1、What is Time Value? We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return In other words, “a dollar received today is worth more than a dollar to be received tomorrow” That is because todays dollar can be invested so that we h

2、ave more than one dollar tomorrow第1頁/共44頁The Terminology of Time Value Present Value - An amount of money today, or the current value of a future cash flow Future Value - An amount of money at some future time period Period - A length of time (often a year, but can be a month, week, day, hour, etc.)

3、 Interest Rate - The compensation paid to a lender (or saver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate)第2頁/共44頁Abbreviations PV - Present value FV - Future value Pmt - Per period payment amount N - Either the total number of cash flows or the

4、number of a specific period i - The interest rate per period第3頁/共44頁Timelines012345PVFVTodayv A timeline is a graphical device used to clarify the timing of the cash flows for an investmentv Each tick represents one time period第4頁/共44頁Calculating the Future Value Suppose that you have an extra $100

5、today that you wish to invest for one year. If you can earn 10% per year on your investment, how much will you have in one year?012345-100?FV1100 1010110.第5頁/共44頁Calculating the Future Value (cont.) Suppose that at the end of year 1 you decide to extend the investment for a second year. How much wil

6、l you have accumulated at the end of year 2?012345-110?FVorFV222100 1010 1010121100 1010121.第6頁/共44頁Generalizing the Future Value Recognizing the pattern that is developing, we can generalize the future value calculations as follows:FVPViNN1vIf you extended the investment for a third year, you would

7、 have:FV33100 101013310.第7頁/共44頁Compound Interest Note from the example that the future value is increasing at an increasing rate In other words, the amount of interest earned each year is increasing Year 1: $10 Year 2: $11 Year 3: $12.10 The reason for the increase is that each year you are earning

8、 interest on the interest that was earned in previous years in addition to the interest on the original principle amount第8頁/共44頁Compound Interest Graphically第9頁/共44頁The Magic of Compounding On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for $24 worth of be

9、ads and other trinkets(珠子和其他飾品). Was this a good deal for the Indians? This happened about 371 years ago, so if they could earn 5% per year they would now (in 1997) have:vIf they could have earned 10% per year, they would now have:$54,562,898,811,973,500.00 = 24(1.10)371$1,743,577,261.65 = 24(1.05)3

10、71Thats about 54,563 Trillion(萬億) dollars!第10頁/共44頁The Magic of Compounding (cont.) The Wall Street Journal (17 Jan. 92) says that all of New York city real estate is worth about $324 billion. Of this amount, Manhattan is about 30%, which is $97.2 billion At 10%, this is $54,562 trillion! Our U.S. G

11、NP is only around $6 trillion per year. So this amount represents about 9,094 years worth of the total economic output of the USA!.第11頁/共44頁Calculating the Present Value So far, we have seen how to calculate the future value of an investment But we can turn this around to find the amount that needs

12、to be invested to achieve some desired future value: PVFViNN1第12頁/共44頁Present Value: An Example Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about $100,000 on her 18th birthday to pay for four years of

13、 college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal?PV 100 0001087697913,.$36,.第13頁/共44頁Annuities An annuity is a series of nominally equal payments equally spaced in time(等時間間隔) Annuities are very common: Rent Mortgage payments Car pa

14、yment Pension income The timeline shows an example of a 5-year, $100 annuity012345100100100100100第14頁/共44頁The Principle of Value Additivity How do we find the value (PV or FV) of an annuity? First, you must understand the principle of value additivity: The value of any stream of cash flows is equal

15、to the sum of the values of the components In other words, if we can move the cash flows to the same time period we can simply add them all together to get the total value 價值相加第15頁/共44頁Present Value of an Annuity We can use the principle of value additivity to find the present value of an annuity, b

16、y simply summing the present values of each of the components: PVPmtiPmtiPmtiPmtiAtttNNN111111122第16頁/共44頁Present Value of an Annuity (cont.) Using the example, and assuming a discount rate of 10% per year, we find that the present value is:PVA1001101001101001101001101001103790812345.012345100100100

17、10010062.0968.3075.1382.6490.91379.08第17頁/共44頁Present Value of an Annuity (cont.) Actually, there is no need to take the present value of each cash flow separately We can use a closed-form of the PVA equation instead:PVPmtiPmtiiAtttNN11111第18頁/共44頁Present Value of an Annuity (cont.) We can use this

18、equation to find the present value of our example annuity as follows:PVPmtA11110010379 085.vThis equation works for all regular annuities, regardless of the number of payments第19頁/共44頁The Future Value of an Annuity We can also use the principle of value additivity to find the future value of an annu

19、ity, by simply summing the future values of each of the components:FVPmtiPmtiPmtiPmtAtN ttNNNN11111122第20頁/共44頁The Future Value of an Annuity (cont.) Using the example, and assuming a discount rate of 10% per year, we find that the future value is:FVA100 110100 110100 110100 110100610514321.10010010

20、0100100012345146.41133.10121.00110.00= 610.51at year 5第21頁/共44頁The Future Value of an Annuity (cont.) Just as we did for the PVA equation, we could instead use a closed-form of the FVA equation:FVPmtiPmtiiAtN ttNN1111vThis equation works for all regular annuities, regardless of the number of payment

21、s第22頁/共44頁The Future Value of an Annuity (cont.) We can use this equation to find the future value of the example annuity:FVA1001101010610515.第23頁/共44頁Annuities Due預付年金 Thus far, the annuities that we have looked at begin their payments at the end of period 1; these are referred to as regular annuit

22、ies A annuity due is the same as a regular annuity, except that its cash flows occur at the beginning of the period rather than at the end0123451001001001001001001001001001005-period Annuity Due5-period Regular Annuity第24頁/共44頁Present Value of an Annuity Due We can find the present value of an annui

23、ty due in the same way as we did for a regular annuity, with one exception Note from the timeline that, if we ignore the first cash flow, the annuity due looks just like a four-period regular annuity Therefore, we can value an annuity due with:PVPmtiiPmtADN1111第25頁/共44頁Present Value of an Annuity Du

24、e (cont.) Therefore, the present value of our example annuity due is:PVAD10011110010100416985 1.vNote that this is higher than the PV of the, otherwise equivalent, regular annuity第26頁/共44頁Future Value of an Annuity Due To calculate the FV of an annuity due, we can treat it as regular annuity, and th

25、en take it one more period forward:FVPmtiiiADN111012345PmtPmtPmtPmtPmt第27頁/共44頁Future Value of an Annuity Due (cont.) The future value of our example annuity is:FVAD1001101010110671565.vNote that this is higher than the future value of the, otherwise equivalent, regular annuity第28頁/共44頁Deferred Annu

26、ities遞延年金 A deferred annuity is the same as any other annuity, except that its payments do not begin until some later period The timeline shows a five-period deferred annuity01234510010010010010067第29頁/共44頁PV of a Deferred Annuity We can find the present value of a deferred annuity in the same way a

27、s any other annuity, with an extra step required Before we can do this however, there is an important rule to understand:When using the PVA equation, the resulting PV is always one period before the first payment occurs第30頁/共44頁PV of a Deferred Annuity (cont.) To find the PV of a deferred annuity, w

28、e first find use the PVA equation, and then discount that result back to period 0 Here we are using a 10% discount rate0123450010010010010010067PV2 = 379.08PV0 = 313.29第31頁/共44頁PV251001111001037908.PV023790811031329.Step 1:Step 2:第32頁/共44頁FV of a Deferred Annuity The future value of a deferred annui

29、ty is calculated in exactly the same way as any other annuity There are no extra steps at all第33頁/共44頁Uneven Cash Flows Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity We can find the present or future value of such a stream by using the principle

30、 of value additivity第34頁/共44頁Uneven Cash Flows: An Example (1) Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment?012345100200300PV 10010720010730010751304123.第35頁/共44頁Uneven Cash Flows: An Exampl

31、e (2) Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year?012345300500700FV300 105500 10570015557521.,.第36頁/共44頁Non-annual Compounding So far we have assumed that the time period is equal to a

32、year However, there is no reason that a time period cant be any other length of time We could assume that interest is earned semi-annually, quarterly, monthly, daily, or any other length of time The only change that must be made is to make sure that the rate of interest is adjusted to the period len

33、gth第37頁/共44頁Non-annual Compounding (cont.) Suppose that you have $1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds?BankInterest RateCompoundingFirst National10%AnnualSecond National1

34、0%MonthlyThird National10%Daily第38頁/共44頁Non-annual Compounding (cont.) To solve this problem, you need to determine which bank will pay you the most interest In other words, at which bank will you have the highest future value? To find out, lets change our basic FV equation slightly:FVPVimNm1In this version of the equation m is the number of compounding periods per year第39頁/共44頁Non-annual Compounding (cont.) We can find the FV for each bank as follows:FV 1000 11011001,.,FV 1000 10101211047112,.,.FV 1000 1010365110