5.4
Summary and Conclusions
This chapter has introduced you to the basic principles of present value and discounted cash flow valuation.In it, we explained a number of things about the time value of money, including these:
For a given rate of return, we can determine the value at some point in the future of an investmentmade today by calculating the future value of that investment.
1.
We can determine the current worth of a future cash flow or series of cash flows for a given rate ofreturn by calculating the present value of the cash flow(s) involved.
2.
The relationship between present value (PV) and future value (FV) for a given rate r and time t is givenby the basic present value equation:
PV = FVt
/(1 + r)
t
As we have shown, it is possible to find any one of the four components (PV, FVt
, r, or t) given theother three.
3.
The principles developed in this chapter will figure prominently in the chapters to come. The reason for thisis that most investments, whether they involve real assets or financial assets, can be analyzed using thediscounted cash flow (DCF) approach. As a result, the DCF approach is broadly applicable and widely usedin practice. Before going on, therefore, you might want to do some of the problems that follow.
CONNECT TO FINANCE
Connect Finance offers you plenty of opportunities to practice mastering theseconcepts. Log on to connect.mheducation.com to learn more. If you like what you see, ask your professorabout using Connect Finance!
Can you answer the following Connect Quiz questions?
Section 5.1
You deposited $2,000 in a bank account that pays 5 percent simple interest. How much will you havein this account after three years?
Section 5.2
What is the present value of $11,500 discounted at 9 percent for 11 years?
Section 5.3
Charlie invested $6,200 in a stock last year. Currently, this investment is worth $6,788.38. What is therate of return on this investment?
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CHAPTER REVIEW AND SELF-TEST PROBLEMS
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5.1
Calculating Future Values Assume you deposit $10,000 today in an account that pays 6 percentinterest. How much will you have in five years?
5.2
Calculating Present Values Suppose you have just celebrated your 19th birthday. A rich uncle has setup a trust fund for you that will pay you $150,000 when you turn 30. If the relevant discount rate is 9percent, how much is this fund worth today?
5.3
Calculating Rates of Return Youve been offered an investment that will double your money in 10years. What rate of return are you being offered? Check your answer using the Rule of 72.
5.4
Calculating the Number of Periods Youve been offered an investment that will pay you 9 percentper year. If you invest $15,000, how long until you have $30,000? How long until you have $45,000?ANSWERS TO CHAPTER REVIEW AND SELF-TEST PROBLEMS
5.1
We need to calculate the future value of $10,000 at 6 percent for five years. The future value factor is:1.065
= 1.3382
The future value is thus $10,000 1.3382 = $13,382.26. 5.2
We need the present value of $150,000 to be paid in 11 years at 9 percent. The discount factor is:1/1.0911 = 1/2.5804 = .3875
The present value is thus about $58,130. 5.3
Suppose you invest $1,000. You will have $2,000 in 10 years with this investment. So, $1,000 is theamount you have today, or the present value, and $2,000 is the amount you will have in 10 years, or thefuture value. From the basic present value equation, we have:
From here, we need to solve for r, the unknown rate. As shown in the chapter, there are severaldifferent ways to do this. We will take the 10th root of 2 (by raising 2 to the power of 1/10):
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Using the Rule of 72, we have 72/t = r%, or 72/10 = 7.2%; so, our answer looks good (remember thatthe Rule of 72 is only an approximation).
5.4
The basic equation is this:
If we solve for t, we find that t = 8.04 years. Using the Rule of 72, we get 72/9 = 8 years, so once againour answer looks good. To get $45,000, verify for yourself that you will have to wait 12.75 years.Page 146
CONCEPTS REVIEW AND CRITICAL THINKING QUESTIONS
1. Present Value [LO2] The basic present value equation has four parts. What are they?2. Compounding [LO1, 2 What is compounding? What is discounting?
Compounding and Period [LO1, 2] As you increase the length of time involved, what happens tofuture values? What happens to present values?
3.
Compounding and Interest Rates [LO1, 2] What happens to a future value if you increase the rate r?What happens to a present value?
4.
Ethical Considerations [LO2] Take a look back at Example 5.7. Is it deceptive advertising? Is itunethical to advertise a future value like this without a disclaimer?
Use the following information for the next five questions:
On March 28, 2008, Toyota Motor Credit Corporation (TMCC), a subsidiary of Toyota Motor, offeredsome securities for sale to the public. Under the terms of the deal, TMCC promised to repay the ownerof one of these securities $100,000 on March 28, 2038, but investors would receive nothing until then.Investors paid TMCC $24,099 for each of these securities; so they gave up $24,099 on March 28,2008, for the promise of a $100,000 payment 30 years later.
5.
Time Value of Money [LO2] Why would TMCC be willing to accept such a small amount today($24,099) in exchange for a promise to repay about four times that amount ($100,000) in the future?6.
Call Provisions [LO2] TMCC has the right to buy back the securities on the anniversary date at a priceestablished when the securities were issued (this feature is a term of this particular deal). What impactdoes this feature have on the desirability of this security as an investment?
7.
Time Value of Money [LO2] Would you be willing to pay $24,099 today in exchange for $100,000 in30 years? What would be the key considerations in answering yes or no? Would your answer depend8.
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on who is making the promise to repay?
Investment Comparison [LO2] Suppose that when TMCC offered the security for $24,099, the U.S.Treasury had offered an essentially identical security. Do you think it would have had a higher or lowerprice? Why?
9.
Length of Investment [LO2] The TMCC security is bought and sold on the New York StockExchange. If you looked at the price today, do you think the price would exceed the $24,099 originalprice? Why? If you looked in the year 2019, do you think the price would be higher or lower thantodays price? Why?
10.
QUESTIONS AND PROBLEMS
BASIC
(Questions 114)
Simple Interest versus Compound Interest [LO1] First City Bank pays 8 percent simple intereston its savings account balances, whereas Second City Bank pays 8 percent interest compoundedannually. If you made a deposit of $9,000 in each bank, how much more money would you earn fromyour Second City Bank account at the end of seven years?
1.
Calculating Future Values [LO1] For each of the following, compute the future value:
Present ValueYearsInterest RateFuture Value
$1,975 11 13%
6,734 7 9
81,346 14 12
192,050 8 6
2.
Page 147
Calculating Present Values [LO2] For each of the following, compute the present value:
Present ValueYearsInterest RateFuture Value
13 9% $ 15,451
4 7 51,557
29 24 886,073
40 35 550,164
3.
Calculating Interest Rates [LO3] Solve for the unknown interest rate in each of the following:Present ValueYearsInterest RateFuture Value
$ 181 4 $ 297
335 18 1,080
4.
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48,000 19 185,382
40,353 25 531,618
Calculating the Number of Periods [LO4] Solve for the unknown number of years in each of thefollowing:
Present ValueYearsInterest RateFuture Value
$ 560 6% $ 1,389
810 9 1,821
18,400 11 289,715
21,500 13 430,258
5.
Calculating Interest Rates [LO3] Assume the total cost of a college education will be $320,000 whenyour child enters college in 18 years. You presently have $67,000 to invest. What annual rate of interestmust you earn on your investment to cover the cost of your childs college education?6.
Calculating the Number of Periods [LO4] At 7.3 percent interest, how long does it take to doubleyour money? To quadruple it?
7.
Calculating Interest Rates [LO3] According to the Census Bureau, in January 2013, the averagehouse price in the United States was $306,900. In January 2000, the average price was $200,300. Whatwas the annual increase in selling price?
8.
Calculating the Number of Periods [LO4] Youre trying to save to buy a new $225,000 Ferrari. Youhave $45,000 today that can be invested at your bank. The bank pays 4.8 percent annual interest on itsaccounts. How long will it be before you have enough to buy the car?
9.
Calculating Present Values [LO2] Imprudential, Inc., has an unfunded pension liability of $475million that must be paid in 20 years. To assess the value of the firms stock, financial analysts want todiscount this liability back to the present. If the relevant discount rate is 6.1 percent, what is the presentvalue of this liability?
10.
Page 148
Calculating Present Values [LO2] You have just received notification that you have won the $2million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday(assuming youre around to collect), 80 years from now. What is the present value of your windfall ifthe appropriate discount rate is 7.5 percent?
11.
Calculating Future Values [LO1] Your coin collection contains fifty 1952 silver dollars. If yourgrandparents purchased them for their face value when they were new, how much will your collectionbe worth when you retire in 2063, assuming they appreciate at an annual rate of 4.3 percent?12.
Calculating Interest Rates and Future Values [LO1, 3] In 1895, the first U.S. Open GolfChampionship was held. The winners prize money was $150. In 2014, the winners check was$1,620,000. What was the percentage increase per year in the winners check over this period? If thewinners prize increases at the same rate, what will it be in 2040?
13.
Calculating Rates of Return [LO3] Although appealing to more refined tastes, art as a collectible hasnot always performed so profitably. During 2003, Sothebys sold the Edgar Degas bronze sculpture14.
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Petite Danseuse de Quatorze Ans at auction for a price of $10,311,500. Unfortunately for the previousowner, he had purchased it in 1999 at a price of $12,377,500. What was his annual rate of return on thissculpture?
INTERMEDIATE
(Questions 1520)
Calculating Rates of Return [LO3] The Brasher doubloon, which was featured in the plot of theRaymond Chandler novel The High Window, was sold at auction in 2014 for $4,582,500. The coin hada face value of $15 when it was first issued in 1787 and had been previously sold for $430,000 in 1979.At what annual rate did the coin appreciate from its minting to the 1979 sale? What annual rate did the1979 buyer earn on his purchase? At what annual rate did the coin appreciate from its minting to the2014 sale?
15.
Calculating Rates of Return. [LO3] Refer back to the Series EE savings bonds we discussed at thevery beginning of the chapter.
Assuming you purchased a $50 face value bond, what is the exact rate of return you would earnif you held the bond for 20 years until it doubled in value?
1.
If you purchased a $50 face value bond in early 2014 at the then current interest rate of .10percent per year, how much would the bond be worth in 2024?
2.
In 2024, instead of cashing the bond in for its then current value, you decide to hold the bonduntil it doubles in face value in 2034. What rate of return will you earn over the last 10 years?3.
16.
Calculating Present Values [LO2] Suppose you are still committed to owning a $225,000 Ferrari (seeProblem 9). If you believe your mutual fund can achieve a 12 percent annual rate of return and youwant to buy the car in 9 years on the day you turn 30, how much must you invest today?17.
Calculating Future Values [LO1] You have just made your first $5,000 contribution to yourretirement account. Assuming you earn a return of 10 percent per year and make no additionalcontributions, what will your account be worth when you retire in 45 years? What if you wait 10 yearsbefore contributing? (Does this suggest an investment strategy?)
18.
Calculating Future Values [LO1] You are scheduled to receive $20,000 in two years. When youreceive it, you will invest it for six more years at 7.3 percent per year. How much will you have in eightyears?
19.
Calculating the Number of Periods [LO4] You expect to receive $15,000 at graduation in two years.You plan on investing it at 9 percent until you have $75,000. How long will you wait from now?20.
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