1、CHAPTER 5 INTRODUCTION TO VALUATION:THE TIME VALUE OF MONEY(FORMULAS)Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Determine the future value of an investment made today Determine the present value
2、of cash to be received at a future date Find the return on an investment Calculate how long it takes for an investment to reach a desired valueKey Concepts and SkillsCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-H
3、ill Education.Future Value and Compounding Present Value and Discounting More about Present and Future ValuesChapter OutlineCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Present Value earlier money
4、on a time line Future Value later money on a time line Interest rate “exchange rate”between earlier money and later money Discount rate Cost of capital Opportunity cost of capital Required returnBasic DefinitionsCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution
5、 without the prior written consent of McGraw-Hill Education.Suppose you invest$1,000 for one year at 5%per year.What is the future value in one year?Interest=1,000(.05)=50 Value in one year=principal+interest=1,000+50=1,050 Future Value(FV)=1,000(1+.05)=1,050 Suppose you leave the money in for anoth
6、er year.How much will you have two years from now?FV=1,000(1.05)(1.05)=1,000(1.05)2=1,102.50Future ValueCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.FV=PV(1+r)t FV=future value PV=present value r=p
7、eriod interest rate,expressed as a decimal t=number of periods Future value interest factor=(1+r)tFuture Values:General FormulaCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Simple interest vs.Compou
8、nd interest Consider the previous example FV with simple interest=1,000+50+50=1,100 FV with compound interest=1,102.50 The extra 2.50 comes from the interest of.05(50)=2.50 earned on the first interest payment.Effects of CompoundingCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduc
9、tion or distribution without the prior written consent of McGraw-Hill Education.Texas Instruments BA-II Plus FV=future value PV=present value I/Y=period interest rate P/Y must equal 1 for the I/Y to be the period rate.Interest is entered as a percent,not a decimal.N=number of periods Remember to cle
10、ar the registers(CLR TVM)after each problem.Other calculators are similar in format.Calculator KeysCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Suppose you invest the$1,000 from the previous exampl
11、e for 5 years.How much would you have?FV=1,000(1.05)5=1,276.28 The effect of compounding is small for a small number of periods,but increases as the number of periods increases.(Simple interest would have a future value of$1,250,for a difference of$26.28.)Future Value Example 2Copyright 2019 McGraw-
12、Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Suppose you had a relative deposit$10 at 5.5%interest 200 years ago.How much would the investment be worth today?FV=10(1.055)200=447,189.84 What is the effect of compounding?
13、Simple interest=10+200(10)(.055)=120.00 Compounding added$447,069.84 to the value of the investment.Future Value Example 3Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Suppose your company expects t
14、o increase unit sales of widgets by 15%per year for the next 5 years.If you currently sell 3 million widgets in one year,how many widgets do you expect to sell in 5 years?FV=3,000,000(1.15)5=6,034,072Future Value as a General Growth FormulaCopyright 2019 McGraw-Hill Education.All rights reserved.No
15、reproduction or distribution without the prior written consent of McGraw-Hill Education.What is the difference between simple interest and compound interest?Suppose you have$500 to invest and you believe that you can earn 8%per year over the next 15 years.How much would you have at the end of 15 yea
16、rs using compound interest?How much would you have using simple interest?Quick Quiz Part ICopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.How much do I have to invest today to have some amount in the
17、future?FV=PV(1+r)t Rearrange to solve for PV=FV/(1+r)t When we talk about discounting,we mean finding the present value of some future amount.When we talk about the“value”of something,we are talking about the present value unless we specifically indicate that we want the future value.Present ValueCo
18、pyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Suppose you need$10,000 in one year for the down payment on a new car.If you can earn 7%annually,how much do you need to invest today?PV=10,000/(1.07)1=9,
19、345.79 Calculator 1 N 7 I/Y 10,000 FV CPT PV=-9,345.79Present Value Example 1Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.You want to begin saving for your daughters college education and you estim
20、ate that she will need$150,000 in 17 years.If you feel confident that you can earn 8%per year,how much do you need to invest today?PV=150,000/(1.08)17=40,540.34Present Value Example 2Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written co
21、nsent of McGraw-Hill Education.Your parents set up a trust fund for you 10 years ago that is now worth$19,671.51.If the fund earned 7%per year,how much did your parents invest?PV=19,671.51/(1.07)10=10,000Present Value Example 3Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction
22、or distribution without the prior written consent of McGraw-Hill Education.For a given interest rate the longer the time period,the lower the present value What is the present value of$500 to be received in 5 years?10 years?The discount rate is 10%5 years:PV=500/(1.1)5=310.46 10 years:PV=500/(1.1)10
23、=192.77Present Value Important Relationship ICopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.For a given time period the higher the interest rate,the smaller the present value What is the present valu
24、e of$500 received in 5 years if the interest rate is 10%?15%?Rate=10%:PV=500/(1.1)5=310.46 Rate=15%;PV=500/(1.15)5=248.59Present Value Important Relationship IICopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Ed
25、ucation.What is the relationship between present value and future value?Suppose you need$15,000 in 3 years.If you can earn 6%annually,how much do you need to invest today?If you could invest the money at 8%,would you have to invest more or less than at 6%?How much?Quick Quiz Part IICopyright 2019 Mc
26、Graw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.PV=FV/(1+r)t There are four parts to this equation:PV,FV,r and t If we know any three,we can solve for the fourth.If you are using a financial calculator,be sure to reme
27、mber the sign convention or you will receive an error(or a nonsense answer)when solving for r or t.The Basic PV Equation-RefresherCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Often we will want to
28、know what the implied interest rate is on an investment.Rearrange the basic PV equation and solve for r.FV=PV(1+r)t r=(FV/PV)1/t 1 If you are using formulas,you will want to make use of both the yx and the 1/x keys.Discount RateCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction
29、 or distribution without the prior written consent of McGraw-Hill Education.You are looking at an investment that will pay$1,200 in 5 years if you invest$1,000 today.What is the implied rate of interest?r=(1,200/1,000)1/5 1=.03714=3.714%Calculator the sign convention matters!N=5 PV=-1,000(you pay 1,
30、000 today)FV=1,200(you receive 1,200 in 5 years)CPT I/Y=3.714%Discount Rate Example 1Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Suppose you are offered an investment that will allow you to double
31、 your money in 6 years.You have$10,000 to invest.What is the implied rate of interest?r=(20,000/10,000)1/6 1=.122462=12.25%Discount Rate Example 2Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Suppos
32、e you have a 1-year old son and you want to provide$75,000 in 17 years towards his college education.You currently have$5,000 to invest.What interest rate must you earn to have the$75,000 when you need it?r=(75,000/5,000)1/17 1=.172688=17.27%Discount Rate Example 3Copyright 2019 McGraw-Hill Educatio
33、n.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.What are some situations in which you might want to know the implied interest rate?You are offered the following investments:You can invest$500 today and receive$600 in 5 years.The invest
34、ment is considered low risk.You can invest the$500 in a bank account paying 4%.What is the implied interest rate for the first choice and which investment should you choose?Quick Quiz Part IIICopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior w
35、ritten consent of McGraw-Hill Education.Start with the basic equation and solve for t(remember your logs).FV=PV(1+r)t t=ln(FV/PV)/ln(1+r)You can use the financial keys on the calculator as well;just remember the sign convention.Finding the Number of PeriodsCopyright 2019 McGraw-Hill Education.All ri
36、ghts reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.You want to purchase a new car,and you are willing to pay$20,000.If you can invest at 10%per year and you currently have$15,000,how long will it be before you have enough money to pay cash for th
37、e car?t=ln(20,000/15,000)/ln(1.1)=3.02 yearsNumber of Periods Example 1Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Suppose you want to buy a new house.You currently have$15,000,and you figure you
38、need to have a 10%down payment plus an additional 5%of the loan amount for closing costs.Assume the type of house you want will cost about$150,000 and you can earn 7.5%per year.How long will it be before you have enough money for the down payment and closing costs?Number of Periods Example 2Copyrigh
39、t 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.How much do you need to have in the future?Down payment=.1(150,000)=15,000 Closing costs=.05(150,000 15,000)=6,750 Total needed=15,000+6,750=21,750 Compute the
40、number of periods Using the formula:t=ln(21,750/15,000)/ln(1.075)=5.14 years Using a financial calculator:PV=-15,000,FV=21,750,I/Y=7.5,CPT N=5.14 yearsNumber of Periods Example 2(CTD.)Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written c
41、onsent of McGraw-Hill Education.When might you want to compute the number of periods?Suppose you want to buy some new furniture for your family room.You currently have$500,and the furniture you want costs$600.If you can earn 6%,how long will you have to wait if you dont add any additional money?Quic
42、k Quiz Part IVCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Use the following formulas for TVM calculations FV(rate,nper,pmt,pv)PV(rate,nper,pmt,fv)RATE(nper,pmt,pv,fv)NPER(rate,pmt,pv,fv)The formul
43、a icon is very useful when you cant remember the exact formula.Click on the Excel icon to open a spreadsheet containing four different examples.Spreadsheet ExampleCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill
44、 Education.Many financial calculators are available online.Go to Investopedias website and work the following example:You need$50,000 in 10 years.If you can earn 6%interest,how much do you need to invest today?You should get$27,919.74Work the Web ExampleCopyright 2019 McGraw-Hill Education.All right
45、s reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.Table 5.4Copyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.You have$10,000 to invest for five years.
46、How much additional interest will you earn if the investment provides a 5%annual return,when compared to a 4.5%annual return?How long will it take your$10,000 to double in value if it earns 5%annually?What annual rate has been earned if$1,000 grows into$4,000 in 20 years?Comprehensive ProblemCopyrig
47、ht 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.End of Chapter Chapter 5-formulasCopyright 2019 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.
