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```Quiz 2 solution sketches
1:00 Lecture, Version A
Note for multiple-choice questions:
Growing Annuity

Aubrey will receive \$50 today. She will
receive 6% more each subsequent year.
The last payment she will receive will be 30
years from today. What is the total present
value of all payments if the effective annual
discount rate is 18%?



PV = 50 + 53/(.18-.06) * [1 – (1.06/1.18)30]
PV = 50 + 53 * 7.99950
PV = \$473.97
Interest-Only Loan

Carrie borrows \$5,000 today from the
Isla Vista Monster Bank. She makes
monthly payments of \$100 for 48
months, starting one month from today.
She makes one additional payment 50
months from today to completely pay
off the loan. How much will the
payment be if the stated annual interest
rate is 24%, compounded monthly?
Interest-Only Loan

Interest-only loan for first 48 months



(1st month’s interest = 5000 * .02 = \$100)
Balance in 48 months = \$5000
Payment in 50 months:

5000 * (1.02)2 = \$5,202
Time to Double

The stated annual interest rate is 0.9%,
compounded continuously. How many
years will it take to double an initial
deposit made today? Assume that any
interest gets re-invested.



e(.009T) = 2
.009 * T = ln 2
T = (ln 2)/.009 = 77.016 years
Profitability Index

Seamus invests \$8,000 today. He is set
to receive \$2,000 per year, forever,
starting six years from now. What is the
profitability index for this investment if
the effective annual interest rate is
20%?
Profitability Index




PVbenefits = 2000/.2 * 1/(1.2)5
PVbenefits = \$4,018.78
P.I. = PVbenefits / Today’s cost
P.I. = 4018.78 / 8000 = 0.50235
Internal Rate of Return

Maxton invests \$5,000 today and will
receive \$500 per year forever, starting
later today. What is the internal rate of
return for this investment?



-5000 + 500 + 500/IRR = 0
500/IRR = 4500
IRR = 11.11%
Payback Period Method

Use the undiscounted payback period
method, with the cutoff date 8 years, 4
months from now. (In other words, the
payback period is 8 years, 4 months.)
The effective annual discount rate is
14%. Which of the following offers
should be picked if someone uses this
method?
Payback Period Method





A: \$1,000 per year forever, starting 4
years from now
B: \$510 per year forever, starting today
C: \$4,800 every 8 years forever,
starting 8 years from now
D: \$10,000 every 10 years forever,
starting 10 years from now
E: A one-time payment of \$4,500 today
Payback Period Method

Undiscounted future value at 8y 4m:






A: \$1,000 * 5 = \$5,000
B: \$510 * 9 = \$4,590
C: \$4,800 (one payment at year 8)
D: \$0 (first payment not until year 10)
E: \$4,500
Using the undiscounted payback period
method, one should choose option A
PV of Perpetuity



Bayleigh will receive \$25,000 per year,
forever, starting 18 months from now.
What is the present value of this
perpetuity if her effective annual
discount rate is 12.36%?
PV = 25,000/.1236 * 1/(1.1236)1/2
PV = \$190,816
PV of a Single Payment

Orlando will receive \$30,000 from his
grandfather in 10 years. What is the
present value of this payment if his
stated annual discount rate is 12%,
compounded every 30 minutes?
PV of a Single Payment


2 ways to solve
Approximate with continuous
compounding:


PV = 30,000/e(.12*10) = \$9,035.83
Compound 48x/day, or 17,520x/year


EAR = (1 + .12/17520)17520 – 1 = 12.7496%
PV = 30,000/(1.127496)10 = \$9,035.86
Credit Card Balance Transfers

Paloma has just charged \$60,000 of
construction equipment on her Vampire
Express credit card, which charges 15%
stated annual interest, compounded
monthly. After getting home, she finds
she has just been approved for a new
Mummy Express credit card, which
charges 12% stated annual interest,
compounded monthly.
Credit Card Balance Transfers

The Mummy Express charges a 2% fee
for any balances that are transferred. If
Paloma makes monthly payments of
\$2,000 to pay off the equipment,
should she pay off her Vampire Express
card, or transfer her balance to the
Mummy Express card to pay off the
higher balance? Assume the first \$2,000
will be one month from today.
Credit Card Balance Transfers

Vampire: monthly rate = .15/12 = .0125




60,000 = 2,000/.0125 * [1 – 1/(1.0125)T]
60,000 = 160,000 – 160,000/(1.0125)T
160,000/100,000 = (1.0125)T
T = (ln 1.6)/(ln 1.0125) = 37.8348 months
Credit Card Balance Transfers

Mummy: monthly rate = .12/12 = .01





60,000 * 1.02 = 2,000/.01 * [1 – 1/(1.01)T]
61,200 = 200,000 – 200,000/(1.01)T
200,000/138,800 = (1.01)T
T = (ln 1.4409)/(ln 1.01) = 36.7107 months
Choose the Mummy Express card
because the number of months to pay
off is lower.
```