LG5 5-27 Present Value of a Perpetuity A perpetuity pays $100 per year and interest rates are 7.5
percent. How much would its value change if interest rates increased to 9 percent? Did the
value increase or decrease?
Use equation 5-5:
$100
of a perpetuity $1,333.33
0.075
PV = =
$100
of a perpetuity $1,111.11
0.09
PV = =
The difference between these perpetuities is $222.22. The value of the perpetuity decreased with
an increase in the interest rate.
LG5 5-28 Present Value of a Perpetuity A perpetuity pays $50 per year and interest rates are 9
percent. How much would its value change if interest rates decreased to 7.5 percent? Did the
value increase or decrease?
Use equation 5-5:
$50
of a perpetuity $555.56
0.09
PV = =
$50
of a perpetuity $666.67
0.075
PV = =
The difference between these perpetuities is $111.11. The value of the perpetuity increased with
a decrease in the interest rate.
LG6 5-29 Future and Present Value of an Annuity Due If you start making $50 monthly
contributions today and continue them for five years, what’s their future value if the
compounding rate is 10 percent APR? What is the present value of this annuity?
Compute the future value using equation 5-2:
( ) ( )
60
60
1 0.10 /12 1
$50 1 0.10 /12 $50 77.437072 1.008333 $3,904.12
0.10/12
FVA + –
= ´ ´ + = ´ ´ =
Compute the present value using equation 5-4:
( )
60
60
1
11 0.10 /12
$50 1.008333 $50 47.065369 1.008333 $2,372.88
0.10/12
PVA
é ù
–
ê ú
+
ê ú
= ´ ´ = ´ ´ =
ê ú
ê ú
ë û
LG6 5-30 Future and Present Value of an Annuity Due If you start making $75 monthly
contributions today and continue them for four years, what is their future value if the
compounding rate is 12 percent APR? What is the present value of this annuity?
First calculate the future values and present values, using equations 5-2 and 5-4, respectively.
Using these results, the annuity due values can be computed using equations 5-6 and 5-7,
respectively.
( )
48
48
1 0.12 /12 1
$75 $75 61.2226 $4,591.695 $4,591.70 (1 0.12 /12) $4,647.61
0.12/12
FVA + –
= ´ = ´ = = ´ + =
( )
48
48
1
11 0.12 /12
$75 $75 37.973959 $2,848.05 (1 0.12 /12) $2,876.53
0.12/12
PVA
é ù
–
ê ú
+
ê ú
= ´ = ´ = ´ + =
ê ú
ê ú
ë û
Or N=4 x 12, I=12/12, PV=0, PMT=−75, CPT FV == 4,591.70 use DUE or BGN setting
and N=4 x 12, I=12/12, PMT=−75, FV=0, CPT PV == 2,848.05 use DUE or BGN setting
LG7 5-31 Compound Frequency Payday loans are very short-term loans that charge very high
interest rates. You can borrow $225 today and repay $300 in two weeks. What is the
compounded annual rate implied by this 33.33 percent rate charged for only two weeks?
33.33 percent for two weeks needs to be compounded 26 times to form a year:
LG7 5-32 Compound Frequency Payday loans are very short-term loans that charge very high
interest rates. You can borrow $500 today and repay $590 in two weeks. What is the
compounded annual rate implied by this 18 percent rate charged for only two weeks?
LG8 5-33 Annuity Interest Rate What is the interest rate of a 6-year, annual $5,000 annuity with
present value of $20,000?
Use equation 5-4 and solve for i:
$20 ,000=$5,000×
1−1
(1+i)6
i⇒i=12. 98
or: N=6, PV=–20000, PMT=5,000, FV=0, CPT I = 12.98%
LG8 5-34 Annuity Interest Rate What’s the interest rate of a 7-year, annual $4,000 annuity with
present value of $20,000?
Use equation 5-2 and solve for i:
$20 ,000=$4, 000×
1−1
(1+i)7
i⇒i=9. 20
or: N=7, PV=-20,000, PMT=4,000, FV=0, CPT I = 9.20%
LG8 5-35 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a
$1,000 per month contribution to grow to $75,000 in six years?
LG8 5-36 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a
$600 per month contribution to grow to $45,000 in six years?
Use equation 5-2 and solve for i:
$45,000=$600×
(
1+i/12
)
72−1
i/12 ⇒i=1. 37 .
or: N=72, PV=0, PMT=-600, FV=45,000, CPT I = 0.1143%
Now convert the monthly interest rate to an annual rate by multiplying by 12 which yields 1.37
percent.
LG8 5-37 Add-on Interest Payments To borrow $500, you are offered an add-on interest loan at 8
percent. Two loan payments are to be made, one at six months and the other at the end of the
year. Compute the two equal payments.
LG8 5-38 Add-on Interest Payments To borrow $800, you are offered an add-on interest loan at 7
percent. Three loan payments are to be made, one at four months, another at eight months, and
the last one at the end of the year. Compute the three equal payments.
LG9 5-39 Loan Payments You wish to buy a $25,000 car. The dealer offers you a 4-year loan with a
9 percent APR. What are the monthly payments? How would the payment differ if you paid
interest only? What would the consequences of such a decision be?
Use equation 5-9:
( )
48
48
0.09 /12
$25,000 $25,000 0.0248850 $622.13
1
11 0.09 /12
PMT
é ù
ê ú
ê ú
= ´ = ´ =
ê ú
–
ê ú
+
ë û
or: N=4 x 12, I=9/12, PV=25,000, FV=0, CPT PMT = −622.13
If you only paid interest over the length of the loan and your principal balance was repaid at the
end of the 48 months, your payment would be $187.50 per month (= $25,000 × 0.09 ÷ 12) for
interest only and you would owe $25,000 at the end of the 48 months, too.
LG9 5-40 Loan Payments You wish to buy a $10,000 dining room set. The furniture store offers you
a 3-year loan with an 11 percent APR. What are the monthly payments? How would the
payment differ if you paid interest only? What would the consequences of such a decision be?
Use equation 5-9:
( )
36
36
0.11/12
$10,000 $10,000 0.03279 $327.39
1
11 0.11/12
PMT
é ù
ê ú
ê ú
= ´ = ´ =
ê ú
–
ê ú
+
ë û
or: N=3 x 12, I=11/12, PV=10,000, FV=0, CPT PMT = −327.39
LG10 5-41 Number of Annuity Payments Joey realizes that he has charged too much on his credit
card and has racked up $5,000 in debt. If he can pay $150 each month and the card charges 17
percent APR (compounded monthly), how long will it take him to pay off the debt?
Rewrite equation 5-9 in terms of N:
N=
ln
(
$150
$
(
150−$5,000×0 . 17/12
)
)
ln
(
1+0 . 17/12
)
=45 .43 months
or: PV=5000, PMT=–150, FV= 0, I=1.417; CPT N = 45.43 months
LG10 5-42 Number of Annuity Payments Phoebe realizes that she has charged too much on her credit
card and has racked up $6,000 in debt. If she can pay $200 each month and the card charges 18
percent APR (compounded monthly), how long will it take her to pay off the debt?
Rewrite equation 5-9 in terms of N:
N=
ln
(
$200
(
$200−$6, 000×0 . 18/12
)
)
ln
(
1+0 . 18/12
)
=40. 15 months
or: PV=6000, PMT=–200, FV=0, I=1.50; CPT N = 40.15 months
advanced problems
LG1 5-43 Future Value Given an 8 percent interest rate, compute the year 7 future value if deposits
of $1,000 and $2,000 are made in years 1 and 3, respectively, and a withdrawal of $700 is made
in year 4.
LG1 5-44 Future Value Given a 9 percent interest rate, compute the year 6 future value if deposits of
$1,500 and $2,500 are made in years 2 and 3, respectively, and a withdrawal of $600 is made in
year 5.
Use equation 5-1:
LG7 – LG8 5-45 EAR of Add-on Interest Loan To borrow $2,000, you are offered an add-on
interest loan at 10 percent with 12 monthly payments. First, compute the 12 equal payments and
then compute the EAR of the loan:
LG9 can receive $500 cash back on the purchase, or a 3 percent APR, 4-year loan. The price of the car
is $15,000 and you could obtain a 4-year loan from your credit union, at 6 percent APR. Which
deal is cheaper?
LG4 5-48 Low Financing or Cash Back? A car company is offering a choice of deals. You
LG9 can receive $1,000 cash back on the purchase, or a 2 percent APR, 5-year loan. The price of the
car is $20,000 and you could obtain a 5-year loan from your credit union, at 7 percent APR.
Which deal is cheaper?
Compare two cases. The first case is to elect the 2 percent APR and fully finance $20,000 over
60 months. Using equation 5-9, the payment under this scenario would be:
( )
60
60
0.02 /12
$20,000 $350.56
1
11 0.02/12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
–
ê ú
+
ë û
or: N=5 x 12, I=2/12, PV=20000, FV=0, CPT PMT = −350.56
The second case is to take the $1,000 cash back, apply it to the purchase and finance only
$19,000 through your credit union at 7 percent. The payment under this scenario would be:
( )
60
60
0.07 /12
$19,000 $376.22
1
11 0.07/12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
–
ê ú
+
ë û
or: N=5 x 12, I=7/12, PV=19000, FV=0, CPT PMT = −376.22
The lower payment represents the more advantageous scenario that you should choose, electing
the 2 percent financing through the car dealer.
LG9 5-49 Amortization Schedule Create the amortization schedule for a loan of $15,000, paid
monthly over three years using a 9 percent APR.
PMT 36=$15 ,000×
[
0 . 09/12
1−1
(1+0 .09 /12 )36
]
=$477. 00
Mont
h
Beginning
Balance
Total
Payment
Interes
t Paid
Principal
Paid
Ending
Balance
1 $15,000.00 $477.00
$112.5
0 $364.50 $14,635.50
2 14,635.50 477.00 109.77 367.23 14,268.27
3 14,268.27 477.00 107.01 369.98 13,898.29
LG9 5-50 Amortization Schedule Create the amortization schedule for a loan of $5,000, paid
monthly over two years using an 8 percent APR.
PMT 24=$5, 000×
[
0 .08 /12
1−1
(1+0 . 08/12)24
]
=$226 . 14
Month
Beginning
Balance
Total
Payment
Interest
Paid
Principal
Paid
Ending
Balance
1 $5,000.00 $226.14 $33.33 $192.80 $4,807.20
2 4,807.20 $226.14 $32.05 194.09 4,613.11
LG4 5-51 Investing for Retirement Monica has decided that she wants to build enough
LG9 retirement wealth that, if invested at 8 percent per year, will provide her with $3,500 of monthly
income for 20 years. To date, she has saved nothing, but she still has 30 years until she retires.
How much money does she need to contribute per month to reach her goal?
LG4 5-52 Investing for Retirement Ross has decided that he wants to build enough retirement
LG9 wealth that, if invested at 7 percent per year, will provide him with $3,000 of monthly income for
30 years. To date, he has saved nothing, but he still has 20 years until he retires. How much
money does he need to contribute per month to reach his goal?
LG9 5-53 Loan Balance Rachel purchased a $15,000 car three years ago using an 8 percent, 4-year
loan. She has decided that she would sell the car now, if she could get a price that would pay off
the balance of her loan. What is the minimum price Rachel would need to receive for her car?