Chapter 6
Discounted Cash Flows and Valuation
Learning Objectives
1. Explain why cash flows occurring at different times must be discounted to a common
date before they can be compared, and be able to compute the present value and future
value for multiple cash flows.
2. Describe how to calculate the present value of an ordinary annuity and how an
ordinary annuity differs from an annuity due.
3. Explain what a perpetuity is and how it is used in business, and be able to calculate the
value of a perpetuity.
4. Discuss growing annuities and perpetuities, as well as their application in business, and
be able to calculate their value.
5. Discuss why the effective annual interest rate (EAR) is the appropriate way to annualize
interest rates, and be able to calculate EAR.
I.
Chapter Outline
6.1
Multiple Cash Flows
A.
Future Value of Multiple Cash Flows

In contrast to Chapter 5, we now consider situations in which there are multiple cash
flows. Solving future value problems with multiple cash flows involves a simple
process.
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
First, draw a time line to make sure that each cash flow is placed in the correct time
period.

Second, calculate the future value of each cash flow for its time period.

Third, add up the future values.
♦ Future Value with Multiple Cash Flows – Use a timeline to illustrate the time period in
which cash flows occur. In almost all such calculations, it is implicitly assumed that the cash
flows occur at the end of each period.
Future Value of Uneven Cash Flow Streams
Considering the example above, what is the future value of the cash flows at the end of year
three if the interest rate is 6% compounded annually?
Note: Treat each cash flow as a lump sum amount and compound for the appropriate
number of periods.

FVn = PV( 1 + i )n
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
FV3 = $100(1.06)2 + $200(1.06)1 + $300 = $100(1.1236) + $200(1.06) + $300 = $624.36
What is the future value of the cash flows if the interest rate is 6% compounded monthly?

FVn = PV  1  i/m  (m x n)

FV3 = $100(1 + .06/12)(12)(2) + $200(1 + .06/12)(12)(1) + $300(12)(0)
FV3 = $100(1.12716) + $200(1.061678) + $300 = $625.05
What is the future value of the cash flows at the end of year ten if the interest rate is 6%
compounded annually?
0
Period
CF

1
2
3
$100 $200 $300
4
5
6
7
8
9
10
FV10 =
or
FV10 =
B.
Present Value of Multiple Cash Flows

Many situations in business call for computing the present value of a series of
expected future cash flows. This could be to determine the market value of a security
or business or to decide whether a capital investment should be made.

The process is similar to determining the future value of multiple cash flows.

First, prepare a time line to identify the magnitude and timing of the cash flows.

Next, calculate the present value of each cash flow using Equation 5.4 from the
previous chapter.

Finally, add up all the present values.

The sum of the present values of a stream of future cash flows is their current market
price, or value.
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♦ Present Value with Multiple Cash Flows
Present Value of Uneven Cash Flow Streams
Considering the example above, what is the present value of the cash flows at time zero
(today) if the interest rate is 6% compounded annually?
Note: Treat each cash flow as a lump sum amount and discount for the appropriate number
of periods.

PV = FVn ( 1 + i )-n

PV0 = $100(1.06)-1 + $200(1.06)-2 + $300(1.06)-3
PV0 = $100(.943396) + $200(.889996) + $300(.839619) = $524.22
What is the present value of the cash flows if the interest rate is 6% compounded monthly?

PV = FVn ( 1 + i/m )-(m x n)

PV0 = $100(1 + .06/12)-(12)(1) + $200(1 + .06/12)-(12)(2) + $300(1 + .06/12)-(12)(3)
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PV0 = $100(.941905) + $200(.887186) + $300(.835645) = $522.32
What is the present value of the following cash flows if the interest rate is 6% compounded
annually?
Period
CF
0
1
2
3
4
5
6
7
$100 $200 $300
8
9
10
PV0 = $100( ) + $200( ) + $300( ) =
6.2
Level Cash Flows: Annuities and Perpetuities

There are many situations in which both businesses and individuals would be faced with
either receiving or paying a constant amount for a length of period.

When a firm faces a stream of constant payments on a bank loan for a period of time, we
call that stream of cash flows an annuity.

Individual investors may make constant payments on their home or car loans, or
invest a fixed amount year after year to save for their retirement.

Any financial contract that calls for equally spaced and level cash flows over a finite
number of periods is called an annuity.

If the cash flow payments continue forever, the contract is called a perpetuity.

Constant cash flows that occur at the end of each period are called ordinary annuities.
A.
Present Value of an Annuity
 We can calculate the present value of an annuity the same way as we calculated the
present value of multiple cash flows. However, if the number of payments were to be
very large, then this process will be tedious.
 Instead we can simplify Equation 5.4 to obtain an annuity factor. This results in
Equation 6.1, which can be used to calculate the present value of an annuity.
Prepared by Jim Keys
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1

1  (1  i) n
PVA n  CF 
i





1  (1  i) -n 

CF



i




In addition to using this annuity equation to solve for the present value of an annuity,
financial calculators and spreadsheets may be used. Present value and annuity tables
created with the help of Equation 6.1 have limited use outside of a classroom setting.

One problem that is widely solved using a financial calculator is finding the monthly
payment on a car loan or home loan.
♦ Present Value for Annuity Cash Flows
Present Value of an Annuity with Annual Payments or Deposits
If you deposit $1,000 at the end of each of the next five years, what is the present value of
these deposits if the interest rate is 6%?
or
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or

1 - ( 1 + .06 )-5 
1  (1  i) -n 
=
$1,000
PVAn  CF 

 = $1,000(4.212363786) = $4,212.36

.06
i




Note: The cash flow, CF, is defined as the periodic level cash flow.
or

PVAn = CF(PVIFA 6%,5) = $1,000(4.21236) = $4,212.36

Financial calculator:
n
N
5
i
I/YR
6
PVA
PV
4,212.36
CF
PMT
-1,000
FV
FV
0
Present Value of an Annuity with Non-annual Payments or Deposits
If you deposit $500 every six months over the next five years, what is the present value of
these deposits if the interest rate is 6%?
Note: By default, the frequency of the deposits equals the frequency of the
compounding/discounting. In this case, there are two cash flows per year, therefore, the
discounting frequency is semi-annual.
.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Period 0
$500 $500 $500 $500 $500 $500 $500 $500 $500 $500
CF


1 -  1 + i/m  - (m x n) 
1 -  1 + .06/2  - (5)(2) 
=
PVAn = CF 
$500


 = $500(8.530202837) =
i/m
.06/2




$4,265.10
Financial calculator:
n
N
5 x 2 = 10
i
I/YR
6/2 = 3
PVA
PV
4,265.10
CF
PMT
-500
FV
FV
0
Solving for the Annual Payment/Deposit when the Present Value of the Annuity is Given
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You have $10,000 in your bank account earning a 7% rate of interest. You would like to
withdraw an equal amount of money at the end of the next eight years. What is the cash flow
amount that you can withdraw each year? Note: The loan amount is the present value of an
annuity.
PVAn

CF =

Financial calculator:
=
1 - ( 1 + i ) 


i


-n
n
N
8
$10,000
1 - ( 1 + .07 ) 


.07


-8
i
I/YR
7
=
$10,000
= $1,674.68
5.971298506
PVA
PV
-10,000
CF
PMT
1,674.68
FV
FV
0
Solving for the Non-annual Payment/Deposit when the Present Value of the Annuity is Given
If you can obtain a 30-year mortgage at 6.5%, what is your monthly payment on a $250,000
mortgage?


CF =
PVAn
1 - ( 1 + i/m )

i/m

$1,580.17
- (m x n)
=



$250,000
1 - ( 1 + .065/12 )

.065/12

- (30)(12)



=
$250,000
=
158.21081957
Financial calculator:
n
N
30 x 12
i
I/YR
6.5/12
PVA
PV
250,000
CF
PMT
-1,580.17
FV
FV
0
Total paid over the life of the loan = $1,580.17 x 360 payments = $568,861.20
Interest = $568,861.20 - $250,000 = $318,861.20
If you can obtain a 15-year mortgage at 6.5%, what is your monthly payment on a $250,000
mortgage?


CF =
PVAn
1 - ( 1 + i/m )

i/m

$2,177.77
- (m x n)



=
$250,000
1 - ( 1 + .065/12 )

.065/12

- (12)(15)



=
$250,000
=
114.796412038
Financial calculator:
Prepared by Jim Keys
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n
N
15 x 12
i
I/YR
6.5/12
PVA
PV
250,000
CF
PMT
-2,177.77
FV
FV
0
Total paid over the life of the loan = $2,177.77 x 180 payments = $391,998.60
Interest = $391,998.60 - $250,000 = $141,998.60
B.
Preparing a Loan Amortization Schedule

Amortization refers to the way the borrowed amount (principal) is paid down over
the life of the loan.

The monthly loan payment is structured so that each month a portion of the principal
is paid off and at the time the loan matures, the loan is entirely paid off.

With an amortized loan, each loan payment contains some payment of principal and
an interest payment.

A loan amortization schedule is just a table that shows the loan balance at the
beginning and end of each period, the payment made during that period, and how
much of that payment represents interest and how much represents repayment of
principal.

With an amortized loan, a bigger proportion of each month’s payment goes toward
interest in the early periods. As the loan gets paid down, a greater proportion of each
payment is used to pay down the principal.
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
Amortization schedules are best done on a spreadsheet (see Exhibit 6.5) or on the
web:
http://www.webmath.com/amort.html
Suppose a business takes out a $5,000, five-year loan at 9 percent. The loan will be
amortized by making equal end-of-year payments. Determine the amount of each payment
and create an amortization showing the interest and principal paid for each period.

CF =
PVAn
1 - ( 1 + i )-n 


i


=
$5,000
1 - ( 1 + .09 )-5 


.09


=
$5,000
= $1,285.46
3.889651263
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
Loan Amortization Using a Spreadsheet - Loan amortization is a very common
spreadsheet application. To illustrate, we will set up the problem that we have just
examined, a five-year, $5,000, 9 percent loan with constant payments. Our spreadsheet
looks like this:
C.
Finding the Interest Rate

The annuity equation can also be used to the find the interest rate or discount rate for
an annuity.
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
To determine the rate of return for the annuity, we need to solve the equation for the
unknown value i.

Other than using a trial-and-error approach, it is easier to solve using this with a
financial calculator.
Solving for Time Periods and Interest Rate in Annuities (Present Value Given)
How long will it take you to pay off a $28,000 auto loan if you make yearly payments of
$6,000 and the bank is charging you 5.9% interest?

Financial calculator:
n
i
N
I/YR
5.6178*
5.9
PVA
PV
28,000
CF
PMT
-6,000
FV
FV
0
*5.6178 years (5 years, 226 days)
How long will it take you to pay off a $28,000 auto loan if you make monthly payments of
$500 and the bank is charging you 5.9% interest?
n
N
65.6613*
i
I/YR
5.9/12
PVA
PV
28,000
CF
PMT
-500
FV
FV
0
*65.6613 months = 65.6613 / 12 = 5.4718 years (5 years, 172 days)
You won the Florida Lottery when the advertised prize was $3,000,000. If you choose the
“lump sum” option you will be paid $1,500,000 immediately. Instead, you choose to receive
thirty annual payments in the amount of $100,000. What is the implied rate of return?
n
N
30
i
I/YR
5.2166
PVA
PV
-1,500,000
CF
PMT
100,000
FV
FV
0
If the Lottery Commission agreed to pay you $25,000 each quarter, how would this affect
your rate of return?
n
N
30 x 4 = 120
i
I/YR
1.3320*
PVA
PV
-1,500,000
CF
PMT
25,000
FV
FV
0
*1.3320 is the quarterly (periodic) rate; annual rate = 1.3320 x 4 = 5.288%
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D.
Future Value of an Annuity

Future value annuity calculations usually involve finding what a savings or an
investment activity is worth at some point in the future.

This could be saving periodically for a vacation, car, or house, or even retirement.

We can derive the future value annuity equation from the present value annuity
equation (Equation 6.1). This results in Equation 6.2, as follows.
 (1  i) n  1
FVA n  CF  

i



As with present value annuity calculations, future value calculations are made easier
when financial calculators or spreadsheets are used, especially when lengthy
investment periods are involved.
Future Value of an Annuity with Annual Payments or Deposits
If you deposit $2,000 at the end of each of the next five years, what is the future value of
these deposits if the interest rate is 10%?
Time line for $2,000 per year for five years
Future value calculated by compounding forward one period at a time
Future value calculated by compounding each cash flow separately
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
 ( 1 + i ) n  1
 ( 1 + .10 )5  1
=
FVAn = CF 
$2,000


 = $2,000(6.1051) = $12,210.20
i
.10




Note: The cash flow, CF, is defined as the periodic level cash flow.
or

FVAn = CF(FVIFA 10%,5) = $2,000(6.10510) = $12,210.20

Financial calculator:
n
N
5
i
I/YR
10
PV
PV
0
CF
PMT
-2,000
FVA
FV
12,210.20
Future Value of an Annuity with Non-annual Payments or Deposits
If you deposit $500 every three months over the next five years, what is the future value of
these deposits if the interest rate is 10%?
Note: By default, the frequency of the deposits equals the frequency of the
compounding/discounting. In this case, there are four cash flows per year, therefore, the
compounding frequency is quarterly.
.50
.75
1
4
4.25 4.50 4.75
5
Period 0 .25
… … …
$500 $500 $500 $500 … … … $500 $500 $500 $500 $500
CF

  1 + i/m  (m x n) - 1
  1 + .10/4  (4)(5) - 1
=
FVAn = CF 
$500


 = $500(25.54465761) =
i/m
.10/4




$12,772.33
Prepared by Jim Keys
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
Financial calculator:
n
N
5x4
i
I/YR
10/4
PV
PV
0
CF
PMT
-500
FVA
FV
12,772.33
Solving for the Annual Payment/Deposit when the Future Value of the Annuity is Given
You wish to have $2,500,000 in your investment account forty years from now when you
retire. You plan to accumulate this sum by making end-of-year deposits into a mutual fund.
If the fund earns a rate of return of 12%, how much must you contribute each year?
FVAn

CF =

Financial calculator:
 ( 1 + i )n  1


i


n
N
40
=
$2,500,000
 ( 1 + .12 )40  1


.12


i
I/YR
12
$2,500,000
= $3,259.06
767.0914203
=
PV
PV
0
CF
PMT
-3,259.06
FVA
FV
2,500,000
Solving for the Non-annual Payment/Deposit when the Future Value of the Annuity is Given
If you make weekly deposits to the retirement fund mentioned above, how much must you
contribute each week?


CF =
FVAn
 ( 1 + i/m )

i/m

per week
(m x n)
 1


=
$2,500,000
 ( 1 + .12/52 )

.12/52

(52)(40)
 1


=
$2,500,000
= $48.14
51,930.8076686
Financial calculator:
n
N
40 x 52 =
2080
i
I/YR
12/52 =
.230769231…
PV
PV
CF
PMT
FVA
FV
0
-48.14*
2,500,000
*$48.14 per week totals $2,503.33 per year
Solving for Time Periods and Interest Rate in Annuities (Future Value Given)
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How long will it take you to save for a $20,000 down payment on a home if you make
yearly deposits of $7,500 and the bank is paying you 5% interest?

Financial calculator:
n
N
2.5653*
i
I/YR
5
PV
PV
0
CF
PMT
-7,500
FVA
FV
20,000
*2.5653 years (2 years, 206 days)
How long will it take you to save for a $20,000 down payment on a home if you make three
deposits a year of $2,500 and the bank is paying you 5% interest?
n
N
7.5722*
i
I/YR
5/3
PV
PV
0
CF
PMT
-2,500
FVA
FV
20,000
*7.5722 four month periods = 7.5722 / 3 = 2.5241 years (2 years, 191 days)
You contributed $3,000 per year to your Roth IRA account over the past twenty-five years.
The balance in your account is now $500,000. What compound annual rate of return have
you achieved?
n
N
25
i
I/YR
13.4409
PV
PV
0
CF
PMT
-3,000
FVA
FV
500,000
What would your rate of return have been if you made quarterly deposits of $750?
n
N
25 x 4 = 100
i
I/YR
3.1344*
PV
PV
0
CF
PMT
-750
FVA
FV
500,000
*3.1344% per quarter = 3.1344 x 4 = 12.5377% per year
E.
Perpetuities

A perpetuity is a constant stream of cash flows that goes on for an infinite period.

In the stock markets, preferred stock issues are considered to be perpetuities, with the
issuer paying a constant dividend to holders.
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
The equation for the present value of a perpetuity can be derived from the present
value of an annuity equation with n tending to infinity.
PVA  CF  Present va lue factor for anannuity
1

1  (1  i ) 
 CF  
i


CF

i



(1  0)
  CF 
i


One thing that should be emphasized in the relationship between the present value of
an annuity and a perpetuity is that just as a perpetuity equation was derived from the
present value annuity equation, we could also derive the present value of an annuity
from the equation for a perpetuity.
♦ Perpetuities - a level stream of cash flows that occur at equal intervals for an infinite
period of time.
Present Value of a Perpetuity
The University wishes to establish an endowment fund that will generate $50,000 per year in
scholarships. If they can earn a rate of return of 5.5%, how much will they have to raise?
PVA  PVperpetuity 
CF
$50,000

= $909,090.91
i
.055
How much would they have to raise if they could earn a rate of return of 5.5% compounded
daily?
PVA  PVperpetuity 
F.
$50,000
CF
$50,000


= $884,388.54
n
365
.056536237
[(1  i/m)  1]
[(1  .055/365)  1]
Annuity Due

When you have an annuity with the payment being incurred at the beginning of each
period rather than at the end, the annuity is called an annuity due.
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
Rent or lease payments are typically made at the beginning of each period rather than
at the end of each period.

The annuity transformation method (Equation 6.4) shows the relationship between
the ordinary annuity and the annuity due.

Each period’s cash flow thus earns an extra period of interest compared to an
ordinary annuity. Thus, the present value or future value of an annuity due is always
higher than that of ordinary annuity.
Annuity due value = Ordinary annuity value × (1 + i)
6.3
Cash Flows That Grow at a Constant Rate

In addition to constant cash flow streams, one may have to deal with cash flows that
grow at a constant rate over time.

These cash flow streams are called growing annuities or growing perpetuities.
A.
Growing Annuity

Business may need to compute the value of multiyear product or service contracts
with cash flows that increase each year at a constant rate.

These are called growing annuities.

An example of a growing annuity could be the valuation of a growing business
whose cash flows are increasing every year at a constant rate.

This equation to evaluate the present value of a growing annuity (Equation 6.5) can
be used when the growth rate is less than the discount rate.
→ Use this form of the equation if the next period’s cash flow (CF1) is given.
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PVAgrowing (n)
n
CF1   1  g  

 
1  
(i  g)   1  i  
→ Use this form of the equation if the current period’s cash flow (CF0) is
given.
PVAgrowing (n)
n
CF0 (1  g)   1  g  

 
1  
(i  g)   1  i  
Example: Modern Energy Company owns several gas stations. Management is looking to open a new station
in the western suburbs of Baltimore. One possibility they are evaluating is to take over a station located at a
site that has been leased from the county. The lease, originally for 99 years, currently has 73 years before
expiration. The gas station generated a net cash flow of $92,500 last year, and the current owners expect an
annual growth rate of 6.3 percent. If Modern Energy uses a discount rate of 14.5 percent to evaluate such
Prepared by Jim Keys
19
businesses, what is the present value of this growing annuity?
Time for lease to expire = n = 73 years
Last year’s net cash flow = CF0 = $92,500
Expected annual growth rate = g = 6.3% (constant)
Firm’s required rate of return = i = 14.5%
Expected cash flow next year = CF1 = $92,500(1 + g) = $92,500(1.063) = $98,327.50
Present value of growing annuity = PVAgrowing (n)
n
  1.063  73 
CF1   1  g  
$98,327.50
PVA n 
 1  
 1  
 
 
(i  g)   1  i   (0.145  0.063)   1.145  
 $1,199,115.85  0.9955931549
 $1,193,831.53
B.
Growing Perpetuity

When the cash flow stream features a constant growing annuity forever, it is called a
growing perpetuity.

This can be derived from Equation 6.5 when n tends to infinity and results in
Equation 6.6.
→ Use this form of the equation if the next period’s cash flow (CF1) is given.
PVA 
CF1
(i  g)
→ Use this form of the equation if the current period’s cash flow (CF0) is
given.
PVA 
CF0 (1  g)
(i  g)
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20
Example: You are offered an investment that will pay perpetual cash flows. The first cash flow of
$12,000, beginning one year from today, will increase at a constant rate of 3.5%. What are you
willing to pay for this investment if your required rate of return is 7.25%?
Term of the cash flows = n = ∞
Next year’s net cash flow = CF1 = $12,000
Expected annual growth rate = g = 3.5% (constant)
Your required rate of return = i = 7.25%
Present value of growing perpetuity = PVAgrowing (∞)
PVA 
CF1
$12,000
$12,000


 $320,000
(i  g) (.0725  .035)
.0375
What is the value of the cash flow to be received in year 2?
CF2 = CF1(1 + g) = $12,000(1.035) = $12,420
6.4
The Effective Annual Interest Rate

Interest rates can be quoted in the financial markets in a variety of ways.

The most common quote, especially for a loan, is the annual percentage rate (APR).

The APR is a rate that represents the simple interest accrued on a loan or an investment
in a single period. This is annualized over a year by multiplying it by the appropriate
number of periods in a year.
A.
Calculating the Effective Annual Interest Rate (EAR)

The correct way to compute an annualized rate is to reflect the compounding that
occurs. This involves calculating the effective annual rate (EAR).
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21

The effective annual interest rate (EAR) is defined as the annual growth rate that
takes compounding into account.

Equation 6.7 shows how the EAR is computed.
EAR = (1 + Quoted rate/m)m – 1,
where, m is the number of compounding periods during a year.

The EAR conversion formula accounts for the number of compounding periods and,
thus, effectively adjusts the annualized interest rate for the time value of money.

The EAR is the true cost of borrowing and lending.
Example: Your credit card company charges you 18.99%, but requires you to make monthly
payments. What is the effective annual rate you are paying?

EAR = (1  i/m) m - 1 = (1  .1899/12)12 - 1 = .207332 = 20.73%
♦ Calculating and Comparing Effective Annual Rates
You have compiled the following information on 1-year CD rates from three local banks.
Which bank will you choose to deposit your money?
Bank
A
B
C
APR
4.10%
4.08%
4.06%
m
Annual
Quarterly
Continuously
EAR
1
EARA = (1  .041/1) - 1 =
EARB = (1  .0408/4) 4 - 1 =
EARC = ei - 1 = (2.71828).0406 -1 =

Financial calculator:
Your credit card company charges you 18.99%, but requires you to make monthly
payments. What is the effective annual rate you are paying?
m
P/YR
12
APR
NOM%
18.99
Prepared by Jim Keys
EAR
EFF%
20.7332
22
Spreadsheet: EFFECT(nominal_rate,npery)
B.
Consumer Protection Acts and Interest Rate Disclosure

Congress passed the Truth-in-Lending Act in 1968 to ensure that the true cost of
credit was disclosed to consumers so that they could make sound financial decisions.

Similarly, another piece of legislation called the Truth-in-Savings Act was passed
to provide consumers with an accurate estimate of the return they would earn on an
investment.

These two pieces of legislation require by law that the APR be disclosed on all
consumer loans and savings plans and that it be prominently displayed on advertising
and contractual documents.

It is important to note that the EAR, not the APR, is the appropriate rate to use in
present and future value calculations.
Prepared by Jim Keys
23
Chapter 6 Sample Questions
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. FV of multiple cash flows: Chandler Corp. is expecting a new project to start producing cash
flows, beginning at the end of this year. They expect cash flows to be as follows:
1
$458,357
2
$527,111
3
$585,093
4
$672,857
5
$787,243
If they can reinvest these cash flows to earn a return of 5.2 percent, what is the future value of this
cash flow stream at the end of five years?
a.
b.
c.
d.
$3,616,289.15
$3,052,280.75
$3,317,696.47
$4,379,359.34
2. PV of multiple cash flows: Ferris, Inc., has borrowed from their bank at a rate of 7 percent and will
repay the loan with interest over the next five years. Their scheduled payments, starting at the end
of the year are as follows—$550,000, $640,000, $730,000, $840,000, and $960,000. What is the
present value of these payments?
a.
b.
c.
d.
$2,425,314.67
$3,113,984.26
$2,994,215.64
$2,814,562.70
3. Present value of an annuity: Lorraine Jackson won a lottery. She will have a choice of receiving
$50,000 at the end of each year for the next 40 years, or a lump sum today. If she can earn a return
of 4 percent on any investment she makes, what is the minimum amount she should be willing to
accept today as a lump-sum payment?
a.
b.
c.
d.
$870,882.05
$989,638.69
$1,345,908.62
$1,296,426.69
4. Future value of an annuity: Jayadev Athreya has started on his first job. He plans to start saving
for retirement early. He will invest $3,500 at the end of each year for the next 40 years in a fund that
will earn a return of 9 percent. How much will Jayadev have at the end of 40 years?
a.
b.
c.
d.
$1,076,155.59
$1,182,588.56
$969,722.62
$1,620,146.32
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24
5. Computing annuity payment: Maricela Sanchez needs to have $34,000 in three years. If she can
earn 5 percent on any investment, what is the amount that she will have to invest at the end of every
year for the next three years?
a.
b.
c.
d.
$14,452.02
$9,922.28
$12,187.15
$10,785.09
6. Computing annuity payment: Jackson Electricals has borrowed $39,150 from its bank at an
annual rate of 5.2 percent. It plans to repay the loan in three equal installments, beginning at the end
of next year. What is its annual loan payment?
a.
b.
c.
d.
$12,554.21
$14,430.12
$16,883.24
$17,604.75
7. Computing annuity payment: Trevor Smith wants to have $2,750,000 at retirement, which is 31
years away. He already has $200,000 in an IRA earning 5 percent annually. How much does he
need to save each year, beginning at the end of this year, to reach his target? Assume he could earn
5 percent on any investment he makes.
a.
b.
c.
d.
$26,036.91
$23,172.85
$18,486.20
$34,108.35
8. Perpetuity: Roger Barkley wants to set up a scholarship at his alma mater. He is willing to invest
$100,000 in an account earning 9.25 percent. What will be the annual scholarship that can be given
from this investment?
a.
b.
c.
d.
$9,750
$1,081,081
$6,750
$9,250
9. Perpetuity: Chris Collinge has funded a retirement investment with $250,000 earning a return of
5.75 percent. What is the value of the payment that he can receive in perpetuity? (Round to the
nearest dollar.)
a. $12,150
b. $15,250
c. $14,375
d. $14,900
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25
10. Growing perpetuity: Jack Benny is planning to invest in an insurance company product. The
product will pay $10,000 at the end of this year. Thereafter, the payments will grow annually at a 3
percent rate forever. Jack will be able to invest his cash flows at a rate of 6.5 percent. What is the
present value of this investment cash flow stream? (Round to the nearest dollar.)
a. $326,908
b. $312,766
c. $285,714
d. $258,133
11. Growing annuity: Hill Enterprises is expecting tremendous growth from its newest boutique store.
Next year the store is expected to bring in net cash flows of $675,000. The company expects its
earnings to grow annually at a rate of 13 percent for the next 15 years. What is the present value of
this growing annuity if the firm uses a discount rate of 18 percent on its investments? (Round to the
nearest dollar.)
a. $6,448,519
b. $6,750,000
c. $7,115,449
d. $5,478,320
12. Effective annual rate: Desire Cosmetics borrowed $152,300 from a bank for three years. If the
quoted rate (APR) is 11.75 percent, and the compounding is daily, what is the effective annual rate
(EAR)? (Round to one decimal place.)
a. 11.75%
b. 14.3%
c. 12.5%
d. 11.6%
13. The condominium at the beach that you want to buy costs $249,500. You plan to make a cash down
payment of 20 percent and finance the balance over 10 years at 6.75 percent. What will be the
amount of your monthly mortgage payment?
a.
b.
c.
d.
$2,291.89
$2,809.10
$3,287.46
$3,412.67
Prepared by Jim Keys
26
Chapter 6 Sample Questions
Answer Section
MULTIPLE CHOICE
1. ANS: C
Learning Objective: LO 1
Level of Difficulty: Medium
Feedback:
0
1
$458,357
2
$527,111
n = 5;
3
$585,093
i = 5.2%
4
$672,857
5
$787,243
FV5 = $458,357(1.052)4 + $527,111(1.052)3 + $585,093(1.052)2
+ $672,857(1.052)1 + $787,243(1.052)0
FV5 = $561,392.79 + $613,690.36 + $647,524.76 + $707,845.56 + $787,243.00
FV5 = $3,317,696.47
PTS: 1
2. ANS: C
MSC: JDK
Learning Objective: LO 1
Level of Difficulty: Medium
Feedback:
0
1
$550,000
2
$640,000
n = 5;
3
$730,000
i = 7%
4
$840,000
5
$960,000
PV = $550,000(1.07)-1 + $640,000(1.07)-2 + $730,000(1.07)-3
+ $840,000(1.07)-4 + $960,000(1.07)-5
PV = $514,018.69 + $559,000.79 + $595,897.45 + $640,831.98 + $684,466.73
PV = $2,994,215.64
PTS: 1
3. ANS: B
MSC: JDK
Learning Objective: LO 2
Level of Difficulty: Medium
Feedback:
0
1
$50,000
2
3
$50,000
$50,000
n = 40;
i = 4%
...
...
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39
$50,000
40
$50,000
27
= $989,638.69
Key:
Enter:
Solve For:
n
N
40
i
I/YR
4
PVA
PV
CF
PMT
50,000
FV
FV
0
-989,638.69
PTS: 1
4. ANS: B
MSC: JDK
Learning Objective: LO 2
Level of Difficulty: Medium
Feedback:
0
1
$3,500
2
$3,500
3
$3,500
n = 40;
i = 9%
...
...
39
$3,500
40
$3,500
= $1,182,588.56
Key:
Enter:
Solve For:
n
N
40
i
I/YR
9
PV
PV
0
CF
PMT
-3,500
FVA
FV
1,182,588.56
PTS: 1
5. ANS: D
MSC: JDK
Learning Objective: LO 2
Level of Difficulty: Medium
Feedback:
0
1
$CF
2
$CF
n = 3;
3
$CF
i = 5%
4
$0
5
$0
= $10,785.09
Key:
Enter:
Solve For:
n
N
3
i
I/YR
5
PV
PV
0
CF
PMT
FVA
FV
34,000
-10,785.09
Prepared by Jim Keys
28
PTS: 1
6. ANS: B
MSC: JDK
Learning Objective: LO 2
Level of Difficulty: Medium
Feedback:
0
1
$CF
2
$CF
3
$CF
4
$0
n = 3;
5
$0
i = 5.2%
6
$0
7
$0
8
$0
9
$0
= $14,430.12
Key:
Enter:
Solve For:
n
N
3
i
I/YR
5.2
PV
PV
39,150
CF
PMT
FVA
FV
0
-14,430.12
Each payment made by Jackson Electricals will be $14,430.12, starting at the end of next year.
PTS: 1
7. ANS: A
MSC: JDK
Learning Objective: LO 2
Level of Difficulty: Medium
Feedback:
Retirement investment target in 31 years = $2,750,000
Amount invested in IRA account now = PV = $200,000
Return earned by investment = i = 5%
Value of current investment in 31 years = FV31 = $200,000(1 + .05)31 = $907,607.90
Balance of money needed = $2,750,000 - $907,607.90 = $1,842,392.10 = FVA
Payment needed to reach target = CF = PMT
= $26,036.91
Key:
Enter:
Solve For:
PTS: 1
8. ANS: D
n
N
31
i
I/YR
5
PV
PV
-200,000
CF
PMT
FVA
FV
2,750,000
-26,036.91
MSC: JDK
Learning Objective: LO 3
Level of Difficulty: Medium
Prepared by Jim Keys
29
Feedback:
Annual payment needed = PMT = CF
Present value of investment = PVA = $100,000
Investment rate of return = i = 9.25%
Term of payment = Perpetuity
PMT = CF = PVperpetuity x i = $100,000 x .0925 = $9,250
PTS: 1
9. ANS: C
MSC: JDK
Learning Objective: LO 3
Level of Difficulty: Medium
Feedback: Annual payment needed = PMT
Present value of investment = PVA = $250,000
Investment rate of return = i = 5.75%
Term of payment = Perpetuity
PTS: 1
10. ANS: C
Learning Objective: LO 4
Level of Difficulty: Medium
Feedback: Cash flow at t=1 = CF1 = $10,000
Annual growth rate = g = 3%
Discount rate = i = 6.5%
Present value of growing perpetuity = PVA¥
PTS: 1
11. ANS: A
Learning Objective: LO 4
Level of Difficulty: Medium
Feedback: Time of growth = n = 15 years
Next year's expected net cash flow = CF1 = $675,000
Expected annual growth rate = g = 13%
Firm's required rate of return = i = 18%
Present value of growing annuity = PVAn
Prepared by Jim Keys
30
PTS: 1
12. ANS: C
Learning Objective: LO 5
Level of Difficulty: Medium
Feedback: Loan amount = PV = $152,300
Interest rate on loan = i = 11.75%
Frequency of compounding = m = 365
Effective annual rate = EAR
PTS: 1
13. ANS: A
PTS: 1
MSC: JDK
Prepared by Jim Keys
31
Chapter 6 (Additional Problems)
1. You are given the following cash flow information. The discount rate is 11.15 percent and the payments are
received at the end of each year.
Year
1–5
6–10
Amount
$3,000
$9,000
What would you be willing to pay right now to receive the income stream above?
a. $30,579.40
b. $13,760.73
c. $17,124.46
d. $42,199.57
e. $49,844.42
2. You make deposits to your investment account as shown in the table below. Your rate of return is 11.10 percent
and the deposits are made at the end of each year.
Year
1–5
6–10
Amount
$3,000
$7,000
What is the value of your account at the end of Year 10?
a. $75,369.05
b. $108,531.43
c. $145,462.27
d. $110,792.51
e. $104,009.29
3. You have just taken out a 30-year, $100,000 mortgage on your new home. This mortgage is to be repaid in equal
end-of-month installments. If the nominal interest rate on the mortgage is 5.43 percent, what is the amount of each
monthly payment?
a. $277.78
b. $563.40
c. $642.28
d. $794.40
e. $997.23
4. You have just taken out a 15-year, $375,000 mortgage on your new home. This mortgage is to be repaid in equal
end-of-month installments. If the nominal interest rate on the mortgage is 9.52 percent, what is the total amount of
interest you will pay over the life of the mortgage?
a. $330,666.50
b. $419,946.45
c. $426,559.78
d. $380,266.47
e. $410,026.46
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32
5. Growing perpetuity: Jack Benny is planning to invest in an insurance company product. The product
will pay $2,750 at the end of this year. Thereafter, the payments will grow annually at a 2 percent rate
forever. Jack will be able to invest his cash flows at a rate of 7.1 percent. What is the present value of this
investment cash flow stream?
a.
b.
c.
d.
$61,200.98
$53,921.57
$69,990.20
$44,053.92
6. Growing annuity: Hill Enterprises is expecting tremendous growth from its newest boutique store.
Next year the store is expected to bring in net cash flows of $625,000. The company expects its earnings
to grow annually at a rate of 9 percent for the next 5 years. What is the present value of this growing
annuity if the firm uses a discount rate of 10 percent on its investments?
a.
b.
c.
d.
$2,789,723.64
$3,540,159.30
$3,442,518.97
$3,063,116.56
Prepared by Jim Keys
33
Chapter 6 (Additional Problems)
Answer Section
MULTIPLE CHOICE
1. ANS:
TOP:
2. ANS:
TOP:
3. ANS:
TOP:
4. ANS:
TOP:
5. ANS:
A
PTS: 1
DIF:
PV of an uneven CF stream
A
PTS: 1
DIF:
FV of an uneven CF stream
B
PTS: 1
DIF:
Mortgage Payment
A
PTS: 1
DIF:
Total Interest Paid Over Life of Mortgage
B
Medium
OBJ: TYPE: Problem
Medium
OBJ: TYPE: Problem
Medium
OBJ: TYPE: Financial Calculator
Medium
OBJ: TYPE: Financial Calculator
Learning Objective: LO 4
Level of Difficulty: Medium
Feedback: Cash flow at t = 1 = CF1 = $2,750
Annual growth rate = g = 2%
Discount rate = i = 7.1%
Present value of growing perpetuity = PVA(Growing perpetuity)
PTS: 1
6. ANS: A
MSC: JDK
Learning Objective: LO 4
Level of Difficulty: Medium
Feedback: Time for lease to expire = n = 5 years
Next year's net cash flow = CF1 = $625,000
Expected annual growth rate = g = 9%
Firm's required rate of return = i = 10%
Present value of growing annuity = PVA(Growing annuity)
PTS: 1
MSC: JDK
34