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Chapter 05 - Time Value of Money 2: Analyzing Annuity Cash Flows
CHAPTER 5 – TIME VALUE OF MONEY 2: ANALYZING ANNUITY CASH FLOWS
questions
LG1 5-1 How can you add a cash flow in year 2 and a cash flow in year 4 in year 7?
LG2 5-2 People can become millionaires in their retirement years quite easily if they start saving early
in employer 401(k) or 403(b) programs (or even if their employers don’t offer such programs).
Demonstrate the growth of a $250 monthly contribution for 40 years earning 9 percent APR.
LG6 5-6 Explain why you use the same adjustment factor, (1 + i), when you adjust annuity due
payments for both future value and present value.
LG7 5-7 Use the idea of compound interest to explain why EAR is larger than APR.
LG8 5-8 Would you rather pay $10,000 for a 5-year $2,500 annuity or a 10-year $1,250 annuity?
Why?
LG9 5-9 The interest on your home mortgage is tax deductible. Why are the early years of the
mortgage more helpful in reducing taxes than in the later years?
LG10 5-10 How can you use the concepts illustrated in computing the number of payments in an
annuity to figure how to pay off a credit card balance? How does the magnitude of the payment
impact the number of months?
Chapter 05 - Time Value of Money 2: Analyzing Annuity Cash Flows
Problems
basic problems
LG1 5-1 Future Value Compute the future value in year 9 of a $2,000 deposit in year 1 and
another $1,500 deposit at the end of year 3 using a 10 percent interest rate.
Use equation 5-1:
LG1 5-2 Future Value Compute the future value in year 7 of a $2,000 deposit in year 1 and another
$2,500 deposit at the end of year 4 using an 8 percent interest rate.
Use equation 5-1:
LG2 5-3 Future Value of an Annuity What is the future value of a $900 annuity payment over five
years if interest rates are 8 percent?
Use equation 5-2:
( )
5
5
1 0.08 1
$900 $900 5.8666 $5,279.94
0.08
FVA + -
= ´ = ´ =
Or N=5, I=8, PV=0, PMT=−900, CPT FV == 5,279.94
LG2 5-4 Future Value of an Annuity What is the future value of a $700 annuity payment over six
years if interest rates are 10 percent?
Use equation 5-2:
( )
6
6
1 0.10 1
$700 $700 7.7156 $5, 400.93
0.10
FVA + -
= ´ = ´ =
Or N=6, I=10, PV=0, PMT=−700, CPT FV == 5,400.93
5-3
Chapter 05 - Time Value of Money 2: Analyzing Annuity Cash Flows
LG3 5-5 Present Value Compute the present value of a $2,000 deposit in year 1 and another $1,500
deposit at the end of year 3 if interest rates are 10 percent.
Use equation 5-3:
LG3 5-6 Present Value Compute the present value of a $2,000 deposit in year 1 and another $2,500
deposit at the end of year 4 using an 8 percent interest rate.
LG4 5-7 Present Value of an Annuity What is the present value of a $900 annuity payment over five
years if interest rates are 8 percent?
Use equation 5-4:
( )
5
5
1
11 0.08
$900 $900 3.9927 $3,593.44
0.08
PVA
é ù
-
ê ú
+
ê ú
= ´ = ´ =
ê ú
ê ú
ë û
Or N=5, I=8, PMT=−900, FV=0, CPT PV == 3,593.44
LG4 5-8 Present Value of an Annuity What is the present value of a $700 annuity payment over six
years if interest rates are 10 percent?
Use equation 5-4:
( )
6
6
1
11 0.10
$700 $700 4.355261 $3, 048.68
0.10
PVA
é ù
-
ê ú
+
ê ú
= ´ = ´ =
ê ú
ê ú
ë û
Or N=4, I=10, PMT=−700, FV=0, CPT PV == 3,048.68
5-4
Chapter 05 - Time Value of Money 2: Analyzing Annuity Cash Flows
LG5 5-9 Present Value of a Perpetuity What is the present value, when interest rates are 7.5 percent,
of a $50 payment made every year forever?
Use equation 5-5:
$50
of a perpetuity $666.67
0.075
PV = =
LG5 5-10 Present Value of a Perpetuity What is the present value, when interest rates are 8.5
percent, of a $75 payment made every year forever?
Use equation 5-5:
$75
of a perpetuity $882.35
0.085
PV = =
LG6 5-11 Present Value of an Annuity Due If the present value of an ordinary, 7-year annuity is
$6,500 and interest rates are 7.5 percent, what’s the present value of the same annuity due?
Use equation 5-7:
LG6 5-12 Present Value of an Annuity Due If the present value of an ordinary, 6-year annuity is
$8,500 and interest rates are 9.5 percent, what’s the present value of the same annuity due?
Use equation 5-7:
LG6 5-13 Future Value of an Annuity Due If the future value of an ordinary, 7-year annuity is
$6,500 and interest rates are 8.5 percent, what is the future value of the same annuity due?
Use equation 5-6:
LG6 5-14 Future Value of an Annuity Due If the future value of an ordinary, 6-year annuity
is $8,500 and interest rates are 9.5 percent, what’s the future value of the same annuity due?
Use equation 5-6:
5-5
Chapter 05 - Time Value of Money 2: Analyzing Annuity Cash Flows
LG7 5-15 Effective Annual Rate A loan is offered with monthly payments and a 10 percent APR.
What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
EAR=
(
1+0 . 10
12
)
12
−1=0. 1047=10. 47
LG7 5-16 Effective Annual Rate A loan is offered with monthly payments and a 13 percent APR.
What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
EAR=
(
1+0 . 13
12
)
12
−1=0. 1380=13 . 80
intermediate problems
LG1 5-17 Future Value Given a 4 percent interest rate, compute the year 6 future value of deposits
made in years 1, 2, 3, and 4 of $1,100, $1,200, $1,200, and $1,500.
Use equation 5-1:
LG1 5-18 Future Value Given a 5 percent interest rate, compute the year 6 future value of deposits
made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400.
Use equation 5-1:
Chapter 05 - Time Value of Money 2: Analyzing Annuity Cash Flows
LG2 5-19 Future Value of Multiple Annuities Assume that you contribute $200 per month to a
retirement plan for 20 years. Then you are able to increase the contribution to $300 per month
for another 30 years. Given a 7 percent interest rate, what is the value of your retirement plan
after the 50 years?
Break the annuity streams into a level stream of payments of $200 for 50 years and another level
stream of payments of $100 for the last 30 years. Use equation 5-2 for each payment stream and
add the results:
( ) ( )
600 360
50 30
1 0.07 /12 1 1 0.07 /12 1
$200 $100 $200 5, 448.0709 $100 1,219.9710 $1, 211,611.28
0.07/12 0.07/12
FVA FVA + - + -
+ = ´ + ´ = ´ + ´ =
Or N=50 x 12, I=7/12, PV=0, PMT=−200, CPT FV == 1,089,614.18
and N=30 x 12, I=7/12, PV=0, PMT=−100, CPT FV == 121,997.10
sum the FVs to get $1,211,611.28
LG2 5-20 Future Value of Multiple Annuities Assume that you contribute $150 per month to a
retirement plan for 15 years. Then you are able to increase the contribution to $350 per month
for the next 25 years. Given an 8 percent interest rate, what is the value of your retirement plan
after the 40 years?
Break the annuity streams into a level stream of payments of $150 for 40 years and another level
stream of payments of $200 for the last 25 years. Use equation 5-2 for each payment stream and
add the results:
( ) ( )
480 300
40 25
1 0.08 /12 1 1 0.08 /12 1
$150 $200 $150 3, 491.0078 $200 951.0264 $713,856.45
0.08/12 0.08/12
FVA FVA + - + -
+ = ´ + ´ = ´ + ´ =
Or N=40 x 12, I=8/12, PV=0, PMT=−150, CPT FV == 523,651.17
and N=2 5x 12, I=8/12, PV=0, PMT=−200, CPT FV == 190,205.28
sum the FVs to get $713,856.45
LG3 5-21 Present Value Given a 6 percent interest rate, compute the present value of payments made
in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500.
Use equation 5-3:
LG3 5-22 Present Value Given a 7 percent interest rate, compute the present value of payments made
in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400.
Use equation 5-3:
LG4 5-23 Present Value of Multiple Annuities A small business owner visits her bank to ask for a
loan. The owner states that she can repay a loan at $1,000 per month for the next three years and
then $2,000 per month for two years after that. If the bank is charging customers 7.5 percent
APR, how much would it be willing to lend the business owner?
Break the annuity streams into a level stream of payments of $2,000 for 5 years and another level
stream of payments of $1,000 for the first 3 years. Use equation 5-4 for each payment stream
and subtract the results:
( ) ( )
60 36
60 36
1 1
1 1
1 0.075 /12 1 0.075 /12
$2,000 $1, 000 $99,810.6164 $32,147.9132 $67, 662.70
0.075/12 0.075/12
PVA PVA
é ù é ù
- -
ê ú ê ú
+ +
ê ú ê ú
- = ´ - ´ = - =
ê ú ê ú
ê ú ê ú
ë û ë û
Or N=5 x 12, I=7.5/12, PMT=−2000, FV=0, CPT PV == 99,810.616
and N=3 x 12, I=7.5/12, PMT=1000, FV=0, CPT PV == −32,147.913
sum the FVs to get $67,662.70
LG4 5-24 Present Value of Multiple Annuities A small business owner visits his bank to ask for a
loan. The owner states that he can repay a loan at $1,500 per month for the next three years and
then $500 per month for two years after that. If the bank is charging customers 8.5 percent APR,
how much would it be willing to lend the business owner?
Break the annuity into two streams of payments: $500 monthly for five years and $1,000 for
three years. Use equation 5-4 for each annuity and add the results:
5-8
LG4 5-25 Present Value You are looking to buy a car. You can afford $450 in monthly payments for
four years. In addition to the loan, you can make a $1,000 down payment. If interest rates are 5
percent APR, what price of car can you afford?
Find the loan value of the monthly payments and add the down payment:
( )
48
48
1
11 0.05 /12
$450 $1, 000 $19,540.33 $1, 000 $20,540.33
0.05/12
PVA
é ù
-
ê ú
+
ê ú
= ´ + = + =
ê ú
ê ú
ë û
Or N=4 x 12, I=5/12, PMT=−450, FV=0, CPT PV == 19,540.33
Add the down payment of $1,000 to get $20,540.33
LG4 5-26 Present Value You are looking to buy a car. You can afford $650 in monthly payments for
five years. In addition to the loan, you can make a $750 down payment. If interest rates are 8
percent APR, what price of car can you afford?
Find the loan value of the monthly payments and add the down payment:
( )
60
60
1
11 0.08 /12
$650 $750 $32, 056.98 $750 $32,806.98
0.08/12
PVA
é ù
-
ê ú
+
ê ú
= ´ + = + =
ê ú
ê ú
ë û
Or N=5 x 12, I=8/12, PMT=−650, FV=0, CPT PV == 32,056.98
Add the down payment of $750 to get $32,806.98
5-9
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