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Managerial Economics (ARE) 171A
University of California, Davis, Summer Session I, 2013
Dr. John H. Constantine
Midterm, Thursday, July 11, 2013
75 TOTAL POINTS
1
Part I: Multiple Choice (45 total points; each question is worth 3 points.) There are 15 multiple choice questions,
each having only one correct answer.
1) Which of the following statements is correct?
a) One of the advantages of the corporate form of organization is that it avoids double taxation.
b) It is easier to transfer one’s ownership interest in a partnership than in a corporation.
c) One of the disadvantages of a sole proprietorship is that the proprietor is exposed to unlimited liability.
d) One of the advantages of a corporation from a social standpoint is that every stockholder has equal voting
rights, i.e., “one person, one vote.”
e) Corporations of all types are subject to the corporate income tax.
2) Which of the following bonds will have the greatest percentage increase in value if all interest rates decrease by
1%?
a) 20-year, zero coupon bond.
b) 10-year, zero coupon bond.
c) 20-year, 10% coupon bond.
d) 20-year, 5% coupon bond.
e) 1-year, 10% coupon bond.
3) You recently sold to your brother 200 shares of Disney stock, and the transfer was made through a broker. This is
an example of:
a) A money market transaction.
b) A primary market transaction.
c) A secondary market transaction.
d) A futures market transaction.
e) An over-the-counter market transaction.
4) The increased volatility of longer term bonds in response to interest rate movements is reflected in the
a) pure interest rate
b) default risk premium
c) liquidity risk premium
d) maturity risk premium
5) The primary function of financial markets is to:
a) ensure that interest and dividend payments are made to stockholders and bondholders.
b) facilitate the movement of cash from savers to companies that need money.
c) facilitate the payment for goods and services between producers and consumers.
d) financial markets perform all of these functions
6) You are analyzing the value of an investment by calculating the present value of its expected cash flows. Which of
the following would cause the investment to look better?
a) The discount rate decreases.
b) The cash flows are extended over a longer period of time, but the total amount of the cash flows remains the
same.
c) The riskiness of the project’s cash flows increases.
d) The total amount of cash flows remains the same, but more of the cash flows are received in the later years and
less are received in the earlier years.
e) All of the above are true.
2
7) Which of the following could explain why a business might choose to operate as a corporation rather than as a sole
proprietorship or a partnership?
a) Corporations generally face fewer regulations.
b) Less of a corporation’s income is generally subject to taxes.
c) Corporate shareholders are exposed to reduced liability, but this factor is offset by the tax advantages of
incorporation.
d) Corporate investors are exposed to unlimited liability.
e) Corporations generally find it easier to raise capital.
8) Which of the following is a false statement?
a) Future values and interest rates move in the same direction.
b) The EFF (or APY) is generally less than the nominal or quoted rate of interest.
c) Compounding means earning interest on interest.
d) Future values decrease with decreases in interest rates
9) Which of the following events would make it more likely that a company would choose to call its outstanding
callable bonds?
a) The company’s bonds are downgraded.
b) Market interest rates rise sharply.
c) Inflation increases significantly.
d) The company's financial situation deteriorates significantly.
e) Market interest rates decline sharply.
10) The interest rates we observe in the economy differ from the risk-free rate because of
a) the real rate of interest.
b) diversification.
c) risk premiums.
d) all the above
11) Suppose someone offered you the choice of two equally risky annuities, each paying $10,000 per year for five
years. One is an ordinary (or deferred) annuity, while the other is an annuity due. Which of the following
statements is correct?
a) The present value of the ordinary annuity must exceed the present value of the annuity due, but the future
value of an ordinary annuity may be less than the future value of the annuity due.
b) The present value of the annuity due exceeds the present value of the ordinary annuity, while the future value
of the annuity due is less than the future value of the ordinary annuity.
c) The present value of the annuity due exceeds the present value of the ordinary annuity, and the future
value of the annuity due also exceeds the future value of the ordinary annuity.
d) If interest rates increase, the difference between the present value of the ordinary annuity and the present value
of the annuity due remains the same.
e) The present value of the ordinary annuity exceeds the present value of the annuity due, and the future value of
an ordinary annuity also exceeds the future value of the annuity due.
12) The yield curve is
a) inverted when short term rates are higher than long term rates.
b) normal when it slopes upward to the right
c) a plot of interest versus term also called the term structure of interest rates
d) all of the above
3
13) More frequent compounding results in
future values and
frequent compounding at the same nominal interest rate.
a) higher, lower
b) higher, higher
c) lower, higher
d) lower, lower
present values than less
14) Which of the following statements is correct?
a) A hostile takeover is the main method of transferring ownership interest in a corporation.
b) A corporation is a legal entity created by a state, and it has a life and existence that is separate from the
lives and existence of its owners and managers.
c) Unlimited liability and limited life are two key advantages of the corporate form over other forms of business
organization.
d) Limited liability is an advantage of the corporate form of organization to its owners (stockholders), but
corporations have more trouble raising money in financial markets because of the complexity of this form of
organization.
e) Although the stockholders of the corporation are insulated by limited legal liability, the legal status of the
corporation does not protect the firm’s managers in the same way, i.e., bondholders can sue its managers if the
firm defaults on its debt.
15) You are considering two bonds. Bond A has a 9% annual coupon while Bond B has a 6% annual coupon. Both
bonds have a 7% yield to maturity, which is expected to remain constant. Which of the following statements is
correct?
a) The prices of both bonds will increase by 7% per year.
b) The prices of both bonds will increase over time, but the price of Bond A will increase by more.
c) The prices of both bonds will remain unchanged.
d) The price of Bond A will decrease over time, but the price of Bond B will increase over time.
e) The price of Bond B will decrease over time, but the price of Bond A will increase over time.
Part II: True/False (15 total points; each question is worth 3 points.) There are 5 True/False questions, each
having only one correct answer. Mark answers on Scantron—True is “A” and False is “B”.
16) Double taxation of earnings is the primary financial disadvantage of the corporate form of business organization.
True.
17) Bond ratings are the primary measure of default risk. True.
18) Call provisions are most likely exercised by the borrower when interest rates are falling. True.
19) If inflation is expected to increase in the future and the maturity risk premium (MRP) is greater than zero, the yield
curve will be upward sloping. True.
20) The yield on a 3-year Treasury bond cannot exceed the yield on a 10-year Treasury bond. False.
4
Part III: Short Answer/Problem Solving (15 total points.)
You must show ALL work for these problems. Simply entering numbers into the
calculator without the associated work will earn NO POINTS.
Problem 1 (10 points):
You are considering buying a new, $15,000 car, and you have $2,000 to put toward a down payment. If you can
negotiate a nominal annual interest rate of 10% and finance the car over 60 months, how much total interest will you
have paid once the debt is paid off?


PMT  1
1

PV 
mN

m  I   I 
     1  I 
  m   m  m 








PMT  1
1
13,000 


12  0.10   0.10  0.10  12( 5 )

 

 1 
  12   12 
12 






Monthly payments = PMT/12 = $276.21.
Total payment = PMT(5 years) = $276.21(12)(5) = $16,572.60
Interest payments = $16,572.60 – $13,000 = $3,572.60
5
Problem 2 (5 points):
Longly Trucking is issuing a 20-year bond with a $2,000 face value tomorrow. The issue is to pay an 8% coupon rate,
because that was the interest rate while it was being planned. However rates have increased suddenly and are expected
to be 9% when the bond is marketed. What will Longly receive for each bond tomorrow?
r = 0.09; k = 0.08; C = (0.08)*($2,000) = $160.


1

PB = (CPN/2) A r2T/ 2 + PAR 
2T 
(
1
r
)



= $80 (18.4016) + $2,000 (0.1719)
= $1,815.93
1
Managerial Economics (ARE) 171A
University of California, Davis
Summer Session
Instructor: John H. Constantine
KEY—Homework 1
Problem 1:
Compare and contrast the forms of business organizations with respect to: (a) the legal liability of their
owners, (b) the life of the entity, and (c) their ability to raise capital.
Legal liability of owners
Life of the entity
Ability to raise capital
Sole Proprietorship
Unlimited Personal
Liability
Limited to life of
individual who created
it
Inability to raise large
amounts of capital:
Equity limited to
owner’s personal
assets; ability to issue
debt is limited by
owner’s equity
investment
Partnership
Partners have
unlimited Liability
Typically limited to
life of partners
Corporation
Limited to amount of
investment
Theoretically
unlimited life
Easier to raise money
but ability to raise
large amounts of
capital is limited by
the partners’ personal
wealth
Can potentially raise
vast amounts of capital
2
Problem 2:
(a)
What role do financial markets play in our economy? Give examples of individuals/groups who
supply funds to these markets. Give examples of individuals/groups who demand funds from
these markets.
Financial markets provide a forum in which suppliers of funds and demanders of funds can
transact business directly.
As it turns out, the suppliers and demanders of funds are the same groups of people, and at times,
the same people. The four main groups are:
(i)
(ii)
(iii)
(iv)
households
businesses
government
foreigners
Households are big suppliers of funds, either through direct savings, investments, retirement funds,
etc. But at the same time, households borrow money to purchase cars, homes, etc.
Firms are big demanders of funds, especially when they want to expand or upgrade their plants,
expand into new markets, etc. But firms also supply funds so that they may have access to ready
cash if needed.
Government is a supplier of funds if there is a budget surplus and a demander of funds if there is a
budget deficit. This year (2004) the government’s deficit will exceed $400 billion, all of which needs
to be financed by borrowing. (By the way, this high level of government borrowing can lead to
increased interest rates and thus lower long term economic growth. Why? What is this
phenomenon called?
(b)
What are primary and secondary markets? Are these two markets closely or loosely related to
each other? Explain.
Primary Market:
Denotes the fact that the demander of funds issues a security directly to the supplier of funds. An
example is when a firm (Microsoft) sells new shares common stock to the public. (Often, this is
referred to as an IPO—initial public offering.)
Secondary Market:
The trading of securities subsequent to the primary market issuance. In the secondary market, no
new funds are being raised by the demander. The security is trading ownership among investors. An
example of this is when you buy Microsoft stock through your broker or from your uncle.
Relationship between the primary and secondary markets:
While the secondary market generates no new funds for the firm, this market plays a critical role
for the firm. First, it provides liquidity to investors who may want to sell stocks. Second, it helps
the firm set the stock price on any potential new issues. (You need to know why the last two
comments are true.)
3
Problem 3:
Find the amount to which $45,000 will grow under each of these conditions:
I


FVN = PV  1  NOM 
M 

To this problem, you need to use:
NM
(a)
6 percent compounded annually for 12 years.
FV12 = $45,000(1 + 0.06/1)1(12) = $90,548.84
(b)
6 percent compounded semiannually for 12 years.
FV12 = $30,000(1 + 0.06/2)2(12) = $91,475.73
(c)
6 percent compounded quarterly for 12 years.
(d)
6 percent compounded monthly for 12 years.
(e)
6 percent compounded daily (365 days) for 12 years.
(f)
6 percent compounded continuously for 12 years.
(g)
Why does the observed pattern of FV’s occur?
The FVs increase because as the compounding periods increase, interest is earned on interest more
frequently.
B
2
3
4
5
6
7
8
9
10
PV
i
N
m
m
m
m
m
m
C
$45,000
0.06
12
1
2
4
12
365
continuous
D
E
FV
FV
FV
FV
FV
FV
F
$90,548.84
$91,475.73
$91,956.52
$92,283.79
$92,444.02
G
H
$90,548.84 =FV($C$3/C5,C5*$C$4,0,-$C$2)
$91,475.73 =FV($C$3/C6,C6*$C$4,0,-$C$2)
$91,956.52
$92,283.79
$92,444.02
$92,449.49 =C2*EXP(C3*C4)
4
Problem 4:
Find the present value of $45,000 due in the future under each of the following conditions:




1


To this problem, you need to use: PV  FVN
mN 

  1  r  
 
m  
(a)
(b)
(c)
(d)
(e)
(f)
(g)
6 percent nominal rate, compounded annually, discounted 12 years.
6 percent nominal rate, compounded semiannually, discounted 12 years.
6 percent nominal rate, compounded quarterly, discounted 12 years.
6 percent nominal rate, compounded monthly, discounted 12 years.
6 percent nominal rate, compounded daily, discounted 12 years.
6 percent nominal rate, compounded continuously, discounted 12 years.
Why does the observed pattern of PV’s occur?
B
15
16
17
18
19
20
21
22
23
FV
i
N
m
m
m
m
m
m
C
$45,000
0.06
12
1
2
4
12
365
continuous
D
E
PV
PV
PV
PV
PV
PV
F
$22,363.62
$22,137.02
$22,021.28
$21,943.18
$21,905.15
G
H
$22,363.62 =PV($C$16/C18,C18*$C$17,0,-$C$15)
$22,137.02 =PV($C$16/C19,C19*$C$17,0,-$C$15)
$22,021.28
$21,943.18
$21,905.15
$21,903.85 =C15/EXP(C16*C17)
5
Problem 5:
How many years will the following take?
In general, use the following equation:
FVT = PV(1 + r)T.
Solve for T. This will require the use of logs.
(a)
$500 to grow to $1,039.5 if invested at 5 percent compounded annually.
$1,039.50 = $500*(1 + 0.05)T
2.079
= (1 + 0.05)T.
ln(2.079) = T*ln(1.05)
T = 15.00 years.
(b)
$35 to grow to $53.87 if invested at 9 percent compounded annually.
T = 5.00 years.
(c)
$100 to grow to $298.60 if invested at 20 percent compounded annually.
T = 6 years.
(d)
$53 to grow to $78.76 if invested at 2 percent compounded annually.
T = 20 years
Problem 6:
At what annual rate would the following have to be invested?
In general, use the following equation:
FVT = PV(1 + r)T.
Solve for r. Depending on your strategy, this can be solved using logs, or not.
(a)
$500 to grow to $1,948.00 in 12 years.
FV12 = $500(1 + r)12
$1,948 = $500(1 + r)12
(1 + r)12 = $1,948/$500
1 + r = (3.896)1/12
r = 0.1200 = 12.00%
(b)
$300 to grow to $422.10 in 7 years.
r = 0.0500 = 5.00%
(c)
$50 to grow to $280.20 in 20 years.
r = 0.0900 = 9.00%
(d)
$200 to grow to $497.60 in 5 years.
r = 0.2000 = 20.00%
6
Problem 7:
You are given three investment alternatives to analyze. The nominal dollar cash flows from each of the
investments are as follows:
End of Year
1
2
3
4
5
Investment A
15,000
15,000
15,000
-8,500
-8,500
Investment B
14,000
8,000
12,000
20,000
-20,000
Investment C
25,000
25,000
25,000
25,000
-25,000
What is the present value of each investment alternative if 6% is the appropriate discount rate? Assume
m = 1.
Annual PV Values
End of Year Investment A Investment B Investment C
1
14,150.94
13,207.55
23,584.91
2
13,349.95
7,119.97
22,249.91
3
12,594.29
10,075.43
20,990.48
4
(6,732.80)
15,841.87
19,802.34
5
(6,351.69)
(14,945.16)
(18,681.45)
Sum
27,010.69
31,299.66
67,946.19
Problem 8:
You are given three investment alternatives to analyze. The cash flows from each of the investments are
as follows:
Investment A
15,000
15,000
15,000
-8,500
-8,500
Investment B
14,000
8,000
12,000
20,000
-20,000
Investment C
25,000
25,000
25,000
25,000
-25,000
What is the future value of each investment alternative if 10% is the appropriate discount rate? Assume
m = 1.
Annual FV Values
End of Year Investment A Investment B Investment C
1
21,961.50
20,497.40
36,602.50
2
19,965.00
10,648.00
33,275.00
3
18,150.00
14,520.00
30,250.00
4
(9,350.00)
22,000.00
27,500.00
5
(8,500.00)
(20,000.00)
(25,000.00)
Sum
42,226.50
47,665.40
102,627.50
1
Managerial Economics (ARE) 171A
University of California, Davis
Summer Session
Instructor: John H. Constantine
KEY—Homework 2
Problem 1:
Calculate the APY for each of the following investments. Which of the following investments has the
highest effective return (APY)? Assume that all CDs are of equal risk.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
A bank CD which pays 10 percent interest quarterly.
A bank CD which pays 10 percent monthly.
A bank CD which pays 10.2 percent annually.
A bank CD which pays 10 percent semiannually.
A bank CD which pays 9.6 percent daily (on a 365-day basis).
A bank CD which pays 9.6 percent continuously.
In general (r = APR):
Discrete:
Continuous:
m


r
APY   1    1

 m 
r
APY  e  1
Convert each of the alternatives to an effective annual rate (APY) for comparison.
(i)
APY = 10.38%.
(ii)
APY = 10.47%.
(iii)
APY = 10.20%.
(iv)
APY = 10.25%.
(v)
APY = 10.0745%.
(vi)
APY = 10.0759%
Therefore, the highest effective return is choice (ii).
2
Problem 2:
The following yields on U.S. Treasury securities were taken from a recent financial publication:
Term (Years) Rate (Percent)
0.5
5.10
1
5.50
2
5.60
3
5.70
4
5.80
5
6.00
10
6.10
20
6.50
30
6.30
(a)
Plot a yield curve based on these data.
Yield Curve
7.00
6.00
5.00
Yield
4.00
Yield Curve
3.00
2.00
1.00
0.00
0
5
10
15
20
25
30
35
Years to Maturity
(b)
What information does this graph tell you?
This yield curve tells us generally that either (i) inflation is expected to increase or (ii) there is an
increasing maturity risk premium. Since we stated there are “Government Securities”, when assume
LP = DP = 0.
(c)
Suppose you need to borrow for 30 years to finance a house. Why not simply borrow short-term
money each year for the next 30 years, rather than one a 30 year loan, to save money?
It would make sense to borrow long term because each year the loan is renewed interest rates are
higher. This exposes you to rollover risk. If you borrow for 30 years outright you have locked in a
6.3% interest rate each year.
3
Problem 3:
You read in the financial pages of your newspaper that 30-day T-bills are currently yielding 5.5 percent.
Your brother-in-law, a broker with XYZ Securities, has given you the following estimates of current
interest rate premiums:




Inflation premium = 3.25%
Liquidity premium = 0.6%
Maturity risk premium = 1.8%
Default risk premium = 2.15%
On the basis of these data, what is the real risk-free rate of return?
T-bill rate
= r* + IP
5.5%
= r* + 3.25%
r*
= 2.25%.
Problem 4:
The real risk-free rate is 3 percent. Inflation is expected to be 3 percent for the next two years. A 2-year
Treasury security yields 6.2 percent. What is the maturity risk premium for the 2-year security?
r* = 3%; IP2 = 3%; rT2 = 6.2%; MRP2 = ?
rT2
= r* + IP2 + MRP2 = 6.2%
rT2
= 3% + 3% + MRP2 = 6.2%
MRP2 = 0.2%.
Problem 5:
What are premium, discount and par bonds?
Premium (par, discount) bonds are bonds that sell for more than (the same as, less than) their face
or par value.
Problem 6:
XYZ Inc. has 8 percent coupon bonds on the market that have 14 years left to maturity. If the YTM on
these bonds is 9.1 percent, what is the current bond price?
P = $913.90
Problem 7:
A bond with a maturity of 19.5 years sells for $1,047. If the coupon rate is 6.5 percent, what is the yield
to maturity of the bond?
YTM = 6.09%
4
Problem 8:
The pension plan at your new company will set aside 25 percent of your annual salary of $32,000 every
year until you retire in 25 years. Interest rates are currently 8 percent.
(a)
How much can you expect to have at retirement?
Assuming the first payment is made one year from today, the present value of the contributions is
(r=8%, n=25 years): PVA = L*PVIFAr,n = 8000  10.6748 = $85,398.21. The future value is 85,398.21
 1.0825 = $584,847.52.
(b)
If your company's plan begins setting aside this amount for you after five years of employment,
how much can you expect to have at retirement?
Assuming the first payment is made six years from today, the value of the pension plan can be viewed
as the future value of a 20-year annuity whose first payment begins in one year: (r=8%, n=20 years)
FVA = L*FVIFAr,n = 8,000 x 45.7620 = $366,096.
5
Problem 9:
The Early Saver invests $1,000 a year for ten years, at 9 percent compounded annually, and then stops.
The Late Saver starts saving $1,000 a year ten years from now, also at 9 percent.
(a)
How many years will it take the Late Saver, depositing $1,000 a year, to catch up with the Early
Saver?
We need to solve for n, where n is the number of years it takes the Late Saver to catch up with the
Early Save. We assume that the Early Saver’s first investment is made immediately and the Late
Saver’s first investment is made at the end of year 10. The present value today of the Early Saver’s
investment is $1,000(1 + PVIFA9,9) = $6,995. This figure represents the present value of an annuity
due of 10 year’s duration, with the first payment made immediately. In n + 9 years, the Early Saver’s
investments will be worth $6,995 * (1.09)n+9 = $15,193 * (1.09)n. The reason we are interested in n + 9
years is that if the Late Saver is investing for n years, with n beginning at the end of year 9 (with the
first investment at the end of the year), then the Early Investor’s investments will have been
compounding for 9 years already. As of the start of year 10, we can treat the Late Saver’s investments
as an ordinary n-year annuity, with a value at the end of year n (using Equation 4.12) equal to
FVIFA9,n = $1,000[
(1.09 )n - 1
]
0.09
Note that the future value of the Late Saver’s investments as of year 10 is the same as its future value
as of year 0 because there were no intervening cash flows. We can now solve for n by setting these two
terms equal, as follows:
(1.09 )n - 1
15,193(1.09 ) = 1,000[
]
0.09
n
As it turns out, this equation has no solution. In other words, the Late Saver can never catch up to the
Early Saver.
(b)
How would your answer to part a change if the interest rate was 7 percent instead of 9 percent?
Using the same reasoning as above, the equation to solve in this case would be
6,247(1.07 )n = 1,000[
(1.07 )n - 1
]
0.07
The solution to this equation is n = 51.61 years. Lowering the interest rate now makes catch up
feasible. The related purposes of this example are to show the powerful effects of compounding and the
advantages of starting to save early.
6
Problem 10:
During a late-night television commercial, the Kwan-tzu Metalcraft Company offers its hardware set for
the unbelievably low price of $200, or at its easy payment plan of $9 down and $9.50 a month for 25
months. What is the finance rate charged by Kwan-tzu on its easy payment plan?
The present value of the payments must be equal to the dollar value of the purchase. In this case, r
is unknown and n=25 months. The equation to solve is:
$200.00 = $9.00 + Present Value of an Annuity.
C/m = $9.50; mT = 25; m = 12; need to find (r/m) and r.
r/m = 1.75% (that is, 1.75% per month)
r = 23.14% (that is, 23.14% compounded annually)
Problem 11:
Susan is retiring this year at age 65, with a life expectancy of 20 years. Suppose she has saved $300,000.
(a)
Assume Susan can earn a 10 percent annual return on her money. What level amount of capital
can she withdraw each year so that her money holds out for at least 20 years? Assume payments
are received at the end of the period.
This is an ordinary annuity problem. Her payments will be $35,237.89.
(b)
Assume Susan can earn a 10 percent annual return on her money. What level amount of capital
can she withdraw each year so that her money holds out for at least 20 years? Assume payments
are received at the beginning of the period.
This is an annuity due problem. Her payments will be $32,034.44.
Problem 12:
RCA made a coupon payment yesterday on its "6.25s12" bonds that mature on October 9, 2012. The
required return on these bonds is 9.2% APR, and today is April 10, 2003. What should be the market
price of these bonds?
With a coupon rate of 6.25%, the annual coupon payment is $62.50; thus, the semi-annual payment
of CPN/2 = $62.50/2 = $31.25. The time to maturity (N) is 9.5 years. Thus, the number of periods is
2N = 2(9.5) = 19 periods. The semi-annual APR is 9.2%/2 = 4.6% or 0.046. We use the bond value
formula with semi-annual payments. PV = 390.286 + 425.498 = $815.78
7
Problem 13:
Clyde Atherton wants to buy a car when he graduates college in two years. He has the following sources
of money:
(i)
He has $5,000 now in the bank in an account paying 8% compounded quarterly.
(ii)
He will receive $2,000 in one year from a trust.
(iii)
He will take out a car loan at the time of purchase on which he'll make $500 monthly payments at
18% compounded monthly over four years.
(iv)
Clyde's uncle is going to give him $1,500 a quarter starting today for one year.
In addition Clyde will save up money in a credit union through monthly payroll deductions at his parttime job. The credit union pays 12% compounded monthly. If the car is expected to cost $40,000 how
much must he save each month?
Current bank account:
FV = $5,000(1 + 0.08/4)4(2) = $5,000(1.1717)
FV = $5,858.50
Trust:
FV = $2,000(1 + 0.08/4)4(1) = $2,000(1.0824)
FV = $2,164.80
Loan:
PVANN = $500 A12(4)
0.18/12 = $500 (34.0426)
PVANN= $17,021.30
Uncle:
  1.02  4(1)  1 
 (1.02) = $1,500(4.1216)(1.02)
FVANN DUE = $1,500 
0.02


FVANN DUE = $6,306.05
Bring this forward as an amount:
FV = $6,306.05(1.02)4 = $6,306.05 (1.0824)
FV = $6,825.67
Sources:
$ 5,858.50
2,164.80
17,021.30
6,825.67
$31,870.27
Shortfall:
Save-up:
FVA = (PMT/m)*(FV Annuity Factor)
  1.0112(2)  1 


$8,129.73 = (PMT/12)
0.01


PMT/12 = $301.40
$40,000.00
31,870.27
$ 8,129.73—Needs to Save
8
Problem 14:
Find the missing information for each of the following bonds. The coupons are paid in semiannual
installments.
(A)
BOND
1
T (YEARS)
10
YIELD TO
MATURITY
(YTM)
7.8%
2
5
10.5%
B
9.5%
$1,000
3
25
8.2%
C
5.5%
$1,000
4
15
D
$1,050.00
7.4%
$1,000
5
E
9.0%
$977.20
8.5%
$1,000
6
8
7.0%
$1,120.94
F
$1,000
PRESENT
VALUE
A
COUPON
RATE
7.8%
FACE
VALUE
$1,000
N = 10 x 2 = 20; r/2 = 7.8%/2 = 3.9%; CPN = 7.8%*(1000/2) = $39; PAR = $1000;
PBOND = $1,000.00
(B)
N = 5 x 2 = 10; r/2 = 10.5%/2 = 5.25%; CPN = 9.5%*(1000/2) = $47.50; PAR = $1000
PBOND = $961.86
(C)
N = 25 x 2 = 50; r/2 = 8.2%/2 = 4.1%; CPN = 5.5% *(1000/2) = $27.50; PAR = $1000
PBOND = $714.89
(D)
N = 15 x 2 = 30; PBOND = $1050; CPN = 7.4%*(1000/2) = $37; PAR = $1000
r/2 = 3.43%; YTM = 3.43% x 2 = 6.86%
(E)
r/2 = 9.0%/2 = 4.5%; PBOND = $977.20; CPN = 8.5%*(1000/2) = $42.50; PAR = $1000
T = 12.00 semiannual periods; T = 12 / 2 = 6 years
(F)
N = 8 x 2 = 16; r/2 = 7.0% / 2 = 3.5%; PBOND = $1120.94; PAR = $1000; CPN = $45.00
Coupon Rate = 45 x 2 / 1000 = 9.0%
9
Problem 15:
Metropolitan Life will sell you a 20-year annuity, paying $45,300 annually in regular monthly checks, for
$500,000.
(a)
At what effective annual interest rate is this a fair deal?
T = 20; C = $45,300; m = 12; r = ?
PV = (C/m)*Ar/mmT
$500,000 = ($45,000/12)*Ar/12(12)(20)
Ar/12240 = 132.50
By trial and error or financial calculator, you find that:
r/m = 0.00555, or
r = 0.0666
This is equivalent to an effective annual rate of:
6.8667% = (1.00555)12 - 1.
(b)
Alternatively, for $500,000, you can buy a package of 40 zero-coupon Treasury bonds paying
$25,000 every six months. What is the annualized return on your money in this case?
Again, using a calculator, the solution is a semiannual rate of 3.9302%, which is equivalent to
8.0149% annually.
Problem 16:
Your firm offers a 10-year, zero coupon bond. The yield to maturity is 8.8 %. What is the current market
price of a $1,000 face value bond?
P = $430.24.
Problem 17:
Mitzi’s, II. Bonds offer a 6 % coupon at a current market price of $989. The bonds have a face value of
$1,000 and a call price of $1,020. What is the current yield on these bonds?
YTM = 6.07 %.
1
Managerial Economics (ARE) 171A
University of California, Davis
Summer Session I, 2011
Instructor: John H. Constantine
KEY—Homework 3: Due Monday, July 11, 2011
Problem 1: Tables are given at the end of the assignment to fill out.
State of the Economy
Recession
Stable
Moderate Growth
Boom
Stagnation
(a)
(b)
(c)
(d)
Probability
of
Occurrence
0.15
0.25
0.30
0.20
0.10
X=
Mature
Stock
0.10
0.11
0.14
0.15
-0.02
Y=
Growth
Stock
-0.05
0.15
0.20
0.30
0.01
For each security, calculate (i) the expected returns, (ii) the variance of returns, and (iii) the
standard deviation of returns.
Calculate (i) the covariance of returns between the two stocks and (ii) the correlation coefficient
for the returns between the two stocks.
Given the data in parts (a) and (b), calculate the efficient frontier for a portfolio containing these
two securities. You are to graph the efficient frontier, clearly labeling the minimum variance
portfolio.
Clearly explain why the efficient frontier assumes the shape that it does.
2
State of the Economy
Recession
Stable
Moderate Growth
Boom
Stagnation
Probability
of
Occurrence
0.15
0.25
0.30
0.20
0.10
X=
Mature
Stock
0.10
0.11
0.14
0.15
-0.02
Y=
Growth
Stock
-0.05
0.15
0.20
0.30
0.01
0.1125
0.002289
0.0478
0.1510
0.013209
0.1149
1.00
Sum
Expected Return
Variance
Standard Deviation
Covariance
Correlation Coefficient
0.003768
0.685203
RM
11.25%
SDA
4.78%
RG
15.10%
SDB
11.49%
Portfolio
1
2
3
4
5
6
7
8
9
10
11
Proportion
Security
M
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Proportion
Security G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Corr(RM,RG)
0.69
RP
11.3%
11.6%
12.0%
12.4%
12.8%
13.2%
13.6%
13.9%
14.3%
14.7%
15.1%
SDP(p=0.82)
4.78%
5.16%
5.66%
6.24%
6.89%
7.59%
8.32%
9.09%
9.87%
10.68%
11.49%
3
Efficient Frontier(p=-0.6852)
16.0%
Portfolio Expected Returns
15.0%
14.0%
13.0%
12.0%
11.0%
10.0%
0.00%
2.00%
4.00%
6.00%
8.00%
Portfolio Risk
10.00%
12.00%
14.00%
4
Problem 2:
Suppose securities D, E, and F have the following characteristics with respect to expected return, standard
deviation, and the correlation between them:
Correlation Coefficients (pij)
Security
D
E
F
Rj-bar
0.08
0.15
0.12
SDj
0.02
0.16
0.08
D-E
0.3
0.3
D-F
0.7
0.7
E-F
0.5
0.5
Suppose the portfolio weights are wD = 40%, wE = 20%, and wF = 40%. Calculate the portfolio expected
return (RP) and portfolio standard deviation (P).
E(RP) = wDRD + wERE + wFRF = 0.11 (= 11%).
Var(RP) = wD2D2 + wE2E2 + wF2F2 + 2wDwEDEDE + 2wDwFDFDF + 2wEwFEFEF
= 0.0604 (= 6.04%)
5
Problem 3:
The standard deviation of return for asset j (σj) is 15%, the expected return on the market E(rM) is 8%, the
riskless rate (rRF) is 4%, the standard deviation of the market (σM) is 25%, and the correlation coefficient,
Corr(rj, rM), is 0.5. What is security j’s risk premium?
Security j’s risk premium = [Corr(rj, rM)] (σj) [( r̄ M – rf) / (σM)]
= [0.5]15%[8% – 4% ] / 25%
= 1.2%
Problem 4:
You are considering investing in a technology stock but are concerned that the required return may be too
low. You will not purchase the stock if the required return is below 13%. Will you purchase the stock
given the following information? The standard deviation of return for asset j (σj) is 25%, the expected
return on the market (r̄ M) is 10%, the riskless rate (rf) is 5%, the standard deviation of the market (σM) is
15%, and beta is 1.5.
Security j’s required return is the sum of the riskless return and its risk premium:
rj = rRF + βj((EM) – rRF)] = 5% + 1.5(10% – 5%) = 12.5%
Problem 5:
What is the expected return on asset A if it has a beta of 0.3, the expected market return is 14%, and the
riskless rate is 5%?
The capital asset pricing model (CAPM) gives the expected return for any asset j. Inserting the given
values, we have:
rj = rRF + βj(rM – rRF) = 5% + 0.3(14% – 5%) = 5% + 0.3(9%) = 5% + 2.7% = 7.70%
Multiple Choice:
1) Stock A and Stock B each have an expected return of 15 percent, a standard deviation of 20 percent,
and a beta of 1.2. The returns of the two stocks are not perfectly correlated; the correlation
coefficient is 0.6. You have put together a portfolio which is 50 percent Stock A and 50 percent
Stock B. Which of the following statements is most correct?
a) The portfolio’s expected return is 15 percent.
b) The portfolio’s beta is less than 1.2.
c) The portfolio’s standard deviation is 20 percent.
d) Statements a and b are correct.
e) All of the statements above are correct.
Statement a is true; the others are false. Since both stocks’ betas are equal to 1.2, the beta of the
portfolio will equal 1.2. Because the stocks’ correlation coefficient is less than one, the portfolio’s
standard deviation will be lower than 20 percent.
6
2) The risk-free rate, rRF, is 6 percent and the market risk premium, (rM – rRF), is 5 percent. Assume that
required returns are based on the CAPM. Your $1 million portfolio consists of $700,000 invested in a
stock that has a beta of 1.2 and $300,000 invested in a stock that has a beta of 0.8. Which of the
following statements is most correct?
a) The portfolio’s required return is less than 11 percent.
b) If the risk-free rate remains unchanged but the market risk premium increases by 2
percentage points, the required return on your portfolio will increase by more than 2
percentage points.
c) If the market risk premium remains unchanged but expected inflation increases by 2 percentage
points, the required return on your portfolio will increase by more than 2 percentage points.
d) If the stock market is efficient, your portfolio’s expected return should equal the expected return
on the market, which is 11 percent.
e) None of the above answers is correct.
Statement b is correct; all the other statements are false. If the market risk premium increases by 2
percent and rRF remains unchanged, then the portfolio’s return will increase by 2%(1.08) = 2.16%.
Statement a is false, since rp = 6% + (5%)bp. The portfolio’s beta is calculated as 0.7(1.2) + 0.3(0.8)
= 1.08. Therefore, rp = 6% + 5%(1.08) = 11.4%. Statement c is false. If rRF increases by 2 percent,
but RPM remains unchanged, the portfolio’s return will increase by 2 percent. Statement d is false.

Market efficiency states that the expected return should equal the required return; therefore, r p =
rp = 11.4%.
3) Which of the following statements is most correct?
a) The slope of the security market line is beta.
b) The slope of the security market line is the market risk premium, (r M – r R F ).
c) If you double a company’s beta its required return more than doubles.
d) Statements a and c are correct.
e) Statements b and c are correct.
4) Stock A has a beta = 0.8, while Stock B has a beta = 1.6. Which of the following statements is most
correct?
a) Stock B’s required return is double that of Stock A’s.
b) An equally weighted portfolio of Stock A and Stock B will have a beta less than 1.2.
c) If market participants become more risk averse, the required return on Stock B will
increase more than the required return for Stock A.
d) All of the answers above are correct.
e) Answers a and c are correct.
5) Which of the following statements is most correct?
a) The beta coefficient of a stock is normally found by running a regression of past returns on
the stock against past returns on a stock market index. One could also construct a scatter
diagram of returns on the stock versus those on the market, estimate the slope of the line of
best fit, and use it as beta.
b) It is theoretically possible for a stock to have a beta of 1.0. If a stock did have a beta of 1.0, then,
at least in theory, its required rate of return would be equal to the riskless (default-free) rate of
return, rRF.
c) If you found a stock with a zero beta and held it as the only stock in your portfolio, you would by
definition have a riskless portfolio. Your 1-stock portfolio would be even less risky if the stock
had a negative beta.
d) The beta of a portfolio of stocks is always larger than the betas of any of the individual stocks.
e) All of the statements above are true.
7
Tables for Problem 1:
State of the Economy
Recession
Stable
Moderate Growth
Boom
Stagnation
Probability
of
Occurrence
0.15
0.25
0.30
0.20
0.10
X=
Mature
Stock
0.10
0.11
0.14
0.15
-0.02
Y=
Growth
Stock
-0.05
0.15
0.20
0.30
0.01
Expected Return
Variance
Standard Deviation
Covariance
Correlation Coefficient
Portfolio
1
Proportion Proportion
Security M Security G
1.0
0.0
2
0.9
0.1
3
0.8
0.2
4
0.7
0.3
5
0.6
0.4
6
0.5
0.5
7
0.4
0.6
8
0.3
0.7
9
0.2
0.8
10
0.1
0.9
11
0.0
1.0
RP
SDP
1
Managerial Economics (ARE) 171A
University of California, Davis
Summer Session I, 2011
Instructor: John H. Constantine
KEY—Homework 4: Due Thursday, July 14, 2011
Problem 1:
Note: You can do these regressions in Excel. You will need to enter the data. If you know what you
are doing, running the regressions for this problem will be very quick. Also, ARE 106 is required for
this course; the TAs and I will NOT help students with basic ARE 106-type questions.
Given below the raw returns data for two firms A and B, the proxy Market Portfolio (S&P 500) and the
risk-free rate. You are to estimate the characteristic lines for stocks A and B. The two equations to be
estimated are:
(I)
(II)
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
(a)
(iii)
(iv)
(v)
(vi)
(vii)
(d)
rM
6%
2%
-13%
-4%
-8%
16%
10%
15%
8%
13%
(i = A, B)
(i = A, B)
Actual Return
rA
11%
8%
-4%
3%
0%
19%
14%
18%
12%
17%
rB
16%
11%
-10%
3%
-3%
30%
22%
29%
19%
26%
rRF
7.10%
7.30%
7.70%
7.60%
8.00%
7.00%
7.80%
8.00%
8.10%
8.60%
For each regression equation (I) and (II), and for each stock, A and B, you are to:
(i)
(ii)
(b)
(c)
(ri – rRF) = αi + βi(rM – rRF) + εi.
ri = αi + βirM + εi.
State the value of beta, i.
Conduct hypothesis tests on each i (State the null and alternative hypothesis and conduct
the test.)
For each security, how does the estimating beta differ when using equation (I) or (II)?
State the value of beta, i.
Conduct hypothesis tests on each i (State the null and alternative hypothesis and conduct
the test.)
For each security, how does the estimating beta differ when using equation (I) or (II)?
Are the i for both (I) and (II) valid? Explain.
What is the R2 value, and clearly provide an interpretation of R2 in the context of CAPM.
Graph the data in terms of the Characteristic Line and draw a trend line based on the regression
results. You can do this in Excel quite easily. You can graph both sets of returns on a single
graph, and include a trend line for each security as well.
Some practitioners statistically estimate beta by suppressing the intercept to equal zero. Why do
they do this, and why is it probably a bad idea to do so?
2
Estimating βi using the Risk Premium:
Stock A:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.999430601
R Square
0.998861527
Adjusted R Square
0.998719218
Standard Error
0.002832057
Observations
10
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
8
9
SS
MS
F
Significance F
0.056295836 0.056295836 7018.954656 4.59565E-13
6.41644E-05 8.02054E-06
0.05636
Coefficients Standard Error
t Stat
P-value
0.047557468
0.000928582 51.21516272 2.34028E-11
0.790671849
0.009437567 83.77920181 4.59565E-13
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.045416155 0.049698782 0.045416155 0.049698782
0.76890878 0.812434918 0.76890878 0.812434918
Stock B:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.999862549
R Square
0.999725117
Adjusted R Square
0.999690756
Standard Error
0.002425457
Observations
10
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
8
9
SS
MS
F
Significance F
0.171162937 0.171162937 29095.27607 1.56134E-15
4.70627E-05 5.88284E-06
0.17121
Coefficients Standard Error
t Stat
P-value
0.107845641
0.000795265 135.6096896 9.77715E-15
1.378678512
0.008082613 170.5733745 1.56134E-15
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.106011757 0.109679526 0.106011757 0.109679526
1.360039974 1.39731705 1.360039974 1.39731705
3
Estimating βi using ri (that is, without the Risk Premium):
Stock A:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.999430601
R Square
0.998861527
Adjusted R Square
0.998719218
Standard Error
0.002832057
Observations
10
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
8
9
SS
MS
F
Significance F
0.056295836 0.056295836 7018.954656 4.59565E-13
6.41644E-05 8.02054E-06
0.05636
Coefficients Standard Error
t Stat
P-value
0.062419767
0.000991169 62.97589198 4.49441E-12
0.790671849
0.009437567 83.77920181 4.59565E-13
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.060134126 0.064705407 0.060134126 0.064705407
0.76890878 0.812434918 0.76890878 0.812434918
Stock B:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.999862549
R Square
0.999725117
Adjusted R Square
0.999690756
Standard Error
0.002425457
Observations
10
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
8
9
SS
MS
F
Significance F
0.171162937 0.171162937 29095.27607 1.56134E-15
4.70627E-05 5.88284E-06
0.17121
Coefficients Standard Error
t Stat
P-value
0.080959467
0.000848867 95.37358735 1.63087E-13
1.378678512
0.008082613 170.5733745 1.56134E-15
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.079001977 0.082916957 0.079001977 0.082916957
1.360039974 1.39731705 1.360039974 1.39731705
4
THERE ARE 2 ANSWERS TO PROBLEM 2. THE FIRST ANSWER HAS rRF = 6% – THIS IS
HOW I ORIGINALLY SOLVED IT. THE SECOND ANSWER HAS rRF = 3% WHICH IS HOW
YOU DID IT. I HAVE THE WORK FOR MY NUMBERS AND ONLY THE ANSWERS FOR
YOUR NUMBERS.
Problem 2:
The Smith Chemical Company’s management conducted a study and concluded that if it expands its
consumer products division (which is less risky than its primary business, industrial chemicals), its beta
would decline from 1.2 to 0.9. However, consumer products have a somewhat lower profit margin, and
this would cause its constant growth in earnings and dividends to fall from 6 percent to 4 percent. The
following also apply:
rM = 9%; rRF = 6%; D0 = $2.00.
(a)
Should management expand its consumer products division? Show work.
Old rs = rRF + (rM – rRF)b
= 6% + (3%)1.2 = 9.6%.
New rs = 6% + (3%)0.9
= 8.7%.
D (1  g )
D1
$2 (1.06 )
 0

 $58.89.
rs  g
rs  g
0.096  0.06
$2(1.04)
New price: P̂0 
 $44.26.
0.087  0.04
Old price: P̂0 
Since the new price is lower than the old price, the expansion in consumer products should be
rejected. The decrease in risk is not sufficient to offset the decline in profitability and the reduced
growth rate.
(b)
Assume all the facts as given above, except that beta coefficient changes. How low would the
beta need to fall to cause the expansion to be a good one?
$2(1.04)
.
POld = $58.89. PNew =
rs  0.04
Solving for rs we have the following:
$2.08
$58.89 =
rs  0.04
$2.08 = $58.89(rs) – $2.3556
rs
= 0.07532.
Solving for b:
7.532%= 6% + 3%(b)
b
= 0.5107.
Check: rs = 6% + (3%)0.5107 = 7.532%.
P̂0 =
$2.08
= $58.89.
0.07532  0.04
Therefore, only if management’s analysis concludes that risk can be lowered to b = 0.5107, should
the new policy be put into effect.
5
Problem 2 with your numbers (answer only):
The Smith Chemical Company’s management conducted a study and concluded that if it expands its
consumer products division (which is less risky than its primary business, industrial chemicals), its beta
would decline from 1.2 to 0.9. However, consumer products have a somewhat lower profit margin, and
this would cause its constant growth in earnings and dividends to fall from 6 percent to 4 percent. The
following also apply:
rM = 9%; rRF = 3% ; D0 = $2.00.
(a)
Should management expand its consumer products division? Show work.
Old price = $50.48
New price = $41.27
No Expansion.
(b)
Assume all the facts as given above, except that beta coefficient changes. How low would the
beta need to fall to cause the expansion to be a good one?
Beta should be 0.853
6
Problem 3:
ABC’s most recent dividend as $1.80 per share (D0 = $1.80), and the firm’s required return is 11%. Find
the market value of ABC’s shares when:
In each part below, the first three years’ calculations are identical. The calculation is based on the
following:
 D T 1 


r  g 2 
D 0 (1  g 1 )

P0 

(1  r ) t
(1  r ) T
t 1
3
t
3
D 0 (1  g 1 ) t

P0 

t 1
P0 = 5.11
(1  r )
t
= 1.944/1.111 + 2.10/1.112 + 2.27/1.113
 Value of all parts (a) – (c) for t = 1  3.
(a)
Dividends are expected to grow at 8% annually for 3 years, followed by a 5% constant annual
growth rate in years 4 to infinity.
 D T 1   2.27(1.05) 

 

r  g 2   0.11  0.05 


= $29.00
P0 
(1  r ) T
(1.11) 3
Thus:
Price = $5.11 + $29.00 = $34.12
(b)
Dividends are expected to grow at 8% annually for 3 years, followed by a 0% constant annual
growth rate in years 4 to infinity.
 D T 1   2.27(1.00) 

 

r  g 2   0.11  0.00 


= $15.09
P0 
(1  r ) T
(1.11) 3
Thus:
Price = $5.11 + $15.09 = $20.21
(c)
Dividends are expected to grow at 8% annually for 3 years, followed by a 10% constant annual
growth rate in years 4 to infinity.
 D T 1   2.27(1.10) 

 

r  g 2   0.11  0.10 


= $18.258
P0 
(1  r ) T
(1.11) 3
Thus:
Price = $5.11 + $182.58 = $182.58
7
Problem 4:
Perry Motors’ common stock currently pays an annual dividend of $1.80 per share. The required return
on the common stock is 12%. Estimate the value of the common stock under each of the assumptions
about the dividend.
For this problem:
r = 12% (= 0.12); D0 = $1.80/share.
(a)
Dividends are expected to grow at an annual rate of 0% to infinity (that is, until t = infinity.)
D
D
$1.80
P0  1  0 
 $15 / share .
r
r
0.12
Note: D1 = D0(1 + g), but since g = 0, then D1 = D0.
(b)
Dividends are expected to grow at a constant annual rate of 5% to infinity.
For this problem, g = 0.5:
D1 = D0(1 + g) = $1.80(1.05) = $1.89/share.
P0 
D1
$1.89

 $27.00 / share .
r  g 0.12  0.05
(c)
Dividends are expected to grow at an annual rate of 5% for each of the next three years, followed
by a constant annual growth rate of 4% in years four to infinity.
Variable growth:
gA = 0.05 (for periods 1, 2, and 3)
gB = 0.04 (for periods 4  )
(i)
Periods 1, 2, and 3:
D1 = D0(1 + gA)1 = $1.80(1.05)1 = $1.89/share.
D2 = D1(1 + gA)1 = D0(1 + gA)2 = $1.80(1.05)2 = $1.98/share.
D3 = D2(1 + gA)1 = D0(1 + gA)3 = $1.80(1.05)3 = $2.08/share.
(ii)
Periods 4  :
D4 = D3(1 + gB) = $2.08(1.04) = $2.16.
Next, need to discount back to current period:
(i)
Periods 1, 2, and 3:
P0 = $1.89/(1.12)1 + $1.98/(1.12)2 + $2.08/(1.12)3
P0 = $4.75
(ii)
Periods 4  :
 D4  

$2.16

 
( r  g B )   (0.12  0.04) 

 $19.22
P0  
(1  r ) 3
(1.12) 3
P0 = $4.75 + $19.22 = $23.97/share.
8
Problem 5:
Stock Y has a beta of 1.50 and an expected return of 17 percent. Stock Z has a beta of 0.80 and an
expected return of 10.5 percent. If the risk-free rate is 5.5 percent and the market risk premium is 7.5
percent, are these stocks correctly priced?
There are two ways to correctly answer this question.
Method 1: We can use the CAPM equation.
For Stock Y we substitute in the value we are given for each stock, we find:
E(rY) = 0.055 + .075(1.50) = 0.1675 or 16.75%
It is given in the problem that the expected return of Stock Y is 17 percent, but according to the
CAPM, the return of the stock based on its level of risk, the expected return should be 16.75
percent. This means the stock return is too high, given its level of risk. Stock Y plots above the SML
and is undervalued. In other words, its price must increase to reduce the expected return to 16.75
percent.
For Stock Z, we find:
E(rZ) = 0.055 + .075(0.80) = 0.1150 or 11.50%
The return given for Stock Z is 10.5 percent, but according to the CAPM the expected return of the
stock should be 11.50 percent based on its level of risk. Stock Z plots below the SML and is
overvalued. In other words, its price must decrease to increase the expected return to 11.50 percent.
Method 2: We can use the equation, called the Treynor Ratio, which is a reward/risk relationship:
^
^
^
rM  rRF rY  rRF rZ  rRF


M
Y
Z
Market Portfolio: 0.075/1 = 0.075 = 7.5%
Security Y: (0.17 – 0.055) / 1.50 = 0.0767 = 7.67%
Security Z: (0.105 – 0.055) / 0.80 = 0.0625 = 6.25%
The reward-to-risk ratio for Stock Y is too high (relative to the Market Portfolio), which means the
stock plots above the SML, and the stock is undervalued. Its price must increase until its reward-torisk ratio is equal to the market reward-to-risk ratio.
The reward-to-risk ratio for Stock Z is too low, which means the stock plots below the SML, and
the stock is overvalued. Its price must decrease until its reward-to-risk ratio is equal to the market
reward-to-risk ratio.
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