### Test 1 solution sketches

```Test 1 solution sketches
Note for multiple-choice
questions: Choose the closest
1) Interest rate conversions

If the effective annual discount rate is
12%, what is the stated annual
discount rate if compounded monthly?




(1 + r/12)12 = 1.12
1 + r/12 = 1.0094888
r/12 = .0094888
r = .1138656
2) PV of future payments

Mauricio will receive \$900 three months
from now and \$1500 nine months from
now. The effective annual discount rate
is 16%. Find the total of the present
value of the two payments.


PV = 900/(1.16)1/4 + 1500/(1.16)3/4
PV = \$2209.20
3) Profitability Index

Shane invests \$6,000 in a new
invention. He will receive a positive
cash flow of \$1,000 in 9 months,
followed by a positive cash flow of
\$1,000 every 12 months thereafter.
What is the profitability index of this
investment if the effective annual
discount rate is 15%?
3) Profitability Index


PV of future payments
= 1000/.15 * (1.15)1/4 = \$6903.72
P.I. = 6903.72/6000 = 1.1506
4 & 5) Future Values

For the next 2 questions, assume that
today is February 4, 2014. You invest
\$3,000 today. Find the future values on
the following dates, given the stated
annual interest rates and frequency of
compounding.
4) Future Value

May 4, 2015, 20% interest rate,
compounded every three months


Rate every 3 months = .20/4 = 5%
FV = 3000 * (1.05)5 = \$3828.84
5) Future Value

February 4, 2054, 3% interest rate,
compounded continuously

FV = 3000 * e.03*40 = \$9960.35
6) Equivalent Annual Cost

Cheyenne buys a machine that will
produce \$5,000 worth of hair dye each
year. The machine must be purchased
for \$20,000 today, and a maintenance
cost of \$6,000 must be paid 4 years
from today. If the machine lasts for 7
years, the equivalent annual cost of the
machine is _____ if the effective annual
discount rate is 8%.
6) Equivalent Annual Cost

PV of costs = 20,000 + 6000/(1.08)4


PV of costs = \$24,410
EAC calculation



24,410 = C/.08 * [1 – 1/(1.08)7]
24,410 = 5.2064 * C
C = \$4,688
7) Annuity factor

What is the annuity factor if Frank
receives \$5,000 per year each year for
20 years, starting one year from today?
Assume the effective annual interest
rate is 10%.


A.F. = 1/.1 * [1 – 1/(1.1)20]
A.F. = 8.5136
8) Real interest rate

If the nominal interest rate is 500% and
the inflation rate is 400%, the exact
real interest rate is ______.




(1 + real)(1 + inflation) = (1 + nominal)
(1 + real)(1 + 4) = 1 + 5
1 + real = 6/5 = 1.2
Real = 20%
9) FV of annuity

Leona puts \$1,000 per year into a Roth
IRA, starting today. The account pays
8% effective annual interest. How much
will Leona have in this account
immediately after the deposit is made 3
years from today?


FV = 1000 * (1.08)3 + 1000 * (1.08)2 +
1000 * 1.08 + 1000
FV = \$4,506.11
10) Formula assumptions

Donte will receive \$500 today, \$550
one year from now, \$600 two years
from now, \$650 three years from now,
etc. Donte will receive these payments
forever. If we knew Donte’s effective
annual discount rate, which of the
following formulas could you use to
calculate the present value of this
stream of payments?
10) Formula assumptions






A) annuity
B) growing annuity
C) perpetuity
D) growing perpetuity
E) none of the above
Because the growth rate is not
constant, we cannot use any formula
1) Loan amortization


Jaelyn borrows \$45,000 today from the Party
Polka National Bank. She is currently
negotiating how to pay back the loan.
(a) If she pays the loan back in 12 equal
monthly installments, starting seven months
from today, how much will each payment be
if the stated annual interest rate is 12%,
compounded monthly?
1) Loan amortization


(a) Note monthly rate = .12/12 = 1%
If paid in months 1-12:




45,000 = C/.01 * [1 – 1/(1.01)12]
45,000 = 11.255 * C
C = 3998.20
But since each payment is 6 months
later (months 7-18):

Payment = 3998.20 * (1.01)6 = \$4,244.17
1) Loan amortization

(b) Suppose instead that Jaelyn pays
back the loan with 3 years payments
such that the principal is reduced by the
same amount each year. The payments
would be made 1 year, 2 years, and 3
years, from today. If the effective
annual interest rate is 16%, how much
total interest will Jaelyn pay?
1) Loan amortization







(b) Principal per year= 45000/3= 15000
1st year: 15000 + 45000(.16)= \$22,200
2nd year: 15000 + 30000(.16)= \$19,800
3rd year: 15000 + 15000(.16)= \$17,400
Total payments = \$59,400
Total interest = 59,400 – 45,000
Total interest = \$14,400
2) Internal Rates of Return


Lucia invests in a company that has the
following guaranteed cash flows: She
must pay \$3,000 today, she will receive
\$6,910 one year from today, and she
must pay \$3,930 two years from today.
(a) Find all internal rates of return for
this investment.
2) Internal Rates of Return







(a) Let x = 1 + r
-3000 + 6910/x – 3930/x2 = 0
-3000x2 + 6910x – 3930 = 0
=
−6910± 69102 −4(−3000)(−3930)
2(−3000)
x = 1.1516675 ± .127813
x = 1.27948, 1.023854
IRR = 27.948%, 2.3854%
2) Internal Rates of Return

(b) What is the net present value of this
investment fi the effective annual
discount rate is 25%?


PV = -3000 + 6910/1.25 – 3930/(1.25)2
PV = \$12.80
2) Internal Rates of Return

explain why the following statement is
true or false: “All effective annual
discount rates of 20-30% for this
project lead to a positive net present
value for this investment.”
2) Internal Rates of Return


(c) False, above 27.948%, NPV is negative
since NPV is 0 at IRRs
Possible explanations:


At r=29%:
PV = -3000 + 6910/1.29 – 3930/(1.29)2
PV = -\$5.05
The NPV equation is a downward-opening
parabola because the coefficient on the squared
term (-3000) is negative. So the range between
IRRs must give positive NPV and outside of that
range must have negative NPV.
```