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CHAPTER 5CHAPTER 5

CHAPTER 5

Introduction to Valuation: The Time Value of Money

I.DEFINITIONS

FUTURE VALUE

a1.The amount an investment will worth after one or more periods of time is the _____ value.

a.future

b.present

c.principal

d.discounted

e.simpleCOMPOUNDING

b2.The process of accumulating interest on an investment over time to earn more interest is called:

a.growth.

b.compounding.

c.aggregation.

d.accumulation.

e.discounting.

INTEREST ON INTEREST

d3.Interest earned on the reinvestment of previous interest payments is called _____ interest.

a.free

b.annual

c.simple

d.interest on

e.compound

COMPOUND INTEREST

e4.Interest earned on both the initial principal and the interest reinvested from prior periods is called _____ interest.

a.free

b.annual

c.simple

d.interest on

e.compound

SIMPLE INTEREST

c5.Interest earned only on the original principal amount invested is called _____ interest.

a.free

b.annual

c.simple

d.interest on

e.compound

FUTURE VALUE INTEREST FACTOR

a6.The future value interest factor is calculated as:

a.(1 + r)t.

b.(1 + rt).

c.(1 + r) ( t.

d.1 + r t.

e.(1+r) x rt.

PRESENT VALUE

c7.The current value of future cash flows discounted at the appropriate discount rate to current time is called the _____ value.

a.principal

b.future

c.present

d.simple

e.compoundDISCOUNTING

b8.The process of finding the present value of some future amount is often called:

a.growth.

b.discounting.

c.accumulation.

d.compounding.

e.reduction.

PRESENT VALUE INTEREST FACTOR

d9.The present value interest factor is calculated as:

a.1 ( (1 + r t).

b.1 ( (1 + rt).

c.1 ( (1 + r) ( (t).

d.1 ( (1 + r)t.

e.1 + r + t.

DISCOUNT RATE

e10.The interest rate used to calculate the present value of future cash flows is called the _____ rate.

a.free

b.annual

c.compound

d.simple

e.discount

II.CONCEPTS

PRESENT VALUE AND DISCOUNT RATE

c11.As the discount rate increases, the present value of $500 to be received six years from

now:

a.remains constant.

b.also increases.

c.decreases.

d.becomes negative.

e.will vary but the direction of the change is unknown.PRESENT VALUE AND TIME

e12.Katie is going to receive $1,000 three years from now. Wilt is going to receive $1,000

five years from now. Which one of the following statements is correct if both Katie

and Wilt apply a 5 percent discount rate to these amounts?

a.The present value of Katie and Wilts money is equal.

b.The value of Wilts money will be greater than the value of Katies money six years

from now.

c.In todays dollars, Wilts money is worth more than Katies.

d.In five years, the value of Katies money will be equal to the value of Wilts money.

e.Katies money is worth more than Wilts money today.SIMPLE VERSUS COMPOUND INTERESTd13.Jamie deposits $1,000 into an account that pays 4 percent interest compounded annually. Chris deposits $1,000 into an account that pays 4 percent simple interest.

Both deposits were made today. Which of the following statements are true concerning

these two accounts?

I.

At the end of one year, both Jamie and Chris will have the same amount in their

accounts.

II.At the end of five years, Chris will have more money in his account than Jamie has in

hers.

III.Chris will never earn any interest on interest.

IV.All else equal, Jamie made the better investment.

a.I and II only

b.III and IV only

c.I, II, and IV only

d.I, III, and IV only

e.II, III, and IV onlyFUTURE VALUE AND TIMEe14.Nadine invests $1,000 at 8 percent when she is 25 years old. Neal invests $1,000 at 8

percent when he is 40 years old. Both investments compound interest annually. Both

Nadine and Neal retire at age 60. Which one of the following statements is correct?

a.Nadine will have less money when she retires than Neal.

b.Neal will earn more interest on interest than Nadine.

c.Neal will earn more compound interest than Nadine.

d.If Neal waits to age 70 to retire, then he will have just as much money as Nadine.

e.Nadine will have more money when she retires than Neal.FUTURE VALUE AND RATE

d15.Sun Lee has $500 today. Which one of the following statements is correct if she

invests this money at a positive rate of interest for five years?

a.The higher the interest rate she earns, the less money she will have in the future.

b.The higher the interest rate, the longer she has to wait for her money to grow to $1,000

in value.

c.If Sun Lee can earn 7 percent, she will have to wait about six years to have $1,000

total.

d.At the end of the five years Sun Lee will have less money if she invests at 5 percent

rather than at 7 percent.

e.At 10 percent interest Sun Lee should expect to have $1,000 in her account at the end

of the five years.III.PROBLEMS

SIMPLE INTEREST

d16.Thomas invests $100 in an account that pays 5 percent simple interest. How much

money will Thomas have at the end of five years?

a.$120.00

b.$123.68

c.$124.92

d.$125.00

e.$127.63SIMPLE INTEREST

c17.Betty invests $500 in an account that pays 3 percent simple interest. How much money

will Betty have at the end of ten years?

a.$630.00

b.$633.33

c.$650.00

d.$671.96

e.$675.00SIMPLE VERSUS COMPOUND INTEREST

c18.Beatrice invests $1,000 in an account that pays 4 percent simple interest. How much

more could she have earned over a five-year period if the interest had compounded

annually?

a.$15.45

b.$15.97

c.$16.65

d.$17.09

e.$21.67SIMPLE VERSUS COMPOUND INTEREST

a19.Dale invests $500 in an account that pays 6 percent simple interest. How much more

could he have earned over a thirty year period if the interest had compounded

annually?

a.$1,471.75

b.$1,532.50

c.$1,621.25

d.$1,804.25

e.$2,371.75FUTURE VALUE

d20.What is the future value of $2,896 invested for twelve years at 6.5 percent

compounded annually?

a.$5,827.32

b.$6,023.44

c.$6,049.45

d.$6,165.86


c21.Today you earn a salary of $28,500. What will be your annual salary fifteen years from

now if you earn annual raises of 3.5 percent?

a.$47,035.35

b.$47,522.89

c.$47,747.44

d.$48,091.91


d22.You own a classic automobile that is currently valued at $39,500. If the value increases

by 6 percent annually, how much will the auto be worth ten years from now?

a.$64,341.34

b.$44,734.42

c.$69,843.06

d.$70,738.48


e23.You hope to buy your dream house six years from now. Today your dream house costs

$189,900. You expect housing prices to rise by an average of 4.5 percent per year over

the next six years. How much will your dream house cost by the time you are ready to

buy it?

a.$240,284.08

b.$246,019.67

c.$246,396.67

d.$246,831.94

e.$247,299.20PRESENT VALUE

b24.Your grandmother invested one lump sum 17 years ago at 4.25 percent interest.

Today, she gave you the proceeds of that investment which totaled $5,539.92. How

much did your grandmother originally invest?

a.$2,700.00

b.$2,730.30

c.$2,750.00

d.$2,768.40


b25.What is the present value of $13,450 to be received four years from today if the

discount rate is 5.25 percent?

a.$10,854.20

b.$10,960.59

c.$10,974.21

d.$10,982.18


c26.You would like to give your daughter $40,000 towards her college education thirteen

years from now. How much money must you set aside today for this purpose if you can

earn 6.3 percent on your funds?

a.$17,750.00

b.$17,989.28

c.$18,077.05

d.$18,213.69

e.$18,395.00INTEREST RATE FOR A SINGLE PERIOD

e27.One year ago, you invested $3,000. Today it is worth $3,142.50. What rate of interest

did you earn?

a.4.63 percent

b.4.68 percent

c.4.70 percent

d.4.73 percent

e.4.75 percentINTEREST RATE FOR MULTIPLE PERIODS

d28.Forty years ago, your father invested $2,500. Today that investment is worth $107,921.

What is the average rate of return your father earned on his investment?

a.8.50 percent

b.9.33 percent

c.9.50 percent

d.9.87 percent

e.9.99 percentINTEREST RATE FOR MULTIPLE PERIODS

d29.Ten years ago, Joe invested $5,000. Five years ago, Marie invested $2,500. Today,

both Joe and Maries investments are each worth $8,500. Which one of the following

statements is correct concerning their investments?

a.Three years from today, Joes investment will be worth more than Maries.

b.Last year, Maries investment was worth more than Joes.

c.Joe has earned more interest on interest than Marie.

d.Marie earned an annual interest rate of 27.73 percent.

e.Joe earned an annual interest rate of 6.45 percent.INTEREST RATE FOR MULTIPLE PERIODS

c30.Alpha, Inc. is saving money to build a new factory. Six years ago they set aside

$250,000 for this purpose. Today, that account is worth $306,958. What rate of interest

is Alpha earning on this money?

a.3.43 percent

b.3.45 percent

c.3.48 percent

d.3.52 percent

e.3.55 percentNUMBER OF TIME PERIODS

d31.Bob bought some land costing $14,990. Today that same land is valued at $55,000.

How long has Bob owned this land if the price of land has been increasing at 6 percent

per year?

a.21.82 years

b.21.98 years

c.22.03 years

d.22.31 years

e.22.44 yearsNUMBER OF TIME PERIODS

e32.On your tenth birthday, you received $100 which you invested at 4.5 percent interest,

compounded annually. That investment is now worth $3,000. How old are you today?

a.age 77

b.age 82

c.age 84

d.age 86

e.age 87PRESENT VALUE AND RATE CHANGES

a33.You want to have $10,000 saved ten years from now. How much less do you have to

deposit today to reach this goal if you can earn 6 percent rather than 5 percent on your

savings?

a.$555.18

b.$609.81

c.$615.48

d.$928.73

e.$1,046.22PRESENT VALUE AND RATE CHANGES

e34.Your older sister deposited $5,000 today at 8 percent interest for five years. You would

like to have just as much money at the end of the next five years as your sister.

However, you can only earn 6 percent interest. How much more money must you

deposit today than your sister if you are to have the same amount at the end of five

years?

a.$201.80

b.$367.32

c.$399.05

d.$423.81

e.$489.84PRESENT VALUE AND TIME CHANGES

d35.When you retire forty years from now, you want to have $1 million. You think you can

earn an average of 8.5 percent on your money. To meet this goal, you are trying to

decide whether to deposit a lump sum today, or to wait and deposit a

lump sum five years from today. How much more will you have to deposit as a lump

sum if you wait for five years before making the deposit?

a.$18,001.06

b.$18,677.78

c.$18,998.03

d.$19,272.81

e.$21,036.83PRESENT VALUE AND TIME CHANGES

b36.Antonette needs $20,000 as a down payment for a house five years from now. She

earns 4 percent on her savings. Antonette can either deposit one lump sum today for

this purpose or she can wait a year and deposit a lump sum. How much additional

money must Antonette deposit if she waits for one year rather than making the deposit

today?

a.$639.19

b.$657.54

c.$658.23

d.$659.04

e.$800.00

FUTURE VALUE AND RATE CHANGES

e37.Alpo, Inc. invested $500,000 to help fund a company expansion project scheduled for

eight years from now. How much additional money will they have eight years from

now if they can earn 9 percent rather than 7 percent on this money?

a.$58,829.69

b.$86,991.91

c.$118,009.42

d.$126,745.19

e.$137,188.23FUTURE VALUE AND RATE CHANGES

a38.You will be receiving $5,000 from your family as a graduation present. You have

decided to save this money for your retirement. You plan to retire thirty-five years

after graduating. How much additional money will you have at that time if you can

earn an average of 8.5 percent on your investment instead of just 8 percent?

a.$12,971.49

b.$13,008.47

c.$13,123.93

d.$13,234.44

e.$13,309.85FUTURE VALUE AND TIME CHANGES

c39.You deposit $3,000 in a retirement account today at 5.5 percent interest. How much

more money will you have if you leave the money invested for forty-five years rather

than forty years?

a.$7,714.91

b.$7,799.08

c.$7,839.73

d.$7,846.52

e.$7,858.19FUTURE VALUE AND TIME CHANGESb40.You collect model cars. One particular model increases in value at a rate of 5

percent per year. Today, the model is worth $29.50. How much additional money can

you make if you wait ten years to sell the model rather than selling it five years from

now?

a.$9.98

b.$10.40

c.$10.86

d.$11.03

e.$11.24IV.ESSAYS

PRESENT VALUE AND DISCOUNTING

41.Write a sentence explaining why present values decrease as the discount rate

increases.

Student answers will vary. Here is one example. When you can earn more interest, you need less of your own money to reach the same future dollar amount.COMPOUNDING

42.Explain what compounding is and the relationship between compound interest earned and

the number of years over which an investment is compounded.

Compounding is earning interest on interest. Compounding is not very significant over short time periods, but greatly increases in importance over a longer time period.FUTURE VALUES

43.Draw a picture illustrating the future value of $1, using five different interest rates (including 0 percent) and maturities ranging from today to 10 years from now. Plot time to maturity on the horizontal axis and dollars on the vertical axis. (Note: you need not make any calculations, draw the figure using your intuition.)

COMPARING LUMP SUMS

44.You are considering two lottery payment streams, choice A pays $1,000 today and choice B pays $1,750 at the end of five years. Using a discount rate of 5 percent, based on present values, which would you choose? Using the same discount rate of 5 percent, based on future values five years from now, which would you choose? What do your results suggest as a general rule for approaching such problems? (Make your choices based purely on the time value of money.)

PV of A = $1,000; PV of B = $1,371; FV of A = $1,276; FV of B = $1,750. Based on both present values and future values, B is the better choice. The student should recognize that finding present values and finding future values are simply reverse processes of one another, and that choosing between two lump sums based on PV will always give the same result as choosing between the same two lump sums based on FV.

RULE OF 72 AND COMPOUNDING

45.At an interest rate of 10 percent and using the Rule of 72, how long will it take to double the value of a lump sum invested today? How long will it take after that until the account grows to four times the initial investment? Given the power of compounding, shouldnt it take less time for the money to double the second time?

It will take 7.2 years to double the initial investment, then another 7.2 years to double it again. That is, it takes 14.4 years for the value to reach four times the initial investment. Compounding doesnt affect the amount of time it takes for an investment to double the second time, but note that during the first 7.2 years, the interest earned is equal to 100 percent of the initial investment. During the second 7.2 years, the interest earned is equal to 200 percent of the initial investment. That is the power of compounding.

THE TIME VALUE OF MONEY

46.Some financial advisors recommend you increase the amount of federal income taxes withheld from your paycheck each month so that you will get a larger refund come April 15th. That is, you take home less today but get a bigger lump sum when you get your refund. Based on your knowledge of the time value of money, what do you think of this idea? Explain.

Some students may slip in a discussion about the benefits of forced savings, etc., but these issues are based on preferences, not the time value of money. Based on the time value of money, the students should recommend just the opposite. That is, withhold as little as possible, while still avoiding tax penalties for underwithholding, and pay the remaining tax bill when it comes due the following year. This is the usual dollar today versus a dollar tomorrow argument.

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