ch06

Chapter 6Discounted 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.

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.

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.

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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.

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

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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.

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.

Amortization schedules are best done on a spreadsheet (see Exhibit 6.5).

C. Finding the Interest Rate

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The annuity equation can also be used to the find the interest rate or discount rate

for an annuity.

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.

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.

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.

E. Perpetuities

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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.

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.

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.

F. 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.

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.

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Annuity due = 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.

B. Growing Perpetuity

When the cash flow stream features a constant growing annuity forever, it is

called a growing perpetuity.

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This can be derived from Equation 6.5 when n tends to infinity and results in

Equation 6.6.

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).

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.

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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.

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II. Suggested and Alternative Approaches to the Material

This chapter begins with a discussion of present value and future value computations when a

stream of cash flows, not all being equal, is involved. This is followed by an analysis of

situations when the recurring cash flows over time are constant—namely, annuities. Both present

value and future value of an annuity are developed in detail. In addition, the cases of a

perpetuity, growing annuity, and growing perpetuity are also covered. Finally, the discussion

evolves around the merits of annual percentage rates and effective annual rates.

As in the last chapter, the instructor has the flexibility to cover all or some of the

concepts. Some may choose to cover the chapter in full, whereas others may focus their

discussion on the computation of the present value and future value of uneven and level cash

flow streams only.

The end of the chapter presents a large number of exercises that can be utilized to help

students learn the basic concepts in this chapter before moving on to other topics.

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III. Summary of 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 present value

and future value for multiple cash flows.

When making decisions involving cash flows over time, we should first identify the

magnitude and timing of the cash flows and then discount each individual cash flow to its

present value. The process of discounting the cash flows adjusts them for the time value

of money, because today’s dollars are not equal in value to dollars in the future. Once all

of the cash flows are in present value terms, they can be compared to make decisions.

Section 6.1 discusses the computation of present values and future values of 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.

An ordinary annuity is a series of equally spaced level cash flows over time. The cash

flows for an ordinary annuity are assumed to take place at the ends of the periods. To find

the value of an ordinary annuity, we start by calculating the annuity factor, which is equal

to (1 – present value factor)/i. Then, we multiply this factor by the constant future

payment. An annuity due is an annuity in which the cash flows occur at the beginnings of

the periods. A lease is an example of an annuity due. In this case, we are effectively

prepaying for the service. To calculate the value of an annuity due, we multiply the

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ordinary annuity value times (1 + i). Section 6.2 discusses the calculation of the present

value of annuity and 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.

A perpetuity is like an annuity except that the cash flows are perpetual—they never end.

British Treasury Department bonds, called consols, were the first widely used securities

of this kind. The most common example of perpetuity today is preferred stock. The issuer

of preferred stock promises to pay fixed rate dividends forever. The preferred

stockholders must be paid before common stockholders. To calculate the present value of

a perpetuity, we simply divide the promised constant dividend payment (CF) by the

interest rate (i).

4. Discuss growing annuities and perpetuities, as well as their application in business,

and be able to calculate their value.

Financial managers often need to value cash flow streams that increase at a constant rate

over time. These cash flow streams are called growing annuities or growing perpetuities.

An example of a growing annuity would be a 10-year lease contract with an annual

adjustment for the expected rate of inflation over the life of the contract. If the cash flows

continue to grow at a constant rate indefinitely, this cash flow stream is called a growing

perpetuity. Application and calculation of cash flows that grow at a constant rate are

discussed in Section 6.3.

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5. Discuss why the effective annual interest rate (EAR) is the appropriate way to

annualize interest rates, and be able to calculate EAR.

The EAR is the annual growth rate that takes compounding into account. Thus, the EAR

is the true cost of borrowing or lending money. When we need to compare interest rates,

we must make sure that the rates to be compared have the same time and compounding

periods. If interest rates are not comparable, they must be converted into common terms.

The easiest way to convert rates to common terms is to calculate the EAR for each

interest rate. The use and calculations of EAR are discussed in Section 6.4.

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IV. Summary of Key Equations

Equation Description Formula

6.1 Present value of an ordinary annuity

PVAn = CF [1 – Present value factor] /i

= CF {1 – [1/(1 + i)n]}/i

= CF PV annuity factor

6.2 Future value of an ordinary annuity

FVAn= CF [Future value factor – 1]/i

= CF [(1 + i)n – 1]/i

= CF FV annuity factor

6.3 Present value of a perpetuityPVA∞ = CF/i

6.4 Value of an annuity dueAnnuity due value = Ordinary annuity value (1 + i)

6.5Present value of a growing annuity

6.6 Present value of a growing perpetuity PVA∞ = CF1/(i – g)

6.7 Effective annual interest rate EAR = (1 + Quoted interest rate/m)m – 1

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V. Before You Go On Questions and Answers

Section 6.1

1. Explain how to calculate the future value of a stream of cash flows.

It would helpful to first construct a time line so that we can identify the timing of each

cash flow. Then you would calculate the future vale of each individual cash flow. Finally,

you would add up the future values of all the individual cash flows to determine the

future value of the cash flow stream.

2. Explain how to calculate the present value of a stream of cash flows.

To calculate the present value of a stream of cash flows, you should first draw a time line

so that you can see that each cash flow is placed in its correct time period. Then you

simply calculate the present value of each cash flow for its time period, and finally you

add up all the present values.

3. Why is it important to adjust all cash flows to a common date?

When making economic decisions, we need to compare “apples to apples.” This is

possible only when we bring all the cash flows to a common date, which can either be a

present time or some future date. The reason is the time value of money: a dollar today is

worth more than a dollar in the future. Thus, when cash flows are converted to the same

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time period, the time value of money concept holds true, and we can concentrate on the

economic aspects of the decision.

Section 6.2

1. How do an ordinary annuity, an annuity due, and a perpetuity differ?

Ordinary annuity assumes that the cash flows occur at the end of a period. Most types of

loans are ordinary annuities. On the other hand, annuity due is an annuity whose payment

is to be made immediately (or at the beginning of a period) instead of at the end of the

period. For example, in many leases the first payment is due immediately, and each

successive payment must be made at the beginning of the month. Perpetuity is a special

case of annuity, and it refers to a constant stream of identical cash flows with no end.

2. Give two examples of perpetuities.

The text gives the example of British government bonds called consols that have no

maturity and have been traded in the markets since the end of the Napoleonic wars.

Another example could be a preferred stock of a company that has no maturity and will

pay a constant dividend forever.

3. What is the annuity transformation method?

The annuity transformation method refers to the conversion of an ordinary annuity to

annuity due. In this process, you first plot all the cash flows on a time line as if the cash

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flows were an ordinary annuity. Then you calculate the present or future value factor as

you would with an ordinary annuity, and finally, you multiple your answer by (1 + i).

Conveniently, this relationship works for both present and future value calculations.

Section 6.3

1. What is the difference between a growing annuity and a growing perpetuity?

A stream of cash flows that is growing at a constant rate over time can be called a

growing annuity or growing perpetuity. If the cash flows extend over a finite length of

time, then we call it a growing annuity and can use Equation 7.5 to compute the present

value. If the growth will continue for a very long time period and perhaps, forever, we

refer to it as the growing perpetuity. We would then use Equation 7.6 to estimate the

present value of this cash flow stream.

Section 6.4

1. What is the APR, and why are lending institutions required to disclose this rate?

APR, or the annual percentage rate, is the annualized interest rate using simple interest. It

is defined as the simple interest charged per period multiplied by the number of

compounding periods per year. Lending institutions are mandated by federal Truth-in-

Lending Act regulations to disclose this rate to essentially make it easier for consumers to

be exposed to the same kind of rate by all businesses.

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2. What is the correct way to annualize an interest rate in financial decisions?

The correct way to annualize interest rates is by computing the effective annual interest

rate (EAR). This is the annual growth rate that allows for compounding, which means

you earn interest on interest. To calculate the EAR, take the quoted rate and divide it by

the number of compounding periods (quoted rate/m). Then take the resulting interest rate,

add 1 to it, and raise it to the power equal to m. Finally, subtract 1 and the result is EAR.

3. Distinguish between quoted interest rate, interest rate per period, and effective annual interest

rate.

Quoted interest rate, such as APR, is the interest rate that has been annualized by

multiplying the rate per period by the number of compounding periods. Interest rate per

period is the quoted rate per period. It can be stated in the form of an APR—in that case,

just divide it by the number of compounding periods to obtain the interest rate per period.

Finally, EAR is the annual rate of interest that accounts for the effects of compounding.

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VI. Self-Study Problems

6.1 Kronka, Inc., is expecting cash flows of $13,000, $11,500, $12,750, and $9,635 over the

next four years. What is the present value of these cash flows if the appropriate discount

rate is 8 percent?

Solution:

The time line for the cash flows and their present value is as follows:

0 8% 1 2 3 4

├─────────┼─────────┼─────────┼─────────┤

$13,000 $11,500 $12,750 $9,635

6.2 Your grandfather has agreed to deposit a certain amount of money each year into an

account paying 7.25 percent annually to help you go to college. Starting next year, he

plans to deposit $2,250, $8,150, $7,675, $6,125, and $12,345 in to the account. How

much will you have at the end of the five years?

Solution:

The time line for the cash flows and their future value is as follows:

0 7.25% 1 2 3 4 5

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├─────────┼─────────┼─────────┼──────────┼─────────┤

$2,250 $8,150 $7,675 $6,125 $12,345

6.3 Mike White is planning to save up for a trip to Europe in three years. He will need $7,500

when he is ready to make the trip. He plans to invest the same amount at the end of each

of the next three years in an account paying 6 percent. What is the amount he will have to

save every year to reach his goal of $7,500 in three years?

Solution:

Amount Mike White will need in three years = FVA3 = $7,500

Number of years = n = 3

Interest rate on investment =. i = 6.0%

Amount needed to be invested every year = PMT = ?

0 6% 1 2 3

├────┼────┼────┤

FVA = $7,500

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Mike will have to save $2,353.82 every year for the next three years.

6.4 Becky Scholes has $150,000 to invest. She wants to be able to withdraw $12,500 every

year forever without using up any of her principal. What interest rate would her

investment have to earn in order for her to be able to so?

Solution:

Present value of investment = $150,000

Amount needed annually = $12,500

This is a perpetuity!

6.5 Dynamo Corp. is expecting annual payments of $34,225 for the next seven years from a

customer. What is the present value of this annuity if the discount rate is 8.5 percent?

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Solution:

0 8.5% 1 2 3 4 5 6 7

├───┼───┼────┼───┼───┼───┼───┤

PVA= ? $34,225 $34,225 $34,225 $34,225 $34,225 $34,225 $34,225

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VII. Critical Thinking Questions

6.1 Identify the steps involved in computing the future value when you have multiple cash

flows.

First, prepare a time line to identify the size and timing of the cash flows. Second,

calculate the present value of each individual cash flow using an appropriate discount

rate. Finally, add up the present values of the individual cash flows to obtain the present

value of a cash flow stream. This approach is especially useful in the real world where

the cash flows for each period are not the same.

6.2 What is the key economic principle involved in calculating the present value and future

value of multiple cash flows?

Regardless of whether you are calculating the present value or the future value of a cash

flow stream, the key idea is to discount or compound the cash flows to the same point in

time.

6.3 What is the difference between a perpetuity and an annuity?

A cash flow stream that consists of the same amount being received or paid on a periodic

basis is called an annuity. If the same payments are made periodically forever, the

contract is called a perpetuity.

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6.4 Define annuity due. Would an investment be worth more if it was an ordinary annuity or

an annuity due? Explain.

When annuity cash flows occur at the beginning of each period, it is called an annuity

due. Annuity due will result in a bigger investment than an ordinary annuity because each

cash flow will accrue an extra interest payment.

6.5 Raymond Bartz is trying to choose between two equally risky annuities, each paying

$5,000 per year for five years. One is an ordinary annuity, and the other is an annuity

due. Which of the following statements is most correct?

a. The present value of the ordinary annuity must exceed the present value of the

annuity due, but the future value of an ordinary annuity may be less than the future

value of the annuity due.

b. The present value of the annuity due exceeds the present value of the ordinary

annuity, while the future value of the annuity due is less than the future value of the

ordinary annuity.

c. The present value of the annuity due exceeds the present value of the ordinary

annuity, and the future value of the annuity due also exceeds the future value of the

ordinary annuity.

d. If interest rates increase, the difference between the present value of the ordinary

annuity and the present value of the annuity due remains the same.

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c. The present value of the annuity due exceeds the present value of the ordinary

annuity, and the future value of the annuity due also exceeds the future value of the

ordinary annuity.

6.6 Which of the following investments will have the highest future value at the end of three

years? Assume that the effective annual rate for all investments is the same.

a. You earn $3,000 at the end of three years (a total of one payment).

b. You earn $1,000 at the end of every year for the next three years (a total of three

payments).

c. You earn $1,000 at the beginning of every year for the next three years (a total of

three payments).

c. Earning $1,000 at the beginning of each year for the next three years will have the

highest future value as it is an annuity due.

6.7 Explain whether or not each of the following statements is correct.

a. A 15-year mortgage will have larger monthly payments than a 30-year mortgage of

the same amount and same interest rate.

This is a true statement. The 15-year mortgage will have higher monthly payments since

more of the principal will have to be paid each month than in the case of a 30-year

mortgage.

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b. If an investment pays 10 percent interest compounded annually, its effective rate will

also be 10 percent.

This is true since the frequency of compounding is annual and hence the rate for a single

period is the same as the rate for a year.

6.8 When will the annual percentage rate (APR) be the same as the effective annual rate

(EAR)?

The annual percentage rate (APR) will be the same as the effective annual rate only if the

compounding period is annual, not otherwise.

6.9 Why is the EAR superior to the APR in measuring the true economic cost or return?

Unlike the APR, which reflects annual compounding, the EAR takes into account the

actual number of compounding periods. For example, suppose there are two investment

alternatives that both pay an APR of 10 percent. Assume that the first pays interest

annually and that the second pays interest quarterly. It would be a mistake to assume that

both investments will provide the same return. The real return on the first one is 10

percent, but the second investment actually provides a return of 10.38 percent because of

the quarterly compounding. Thus, this is the superior investment!

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6.10 Suppose two investments have equal lives and multiple cash flows. A high discount rate

tends to favor:

a. the investment with large cash flow early.

b. the investment with large cash flow late.

c. the investment with even cash flow.

d. neither investment since they have equal lives.

a. The investment with large cash flows early will be worth more compared to the one

with the large cash flows late. The cash flows that come in later will have a heavier

penalty when using a higher discount rate. Thus the investment with large cash flows

early will be favored.

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VIII. Questions and Problems

BASIC

6.1 Future value with multiple cash flows: Konerko, Inc., expects to earn cash flows of

$13,227, $15,611, $18,970, and $19,114 over the next four years. If the company uses an

8 percent discount rate, what is the future value of these cash flows at the end of year 4?

Solution:

0 8% 1 2 3 4

├───────┼────────┼───────┼────────┤

$13,227 $15,611 $18,970 $19,114

6.2 Future value with multiple cash flows: Ben Woolmer has an investment that will pay

him the following cash flows over the next five years: $2,350, $2,725, $3,128, $3,366,

and $3,695. If his investments typically earn 7.65 percent, what is the future value of the

investment’s cash flows at the end of five years?

Solution:

0 7.65% 1 2 3 4 5

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├───────┼────────┼───────┼────────┼───────┤

$2,350 $2,725 $3,128 $3,366 $3,695

6.3 Future value with multiple cash flows: You are a freshman in college and are planning

a trip to Europe when you graduate from college at the end of four years. You plan to

save the following amounts starting today: $625, $700, $700, and $750. If the account

pays 5.75 percent annually, how much will you have at the end of four years?

Solution:

0 5.75% 1 2 3 4

├───────┼────────┼───────┼────────┤

$625 $700 $700 $750

6.4 Present value with multiple cash flows: Saul Cervantes has just purchased some

equipment for his landscaping business. He plans to pay the following amounts at the end

of the next five years: $10,450, $8,500, $9,675, $12,500, and $11,635. If he uses a

discount rate of 10.875 percent, what is the cost of the equipment he purchased today?

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Solution:

0 10.875% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$10,450 $8,500 $9,675 $12,500 $11,635

6.5 Present value with multiple cash flows: Jeremy Fenloch borrowed from his friend a

certain amount and promised to repay him the amounts of $1,225, $1,350, $1,500,

$1,600, and $1,600 over the next five years. If the friend normally discounts investments

at 8 percent annually, how much did Jeremy borrow?

Solution:

0 8% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$1,225 $1,350 $1,500 $1,600 $1,600

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6.6 Present value with multiple cash flows: Biogenesis, Inc., expects the following cash

flow stream over the next five years. The company discounts all cash flows at a 23

percent discount rate. What is the present value of this cash flow stream?

Solution:

0 23%

1

2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

-$1,133,676 -$978,452 $275,455 $878,326 $1,835,444

6.7 Present value of an ordinary annuity: An investment opportunity requires a payment of

$750 for 12 years, starting a year from today. If your required rate of return is 8 percent,

what is the value of the investment today?

Solution:

0 8% 1 2 3 11 12

├───────┼────────┼───────┼………………┼───────┤

$750 $750 $750 $750 $750

1 2 3 4 5

-$1,133,676 -$978,452 $275,455 $878,326 $1,835,444

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Annual payment = PMT = $750

No. of payments = n = 12

Required rate of return = 8%

Present value of investment = PVA12

6.8 Present value of an ordinary annuity: Dynamics Telecommunications Corp. has made

an investment in another company that will guarantee it a cash flow of $22,500 each year

for the next five years. If the company uses a discount rate of 15 percent on its

investments, what is the present value of this investment?

Solution:

0 15% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$22,500 $22,500 $22,500 $22,500 $22,500

Annual payment = PMT = $22,500


Required rate of return = 15%

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6.9 Future value of an ordinary annuity: Robert Hobbes plans to invest $25,000 a year for

the next seven years in an investment that will pay him a rate of return of 11.4 percent.

He will invest at the end of each year. What is the amount that Mr. Hobbes will have at

the end of seven years?

Solution:

0 11.4% 1 2 3 6 7

├───────┼────────┼───────┼………………┼───────┤

$25,000 $25,000 $25,000 $25,000 $25,000

Annual investment = PMT = $25,000


Investment rate of return = 11.4%

Future value of investment = FVA7

33

6.10 Future value of an ordinary annuity: Cecelia Thomas is a sales executive at a

Baltimore firm. She is 25 years old and plans to invest $3,000 every year in an IRA

account, beginning at the end of this year until she turns 65 years old. If the IRA

investment will earn 9.75 percent annually, how much will she have in 40 years when she

turns 65 years old?

Solution:

0 9.75% 1 2 3 39 40

├───────┼────────┼───────┼………………┼───────┤

$3,000 $3,000 $3,000 $3,000 $3,000





34

6.11 Future value of an annuity. Refer to Problem 6.10. If Cecelia Thomas starts saving at

the beginning of each year, how much will she have at age 65?

Solution:

0 9.75% 1 2 3 39 40

├───────┼────────┼───────┼………………┼───────┤

$3,000 $3,000 $3,000 $3,000 $3,000



Type of annuity = Annuity due



6.12 Computing annuity payment: Kevin Winthrop is saving for an Australian vacation in

three years. He estimates that he will need $5,000 to cover his airfare and all other

expenses for a week-long holiday in Australia. If he can invest his money in an S&P 500

35

equity index fund that is expected to earn an average return of 10.3 percent over the next

three years, how much will he have to save every year, starting at the end of this year?

Solution:

0 10.3% 1 2 3

├───────┼────────┼───────┤

PMT PMT PMT

FVAn = $5,000

Future value of annuity = FVA = $5,000

Return on investment = i = 10.3%

Payment required to meet target = PMT

Using the FVA equation:

Kevin has to save $1,506.20 every year for the next three years to reach his target of

$5,000.

6.13 Computing annuity payment: The Elkridge Bar & Grill has a seven-year loan of

$23,500 with Bank of America. It plans to repay the loan by paying in seven equal

36

installments starting today. If the rate of interest is 8.4 percent, how much will each

payment be worth?

37

0 1 2 3 6 7

├───────┼────────┼───────┼………………┼───────┤

PMT PMT PMT PMT PMT PMT

PVAn = $23,500 n = 7; i = 8.4%

Present value of annuity = PVA = $23,500




Using the PVA equation:

Each payment made by Elkridge Bar & Grill will be $4,221.07, starting today.

6.14 Perpetuity: Your grandfather is retiring at the end of next year. Heould like to receive a

payment of $10,000 a year forever, starting when he retires. If he can invest at 6.5

percent, how much does need to invest to receive the desired cash flow?

38

Solution:

Annual payment needed = PMT = $10,000

Investment rate of return = i = 6.5%

Term of payment = Perpetuity

Present value of investment needed = PV

6.15 Perpetuity: Calculate the perpetuity payments for each of the following cases:

a. $250,000 invested at 6%

b. $50,000 invested at 12%

c. $100,000 invested at 10%

Solution:

a. Annual payment = PMT

Investment rate of return = i = 6%


Present value of investment needed = PV = $250,000

b. Annual payment = PMT


39



c. Annual payment = PMT




6.16. Effective annual rate: Raj Krishnan bought a Honda Accord for a price of $17,345. He

put down $6,000 and financed the rest through the dealer at an APR of 4.9 percent for

four years. What is the effective annual rate (EAR) if payments are made monthly?

Solution:

Loan amount = PV = $11,345

Interest rate on loan = i = 4.9%

Frequency of compounding = m = 12

Effective annual rate = EAR

40

6.17 Effective annual rate: Cyclone Rentals borrowed $15,550 from a bank for three years. If

the quoted rate (APR) is 6.75 percent, and the compounding is daily, what is the effective

annual rate (EAR)?

Solution:

Loan amount = PV = $15,550




6.18 Growing perpetuity: You are evaluating a growing perpetuity product from a large

financial services firm. The product promises an initial payment of $20,000 at the end of

this year and subsequent payments that will thereafter grow at a rate of 3.4 percent

annually. If you use a 9 percent discount rate for investment products, what is the present

value of this growing perpetuity?

Solution:

Cash flow at t = 1 = CF1 = $20,000

41

Annual growth rate = g = 3.4%

Discount rate = i = 9%

Present value of growing perpetuity = PVA∞

INTERMEDIATE

6.19 Future value with multiple cash flows: Trigen Corp. is expecting to invest cash flows

of $331,000, $616,450, $212,775, $818,400, $1,239,644, and $1,617,848 in research and

development over the next six years. If the appropriate interest rate is 6.75 percent, what

is the future value of these investment cash flows?

Solution:

0 6.75% 1 2 3 4 5 6

├───────┼────────┼───────┼────────┼───────┼────────┤

$331,000 $616,450 $212,775 $818,400 $1,239,644 $1,617,848

42

6.20 Future value with multiple cash flows: Stephanie Watson plans to adopt the following

investment pattern beginning next year. She will invest $3,125 in each of the next three

years and will then make investments of $3,650, $3,725, $3,875, and $4,000 over the

following four years. If the investments are expected to earn 11.5 percent annually, how

much will she have at the end of the seven years?

Solution:

Expected rate of return = i = 11.5%

Investment period = n = 7 years

Future value of investment = FV

6.21 Present value with multiple cash flows: Carol Jenkins, a lottery winner, will receive the

following payments over the next seven years. If she can invest her cash flows in a fund

that will earn 10.5 percent annually, what is the present value of her winnings?

1 2 3 4 5 6 7

$200,000 $250,000 $275,000 $300,000 $350000 $400,000 $550,000

Solution:

Expected rate of return = i = 10.5%

Investment period = n = 7 years

43


6.22 Computing annuity payment: Gary Whitmore is a high school sophomore. He currently

has $7,500 in a money market account paying 5.65 percent annually. He plans to use this

and his savings over the next four years to buy a car at the end of his sophomore year in

college. He estimates that the car will cost him $12,000 in four years. How much should

he invest in the money market account every year for the next four years if he wants to

achieve his target?

44

Solution:

Cost of car in four years = $12,000

Amount invested in money market account now = PV = $7,500

Return earned by investment = i = 5.65%

Value of current investment in 4 years = FV4

Balance of money needed to buy car = $12,000 – $9,344.14 = $2,655.86 = FVA

Payment needed to reach target = PMT

6.23 Growing annuity: 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 businesses, what is the present value of this growing annuity?

Solution:

Time for lease to expire = n = 73 years

45

Last year’s net cash flow = CF0 = $92,500

Expected annual growth rate = g = 6.3%

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 = PVAn

6.24 Future value of an annuity due: Jeremy Denham plans to save $5,000 every year for

the next eight years, starting today. At the end of eight years, Jeremy will turn 30 years

old and plans to use his savings toward the down payment on a house. If his investment

in a mutual fund will earn him 10.3 percent annually, how much will he have saved in

eight years when he will need the money to buy a house?

Solution:

0 10.3% 1 2 3 7 8

├───────┼────────┼───────┼………………┼───────┤

$5,000 $5,000 $5,000 $5,000 $5,000



46




6.25 Present value of an annuity due: Grant Productions has borrowed a huge sum from the

California Finance Company at a rate of 17.5 percent for a seven-year period. The loan

calls for a payment of $1,540,862.19 each year beginning today. What is the amount

borrowed by this company? Round to the nearest dollar.

Solution:

0 17.5% 1 2 3 6 7

├───────┼────────┼───────┼………………┼───────┤

PMT =$1,540,862.19 at the beginning of each year

Annual payment = PMT = $1,540,862.19



Required rate of return = 17.5%


47

6.26 Present value of an annuity due: Sharon Kabana has won a state lottery and will

receive a payment of $89,729.45 every year, starting today for the next 20 years. If she

invests the proceeds at a rate of 7.25 percent, what is the present value of the cash flows

that she will receive? Round to the nearest dollar.

Solution:

0 7.25% 1 2 3 19 20

├───────┼────────┼───────┼………………┼───────┤

PMT = $89,729.45 at the beginning of each year

Annual payment = PMT = $89,729.45





48

6.27 Perpetuity: Calculate the present value of the following perpetuities:

a. $1,250 discounted back to the present at 7%

b. $7,250 discounted back to the present at 6.33%

c. $850 discounted back to the present at 20%

Solution:

a. Annual payment = PMT =$1,250




b. Annual payment = PMT =$7,250

Investment rate of return = i = 6.33%

Term of payment = Perpetuity.

Present value of perpetuity = PV

49

c. Annual payment = PMT =$850


Term of payment = Perpetuity.


6.28 Effective annual rate: Find the effective annual interest rate (EAR) on each of the

following:

a. 6% compounded quarterly.

b. 4.99% compounded monthly.

c. 7.25% compounded semi-annually.

d. 5.6% compounded daily.

Solution:

a. Interest rate = i = 6%



50

b. Interest rate = i = 4.99%



c. Interest rate = i = 7.25%



d. Interest rate = i = 5.6%



6.29 Effective annual rate: Which of the following investments has the highest effective

annual rate (EAR)?

a. A bank CD that pays 8.25% interest quarterly.

51

b. A bank CD that pays 8.25% monthly.

c. A bank CD that pays 8.45% annually.

d. A bank CD that pays 8.25% semiannually.

e. A bank CD that pays 8% daily (on a 365-day basis).

Solution:

a. Interest rate on CD = i = 8.25%



b. Interest rate on CD = i = 8.25%



c. Interest rate on CD = i = 4.99%



52

d. Interest rate on CD = i = 8.25%



e. Interest rate on CD = i = 8%



The bank CD that pays 8.25 percent monthly has the highest yield.

6.30 Effective annual rate: You are considering three alternative investments: (1) a three-

year bank CD paying 7.5 percent interest compounded quarterly; (2) a three-year bank

CD paying 7.3 percent interest compounded monthly; and (3) a three-year bank CD

paying 7.75 percent interest compounded annually. Which investment has the highest

effective annual rate?

Solution:

(1) Interest rate on CD = i = 75%


53


(2) Interest rate on CD = i = 7.3%



(3) Interest rate on CD = i = 7.75%



The three-year bank CD paying 7.75 percent interest compounded annually has the

highest effective yield.

ADVANCED

54

6.31 Tirade Owens, a professional athlete, currently has a contract that will pay him a large

amount in the first year of his contract and smaller amounts thereafter. He and his agent

have asked the team to restructure the contract. The team, though reluctant, obliged.

Tirade and his agent came up with a counteroffer. What are the present values of each of

the contracts using a 14 percent discount rate? Which of the three contacts has the highest

present value?

Year Current Contract Team’s Offer Counteroffer

1 $8,125,000 $4,000,000.00 $5,250,000.00

2 $3,650,000 $3,825,000.00 $7,550,000.00

3 $2,715,000 $3,850,000.00 $3,625,000.00

4 $1,822,250 $3,925,000.00 $2,800,000.00

Solution:

Current Contract

Team’s Offer

55

Counteroffer

The counteroffer has the best value for the player.

6.32 Gary Kornig will turn 30 years old next year and wants to retire when his is 65. So far he

has saved (1) $6,950 in an IRA account in which his money is earning 8.3 percent

annually and (2) $5,000 in a money market account in which he is earning 5.25 percent

annually. Gary wants to have $1 million when he retires. Starting next year, he plans to

invest a fixed amount every year until he retires in a mutual fund in which he expects to

earn 9 percent annually. How much will Gary have to invest every year to achieve his

savings goal?

Solution:

Investment (1)

Balance in IRA investment = PV = $6,950

Return on IRA account = i = 8.3%

56

Time to retirement = n = 35 years

Value of IRA at age 65 = FVIRA

Investment (2)

Balance in money market investment = PV = $5,000

Return on money market account = i = 5.25%


Value of money market at age 65 = FVMMA

Target retirement balance = $1,000,000

Future value of current savings = $113,235.03 + $29,973.93 = $143,208.96

Amount needed to reach retirement target = FVA = $856,774.04

Annual payment needed to meet target = PMT

Expected return from mutual fund = i = 9%

57

6.33 Babu Baradwaj is planning to save for his son’s college tuition. His son is currently 11

years old and will begin college in seven years. He has an index fund investment of

$7,500 earning 9.5 percent annually. College expenses in a state university in Maryland

currently total $15,000 per year but are expected to grow at roughly 6 percent each year.

Babu plans to invest a certain amount in a mutual fund that will earn 11 percent annually

to make up the difference between the college expenses and his current savings. In total,

Babu will make seven equal investments with the first starting today and with the last

being made a year before his son begins college.

a. What will be the present value of the fours years of college expenses just when the

son starts college? Assume a discount rate of 5.5 percent.

b. What will be the value of the index mutual fund when his son just starts college?

c. What is the amount that Babu will have to have saved when his son turns 18 if Babu

plans to cover all of his son’s college expenses?

d. How much will Babu have to invest every year in order for him to have enough funds

to cover all his son’s expenses?

Solution:

Annual cost of college tuition today (t = 0) = $15,000

Expected increase in annual tuition costs = g = 6%

a. Four-year tuition costs ( t = 7 to t = 10)

Years from now Future value calculation Tuition costs

7 $15,000(1.06)7 $22,554.45

58

8 $15,000(1.06)8 $23,907.72

9 $15,000(1.06)9 $25,342.18

10 $15,000(1.06)10 $26,862.72

Discount rate = i = 5.5%

Present value of tuition costs = PV

b. Future value of the index mutual fund at t = 7

Present value of index fund investment = PV = $7,500

Return on fund = i = 9.5%


c. Target savings needed at t = 7

PV of tuition costs – Future value of investment = $86,124.36 – $14,156.64

= $71,967.72

d. Annual savings needed

Return on fund = i = 11%

Amount that needs to be saved = FVA = $71,967.72

59

Annuity payment needed = PMT

6.34 You are now 50 years old and plan to retire at age 65. You currently have a stock

portfolio worth $150,000, a 401(k) retirement plan worth $250,000, and a money market

account worth $50,000. Your stock portfolio is expected to provide you annual returns of

12 percent, your 401(k) investment will earn you 9.5 percent annually, and the money

market account earns 5.25 percent, compounded monthly.

a. If you do not save another penny, what will be the total value of your investments

when you retire at age 65?

b. Assume you plan to invest $12,000 every year in your 401K plan for the next 15

years (starting one year from now). How much will your investments be worth when

you retire at 65?

c. Assume that you expect to live another 25 years after you retire (until age 90). Today,

at age 50, you now take all of your investments and place them in an account that

pays 8 percent (use the scenario from part b in which you continue saving). If you

start withdrawing funds starting at age 66, how much can you withdraw every year

(e.g., an ordinary annuity) and leave nothing in your account after a 25th and final

withdrawal at age 90?

60

d. You want your current investments, which are described in the problem statement, to

support a perpetuity that starts a year from now. How much can you withdraw each

year without touching your principal?

Solution:

a. Stock Portfolio

Current value of stock portfolio = $150,000

Expected return on portfolio = i = 12%


Expected value of portfolio at age 65 = FVStock

410(k) Investment

Current value of 410(k) portfolio = $250,000

Expected return on portfolio = i = 9.5%


Expected value of portfolio at age 65 = FV401k

Money market account

Current value of savings = $50,000

Expected return on portfolio = i = 5.25%


61


Expected value of portfolio at age 65 = FVMMA

Total value of all three investments = $821,034.86 + $975,330.48 + $109,706.14

= $1,906,071.48

b. Planned annual investment in 401k plan = $12,000

Future value of annuity = FVA

Total investment amount at retirement = $1,906,071.48 + $366,482.77

= $2,272,554.25

c. Amount available at retirement = PVA = $2,272,554.25

Length of annuity = n = 25

Expected return on investment = i = 8%

Annuity amount expected = PMT

Using the PVA equation:

62

Each payment received for the next 25 years will be $212,889.63.

d. Type of payment = Perpetuity

Present value of perpetuity = PVA = $2,272,554.25

Expected return on investment = i = 8%

You could receive an annual payment of $181,804.34 forever.

6.35 Trevor Diaz is looking to purchase a Mercedes Benz SL600 Roadster, which has an

invoice price of $121,737 and a total cost of $129,482. Trevor plans to put down $20,000

and will pay the rest by taking on a 5.75 percent five-year bank loan. What is the monthly

payment on this auto loan? Prepare an amortization table using Excel.

Solution:

Cost of new car = $129,482

Down payment = $20,000

63

Loan amount = $129,482 – $20,000 = $109,482


Term of loan = n = 5 years

Frequency of payment = m = 12

Monthly payment on loan = PMT

6.36 The Sundarams are buying a new 3,500-square-foot house in Muncie, Indiana, and will

borrow $237,000 from Bank One at a rate of 6.375 percent for 15 years. What is their

monthly loan payment? Prepare an amortization schedule using Excel.

Solution:

Home loan amount = $237,000


Term of loan = n = 15 years

Frequency of payment = m = 12

Monthly payment on loan = PMT

64

6.37 Assume you will start on a job as soon as you graduate. You plan to start saving for your

retirement when you turn 25 years of age. Assume you are 21 years of age at the time of

graduation. Everybody needs a break! Currently, you plan to retire when you turn 65

years old. After retirement, you expect to live at least until you are 85. You wish to be

able to withdraw $40,000 (in today’s dollars) every year from the time of your retirement

until you are 85 years old (i.e., for 20 years). You can invest, starting when you turn 25

years old, in a portfolio fund. The average inflation rate is likely to be 5 percent.

a. Calculate the lump sum you need to have accumulated at age 65 to be able to draw

the desired income. Assume that your return on the portfolio investment is likely to

be 10 percent.

b. What is the dollar amount you need to invest every year, starting at age 26 and ending

at age 65 (i.e., for 40 years) to reach the target lump sum at age 65?

c. Now answer questions a and b assuming your rate of return to be 8 percent per year,

and then 15 percent per year.

65

d. Now assume you start investing for your retirement when you turn 30 years old and

analyze the situation under rate of return assumptions of (i) 8 percent, (ii) 10 percent,

and (iii) 15 percent.

e. Repeat the analysis by assuming that you start investing only when you are 35 years

old.

66

Solution:

RETIREMENT ANALYSIS SUMMARY

INVESTMENT AGE = 25 INVESTMENT AGE = 30 INVESTMENT AGE = 35INVESTMENT AGE = 35

Rate of

Return

8% 10% 15% 8% 10% 15% 8%8% 10%10% 15%15%

Inflation rate 5%

Retirement

Income Level

$40,000

Lump sum

needed at age

67

65 $5,160,266 $4,353,087 $3,011,353 $5,160,266 $4,353,087 $3,011,353 $5,160,266$5,160,266 $4,353,087$4,353,087 $3,011,353$3,011,353

Annuity

payment

needed

$19,919 $9,835 $1,693 $29,946 $16,062 $3,417 $45,552$45,552 $26,463$26,463 $6,927$6,927

68

Appendix: Deriving the Formula for the Present Value of an

Ordinary Annuity

In this chapter we showed that the formula for a perpetuity can be obtained from the formula for

the present value of an ordinary annuity if n is set equal to ∞. It is also possible to go the other

way. In other words, the present value of an ordinary annuity formula can be derived from the

formula for a perpetuity. In fact, this is how the annuity formula was originally obtained. To see

how this was done, assume that someone has offered to pay you $1 per year forever, beginning

next year, but that, in return, you will have to pay that person $1 per year forever, beginning in

year n + 1.

Solution:

In this problem, you will receive an annual payment that grows at a rate of g forever. In return,

you will have to pay that person $1(1 + g)n each year forever, beginning in year n + 1. The cash

flows that you will receive can be represented as in the following time line.

0 1 2 3 n-1n n+1 n+2 ∞

Receive $1 $1(1+g) $1(1+g)2……$1(1+g)n-2 $1(1+g)n-1 $1(1+g)n

Pay $0 $0 $0 $0 $0 $1(1+g)n

The first time line shows the cash flows for the perpetuity that you will receive. This perpetuity

is worth:

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The second time line shows the cash flows for the perpetuity that you will pay. The present value

of what you owe is:

Notice that if you subtract, year-by-year, the cash flows you would pay from the cash flows you

would receive, you get the cash flows for an n-year annuity.

0 1 2 3 n – 1 n n + 1 n + 2 ∞

Difference $1 $1(1 + g) $1(1 + g)2………$1(1 + g)n-2 $1(1 + g)n-1 $0

Therefore, the value of the offer equals the value of an n-year growing annuity. Solving for the

difference between from ,we see that this is the same as Equation 6.5.

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Sample Test Problems

6.1 Groves Corp. is expecting cash flows of $225,000, $278,000, $312,500, and $410,000 at

the end of the next four years. If it uses a discount rate of 6.25 percent, what will be the

present value of this cash flow stream?

Solution:

0 6.25% 1 2 3 4

├───────┼────────┼───────┼────────┤

$225,000 $278,000 $312,500 $410,000

6.2 Freisinger, Inc., is expecting a new project to start paying off, beginning at the end of

next year. It expects cash flows to be as follows:

1 2 3 4 5

$433,676 $478,452 $475,455 $478,326 $535,444

If Freisinger can reinvest these cash flows to earn a return of 7.8 percent, what is the

future value of this cash flow stream at the end of five years?

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Solution:

0 7.8% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$433,676 $478,452 $475,455 $478,326 $535,444

6.3 Vancouver, Canada, is the site of the next Winter Olympics in 2010. City officials plan to

build a new multipurpose stadium. The projected cost of the stadium in 2010 dollars is

$7.5 million. Assume that it is 2008 and city officials intend to put away a certain amount

at the end of each of the next three years in an account that will pay 8.75 percent. What is

the annual payment necessary to meet this projected cost of the stadium?

Solution:

0 8.75% 1 2 3

├───────┼────────┼───────┤

PMT PMT PMT

FVAn = $7,500,000

Expected construction costs = FVA = $7,500,000


72


Using the FVA equation:

6.4 You have just won a lottery that promises an annual payment of $118,312 beginning

immediately. You will receiver a total of 10 payments. If you can invest the cash flows in

an investment paying 7.65 percent annually, what is the present value of this annuity?

Solution:

0 7.65% 1 2 3 9 10

├───────┼────────┼───────┼………………┼───────┤

$118,312 $118,312 $$118,312 $$118,312 $118,312

Annual payment = PMT = $118,312





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6.5 Which of the following investments has the highest effective annual rate (EAR)?

a. A bank CD that pays 5.50% interest quarterly

b. A bank CD that pays 5.45% monthly

c. A bank CD that pays 5.65% annually

d. A bank CD that pays 5.55% semiannually

e. A bank CD that pays 5.35% daily (on a 365-day basis)

Solution:

a. Interest rate on CD = i = 5.50%



b. Interest rate on CD = i = 5.45%



74

c. Interest rate on CD = i = 5.65%



d. Interest rate on CD = i = 5.55%



e. Interest rate on CD = i = 5.35%



The bank CD that pays 5.65 percent annually has the highest yield.

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