛽𝑒
= 1.366
4. Which of the following types of efficiency has the most assumptions?
A. Semi-strong efficiency
B. Strong efficiency
C. Weak efficiency
D. All three have the same level of assumptions
E. There is no way to tell which form has the most assumptions
Answer: B. Strong efficiency makes the most assumptions, and implies the weaker forms of efficiency.
5. Bridgette receives a loan of $50,000 today. She must completely pay the loan back in 48 monthly
payments, starting eight months from today. The effective annual interest rate on the loan is 15%. How much
will each payment be?
A. $1,370
B. $1,420
C. $1,470
D. $1,480
E. $1,500
1
Answer: D. Monthly rate is 𝑟𝑚𝑜 = 1.1512 − 1 = 1.17149%
𝑐
1
50,000 =
[1 −
]
0.0117149
1.011714948
50,000 = 36.5557𝑐
𝑐 = 1,367.78
But the payments must be pushed forward 7 months:
𝑐 ′ = 1367.78(1.0117149)7 = $1,483.96
6. If a rare coin sells for $100,000 today and it sold for $100 forty years ago, what is the geometric average
rate of return over the last forty years?
A. 17%
B. 19%
C. 25%
D. 32%
E. Unable to determine the geometric average from the information given
1
Answer: B.
100,000 40
( 100 )
− 1 = 18.8502%
7. A stated annual interest rate of 9.5%, compounded every hour, is equivalent to an effective annual interest
rate of _____. (Assume 365 days in a year.)
A. 9.65%
B. 9.75%
C. 9.85%
D. 9.95%
E. 10.05%
Answer: D. We can use continuous compounding to approximate hourly compounding.
𝑒 0.095 − 1 = 9.9659%
0.095
365∗24
The exact calculation would be (1 + 365∗24)
− 1 = 9.9658%
8. Two stocks, Y and Z, have a correlation coefficient of 1. Suppose that you invest 80% of your money in
stock Y and 20% of your money in stock Z. What is the standard deviation of this portfolio if the standard
deviation of stock Y’s return is 30% and the standard deviation of stock Z’s return is 15%?
A. 15%
B. 18%
C. 21%
D. 24%
E. 27%
𝑉𝑎𝑟(𝑅𝑝 ) = 𝑥𝑦2 𝜎𝑦2 + 𝑥𝑧2 𝜎𝑧2 + 2𝑥𝑦 𝑥𝑧 𝜎𝑦𝑧 = 𝑥𝑦2 𝜎𝑦2 + 𝑥𝑧2 𝜎𝑧2 + 2𝑥𝑦 𝑥𝑧 𝜌𝑦𝑧 𝜎𝑦 𝜎𝑧
= 0.82 (0.3)2 + 0.22 (0.15)2 + 2(0.8)(0.2)(1)(0.3)(0.15)
= 0.0729
𝑆𝐷(𝑅𝑝 ) = √𝑉𝑎𝑟(𝑅𝑝 ) = 27%
9. Three stocks have annual returns of 12%, 24%, and 32%. The variance of this sample is _____.
A. 0.0044
B. 0.0064
C. 0.01
D. 0.03
E. 0.08
𝑅̅ =
𝑉𝑎𝑟(𝑅) =
12% + 24% + 32%
= 0.22667
3
1
[(0.12 − 0.22667)2 + (0.24 − 0.22667)2 + (0.32 − 0.22667)2 ] = 0.0101334
2
10. What is the annuity factor of a 50-year annuity with an effective annual interest rate of 18%? Assume
annual payments, with 50 payments made starting one year from today.
A. 5.5
B. 5.25
C. 5
D. 4.75
E. 4.5
𝐴𝐹 =
1
1
[1 −
] = 5.55414
0.18
1.1850
For the following problems, you will need to write out the solution. You must show all work to receive
credit. Each problem (or part of problem) shows the maximum point value. Provide at least four
significant digits to each answer or you may not receive full credit for a correct solution. Show all
work in order to receive credit. You will receive partial credit for incorrect solutions in some
instances. Clearly circle your answer(s) or else you may not receive full credit for a complete and
correct solution.
11. Buzzy will invest in a portfolio, with ¾ of the money invested in Dull Doldrum Dolly Company, and ¼
of money in a risk-free bond. Dull Doldrum Dolly Company stock could have a rate of return of 0%, 15%, or
30%, each with one-third probability. The risk-free bond has a rate of return of 10%.
(a) (3 points) If Dull Doldrum Dolly Company has a beta value of 2, what is the expected return of a stock
with the same beta value as the market portfolio?
1
1
1
𝔼𝑅 = ( ) (0%) + ( ) (15%) + ( ) (30%) = 15%
3
3
3
15% = 10% + 2(𝑅𝑚 − 10%)
𝑅𝑚 = 12.5%
(b) (4 points) What is the standard deviation of Buzzy’s portfolio?
𝜎𝑑2 =
1
[(0 − .15)2 + (.15 − .15)2 + (.3 − .15)2 ] = 0.015
3
𝑉𝑎𝑟(𝑅𝑝 ) = 𝑥𝑑2 𝜎𝑑2 + 𝑥𝑓2 𝜎𝑓2 + 2𝑥𝑑 𝑥𝑓 𝜎𝑑𝑓
3 2
3 1
1 2
= ( ) (0.015) + 2 ( ) ( ) (0) + ( ) (0)
4
4 4
4
= 0.0084375
𝑆𝐷(𝑅𝑝 ) = √0.0084375 = 9.18559%
12. (6 points) Dakota has just received $800,000 for a house mortgage. The loan will be paid back as
follows: The first payment of $1,000 will be paid one month from today. 359 subsequent monthly payments
will be made, each 0.4% higher than the previous payment. A final balloon payment will be made 400
months from today to completely pay off the loan. How much will this balloon payment be? Assume a stated
annual interest rate of 18%, compounded monthly.
1
1
1.004 360
PV of first 360 payments: 𝑃𝑉 = 1000 [0.015−0.004 − 0.015−0.004 (1.015)
] = 89110.35
PV of final payment: 800,000 − 89110.35 = 710,889.65
𝐹𝑉𝑚𝑜𝑛𝑡ℎ400 = 710,889.65(1.015)400 = $274,295,758
13. Solve each of the following:
(a) (3 points) Give an example in which for a given year, the capital gain for a stock is negative but the
percentage return is positive.
Current stock value: $100.
Stock value in one year: $95.
Dividend paid in one year: $15.
$95−$100
Capital gain is $100 = −5% .
Percentage return is
($95+$15)−$100
$100
= 10%.
(b) (3 points) A stated annual interest rate of 15%, compounded 5 times per year is equivalent to what stated
annual interest rate if compounded 6 times per year?
𝐸𝐴𝑅 = 1.035 − 1 = 15.9274%
1
1.1592746 − 1 = 2.49382%
𝑆𝐴𝐼𝑅 = 6(2.49382%) = 14.96293%
(This SAIR is compounded six times per year.)
14. (7 points) Suppose that Stock F and Stock G have a correlation value of ρ = –1. Stock F has an expected
return of 12% and standard deviation 15%. Stock G also has an expected return of 12% and standard
deviation 15%. Today, each stock is valued at $200 per share. Over the next year, Stock F will go up by $9.
How much will Stock G go up by next year? (Please completely justify your answer to get full credit.)
If $200 is invested in each stock, 𝑥𝑓 = 𝑥𝑔 = 0.5.
𝜎𝑓𝑔 = 𝜌𝑓𝑔 𝜎𝑓 𝜎𝑔 = −1 ∗ 0.15 ∗ 0.15 = −0.0225
2 2
2 2
𝑉𝑎𝑟(𝑅𝑝 ) = 𝑥𝑓 𝜎𝑓 + 𝑥𝑔 𝜎𝑔 + 2𝑥𝑓 𝑥𝑔 𝜎𝑓𝑔 = 0.52 ∗ 0.152 + 0.52 ∗ 0.152 + 2 ∗ 0.5 ∗ 0.5 ∗ (−0.0225)
= 0.005625 − 0.01125 = 0
Since variance of this portfolio is zero, we know with certainty that the combined return of one share of each
stock will be 12%. On the portfolio value of $400, the return is $400 ∗ 12% = $48. So if stock F goes up in
value by $9, stock G must go up in value by $48 − $9 = $39.
15. Nola is considering an investment that will require her to receive $500 today and $1,195 two years from
today. She will also need to pay $1,595 one year from today as part of the investment.
(a) (4 points) Find all internal rates of return.
0 = 500 −
1595
1+𝑟
1195
+ (1+𝑟)2 . Let 𝑥 = 1 + 𝑟. Then 0 = 500𝑥 2 − 1595𝑥 + 1195. Using the quadratic formula,
we find that 𝑥 ∈ {1.98746, 1.20254}. Because 𝑟 = 𝑥 − 1, 𝑟 ∈ {98.746%, 20.254%}.
(b) (2 points) For what discount rates will there be positive net present values for this investment? You must
completely justify your answer to receive credit.
Pick a point in each relevant range…
1595
1195
+
= $100
1 + 0 (1 + 0)2
1595
1195
𝑟 = 50% ⇒ 𝑃𝑉 = 500 −
+
= −32.22
1 + 50% (1 + 50%)2
1595
1195
𝑟 = 100% ⇒ 𝑃𝑉 = 500 −
+
= $1.25
1 + 100% (1 + 100%)2
So NPV is positive for all discount rates except between 20.254% and 98.746%.
𝑟 = 0% ⇒ 𝑃𝑉 = 500 −
16. (6 points) Marilyn owns a European call option for TMIWD, Inc. The option has an exercise price of
$130, with an expiration date one year from today. If Marilyn assumes that TMIWD stock’s value on the
expiration date comes from a uniform distribution, with all prices between $100-$140 equally likely, what is
the perceived expected value of the option? Assume that Marilyn is risk neutral, and assume an effective
annual discount rate of 25% for this option. (Note that with this uniform distribution, any price from $100 to
$140 is equally probable in 1-cent increments, but no other price can occur with positive probability. You
can use a continuous distribution as an approximation if you want.)
A price in the range of $130 to $140 occurs with probability of 25%.
The average price in the range of $130 to $140 is $135.
With 25% probability, the average FV will be $135-$130=$5.
1
Then 𝑃𝑉 = 1.25 (25%)$5 = $1.
17. (6 points) Travis buys two put options with an exercise price of $50 (per share) today, three call options
with an exercise price of $150 (per share), and four shares of stock currently valued at $90 (per share). The
expiration date of all of these options is one year from today. Each option is for buying or selling one share.
The effective annual discount rate for these options is 20%. Draw a well-labeled graph that shows the value
of a combination of the five options and four shares of stock, as a function of the value of the stock at
expiration. The vertical intercept should have the present value of the combination of the assets. The
horizontal intercept should have the future value of the stock on the expiration date. Make sure to label your
intercepts and other relevant numbers on each axis, where relevant. (Hint: You may want to look at the front
page of the test to see a well-labeled graph.) Explain your answer in words, math, and/or using additional
graphs. Include enough detail so that everything on the graph is unambiguous.
If 𝑝 < $50, then
FV of assets is 2(50 − 𝑝) + 3(0) + 4(𝑝) = 100 + 2𝑝
100+2𝑝
PV of assets is 1.2 = 83.33 + 1.667𝑝
If $50 < 𝑝 < $150, then
FV of assets is 2(0) + 3(0) + 4(𝑝) = 4𝑝
4𝑝
PV of assets is = 3.333𝑝
1.2
If 𝑝 > $150, then
FV of assets is 2(0) + 3(𝑝 − 150) + 4(𝑝) = 7𝑝 − 450
7𝑝−450
PV of assets is 1.2 = 5.8333𝑝 − 375
18. (5 points) Suppose you start a bank account one year from today. If you make annual deposits of $100
into an account that pays an effective annual interest rate of 15%, when will you have an account balance of
$50,000? (Round to the nearest number of years. The deposits will occur on the same date each year starting
one year from today.)
50,000 100
1
=
[1
−
]
1.15𝑇
0.15
1.15𝑇
100
[1.15𝑇 − 1]
50,000 =
0.15
75 = 1.15𝑇 − 1
76 = 1.15𝑇
log 76
𝑇=
= 30.9865 ⇒ 𝑇 = 31
log 1.15
NOTE: YOU CAN TEAR THIS SHEET OFF
AND USE AS EXTRA SCRATCH PAPER.
PLEASE NOTE THAT ANYTHING ON THIS
SHEET WILL NOT BE GRADED UNLESS
EXPLICITLY SPECIFIED ON THE TEST.
Perpetuity
PV 
C
r
Annuity
C
1 
PV  1 
r  (1  r )T 
Logarithmic rule
ab = c  b = log c / log a
Variance of a sample
1 T
Var 
( Ri  R ) 2

T  1 i 1
Variance of a distribution, with each outcome
having the same probability of occurring
1 T
Var   ( Ri  R ) 2
T i 1
Covariance formula
N
Growing perpetuity
C
PV 
rg
Growing annuity
T
 1
1
1  g  
PV  C 


 
 r  g r  g  1  r  
Quadratic formula
ax2 + bx + c = 0 
x
 b  b 2  4ac
2a
 X ,Y  Cov( X .Y )  
i 1
( xi  x )( yi  y )
N
Correlation of A and B
Cov( A, B)
, where SD stands
Corr ( A, B) 
SD( A)  SD( B)
for standard deviation
Variance of a portfolio
X A2 A2  2 X A X B A,B  X B2 B2