Test 1 solution sketches
Note for multiple-choice
questions: Choose the closest
answer
Loan calculations

Billy’s Pianos receives a loan of
$180,000 today. The stated annual
interest rate is 8.4%, compounded
monthly. Payments are monthly,
starting one month from today. The
loan is amortized over 30 years.
Loan calculations

If Billy pays an equal amount of
principal each month, how much will
the first payment be?




Monthly rate = .0084 / 12 = 0.7%
Amount of principal paid each month =
$180,000 / 360 = $500
Amount of interest accrued in first month
= $180,000 * .007 = $1,260
First payment = 500 + 1,260 = $1,760
Loan calculations

If Billy makes equal month payments
each month, how much will the first
payment be?



180,000 = C / .007 * [1 – 1 / (1.007)360]
180,000 = 131.262 * C
C = $1,371.31
Loan calculations

If Billy pays an equal amount of
principal each month, how much will
the last payment be?



Principal owed in 359 months =
180,000 / 360 = 500
Interest owed = 500 * .007 = 3.50
Last payment = 500 + 3.50 = $503.50
Loan calculations

If Billy makes equal month payments
each month, how much will the last
payment be?




Note: equal payments means first = last
(so same answer as #2)
180,000 = C / .007 * [1 – 1 / (1.007)360]
180,000 = 131.262 * C
C = $1,371.31
Profitability Index

Carly Rae pays $50,000 to open her
dating service. She receives $2,700 per
year in cash flow, starting in two years.
Annual discount rate is 5%. What is
the profitability index?



PV of benefits = 2700 / .05 * 1 / 1.05
= 51,429
PV of costs = 50,000
PI = 51,429 / 50,000 = 1.029
Effective Discount Rates

If the effective annual discount rate is
15%, then what is the effective
discount rate for 8 months?

(1.15)8/12 – 1 = 9.76534%
PV of Annuity

Wolfgang will receive royalty payments of
$500 every year, starting 5 years from
today and ending 25 years from today.
What is the present value of these
payments if the effective annual discount
rate is 15%?


Annuity formula for 21 payments, discounted
by 4 years due to 1st payment in year 5
500/.15 * [1 – 1 / 1.1521] * 1 / 1.154 =
$1,804.59
Real payments

If the inflation rate this year is 5% and
the nominal interest rate is 15%, then
what is the real interest rate?




(1 + real)(1 + inflation) = (1 + nominal)
(1 + real)(1.05) = 1.15
1 + real = 1.15 / 1.05 = 1.0952381
Real = 9.52381%
Discounted vs. undiscounted
payback periods


Reba’s Rabbits invests $50,000 today, and will
earn $10,000 each year starting one year from
today. The effective annual discount rate is
9%.
If Reba uses discounted cash flows, how many
years is the payback period for this investment?




50000 = 10000/.09 (1 – 1/1.09T)
61111 = (10000/.09)/(1.09T)
1.09T = (10000/.09)/61111 = 1.81818
T = ln(1.81818)/ln(1.09) = 6.93726 ≈ 7
Discounted vs. undiscounted
payback periods

If Reba uses undiscounted cash flows,
how many years is the payback period
for this investment?

50000 / 10000 = 5
Pyotr’s Beauty Products

Pyotr’s Beauty Products is considering
buying a new device. This machine
would cost $8,000 today, and require
maintenance costs of $600 every three
years, starting in 2 years and ending in
11 years. The machine lasts 12 years,
and the effective annual discount rate is
14%.
Part (a)

What is the present value of all costs of
the machine over its life?


Purchase cost today and maintenance
costs in years 2, 5, 8, and 11
8000 + 600/(1.142) + 600/(1.145) +
600/(1.148) + 600/(1.1411) = $9,125.61
Part (b)

Pyotr pays $X per year for five years,
starting today. These payments will
have the same present value as the
answer you got from part (a). Find X.



X + X/1.14 + X/(1.142) + X/(1.143) +
X/(1.144) = 9125.61
3.91371 * X = 9125.61
X = $2,331.70
Yield to Maturity

A bond has a face value of $750. It
pays a coupon of 10% today, one year
from today, and two years from today.
Two years from today, the bond
matures. If the current selling price of
the bond is $800, what is the yield to
maturity (expressed as an effective
annual discount rate)?
Yield to Maturity




800 = 75 + 75/(1+r) + 825/(1+r)2
725(1+r)2 – 75(1+r) – 825 = 0
725r2 + 1375r – 175 = 0
29r2 + 55r – 7 = 0
Ignore negative root. r = 0.119716 so
r = 11.97%. Or…
Yield to Maturity




800 = 75 + 75/(1+r) + 825/(1+r)2
725(1+r)2 – 75(1+r) – 825 = 0
Let x = 1+r
29x2 – 3x – 33 = 0
Ignore negative root. x = 1.1197 so
r = 11.97%
Balloon Payment

Michael is taking out a loan of $1,000,000
today and he will pay $22,000 per month for
the next 10 years (120 payments, starting
one month from today). The stated annual
interest rate is 24%, compounded monthly.
13 years from today, Michael will make one
additional payment to pay off the loan. How
much will this payment be?
Balloon Payment

PV of monthly payments:


PV of payment made in 13 years:


22000/.02 * [1 – 1/(1.02120)] = 997,818.55
1,000,000 – 997,818.55 = 2,181.45
FV of payment made in 13 years:

2,181.45 (1.02)12*13 = $47,904.10