Chapter 5, Solutions
Cornett, Adair, and Nofsinger
CHAPTER 5 – Time Value of Money 2: ANALYZING ANNUITY CASH FLOWS
Questions
LG1
5-1 How can you add a cash flow in year two and a cash flow in year four in year seven?
To add cash flows, they need to be moved to the same time period. The cash flows in years
two and four should be moved forward with interest to year seven, then they can be added
together.
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.
Using equation 5-2, we have:
FVA 40  250 
LG3
1  0.09 / 12480  1  $1,170,330.07
0.09/12
5-3 When you discount multiple cash flows, how does the future period that a cash flow is
paid affect its present value and its contribution to the value of all the cash flows?
Discounting reduces a future cash flow to a smaller present value. Cash flows far into the
future become very small when discounted to the present. Thus, cash flows in distant future
periods have small impacts on present values.
LG4
5-4 How can you use the present value of an annuity concept to determine the price of a
house you can afford?
Mortgages are typically for a large enough amount of money that borrowing is required to
purchase a home. The amount that one can afford for a home is a function of their current
state of wealth. Mortgages allow consumers to spread the expense of a home over a longer
period, typically 15 or 30 years. This allows consumers to put a smaller portion of wealth
into the home (for example, a 20% down payment) and borrow the balance over the life of
the loan. Due to the effect of annuity compounding, the payments for such a long lived debt
make the monthly payments of a manageable nature so that they can be paid from current
income.
LG5
5-5 Since perpetuity payments continue for ever, how can a present value be computed?
Why isn’t the present value infinite?
5-1
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
Equation 5-5 is used to calculate the present value of a perpetuity. It is a limiting version of
equation 5-4 in which the period N grows infinitely large. As this occurs the expression
following the “1” in equation 5-4 drives to the value 0 and the numerator simply become
“1.” The present value is not infinite since the terms following the PMT in equation 5-4
converge to a finite limit of 1/i. This also demonstrates how payments far into the future
have infinitesimal value today.
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.
Adjusting an annuity due calculation involves shifting the entire series of payments forward
one period. This is accomplished by multiplying by (1 + i) irrespective of whether it is a
future value or present value calculation.
LG7
5-7 Use the idea of compound interest to explain why EAR is larger than APR.
The annual percentage rate does not take into account the frequency of interest
compounding. Equation 5-8 illustrates the conversion from APR to EAR. The effective
annual rate converts the annual percentage rate to a rate that can be compared to other
annual rates.
LG8
5-8 Would you rather pay $10,000 for a five year $2,500 annuity or a ten-year $1,250
annuity? Why?
The effective annual rates for these two payment streams are 7.93% and 4.28% respectively.
I would rather pay $10,000 for a five year $2,500 annuity as it earns a higher effective
annual rate of interest.
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?
Mortgage payments at the beginning of the amortization schedule are predominantly interest
with little principal. In later years, interest payments decline and principal payments make
up an ever increasing part of the payments. Thus, the tax deductible part (the interest
payment) is larger in the beginning 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?
Utilizing equation 5-2, you can declare the present balance for the credit card and set that
equal to the PVA. The interest can be obtained as an APR and converted to and EAR using
equation 5-8. This is the value to put into “i” in equation 5-2. You then decide when you
want to have the credit card paid off and convert this to a monthly value of N in equation 5-
5-2
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
2. Solving for PMT will yield the amount needed to pay the credit card off in the time
frame you desire, assuming that no additional charges are made to the credit card and that
the interest rate remains level.
Problems
Basic
Problems
LG1
5-1 Future Value Compute the future value in year 8 of a $1,000 deposit in year 1 and
another $1,500 deposit at the end of year 3 using a 10% interest rate.
Use equation 5-1:
FV = $1,000 × (1 + 0.10)7 + $1,500× (1 + 0.10)5 = $1,948.72 + $2,415.77 = $4,364.48
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 a 8% interest rate..
Use equation 5-1:
FV = $2,000 × (1 + 0.08)6 + $2,500 × (1 + 0.08)3 = $3,173.75 + $3,149.28 = $6,323.03
LG2
5-3 Future Value of an Annuity What is the future value of a $500 annuity payment over 5
years if interest rates are 9 percent?
Use equation 5-2:
FVA 5  $500 
LG2
0.09
5-4 Future Value of an Annuity What is the future value of a $700 annuity payment over 4
years if interest rates are 10 percent?
Use equation 5-2:
FVA 4  $500 
LG3
1  0.095  1  $500  5.9847  $2,992.36
1  0.104  1  $700  4.641  $3,248.70
0.10
5-5 Present Value Compute the present value of a $1,000 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:
PV = $1,000 ÷ (1 + 0.10)1 + $1,500 ÷ (1 + 0.10)3 = $909.09 + $1,126.97 = $2,036.06
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% interest rate.
5-3
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
Use equation 5-3:
PV = $2,000 ÷ (1 + 0.08)1 + $2,500 ÷ (1 + 0.08)4 = $1,851.85 + $1,837.57 = $3,689.43
LG4
5-7 Present Value of an Annuity What’s the present value of a $500 annuity payment over
5 years if interest rates are 9 percent?
Use equation 5-4:
1


1  1  0.095 
  $500  3.889651  $1,944.83
PVA 5  $500  
0.09




LG4
5-8 Present Value of an Annuity What’s the present value of a $700 annuity payment over
4 years if interest rates are 10 percent?
Use equation 5-4:
1


1  1  0.104 
  $700  3.1698655  $2,218.91
PVA 4  $700  
0.10




LG5
5-9 Present Value of a Perpetuity What’s the present value, when interest rates are 7.5
percent, of a $50 payment made every year forever?
Use equation 5-5:
PV of a perpetuity 
LG5
$50
 $666.67
0.075
5-10 Present Value of a Perpetuity What’s the present value, when interest rates are 8.5
percent, of a $75 payment made every year forever?
Use equation 5-5:
PV of a perpetuity 
LG6
$75
 $882.35
0.085
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 8.5 percent, what’s the present value of the same annuity
due?
Use equation 5-7:
PVA7 due = 6,500 × (1 +0.085) = $7,052.50
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?
5-4
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
Use equation 5-7:
PVA6 due = 8,500 × (1 +0.095) = $9,307.50
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:
FVA7 due = 6,500 × (1 + 0.085) = $7,052.50
(Note this is the same answer as problem 5-11, as expected)
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:
FVA6 due = 8,500 × (1 + 0.095) = $9,307.50
(Note this is the same answer as problem 5-12, as expected)
LG7
5-15 Effective Annual Rate A loan is offered with monthly payments and an 11 percent
APR. What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
12
 0.11 
EAR  1 
  1  0.115718841  11.57%
12 

LG7
5-16 Effective Annual Rate A loan is offered with monthly payments and a 12 percent
APR. What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
12
 0.12 
EAR  1 
  1  0.12682503  12.68%
12 

Intermediate
Problems 5-17 Future Value Given a 6 percent interest rate, compute the year 6 future value of
deposits made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500.
LG1
Use equation 5-1:
5-5
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
FV = $1,000 × (1 + 0.06)5 + $1,200× (1 + 0.06)4 + $1,200 × (1 + 0.06)3 + $1,500× (1 + 0.06)2
FV = $1,338.23 + $1,514.97 + $1,429.22 + $1,685.40 = $5,967.82
LG1
5-18 Future Value Given a 7 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:
FV = $1,000 × (1 + 0.07)5 + $1,300× (1 + 0.07)4 + $1,300 × (1 + 0.07)3 + $1,400× (1 + 0.07)2
FV = $1,402.55 + $1,704.03 + $1,592.56 + $1,602.86 = $6,302
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 $400 per
month for another 20 years. Given a 7 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 $200 for 40 years and another
level stream of payments of $200 for the last 20 years. Use equation 5-2 for each payment
stream and add the results:
FVA 40  FVA 20  $200 
LG2
1  0.07 / 12480  1  $200  1  0.07 / 12240  1  $200  2,796.2415  $200  520.9267  $629,148.01
0.07/12
0.07/12
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
FVA 40  FVA 25  $150 
LG3
1  0.08 / 12480  1  $200  1  0.08 / 12300  1  $150  3,491.0078  $200  951.0264  $713,856.45
0.08/12
0.08/12
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:
PV = $1,000 ÷ (1 + 0.06)1 + $1,200 ÷ (1 + 0.06)2 + $1,200 ÷ (1 + 0.06)3 + $1,500 ÷ (1 + 0.06)4
PV = $943.40 + $1,068 + $1,007.54 + $1,188.14 = $4,207.08
5-6
Chapter 5, Solutions
LG3
Cornett, Adair, and Nofsinger
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:
PV = $1,000 ÷ (1 + 0.07)1 + $1,300 ÷ (1 + 0.07)2 + $1,300 ÷ (1 + 0.07)3 + $1,400 ÷ (1 + 0.07)4
PV = $934.58 + $1,135.47 + $1,061.19 + $1,068.05 = $4,199.29
LG2
5-23 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 $1,000 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.
1
1




1  1  0.075 / 1260 
1  1  0.075 / 1236 
  $1,000  
  $99,810.62  $32,147.92  $67,662.70
PVA 5  PVA 3  $2,000  
0.075/12
0.075/12








LG2
5-24 Present Value of Multiple Annuities A small business owner visits his bank to ask
for a loan. The owner states that she 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.
1
1




1  1  0.085 / 1260 
1  1  0.085 / 1236 
  $1,000  
  $24,370.59  $31,678.11  $56,048.70
PVA 5  PVA 3  $500  
0.085/12
0.085/12








LG5
5-25 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 8.5 percent?
Did the value increase or decrease?
Use equation 5-5:
PV of a perpetuity 
$100
 $1,333.33
0.075
5-7
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
PV of a perpetuity 
$100
 $1,176.47
0.085
The difference between these perpetuities is $156.86. The value of the perpetuity decreased with
an increase in the interest rate.
LG5
5-26 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 8 percent? Did the
value increase or decrease?
Use equation 5-5:
PV of a perpetuity 
$50
 $555.55
0.09
PV of a perpetuity 
$50
 $625.00
0.08
The difference between these perpetuities is $69.45. The value of the perpetuity increased
with a decrease in the interest rate.
LG6
5-27 Future and Present Value of an Annuity Due If you start making $50 monthly
contributions today and continue them for 5 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:
FVA 60  $50 
1  0.10 / 1260  1  1  0.10 / 12  $50  77.4343706824  1.00833  $3,903.98
0.10/12
Compute the present value using equation 5-4:
PVA 60
LG6
1


1  1  0.10 / 1260 
  1.00833  $50  47.06536757  1.00833  $2,372.88
 $50  
0.10/12




5-28 Future and Present Value of an Annuity Due If you start making $75 monthly
contributions today and continue them for 4 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.
FVA 48  $75 
1  0.12 / 1248  1  $75  61.2226  $4,591.70
0.12/12
5-8
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
FVA48 due = $4,591.70 × (1 + 0.12/12) = $4,637.61
PVA 48
1


1  1  0.12 / 1248 
  $75  37.97395951  $2,848.05
 $75  
0.12/12




PVA48 due = $2,848.05 × (1 +0.12/12) = $2,876.53
LG7
5-29 Compound Frequency Payday loans are very short-term loans that charge very high
interest rates. You can borrow $250 today and repay $300 in two weeks. What is the
compounded annual rate implied by this 20 percent rate charged for only two weeks?
20% for two weeks needs to be compounded 26 times to form a year.
(1 + i)26 – 1 = (1 + 0.20)26 – 1 = 113.475 = 11,347.5%
LG7
5-30 Compound Frequency Payday loans are very short-term loans that charge very high
interest rates. You can borrow $500 today and repay $575 in two weeks. What is the
compounded annual rate implied by this 15 percent rate charged for only two weeks?
15% for two weeks needs to be compounded 26 times to form a year.
(1 + i)26 – 1 = (1 + 0.15)26 – 1 = 36.857 = 3,685.7%
LG8
5-31 Annuity Interest Rate What’s the interest rate of a 5-year, annual $5,000 annuity
with present value of $20,000?
Use equation 5-2 and solve for i:
20,000  $5,000 
1  i 5  1  i  7.93%
i
or TVM calculator: N=5, PV=-20,000, PMT=5,000, FV=0, CPT I = 7.93%
LG8
5-32 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  i 7  1  i  9.20%
i
or TVM calculator: N=7, PV=-20,000, PMT=4,000, FV=0, CPT I = 9.20%
5-9
Chapter 5, Solutions
LG8
Cornett, Adair, and Nofsinger
5-33 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 5 years?
Use equation 5-2 and solve for i:
$75,000  $1,000 
1  i / 1260  1  i  .732%
i/12
Now convert the monthly interest rate to an annual rate by multiplying by 12 which yields
8.78%.
or TVM calculator: N=60, PV=0, PMT=-1,000, FV=75,000, CPT I = 0.732%
LG8
5-34 Annuity Interest Rate What annual interest rate would you need to earn if you
wanted a $500 per month contribution to grow to $45,000 in 6 years?
Use equation 5-2 and solve for i:
$45,000  $500 
1  i / 1272  1  i  .60809%
i/12
Now convert the monthly interest rate to an annual rate by multiplying by 12 which yields
7.30%.
or TVM calculator: N=72, PV=0, PMT=-500, FV=45,000, CPT I = 0.608%
LG9
5-35 Loan Payments You wish to buy a $25,000 car. The dealer offers you a 4-year loan
with a 10 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:




.10 / 12
  $25,000  0.025362584  $634.06
PMT48  $25,000  
1
1 

 1  .10 / 1248 
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 $208.33 per month (= $25,000×0.10÷12)
for interest only and you would owe $25,000 at the end of the 48 months, too.
LG9
5-36 Loan Payments You wish to buy a $10,000 dining room set. The furniture store
offers you a 3-year loan with a 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?
5-10
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
Use equation 5-9:




.11 / 12
  $10,000  0.03273872  $327.39
PMT36  $10,000  
1
1 

 1  .11 / 1236 
If you only paid interest over the length of the loan and your principal balance was repaid at
the end of the 36 months, your payment would be $91.67 per month (= $10,000×0.11÷12)
for interest only and you would owe $10,000 at the end of the 36 months, too.
LG10 5-37 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

 45.4 months
ln 1  0.17 / 12
LG10 5-38 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
Advanced
Problems
LG1

ln  200
200  6,000  0.18 / 12

 40.15 months
ln 1  0.18 / 12
5-39 Future Value Given a 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 $500 is made in year 4.
Use equation 5-1:
FV = $1,000 × (1 + 0.08)6 + $2,000× (1 + 0.08)4 - $500× (1 + 0.08)3 = $1,586.87 + $2,720.98 –
$629.86 = $3,677.99
LG1
5-40 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
$700 is made in year 5.
5-11
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
Use equation 5-1:
FV = $1,500 × (1 + 0.09)4 + $2,500× (1 + 0.09)3 - $700× (1 + 0.09)1 = $2,117.37 + $3,237.57 –
$763 = $4,591.94
LG4
LG9
5-41 Low Financing or Cash Back? A car company is offering a choice of deals. You
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 7 percent
APR. Which deal is cheaper?
Compare two cases. The first case is to elect the 3% APR and fully finance $15,000 over 48
months. Using equation 5-9, the payment under this scenario would be:




0.03 / 12
  $332.01
PMT48  15,000  
1
1 

 1  0.03/1248 
The second case is to take the $500 cash back, apply it to the purchase and finance only
$14,500 through your credit union at 7%. The payment under this scenario would be:




0.07 / 12
  $347.22
PMT48  14,500  
1
1 

 1  0.07/12 48 
The lower payment represents the more advantageous scenario that you should choose,
electing the 3% financing through the car dealer.
LG4
LG9
5-42 Low Financing or Cash Back? A car company is offering a choice of deals. You
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% APR and fully finance $20,000 over 60
months. Using equation 5-9, the payment under this scenario would be:




0.03 / 12
  $350.56
PMT60  20,000  
1
1 

 1  0.02/12 60 
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%. The payment under this scenario would be:
5-12
Chapter 5, Solutions
Cornett, Adair, and Nofsinger




0.07 / 12
  $376.22
PMT60  19,000  
1
1 

 1  0.07/12 60 
The lower payment represents the more advantageous scenario that you should choose,
electing the 2% financing through the car dealer.
LG9
5-43 Amortization Schedule Create the amortization schedule for a loan of $15,000, paid
monthly over 3 years using a 9 percent APR.
Month
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Beginning
Balance
$15,000.00
14,635.50
14,268.27
13,898.29
13,525.53
13,149.98
12,771.61
12,390.40
12,006.33
11,619.38
11,229.53
10,836.76
10,441.03
10,042.35
9,640.67
9,235.98
8,828.25
8,417.47
8,003.60
7,586.63
7,166.54
6,743.29
6,316.87
5,887.25
5,454.41
5,018.32
4,578.96
4,136.31
3,690.33
3,241.02
2,788.33
2,332.24
1,872.74
Total
Payment
$477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
477.00
Interest
Paid
$112.50
109.77
107.01
104.24
101.44
98.62
95.79
92.93
90.05
87.15
84.22
81.28
78.31
75.32
72.31
69.27
66.21
63.13
60.03
56.90
53.75
50.57
47.38
44.15
40.91
37.64
34.34
31.02
27.68
24.31
20.91
17.49
14.05
Principal
Paid
$364.50
367.23
369.98
372.76
375.55
378.37
381.21
384.07
386.95
389.85
392.77
395.72
398.69
401.68
404.69
407.73
410.78
413.86
416.97
420.10
423.25
426.42
429.62
432.84
436.09
439.36
442.65
445.97
449.32
452.69
456.08
459.50
462.95
5-13
Ending
Balance
$14,635.50
14,268.27
13,898.29
13,525.53
13,149.98
12,771.61
12,390.40
12,006.33
11,619.38
11,229.53
10,836.76
10,441.03
10,042.35
9,640.67
9,235.98
8,828.25
8,417.47
8,003.60
7,586.63
7,166.54
6,743.29
6,316.87
5,887.25
5,454.41
5,018.32
4,578.96
4,136.31
3,690.33
3,241.02
2,788.33
2,332.24
1,872.74
1,409.79
Chapter 5, Solutions
34
35
36
LG9
477.00
477.00
477.00
10.57
7.08
3.55
466.42
469.92
473.45
943.37
473.45
0.00
5-44 Amortization Schedule Create the amortization schedule for a loan of $5,000, paid
monthly over 2 years using a 8 percent APR.
Month
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
LG4
LG9
1,409.79
943.37
473.45
Cornett, Adair, and Nofsinger
Beginning
Balance
$5,000.00
4,807.20
4,613.11
4,417.73
4,221.04
4,023.04
3,823.73
3,623.08
3,421.10
3,217.77
3,013.09
2,807.04
2,599.62
2,390.81
2,180.61
1,969.01
1,756.00
1,541.57
1,325.71
1,108.42
889.67
669.46
447.79
224.64
Total
Payment
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
$226.14
Interest
Paid
$33.33
$32.05
$30.75
$29.45
$28.14
$26.82
$25.49
$24.15
$22.81
$21.45
$20.09
$18.71
$17.33
$15.94
$14.54
$13.13
$11.71
$10.28
$8.84
$7.39
$5.93
$4.46
$2.99
$1.50
Principal
Paid
$192.80
194.09
195.38
196.68
198.00
199.32
200.64
201.98
203.33
204.68
206.05
207.42
208.81
210.20
211.60
213.01
214.43
215.86
217.30
218.75
220.21
221.67
223.15
224.64
Ending
Balance
$4,807.20
4,613.11
4,417.73
4,221.04
4,023.04
3,823.73
3,623.08
3,421.10
3,217.77
3,013.09
2,807.04
2,599.62
2,390.81
2,180.61
1,969.01
1,756.00
1,541.57
1,325.71
1,108.42
889.67
669.46
447.79
224.64
0.00
5-45 Investing for Retirement Monica has decided that she wants to build enough
retirement wealth that, if invested at 8 percent per year, will provide her with $3,500 of
monthly income for 30 years. To date, she has saved nothing, but she still has 25 years until
she retires. How much money does she need to contribute per month to reach her goal?
First, calculate the amount you would need to have in 25 years time to yield the $3,500
monthly payments for an additional 30 years. Use equation 5-4 to calculate this present
value:
5-14
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
1


1  1  0.08 / 12360 
  $476,992.23
PVA  $3,500  
0.08/12




This amount will become the future value in the next calculation, assuming 8% interest and
300 level monthly payments. Use equation 5-2 and solve for the monthly payment:
$476,992.23  PMT 
LG4
LG9
1  0.08 / 12300  1  PMT  $501.56
0.08/12
5-46 Investing for Retirement Ross has decided that he wants to build enough retirement
wealth that, if invested at 7 percent per year, will provide him with $4,000 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?
First, calculate the amount you would need to have in 20 years time to yield the $4,000
monthly payments for an additional 30 years. Use equation 5-4 to calculate this present
value:
1


1  1  0.07 / 12360 
  $601,230.27
PVA  $4,000  
0.07/12




This amount will become the future value in the next calculation, assuming 7% interest and
240 level monthly payments. Use equation 5-2 and solve for the monthly payment:
$601,230.27  PMT 
LG9
1  0.07 / 12240  1  PMT  $1,154.16
0.07/12
5-47 Loan Balance Rachel purchased a $15,000 car two years ago using an 8 percent, 4year 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 is the minimum price Rachel would need to receive for
his car?
First calculate the monthly payment that she has been paying using equation 5-9:




0.08 / 12
  $366.19
PMT48  15,000  
1
1 

 1  0.08/12 48 
The loan balance is the principal amount outstanding. The duration of remaining payments
is 24, the interest rate is 8% annual and the monthly payment is $366.19 from the previous
calculation. Use these values to calculate the present value of the loan using equation 5-4:
5-15
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
1


1  1  0.08 / 1224 
  $8,096.66
PVA  $366.19  
0.08/12




This is the minimum price the car needs to be sold for and it represents her break even price.
LG9
5-48 Loan Balance Hank purchased a $20,000 car two years ago using an 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:




0.09 / 12
  $415.17
PMT60  20,000  
1
1 

 1  0.09/12 60 
The loan balance is the principal amount outstanding. The duration of remaining payments
is 36, the interest rate is 9% 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:
1


1  1  0.09 / 1236 
  $13,055.77
PVA  $415.17  
0.09/12




This is the minimum price the car needs to be sold for and it represents his break even price.
LG9
5-49 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:
PMT360




0.04 / 12
  $877.97
 $183,900  
1
1 

360
 1  0.04 / 12 
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
5-16
Chapter 5, Solutions
PVA 336
Cornett, Adair, and Nofsinger
1


1  1  0.04 / 12336 
  $177,291.62
 $877.97  
0.04/12




Now use this amount for the present value, the new interest rate of 7% over the remaining
336 payments in equation to calculate the new payment amount after expiration of the teaser
rate, using equation 5-9:
PMT336
LG9




0.07 / 12
  $1,204.89
 $177,291.62  
1
1 

 1  0.07 / 12336 
5-50 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:
PMT360




0.045 / 12
  $1,413.65
 $279,000  
1
1 

 1  0.045 / 12360 
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
PVA 336
1


1  1  0.045 / 12336 
  $269,791.04
 $1,413.65  
0.045/12




Now use this amount for the present value, the new interest rate of 7.5% over the remaining
336 payments in equation to calculate the new payment amount after expiration of the teaser
rate, using equation 5-9:
PMT336




0.075 / 12
  $1,923.25
 $269,791.04  
1
1 

336
 1  0.075 / 12 
5-51 Excel 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
5-17
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
contributes $3,000 per year. She increases the contribution rate to $5,000 per year in years
11 through 20. This is followed by increases to $10,000 per year in years 21 through 31 and
to $15,000 per year for the last ten years. This money earns a 9 percent return, first
compute the value of the retirement plan when she turns age 65. Then compute the annual
payment she would receive over the next 40 years if the wealth was converted to an annuity
payment at 8 percent.
End of
Age
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
Contribution
$3,000.00
$3,000.00
$3,000.00
$3,000.00
$3,000.00
$3,000.00
$3,000.00
$3,000.00
$3,000.00
$3,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$10,000.00
$10,000.00
$10,000.00
$10,000.00
$10,000.00
$10,000.00
$10,000.00
$10,000.00
$10,000.00
$10,000.00
$15,000.00
$15,000.00
$15,000.00
$15,000.00
$15,000.00
$15,000.00
$15,000.00
$15,000.00
Total Wealth
$3,000.00
$6,270.00
$9,834.30
$13,719.39
$17,954.13
$22,570.00
$27,601.30
$33,085.42
$39,063.11
$45,578.79
$54,680.88
$64,602.16
$75,416.35
$87,203.83
$100,052.17
$114,056.87
$129,321.98
$145,960.96
$164,097.45
$183,866.22
$210,414.18
$239,351.45
$270,893.08
$305,273.46
$342,748.07
$383,595.40
$428,118.99
$476,649.70
$529,548.17
$587,207.50
$655,056.18
$729,011.23
$809,622.25
$897,488.25
$993,262.19
$1,097,655.79
$1,211,444.81
$1,335,474.84
5-18
Chapter 5, Solutions
63
64
$15,000.00
$15,000.00
Cornett, Adair, and Nofsinger
$1,470,667.58
$1,618,027.66
PMT =
($135,688.06)
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://moneycentral.msn.com/investor/calcs/n_retire/main.asp)
SOLUTION: Assuming the principal amount is exhausted and the amount of savings at my
retirement at age 65, here is what the calculator gives.
Summary
Given a certain amount of savings, how much can I spend annually during
retirement?
Your annual income is estimated to be $70,000.
Information entered
1. Savings
Amount saved
Rate of return
Inflation rate
Average effective tax rate
$1,000,000
7%
3.8%
22.6%
2. Retirement years
Retirement age
Life expectancy
Estate amount
65 years
85 years
$0
5-19
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
Using a financial calculator, the following inputs are needed to determine the projected
annual payment:
n = 20 years
i = 1.07/1.038 -1 = 3.083%
PV = $1,000,000
CPT PMT = $67,732.56 (The calculator rounds this amount to the nearest $10,000 for an
estimate of $70,000 gross annually.)
Integrated Mini Case: Paying on your Stafford loan
Consider Gunther, 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 6.8 percent
interest owed (compounded monthly) accrues and is added to the balance of the loan.
Stafford loan limits:
Freshman
$2,625
Sophomore $3,500
Junior
$5,500
Senior
$5,500
After graduation, Gavin gets a six 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 10 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:
a. Show a time line of when the loans will be taken.
Cash Flows $2,625
Period 0
6.8%
$3,500
1
$5,500
6.8% 2
5-20
$5,500
6.8% 3
6.8% 4 6.8%
5 years
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
b. What will be the loan balance when Gavin graduates after his fourth year of school?
Each payment needs to be moved forward with 6.8% interest to the middle of years 4 and 5
to calculate the outstanding accrued loan balance as of the date payments are set to begin:
FV4 = $2,625 × (1.068)4 + $3,500 × (1.068)3 + $5,500 × (1.068)2 + $5,500 × (1.068)1
FV4 = $3,415.19 + $4,263.65 + $6,273.43 + $5,874.00 = $19,826.27
c. What is the loan balance six months after graduation?
Each payment needs to be moved forward with 6.8% interest to the middle of years 4 and 5
to calculate the outstanding accrued loan balance as of the date payments are set to begin:
FV4.5= $2,625 × (1.068)4.5 + $3,500 × (1.068)3.5 + $5,500 × (1.068)2.5 + $5,500 × (1.068)1.5
FV4.5= $3,529.39 + $4,406.23 + $6,483.22 + $6,070.43 = $20,489.27
Note, this can be checked by multiplying $19,826.27 by 1.0680.5.
d. Using the standard repayment plan and a 6.8 percent APR interest rate, compute the
monthly payments Gavin owes after the grace period.
Using equation 5-9, the monthly payments will be:
PMT120




0.068 / 12
  $235.79
 $20,489.27  
1
1 

 1  0.068 / 12120 
Combined Chapter 4 and Chapter 5 Problems
4&5-1 Future Value Consider that you are 35 years old and have just changed to a new job.
You have $80,000 in the retirement plan from your former employer. You can roll that
money into the retirement plan of the new employer. You will also contribute $5,000 each
year into your new employer’s plan. If the rolled-over money and the new contributions
both earn a 7 percent return, how much should you expect to have when you retire in 30
years?
The future value can be calculated by adding the accumulated value of $80,000 brought
forward with interest 30 years to a 30 year annuity of $5,000 per year, both using the 7%
interest assumption. Use equations 5-1 and 5-2 and add the results:
FVA Abe 65  $80,000  1  0.07   $5,000 
30
1  0.07 30  1  $608,980.40  $472,303.93  $1,081,284.33
0.07
5-21
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
4&5-2 Future Value Consider that you are 45 years old and have just changed to a new job.
You have $150,000 in the retirement plan from your former employer. You can roll that
money into the retirement plan of the new employer. You will also contribute $8,000 each
year into your 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?
The future value can be calculated by adding the accumulated value of $150,000 brought
forward with interest 20 years to a 20 year annuity of $8,000 per year, both using the 8%
interest assumption. Use equations 5-1 and 5-2 and add the results:
FVA Abe 65  $150,000  1  0.08  $8,000 
20
1  0.0820  1  $699,143.57  $366,095.71  $1,065,239.28
0.08
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% interest for a
future value:
FVAAbe 65  $1,000,000  1  0.055  $3,813,392.35
25
Now, rewrite equation 5-9 in terms of N:
N

ln  25,000
25,000  3,813,392.35  0.055 / 12

 263 months
ln 1  0.055 / 12
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% interest for a
future value:
FVA Abe 65  $1,500,000  1  0.05 15  $3,118,392.27
Now, rewrite equation 5-9 in terms of N:
5-22
Chapter 5, Solutions
Cornett, Adair, and Nofsinger

ln  20,000
20,000  3,118,392.27  0.05 / 12

N
 253 months
ln 1  0.05 / 12
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?
Use equation 5-1 and solve for the lump sum payment:
PV  $3,000  1  0.05 / 12  $2,715.08
24
Use equation 5-2 and solve for the annuity payment:
$3,000  PMT 
1  0.05/12 24  1  PMT  $119.11 per month
0.05/12
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 3 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 5-1 and solve for the lump sum payment:
PV  $5,000  1  0.04 / 12  $4,435.49
36
Use equation 5-2 and solve for the annuity payment:
$5,000  PMT 
1  0.04/12 36  1  PMT  $130.95
0.04/12
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 10 years? If the house appreciates at 3 percent per year, what
will be the value of the house in 10 years? How much of this value is your equity?
Use equation 5-9 to calculate your monthly payment:
5-23
Chapter 5, Solutions
Cornett, Adair, and Nofsinger




0.07 / 12
  $1,197.54
PMT  $180,000  
1
1 

 1  0.07 / 12360 
In ten years, you will have 240 payments of $1,197.54 left to pay. The present value can be
calculated using equation 5-4:
PVA 240
1


1  1  0.07 / 12240 
  $154,461.71
 $1,197.54  
0.07/12




An appreciation of 3% per year will result in a forecast future value of the home using the
original purchase price in equation 5-1:
FV10 years  $200,000  1  .03  $268,783.28
10
The amount of equity is the difference between the home’s value and the outstanding
balance on the mortgage:
Equity = $268,783.28 - $154,461.17 = $114,321.57
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 5 years? If the house appreciates at 4 percent per year, what
will be the value of the house in 5 years? How much of this value is your equity?
Use equation 5-9 to calculate your monthly payment:




0.07 / 12
  $1,213.42
PMT  $135,000  
1
1 

 1  0.07 / 12180 
In ten years, you will have 120 payments of $1,213.42 left to pay. The present value can be
calculated using equation 5-4:
PVA120
1


1  1  0.07 / 12120 
  $104,507.44
 $1,213.42  
0.07/12




An appreciation of 4% per year will result in a forecast future value of the home using the
original purchase price in equation 5-1:
5-24
Chapter 5, Solutions
Cornett, Adair, and Nofsinger
FV5years  $150,000  1  .04  $182,497.94
5
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 2 years to build
your home. The terms of the loan are that you will borrow $100,000 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 two-year loan, the balance will grow at the 10%
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 paying 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:
FV2  $100,000  1  0.10 / 12  $100,000  1  0.10/12   $232,510.41
24
12
This is the amount that you will need to finance over 30 years at 6%. Use equation 5-9 to
compute the monthly payment:




0.06 / 12
  $1,394.02
PMT  $232,510.41  
1
1 

360
 1  0.06 / 12 
4&5-10 Construction Loan You have secured a loan from your bank for 2 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 two-year loan, the balance will grow. At the
end of the two years, the balance will be converted to a traditional 30-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:
FV2  $100,000  1  0.09 / 12  $50,000  1  0.09/12   $174,331.69
24
12
This is the amount that you will need to finance over 30 years at 7%. Use equation 5-9 to
compute the monthly payment:
5-25
Chapter 5, Solutions
Cornett, Adair, and Nofsinger




0.07 / 12
  $1,159.83
PMT  $174,331.69  
1
1 

 1  0.07 / 12360 
5-26