exam solution 9 fundamentals of corporate finance, 4th edition brealey

AK/ADMS 3530.03 Finance Midterm Exam
Winter 2007
Solutions
Type A Exam
Numerical questions (4 points each)
1. (Q. 2 in B) The Joshua Co. plans on saving money to buy some new
equipment. The company is opening an account today with a deposit of $15,000
and expects To earn 4% interest annually. After 3 years, the firm wants to add
an additional $50,000 to the account. If the account continues and earns 4%
interest compounded semi-annually after 3 years, how much money will the
Joshua Co. have in their account five years from now?
A) $66,872.96
B) $68,249.79
C) $70,952.96
D) $72,385.44
Answer D
Using FV = PV × (1+r)^t formula
FV in 3yrs = $15,000 × (1 + 0.04) ^3 = $16,872.96
FV in 5 yrs = $(16,872.96 + 50,000) × (1 + 0.04 / 2)^ 4 = $72,385.44.
2. (Q. 1 in B) I want to buy a car that I know will cost me $43,860 (before taxes)
in ten years. How much must I save annually, beginning one year from now, in
order to accumulate the purchase price plus all applicable taxes by the end of
Year 10? In this case taxes are 6% GST and 8% PST which are each applied to
the purchase price. Assume that interest is calculated at 9 percent annually.
A) $2,887
B) $3,291
C) $4,500
D) $4,587
Answer B

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Amount required in ten years = $43,860 × (1.14) = $50,000.40
Using the FV annuity formula where
FV= PMT× ((1+r)^t – 1)
r
((1.09)10 − 1)

$50,000 = PMT ×

.09

= PMT × 15.1929

PMT = $3,291.04.
3. (Q. 8 in B) A credit card company sends you a promotion that says it will
charge you an interest rate of 1.25% monthly. In this case the annual
percentage rate (APR) is ____ and the effective annual rate (EAR) is _______
and if I carried a $300 balance throughout the year I would owe _______ at the
end of the year.
A) 16.08%; 15.00%; $348.24
B) 15.00%; 14.55%; $345.00
C) 14.55%; 15.00%; $345.00
D) 15.00%; 16.08%; $348.24
Answer D
APR = 1.25% × 12 = 15%
EAR = (1+1.25%)^12 – 1 = 16.08%
Balance owing = 300 × (1+0.1608) = $348.24.
4. (Q. 9 in B) Prizes are often not “worth” as much as claimed. Place a value on
a prize of $5,000,000 that is to be received in equal annual payments over the

next 20 years, with the first payment beginning today. Assume an interest rate of
7 percent over the 20-year period.
A) $2,212,652
B) $2,648,504
C) $2,833,899
D) $2,950,567
Answer C

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Annual payment = $5,000,000 / 20 = $250,000
1
1 
 −
i i(1 + i)n 
(for annuities due)
PV = PMT + PMT 
 1
1

−

19 
.07 .07(1.07) 
= $250,000 + $250,000 
= $250,000 + $250,000 [10.3356]
=$2,833,898.81.
5. (Q. 3 in B) Which of the following strategies will allow real retirement spending
to remain approximately equal, assuming savings of $1,000,000 invested at 8

percent annually, a 25-year time horizon, and a 4 percent expected annual
inflation rate?
A) Spend approximately $63,000 annually.
B) Spend approximately $78,225 annually.
C) Spend approximately $93,680 annually.
D) Spend approximately $127,500 annually.
Answer A
Using the formula where 1 + real rate =

Real rate =

 1.08 


 1.04.

1 + nominal rate
1 + inflation rate

– 1 = 3.85%.

1

1
Then using the formula where PV of an annuity = C ×  −
t 
 r r(1 + r ) 
 1
1


−

25 
.0385 .0385(1.0385) 
$1,000,000 = pmt 

$63,001 = pmt.
6. (Q. 4 in B) You are saving money to buy a house in ten years. You will need
$75,000 to make the down payment at that time. Due to some other financial
commitments you won’t be able to deposit any money in years 9 and 10 towards
this down payment. How much equal amounts must you deposit in a savings
account at the end of each year (other than years 9 and 10) in order to save
$75,000 if the savings account pays interest at 10 percent per year compounded
annually?
A) $3,875
B) $4,115
C) $5,420
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D) $6,558
Answer C
The timeline of your savings account is as follows:
0
1
2
3
4
5
6

7
8
9
10
I--------I--------I--------I--------I--------I--------I--------I--------I--------I---------I
R
R
R
R
R
R
R
R
$75,000 = R × (FVIFA 10%, 8yrs) × (FVIF 10%, 2yrs)
$75,000 = R x 11.4359 x 1.21
$75,000 = 13.8374R
R = $5,420.09.
7. (Q. 7 in B) You are planning to establish a 30-year scholarship fund at York
University that will pay $12,000 at the end of the first year and then increase by
1.5% per year. The scholarship fund will be managed by a successful fund
manager, Chris Robinson. Chris guarantees that the fund will earn a 6.75%
annual rate of return. How much should you donate to York today in order to
maintain this scholarship?
A) $50,346.20
B) $146,508.84
C) $178,225.23
D) $228,571.43
Answer C
This is a growing annuity.
  1.015  30 

$12,000
 = $178,225.23.
× 1 −
PV =
0.0675 − 0.015  1.0675  
8. (Q. 10 in B) You want to buy a house that costs $400,000. You make a 25%
down payment and finance the rest with a 15 year mortgage. The mortgage has
a five year renewal term for which the annual mortgage rate is 6.5%
compounded semi-annually. What will the remaining principal of the loan be at
the end of the 5-year term?
A) $229,787.77
B) $289,459.98
C) $304,514.42
D) $317,182.54
Answer A

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0.065 2
) − 1 = 6.605625%.
2
Monthly rate is r = (1+0.06605625)1/12 - 1 = 0.005345 = 0.5345%.
Monthly payment:
N=180, I/Y=0.5345%, PV=-$300,000, FV=0, CPTÆ PMT=$2,599.15.
Remaining principal at the end of 5-year term is the PV of remaining payments:
N=120, I/Y=0.5345%, PMT=$2,599.15, FV=0, CPTÆ PV=$229,787.77.
EAR = (1 +

9. (Q. 5 in B) A deposit of $3,500 made three years ago is worth $5,200 today.

The deposit pays interest quarterly. What is the APR with quarterly compounding
on this deposit?
A) 12.97%
B) 13.42%
C) 14.15%
D) 15.54%
Answer B

N=12, PMT=0, FV=$5,200, PV=-$3,500, CPTÆI/Y=3.3541 (this is the quarterly rate),
APR = 3.3541% × 4 = 13.4164%.
10. (Q. 6 in B) How much interest is to be earned between the end of year 4 and
the end of year 6 on a $1,000 deposit made today that earns 7% interest
compounded annually?
A) $189.93
B) $194.11
C) $206.52
D) $246.28
Answer A
The difference between the FV6 and FV4 is the interest you earn over these two
years. With your calculator,
PV= -1000, N=6, i=7%, PMT=0 FV=? FV6 = 1500.73
PV= -1000, N=4, i=7%, PMT=0 FV=? FV4 = 1310.80
So interest earned = $1,500.73 - $1,310.80 = $189.93.

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11. (Q. 16 in B) When an investor purchases a $1,000 par value bond that was
quoted at 97.16, the investor:
A) pays $971.60 for the bond.

B) pays $1,029.23 for the bond.
C) receives $971.60 upon the maturity date of the bond.
D) receives 97.16 percent of the stated coupon payments.
Answer A
Bond prices are quoted as a percentage of their face value. So the bond in
question has a price of 97.16% × $1,000 = $971.60.
12. (Q. 17 in B) What is the yield to maturity for a bond paying $100 coupon
annually that has six years until maturity and sells for $1,074.22? The bond has a
face value of $1,000 and pays semi-annual coupons.
A) 4.20 percent
B) 4.66 percent
C) 8.40 percent
D) 9.31 percent
Answer C
Using your calculator, 50 (PMT), 1000 (FV), 12 (N), -1074.22 (PV), CPT I/Y =
4.2%. So the YTM is 4.2% × 2 = 8.4%.
Alternatively, if you use the approximate formula, you should get:
$(1,000 − 1,074.22)
$100 +
6
= 8.45%.
$(1,000 + 1,074.22) / 2
13. (Q. 18 in B) What happens to the price of a three-year bond with an 8
percent coupon rate, semi-annual coupons, when interest rates change from 7
percent to 6 percent? The bond has a par value of $1,000.
A) A price decrease of $27.03
B) A price decrease of $27.53
C) A price increase of $27.03
D) A price increase of $27.53
Answer D

At the 7% interest rate, the bond price is:

6


1
1
$1,000
]+
−
= $1,026.64.
6
0.035 0.035 × (1.035 ) (1.035 6 )
At the 6% interest rate, the bond price becomes:
1
1
$1,000
$40 × [
−
]+
= $1,054.17.
6
0.03 0.03 × (1.03 ) (1.036 )
So the bond price increases by $1,054.17 - $1,026.64 = $27.53.
$40 × [

14. (Q. 19 in B) What is the rate of return for an investor who pays $1,054.47 for
a three-year bond with a 7 percent coupon and sells the bond one year later for
$1,037.19? Assume the investor can reinvest the coupons at a 9% APR with
semi-annual compounding. The bond pays coupons semi-annually and has a

face value of $1,000.
A) 5.00 percent
B) 5.15 percent
C) 5.30 percent
D) 8.43 percent
Answer B
The rate of return is calculated as:
$35 × (1.045) + $35 + $(1,037.19 − 1,054.47)
= 5.15%.
$1,054.47
15. (Q. 20 in B) Deferred coupon bonds are bonds whose coupon payments are
deferred for a specified number of years. That is, there are no coupon payments
during the deferred period. Consider a 15-year deferred coupon bond with
$1,000 face value. The deferred period is the first 5 years in the life of the bond.
After the deferred period, the issuer will pay X% of the par as annual coupons
(i.e. coupons are paid once per year) until maturity to bond investors, with the
first payment occurring 1 year after the deferred period. The bond is yielding 7%
annually and selling for $926.21. What is the value of X?
A) 10.54%
B) 11.26%
C) 12.05%
D) 12.55%
Answer B
The coupons from this bond are a 10-year annuity of X% of the par ($1,000).
This annuity is delayed for 5 years, i.e. the first coupon payment is to be received
at the end of Year 6.
So the price of the bond is given by:

7

1
1
coupon × [ −
]
r r (1 + r ) t
face value
Pr ice =
+
n
(1 + r )
(1 + r ) m
1
1
X% × $1,000 × [
−
]
0.07 0.07(1.0710 )
$1,000
=
+
5
(1.07 )
(1.0715 )
X% × $1,000 × 7.0236
=
+ $362.4460 = X% × $1,000 × 5.0076 + $362.4460 = $926.21
1.4026
⇒ X = 11.2582%.
16. (Q. 11 in B) What is the expected constant growth rate of dividends for a

stock currently priced at $50, that just paid a dividend of $4, and has a required
rate of return of 18 percent?
A) 3.41 percent
B) 5.50 percent
C) 9.26 percent
D) 12.50 percent
Answer C
$50 = $4(1 + g)/(0.18 – g), so r = 9.26%.
17. (Q. 12 in B) What proportion of earnings is being retained by the firm if the
sustainable growth rate is 8 percent and the firm's ROE is 20 percent?
A) 8%
B) 12%
C) 20%
D) 40%
Answer D
8% = 20% × plowback
40% = plowback
18. (Q. 13 in B) What is the most likely value of the PVGO for a stock with a
current price of $50, expected earnings of $6 per share, and a required rate of
return of 20 percent?
A) $10
B) $20
C) $25
D) $30

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Answer B
With 100 percent payout ratio, the stock would be valued at $30 ($6/0.2 = $30).

Thus, the $20 of additional price must represent the PVGO.
19. (Q. 14 in B) What would be the price of a stock today when dividends are
expected to grow at a 25 percent rate for three years, then grow at a constant
rate of 5 percent forever, if the stock's required rate of return is 13 percent and
next year's dividend will be $4?
A) $61.60
B) $62.08
C) $68.64
D) $79.44
Answer C
$4.00
(1.13)

+

$4.00(1.25)
(1.13)2

+

$4.00(1.25)2
(1.13)3

$4.00(1.25)2 (1.05)
+
.13 −.05
(1.13)3

Po =
$5.00

1.2769

+

$6.25
1.4429

+

6.56
.08

1.4429
= $3.54 +
= $(3.54 + 3.92 + 4.33 + 56.85)
= $68.64.

9


20. (Q. 15 in B) How much should you pay now for a share of stock that offers a
constant growth rate of 10 percent, requires a 16 percent rate of return, and is
expected to sell for $50 one year from now?
A) $42.00
B) $45.00
C) $45.45
D) $47.00
Answer C
The easiest way to solve this problem is to realize:
Expected rate of return = expected dividend yield + expected capital

appreciation
Also:
Expected rate of return = expected dividend yield + constant growth rate
Then:
Expected capital appreciation = constant growth rate (10%)
So: P1 = 110% of Po
$50.00 = 1.1Po, $45.45 = Po.
Conceptual questions (2 points each)
21. (Q. 21 in B) Which of the following is least likely to represent an agency
problem?
A) Lavish spending on expense accounts.
B) Plush remodeling of the executive suite.
C) Excessive investment in "safe" projects.
D) Executive incentive compensation plans.
Answer D
22. (Q. 27 in B) Under which of the following conditions will a future value
calculated with simple interest exceed a future value calculated with compound
interest at the same rate?
A) The interest rate is very high.
B) The investment period is very long.
C) The compounding is annually.
D) This is not possible with positive interest rates.
Answer D

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23. (Q. 28 in B) Which of the following statements best describes the real interest
rate?
A) Real interest rates exceed inflation rates.

B) Real interest rates can decline only to zero.
C) Real interest rates can be negative, zero, or positive.
D) Real interest rates traditionally exceed nominal rates.
Answer C
24. (Q. 29 in B) As long as the interest rate is greater than zero, the present
value of a single sum will always:
A) be less than the future value.
B) decrease as the period of time decreases.
C) equal the future value if the time period is one year.
D) increase as the number of periods increases.
Answer A
25. (Q. 30 in B) Which of the following statements is wrong about the time value
of money (TVM)?
A) Converting an annuity to an annuity due increases the present value.
B) An effective annual rate is always higher than an annual percentage rate,
other things being equal.
C) The more frequent the compounding, the higher the future value, other things
being equal.
D) For a given amount, the higher the discount rate, the lower the present value.
Answer B
26. (Q. 22 in B) Cash flows occurring in different periods should not be compared
unless:
A) interest rates are expected to be stable.
B) the cash flows occur no more than one year from each other.
C) higher rates of interest can be earned on the cash flows.
D) the cash flows have been discounted to a common date.
Answer D
27. (Q. 23 in B) Which of the following will increase the present value of an
annuity, other things being equal?
A) Increasing the interest rate.

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B) Decreasing the interest rate.
C) Decreasing the number of payments.
D) Decreasing the amount of the payments.
Answer B
28. (Q 24 in B) Which of the following statements about a bond’s coupon rate,
current yield, and yield to maturity (YTM) is correct?
A) When the bond sells at a premium, its coupon rate is higher than
yield, and its coupon rate is also higher than the YTM.
B) When the bond sells at a premium, its coupon rate is lower than
yield, and its coupon rate is higher than the YTM.
C) When the bond sells at a discount, its coupon rate is higher than
yield, and its coupon rate is lower than the YTM.
D) When the bond sells at a discount, its coupon rate is lower than
yield, and its coupon rate is higher than the YTM.

the current
the current
the current
the current

Answer A
29. (Q. 25 in B) Reinvesting earnings into a firm will not increase the stock price
unless:
A) the new paradigm of stock pricing is maintained.
B) true depreciation is less than reported depreciation.
C) the firm's dividends are growing also.

D) the ROE of new investments exceeds the firm's required rate of return.
Answer D
30. (Q. 26 in B) The purpose of a sinking fund is to:
A) reduce the par value of stock over time.
B) take advantage of the tax break on preferred stock.
C) periodically retire debt prior to final maturity.
D) allow risky corporations to avoid bankruptcy.
Answer C

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