Quiz 2 solution sketches
11:00 Lecture, Version A
Note for multiple-choice questions:
Choose the closest answer
Internal Rate of Return

Norah invests $1,000 today and will
receive $50 per year forever, starting in
1 hour. What is the internal rate of
return on this investment?



-1000 + 50 + 50/IRR = 0
50/IRR = 950
IRR = 5.26%
Net Present Value

Jacqueline has been asked to determine
the net present value of a business. The
business has costs of $50,000 per year
forever, starting six months from now.
The business receives $40,000 in profits
one year from today, and increases by
8% each year forever. What is the NPV
of the business if the effective annual
discount rate is 12%?
Net Present Value



PVbenefits = 40000/(.12-.08) = $1,000,000
PVcosts = 50000/.12*(1.12)1/2 = $442,959
NPV = $559,041
Semi-annual Perpetuity

Valerie will receive $20,000 per year
forever. However, the first payment will
be made two years from today, and
payments will be made every 6 months.
(In other words, she will receive
$10,000 every 6 months, starting 2
years from today.) If her effective
annual discount rate is 10%, what is
the PV of this stream of payments?
Semi-annual Perpetuity



6-month rate: (1.1)1/2 – 1 = 4.880%
PV = 10,000/.0488 * 1/(1.1)3/2
PV = $177,588
Less-Frequent-than-Annual
Perpetuities

Harrison will receive $50,000 every four
years forever, starting six years from
now. The effective annual interest rate
is 6%. Find the total present value of all
payments.



4-year rate: (1.06)4 – 1 = 26.2477%
FV2 = 50,000/.262477 = $190,493
PV = 190,493/(1.06)2 = $169,538
Equal Loan Payments

Jill Birdwell will borrow $50,000 on
July 1, 2014. She will make 10 equal
yearly payment, on January 1 of years
2015-2024, to completely pay back the
loan. How much will each payment be if
the effect annual interest rate is 8%?
Equal Loan Payments

If payments were made on July 1 of
each year:




50,000 = C/.08 * [1 – 1/(1.08)10]
50,000 = 6.71008 * C
C = $7,451.47
Payments are actually on January 1:


C’ = C/(1.08)1/2
C’= $7,170.18
Equivalent Annual Cost

The Lockyear Machine can be
purchased today for $500. It will incur
maintenance costs of $200 two years
from today and $500 five years from
today. The machine lasts for 10 years.
If the effective annual discount rate is
12%, what is the equivalent annual
costs of the machine?
Equivalent Annual Cost


PVcosts = 500 + 200/(1.12)2 +
500/(1.12)5 = $943.15
EAC:



943.15 = C/.12 * [1 – 1/(1.12)10]
943.15 = 5.6502 * C
C = $166.92
Profitability Index

If you invest $5,000 today in a new
invention, you will receive a positive
cash flow of $800 in 8 months, followed
by a positive cash flow of $800 every 12
months thereafter. What is the
profitability index of this investment if
the effective annual discount rate is
9%?
Profitability Index




PVbenefits = 800/.09 * (1.09)1/3
PVbenefits = $9147.93
P.I. = PVbenefits / Today’s cost
P.I. = 9147.93 / 5000 = 1.82959
Growing Perpetuity

Petey will receive $500 today. He will
receive 5% more each subsequent
year. If his effective annual discount
rate is 15%, what is the present value
of this stream of payments?

PV = 500 + 525/(.15-.05) = $5,750
Long-Answer:
Internal Rates of Return

Rachel will be opening a gold mine this
year. All of the gold will be extracted in
one year, and she must close it in two
years to meet environmental laws. This
leads to the following cash flows: She
knows that she must spend $200 million
today, she will earn $460 million one
year from today, and she must spend
$262 million two years from today.
Long-Answer:
Internal Rates of Return

(a) Find all internal rates of return.



All numbers in $millions. Let X = 1+IRR.
0 = -200 + 460/X – 262/X2
0 = -200X2 + 460X – 262

X=

X=


−460± 4602 −4(−200)(−262)
2(−200)
−460±44.7214
−400
=
X = 1.0381965, 1.2618035
IRR = 3.81965%, 26.18035%
−460± 2000
−400
Long-Answer:
Internal Rates of Return

(b) For what annual discount rates will this
project have positive net present value? You
must show examples to justify your answer
and get full credit.




Pick a discount rate in each range:
r = 0%:
NPV = -200 + 460 – 262 = -2 < 0 NO
r = 10%:
NPV = -200 + 460/1.1 – 262/1.12 = 1.65 > 0
YES
r = 100%: NPV = -200 + 460/2 – 262/22 = -35.5
< 0 NO
Long-Answer:
Internal Rates of Return

(b) Pick a discount rate in each range:




r = 0%:
NPV = -200 + 460 – 262 = -2<0 NO
r = 10%:
NPV = -200 + 460/1.1 – 262/1.12 = 1.65>0
YES
r = 100%:
NPV = -200 + 460/2 – 262/22 = -35.5<0 NO
Positive NPV for r: 0.0381965<r<0.2618035