Econ 134A Test 1
Fall 2012
Solution sketches
Solve each of the following

(a) (5 points) Yongli will receive $750 later
today. He will receive $825, or 10% more,
one year from today. For each of the three
years after that, he will receive $75 more
than the year before. After these five
payments have been made, he will receive
nothing more. Find the present value of
these five payments if the effective annual
discount rate is 4%.
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PV = 750 + 825/1.04 + 900/1.042 +
975/1.043 + 1050/1.044 = $4,139.69
Solve each of the following

(b) (6 points) Yusuf is quoted a price for a
bond of $500. This bond has a face value of
$475 and pays a 10% coupon once per year.
Two coupons will be paid, one later today and
the other one year from today. If the bond
matures one year from today, what is the yield
on this bond (expressed as an effective annual
rate)?
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475 / (1+r) + 47.5 + 47.5 / (1+r) = 500
522.5 / (1+r) = 452.5
1 + r = 1.154696  r = .154696 or 15.4696%
Pepsi and $1 billion
Note that all calculations in this
problem are in millions of dollars

Years ago, Pepsi offered a chance to win $1 billion to many
contestants. Assume that there are 40 yearly payments as
follows:

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For this problem, assume that the first payment is made today
and there is an effective annual discount rate of 7%. Make the
calculations below assuming that the prize is won.
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
$5 million for each of the first 20 payments
$10 million for each of the next 19 payments
$710 million for the final payment
HINTS: Pay careful attention to when each payment is made. A
single mistake could lead to deductions on multiple parts of this
problem. Also note that each payment is made one year apart.
(a) (4 points) What is the present value of the first 20
payments?


Note that we use the annuity formula for 19 years and add in one
more payment today
PV = 5 + (5/0.07)[1 – (1/1.07)19] = 5 + 51.6780 = 56.6780
Pepsi and $1 billion

(b) (6 points) What is the present value
of the next 19 payments?

The 1st $10 million payment will be made
20 years from today

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So we need to discount the annuity formula by
19 years
PV = [1/1.0719] * (10/0.07)[1 – 1/1.0719]
PV = 28.5788
Pepsi and $1 billion

(c) (2 points) What is the present value
of the final payment?
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This payment is made 39 years from now
PV = 710 / 1.0739
PV = 50.7331
Pepsi and $1 billion

(d) (4 points) Pepsi offered a single payment of
$250 million today as an alternative to the
payments mentioned at the beginning of this
problem. Given the information from this problem,
should someone who wins this prize accept the
single payment? Justify your answer with math
and/or a written justification of 40 words or less.


PV of all payments is 56.6780 + 28.5788 + 50.7331, or
135.990
This PV is less than the single payment of 250 made
today

Choose the single payment, because it has a higher PV
C&L Shopping Trolleys

(9 points) C&L Shopping Trolleys, Inc. has just paid out its
annual dividend of $5.70 earlier today. The annual dividend will
go up by 3% each of the next 5 years. This will be followed by
5% growth in the annual dividend every year after that forever.
What will the price of this stock be 4 years from today if the
effective annual discount rate is 8%? (Note: Provide the price
AFTER the dividend has been paid.)
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We need to find the future value 4 years from now
Only dividends paid 5 years or more in the future are included in
this future value
Dividend in year 5: $5.70 * 1.035 = 6.60786
We can use the growing perpetuity formula for payments made 5
or more years into the future
The future value (year 4) of the stock is then 6.60786/(0.08-0.05)

$220.26
Slacking Sean
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
Slacking Sean likes to procrastinate on paying back loans. He is
currently negotiating with Patient Paula on a loan. Today, Paula
is giving Sean $600 for a loan. Assume the effective annual
interest rate is 15%.
(a) (5 points) If Sean pays back a constant amount of the
principal each year for five years to pay off the loan, and
payments are made yearly, how much will the payment 4 years
from today be? Assume that the first payment will be one year
from today.
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Principal must be reduced by $120 per year to pay off the loan
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$240 three years from today
Four years from today, $120 in principal must be paid, along with
interest on $240
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$120 + $240*0.15 = $156
Slacking Sean
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(b) (6 points) If Sean has to make five equal
payments to pay off the loan, how much would
each one have to be if one payment will be
made two years from today, one payment will
be made three years from today, one payment
will be made four years from today, and two
payments will be made five years from today?
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Let X be the amount of each payment
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X/1.152 + X/1.153 + X/1.154 + 2X/1.155 = 600
2.9798X = 600  X = $201.36
A bond with maturity date 4½
years from today…

(7 points) A bond with maturity date 4½ years
from today has a face value of $1500. There are
five coupon payments of $100 each to be made.
These payments will be made annually starting six
months from today. If the effective annual
discount rate is 17%, what is the present value of
future payments that will be paid by this bond?
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Note that we discount the first payment by 6 months
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Each subsequent payment is discounted by an additional
year
100/1.170.5 + 100/1.171.5 + 100/1.172.5 + 100/1.173.5
+ 100/1.174.5 + 1500/1.174.5
$1,086.10