1、 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 have more than one do

2、llar tomorrow 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.) Interest Rate - The compensation paid to a lender (or sa

3、ver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate) PV - Present value FV - Future value Pmt - Per period payment amount N - Either the total number of cash flows or the number of a specific period i - The interest rate per period012345PVFVTodayvA

4、timeline is a graphical device used to clarify the timing of the cash flows for an investmentvEach tick represents one time period Suppose that you have an extra $100 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?01234

5、5-100?FV11001010110. Suppose that at the end of year 1 you decide to extend the investment for a second year. How much will you have accumulated at the end of year 2?012345-110?F Vo rF V2221 0 0101 0101 01 2 11 0 0101 01 2 1. Recognizing the pattern that is developing, we can generalize the future v

6、alue calculations as follows:FVPViNN1vIf you extended the investment for a third year, you would have:FV33100101013310. 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

7、: $12.10 The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amount On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for $24 worth of b

8、eads 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)

9、371Thats about 54,563 Trillion(萬億) dollars! 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. GNP is only around $6 trillion per year.

10、So this amount represents about 9,094 years worth of the total economic output of the USA! . 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 to be invested to achieve some desired future value: PVFViNN1 Suppose that yo

11、ur 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 college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your

12、 goal?PV1000001087697913,.$36,. 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 payment Pension income The timeline shows an example of a 5-year, $100 annuity012345100100100100100 How do we find the value (PV or F

13、V) of an annuity? First, you must understand the principle of value additivity: The value of any stream of cash flows is equal 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

14、值相加 We can use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components: PVPmtiPmtiPmtiPmtiAtttNNN111111122 Using the example, and assuming a discount rate of 10% per year, we find that the present value is:PVA100110100

15、1101001101001101001103790812345.01234510010010010010062.0968.3075.1382.6490.91379.08 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:P VP mtiP mtiiAtttNN11111 We can use this equation to find the present value of

16、our example annuity as follows:P VP m tA111 1 00 1 03 7 90 85.vThis equation works for all regular annuities, regardless of the number of payments We can also use the principle of value additivity to find the future value of an annuity, by simply summing the future values of each of the components:F

17、VPmtiPmtiPmtiPmtAtNttNNNN11111122 Using the example, and assuming a discount rate of 10% per year, we find that the future value is:FVA100110100110100110100110100610514321.100100100100100012345146.41133.10121.00110.00= 610.51at year 5 Just as we did for the PVA equation, we could instead use a close

18、d-form of the FVA equation:FVPmtiPmtiiAtNttNN1111vThis equation works for all regular annuities, regardless of the number of payments We can use this equation to find the future value of the example annuity:F VA1 0 01 1 0101 06 1 05 15. Thus far, the annuities that we have looked at begin their paym

19、ents at the end of period 1; these are referred to as regular annuities 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 We can find t

20、he present value of an annuity 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:P VP m tiiP m tA DN1111 Ther

21、efore, the present value of our example annuity due is:P VAD1 0 0111 1 001 01 0 04 1 69 851.vNote that this is higher than the PV of the, otherwise equivalent, regular annuity To calculate the FV of an annuity due, we can treat it as regular annuity, and then take it one more period forward:F VP mti

22、iiADN111012345PmtPmtPmtPmtPmt The future value of our example annuity is:FVAD1001101010110671565.vNote that this is higher than the future value of the, otherwise equivalent, regular annuity A deferred annuity is the same as any other annuity, except that its payments do not begin until some later p

23、eriod The timeline shows a five-period deferred annuity01234510010010010010067 We can find the present value of a deferred annuity in the same way as 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, th

24、e resulting PV is always one period before the first payment occurs To find the PV of a deferred annuity, we 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 = 313.29P V251 0 0111 1 00 1 03 7

25、 9 0 8.P V023 7 90 81 1 03 1 3 2 9.Step 1:Step 2: The future value of a deferred annuity is calculated in exactly the same way as any other annuity There are no extra steps at all Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity We can find the pre

26、sent or future value of such a stream by using the principle of value additivity 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?012345100200300PV10010720010730010751304123. Suppose that you we

27、re 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?012345300500700FV30010550010570015557521.,. So far we have assumed that the time period is equal to a year However, there is no reason that a time period cant b

28、e 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 length Suppose that you have $1,000 available for investment.

29、 After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds?Ba n kI n t e r e s t Ra t eCo mpo un di n gF i r s t Na t i o n a l1 0 %An n ua lS e c o n d Na t i o n a l1 0 %Mo n t h l yTh i r d Na t i o n a l1 0 %Da i l y To

30、 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:F VP VimN m1In this version of the equation m is the number of compounding periods per year We can find the FV for each bank as follows:FV100011011001,.,FV100010101211047112,.,.FV10001010365110516365