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LG9 5-54 Loan Balance Hank purchased a $20,000 car two years ago using a 9 percent, 5-year loan.
He has decided that he would sell the car now, if he could get a price that would pay off the
balance of his loan. What’s the minimum price Hank would need to receive for his car?
First calculate the monthly payment that he has been paying using equation 5-9:
( )
60
60
0.09 /12
$20,000 $415.17
1
11 0.09/12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
-
ê ú
+
ë û
or: N=5x12, I=9/12, PV=20000, FV=0, CPT PMT = −415.17
The loan balance is the principal amount outstanding. The duration of remaining payments is 36,
the interest rate is 9 percent annual and the monthly payment is $415.17 from the previous
calculation. Use these values to calculate the present value of the loan using equation 5-4:
( )
36
1
11 0.09 /12
$415.17 $13, 055.77
0.09/12
PVA
é ù
-
ê ú
+
ê ú
= ´ =
ê ú
ê ú
ë û
or: N=3x12, I=9/12, PMT = −415.17, FV=0, CPT PV = 13,055.77
This is the minimum price the car needs to be sold for and it represents his break even price.
LG9 5-55 Teaser Rate Mortgage A mortgage broker is offering a $183,900, 30-year mortgage with a
teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on
only a 4 percent APR interest rate. After the second year, the mortgage interest rate charged
increases to 7 percent APR. What are the monthly payments in the first two years? What are the
monthly payments after the second year?
Use equation 5-9 to calculate the payment using the teaser rate:
( )
360
360
0.04 /12
$183, 900 $877.97
1
11 0.04 / 12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
-
ê ú
+
ë û
or: N=30x12, I=4/12, PV=183900, FV=0, CPT PMT = −877.97
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
( )
336
336
1
11 0.04 /12
$877.97 $177, 291.63
0.04/12
PVA
é ù
-
ê ú
+
ê ú
= ´ =
ê ú
ê ú
ë û
or: N=28x12, I=4/12, PMT = −877.97, FV=0, CPT PV = 177,291.63
Now use this amount for the present value, the new interest rate of 7 percent over the remaining
336 payments in equation to calculate the new payment amount after expiration of the teaser rate,
using equation 5-9:
( )
336
336
0.07 /12
$177, 291.63 $1, 204.89
1
11 0.07 /12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
-
ê ú
+
ë û
or: N=28x12, I=7/12, PV=177291.63, FV=0, CPT PMT = −1204.89
LG9 5-56 Teaser Rate Mortgage A mortgage broker is offering a $279,000, 30-year mortgage with a
teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on
only a 4.5 percent APR interest rate. After the second year, the mortgage interest rate charged
increases to 7.5 percent APR. What are the monthly payments in the first two years? What are
the monthly payments after the second year?
Use equation 5-9 to calculate the payment using the teaser rate:
( )
360
360
0.045 /12
$279,000 $1, 413.65
1
11 0.045 /12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
-
ê ú
+
ë û
or: N=30x12, I=4.5/12, PV=279000, FV=0, CPT PMT = −1413.65
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
( )
336
336
1
11 0.045 /12
$1,413.65 $269,791.04
0.045/12
PVA
é ù
-
ê ú
+
ê ú
= ´ =
ê ú
ê ú
ë û
or: N=28x12, I=4.5/12, PMT = −1413.65, FV=0, CPT PV = 269,791.04
Now use this amount for the present value, the new interest rate of 7.5 percent over the
remaining 336 payments in equation to calculate the new payment amount after expiration of the
teaser rate, using equation 5-9:
( )
336
336
0.075 /12
$269,791.04 $1,923.25
1
11 0.075 / 12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
-
ê ú
+
ë û
or: N=28x12, I=7.5/12, PV=269791.04, FV=0, CPT PMT = −1923.25
LG2 LG95-57 Spreadsheet Problem Consider a person who begins contributing to a retirement
plan at age 25 and contributes for 40 years until retirement at age 65. For the first ten years, she
contributes $3,000 per year. She increases the contribution rate to $5,000 per year in years 11
End of
38 5,000.00 87,203.83
39 5,000.00 100,052.17
40 5,000.00 114,056.87
41 5,000.00 129,321.98
42 5,000.00 145,960.96
43 5,000.00 164,097.45
44 5,000.00 183,866.22
45 10,000.00 210,414.18
research it!
Retirement Income Calculators
The Internet provides some excellent retirement income calculators. You can find one by
Googling “retirement income calculator.” Many of the calculators allow you to determine your
predicted annual income from a retirement nest egg under different assumptions. For example,
you can spend only the investment income generated from the nest egg. Most retirees try not to
touch the principal. Or, you can spend both the income and the nest egg itself. These calculators
let you input the size of the retirement wealth and the investment return to be earned. They then
make time value computations to determine the annual income the nest egg will provide.
Go to a retirement income calculator like the one at MSN Money. Use the calculator to create a
retirement scenario. Use the TVM equations or a financial calculator to check the Internet
results.
http://money.msn.com/retirement/retirement-calculator.aspx
Summary
Given a certain amount of savings, how much can I spend annually during
retirement?
Information entered
Using a financial calculator, the following inputs are needed to determine the projected annual
payment:
integrated mini-case: Paying on your Stafford loan
Consider Gavin, a new freshman who has just received a Stafford student loan and started
college. He plans to obtain the maximum loan from Stafford at the beginning of each year.
Although Gavin does not have to make any payments while he is in school, the unsubsidized 6.8
percent interest owed (compounded monthly) accrues and is added to the balance of the loan.
UNSUBSIDIZED Stafford loan limits:
Freshman $6,000
Sophomore 6,000
Junior 7,000
Senior 7,000
After graduation, Gavin gets a 6-month grace period. This means that monthly payments are still
not required, but interest is still accruing. After the grace period, the standard repayment plan is
to amortize the debt using monthly payments for ten years.
a. Show a time line of when the loans will be taken.
b. What will be the loan balance when Gavin graduates after his fourth year of school?
c. What is the loan balance six months after graduation?
d. Using the standard repayment plan and a 6.8 percent APR interest rate, compute the monthly
payments Gavin owes after the grace period.
SOLUTION:
b. What will be the loan balance when Gavin graduates after his fourth year of school?
new employer’s plan. If the rolled-over money and the new contributions both earn an 8 percent
return, how much should you expect to have when you retire in 20 years?
4&5-3 Future Value and Number of Annuity Payments Your client has been given a trust
fund valued at $1 million. He cannot access the money until he turns 65 years old, which is in 25
years. At that time, he can withdrawal $25,000 per month. If the trust fund is invested at a 5.5
percent rate, how many months will it last your client once he starts to withdraw the money?
Using equation 5-1, $1 million will accumulate for 25 more years at 5.5 percent interest for a
future value:
( )
25
Age 65
$1,000,000 1 0.055 $3,813,392.35FVA = ´ + =
or N=25, I=5.5, PV=−1,000,000, PMT=0, CPT FV == 3,813,392.35
Now, rewrite equation 5-9 in terms of N:
N=
ln
(
$25 ,000
(
$25 ,000−$3,813 ,392. 35×0 .055 /12
)
)
ln
(
1+0 . 055/12
)
=262. 65 months
Or: PV = 3813392.35, PMT = −25000, FV = 0, I = 0.458333; CPT N = 262.65 months
4&5-4 Future Value and Number of Annuity Payments Your client has been given a trust fund
valued at $1.5 million. She cannot access the money until she turns 65 years old, which is in 15
years. At that time, she can withdraw $20,000 per month. If the trust fund is invested at a 5
percent rate, how many months will it last your client once she starts to withdraw the money?
Using equation 5-1, $1.5 million will accumulate for 15 more years at 5 percent interest for a
future value:
( )
15
Age 65
$1,500, 000 1 0.05 $3,118,392.27FVA = ´ + =
or N=15, I=5, PV=−1500000, PMT=0, CPT FV == 3,118,392.27
Now, rewrite equation 5-9 in terms of N:
N=
ln
(
$20 ,000
(
$20 ,000−$3,118 ,392 .27×0 . 05/12
)
)
ln
(
1+0 . 05/12
)
=252 .25 months
Or: PV=3118392.27, PMT = −20000, FV = 0, I = 0.416667; CPT N = 252.25 months
4&5-5 Present Value and Annuity Payments A local furniture store is advertising a deal in
which you buy a $3,000 dining room set and do not need to pay for two years (no interest cost is
incurred). How much money would you have to deposit now in a savings account earning 5
percent APR, compounded monthly, to pay the $3,000 bill in two years? Alternatively, how much
would you have to deposit in the savings account each month to be able to pay the bill?
4&5-6 Present Value and Annuity Payments A local furniture store is advertising a deal in
which you buy a $5,000 living room set with three years before you need to make any payments
(no interest cost is incurred). How much money would you have to deposit now in a savings
account earning 4 percent APR, compounded monthly, to pay the $5,000 bill in three years?
Alternatively, how much would you have to deposit in the savings account each month to be able
to pay the bill?
Use equation 5-3 and solve for the lump sum payment:
( )
36
$5,000 1 0.04 / 12 $4, 435.49PV = ¸ + =
or N=36, I=4/12, PMT=0, FV=−5,000, CPT PV == 4,435.49
Use equation 5-2 and solve for the annuity payment:
( )
36
1 0.04/12 1
$5,000 $130.95
0.04/12
PMT PMT
+ -
= ´ Þ =
or: N=3x12, I=4/12, PV=0, FV=5000, CPT PMT = −130.95
4&5-7 House Appreciation and Mortgage Payments Say that you purchase a house for
$200,000 by getting a mortgage for $180,000 and paying a $20,000 down payment. If you get a
30-year mortgage with a 7 percent interest rate, what are the monthly payments? What would
the loan balance be in ten years? If the house appreciates at 3 percent per year, what will be the
value of the house in ten years? How much of this value is your equity?
4&5-8 House Appreciation and Mortgage Payments Say that you purchase a house for
$150,000 by getting a mortgage for $135,000 and paying a $15,000 down payment. If you get a
15-year mortgage with a 7 percent interest rate, what are the monthly payments? What would
the loan balance be in five years? If the house appreciates at 4 percent per year, what will be the
value of the house in five years? How much of this value is your equity?
An appreciation of 4 percent per year will result in a forecast future value of the home using the
original purchase price in equation 5-1:
( )
5
5 years
$150,000 1 0.04 $182, 497.94FV = ´ + =
or N=5, I=4, PV=−150,000, PMT=0, CPT FV == 182,497.94
The amount of equity is the difference between the home’s value and the outstanding balance on
the mortgage:
Equity = $182,497.94 – $104,507.44 = $77,990.50
4&5-9 Construction Loan You have secured a loan from your bank for two years to build your
home. The terms of the loan are that you will borrow $200,000 now and an additional $100,000
in one year. Interest of 10 percent APR will be charged on the balance monthly. Since no
payments will be made during the 2-year loan, the balance will grow at the 10 percent
compounded rate. At the end of the two years, the balance will be converted to a traditional 30-
year mortgage at a 6 percent interest rate. What will you be paying as monthly mortgage
payments (principal and interest only)?
Use equation 5-1 to calculate the capitalized value of your mortgage at the end of year 2:
( ) ( )
24 12
2
$200,000 1 0.10 /12 $100, 000 1 0.10/12 $354,549.50FV = ´ + + ´ + =
or N=2x12, I=10/12, PV=−200000, PMT=0, CPT FV == 244,078.19
and N=1x12, I=10/12, PV=−100000, PMT=0, CPT FV == 110,471.31
sum to get $354,549.50
This is the amount that you will need to finance over 30 years at 6 percent. Use equation 5-9 to
compute the monthly payment:
( )
360
360
0.06 /12
$354,549.50 $2,125.70
1
11 0.06 /12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
-
ê ú
+
ë û
or: N=30x12, I=6/12, PV=354549.50, FV=0, CPT PMT = −2125.70
4&5-10 Construction Loan You have secured a loan from your bank for two years to build your
home. The terms of the loan are that you will borrow $100,000 now and an additional $50,000
in one year. Interest of 9 percent APR will be charged on the balance monthly. Since no
payments will be made during the 2-year loan, the balance will grow. At the end of the two
years, the balance will be converted to a traditional 15-year mortgage at a 7 percent interest rate.
What will you pay as monthly mortgage payments (principal and interest only)?
Use equation 5-1 to calculate the capitalized value of your mortgage at the end of year 2:
( ) ( )
24 12
2
$100,000 1 0.09 /12 $50,000 1 0.09/12 $174,331.70FV = ´ + + ´ + =
or N=2x12, I=9/12, PV=−100000, PMT=0, CPT FV == 119,641.35
and N=1x12, I=9/12, PV=−50000, PMT=0, CPT FV == 54,690.34
sum to get $174,331.70
This is the amount that you will need to finance over 15 years at 7 percent. Use equation 5-9 to
compute the monthly payment:
( )
180
180
0.07 /12
$174,331.70 $1,566.94
1
11 0.07 /12
PMT
é ù
ê ú
ê ú
= ´ =
ê ú
-
ê ú
+
ë û
or: N=15x12, I=7/12, PV=174331.70, FV=0, CPT PMT = −1566.94
