ࡱ> TVQRSy %_bjbj 46hh%W` ` d4!!!#$$$33333335S8b3$$$$$3!#3 . . .$!#3 .$3 . .40\0! @ ^ <X(jL03304T08+J8\0\0 8f3<$$ .$$$$$33 .$$$4$$$$8$$$$$$$$$` i: Simple Interest  HYPERLINK "javascript:popUp('Simple%20Interest:%20Interest%20is%20paid%20on%20beginning%20principal%20only.');" \o "Simple Interest: Interest is paid on beginning principal only." Simple interest is a topic that most people cover in elementary school. Interest may be thought of as rent paid on borrowed money. Simple interest is calculated only on the beginning principal. For instance, if someone were to receive 5% interest on a beginning value of $100, the first year they would get: .05 $100 or $5 in interest If they continued to receive 5% interest on the original $100 amount, over five years the growth in their investment would look like this: Year 1: 5% of $100 = $5 + $100 = $105 Year 2: 5% of $100 = $5 + $105 = $110 Year 3: 5% of $100 = $5 + $110 = $115 Year 4: 5% of $100 = $5 + $115 = $120 Year 5: 5% of $100 = $5 + $120 = $125 Compound Interest  HYPERLINK "javascript:popUp('Compound%20Interest:%20Interest%20paid%20on%20beginning%20principal%20and%20any%20principal%20accumulated%20in%20the%20meantime.');" \o "Compound Interest: Interest paid on beginning principal and any principal accumulated in the meantime." Compound interest is another matter. It's good to receive compound interest, but not so good to pay compound interest. With compound interest, interest is calculated not only on the beginning interest, but on any interest accumulated in the meantime. For instance, if someone were to receive 5% compound interest on a beginning value of $100, the first year they would get the same thing as if they were receiving simple interest on the $100, or $5. The second year, though, their interest would be calculated on the beginning amount in year 2, which would be $105. So their interest would be: .05 $105 or $5.25 in interest This provides a balance at the end of year two of $110.25 If this were to continue for 5 years, the growth in the investment would look like this: Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00 Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25 Year 3: 5% of $110.25 = $5.51 + $110.25 = $115.76 Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55 Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63 Note that in comparing growth graphs of simple and compound interest, investments with simple interest grow in a linear fashion and compound interest results in geometric growth. So with compound interest, the further in time an investment is held the more dramatic the growth becomes. Simple Interest  Compound Interest  Compound Interest Formula Instead of calculating interest year-by-year, it would be simple to see the future value of an investment using a compound interest formula. The formula for compound interest is: Pn = P0(1 + I)n where: Pn = Value at end of n time periods P0 = Beginning Value I = Interest n = Number of years For example, if one were to receive 5% compounded interest on $100 for five years, to use the formula, simply plug in the appropriate values and calculate. Pn = P0(1 + I)n so, Pn = $100(1.05)5 or Pn = $127.63 If there was a factor that summarized the part of the compound interest formula highlighted in red in the equation below, then to find future values all that would be necessary is to multiply that factor by the beginning values. Pn = P0(1 + I)n Future Value Tables Using Tables to Solve Future Value Problems Compound interest tables have been calculated by figuring out the (1+I)n values for various time periods and interest rates. Look at Time Value of Money Table 1: Future Value Factors.   HYPERLINK "javascript:openWin('/common/table1.pdf','550','420');" \o "Display Future Value Factors Table" Time Value of Money Table 1: Future Value Factors Note that all Time Value of Money Tables in this overview are in Adobe Acrobat (PDF) format. To view these tables you will need to download a free copy of  HYPERLINK "http://www.adobe.com/acrobat/readstep.html" \t "new" Adobe Acrobat Reader. This table summarizes the factors for various interest rates for various years. To use the table, simply go down the left-hand column to locate the appropriate number of years. Then go out along the top row until the appropriate interest rate is located. Note there are three pages containing interest rates 1% through 19%. For instance, to find the future value of $100 at 5% compound interest, look up five years on the table, then go out to 5% interest. At the intersection of these two values, a factor of 1.2763 appears. Multiplying this factor times the beginning value of $100.00 results in $127.63, exactly what was calculated using the Compound Interest Formula. Note, however, that there may be slight differences between using the formula and tables due to rounding errors. An example shows how simple it is to use the tables to calculate future amounts. You deposit $2000 today at 6% interest. How much will you have in 5 years? Future Value Tables Using Tables to Solve Future Value Problems Practice Excercise  HYPERLINK "javascript:openWin('/common/table1.pdf','550','420');" \o "Display Future Value Factors Table" Future Value Factors The following excercise should aid in using tables to solve future value problems. Please answer the questions below and proceed by clicking the Check My Answers button located at the bottom of the page. Top of Form 1. You invest $5,000 today. You will earn 8% interest. How much will you have in 4 years? (pick the closest answer)   HTMLCONTROL Forms.HTML:Option.1  $6,802.50   HTMLCONTROL Forms.HTML:Option.1  $6,843.00   HTMLCONTROL Forms.HTML:Option.1  $3,675 2. You have $450,000 to invest. If you think you can earn 7%, how much could you accumulate in 10 years? ? (pick the closest answer)   HTMLCONTROL Forms.HTML:Option.1  $25,415   HTMLCONTROL Forms.HTML:Option.1  $885,240   HTMLCONTROL Forms.HTML:Option.1  $722,610 3. If a commodity costs $500 now and inflation is expected to go up at the rate of 10% per year, how much will the commodity cost in 5 years?   HTMLCONTROL Forms.HTML:Option.1  $805.25   HTMLCONTROL Forms.HTML:Option.1  $3,052.55   HTMLCONTROL Forms.HTML:Option.1  Cannot tell from this information Bottom of Form Annuities Using Tables to Solve Future Value of Annuity Problems An annuity is an equal, annual series of cash flows. Annuities may be equal annual deposits, equal annual withdrawals, equal annual payments, or equal annual receipts. The key is equal, annual cash flows. Annuities work the way shown in the following explanation. Note that the cash flows occur at the end of the year. This makes the cash flow an  HYPERLINK "javascript:popUp('Ordinary%20Annuity:%20End%20of%20year%20cash%20flows.');" \o "Ordinary Annuity: End of year cash flows." ordinary annuity. If the cash flows were at the beginning of the year, they would be an  HYPERLINK "javascript:popUp('Annuity%20Due:%20Beginning%20of%20year%20cash%20flows.');" \o "Annuity Due: Beginning of year cash flows." annuity due. Annuities due will be covered a later. Example: Annuity = Equal Annual Series of Cash Flows Assume annual deposits of $100 deposited at end of year earning 5% interest for three years Year 1: $100 deposited at end of year Year 2: $100 .05 = $5.00 + $100 + $100 Year 3: $205 .05 = $10.25 + $205 + $100 = $100.00 = $205.00 = $315.25 Again, there are tables for working with annuities. Table 2: Future Value of Annuity Factors is the table to be used in calculating annuities due. Basically, this table works the same way as Table 1. Just look up the appropriate number of periods, locate the appropriate interest, take the factor found and multiply it by the amount of the annuity.   HYPERLINK "javascript:openWin('/common/table2.pdf','550','420');" \o "Display Future Value of Annuity Factors Table" Time Value of Money Table 2: Future Value of Annuity Factors For instance, on the three-year, 5% interest annuity of $100 per year. Going down three years, out to 5%, the factor of 3.152 is found. Multiply that by the annuity of $100 yields a future value of $315.20. Another example of calculating the future value of an annuity is illustrated. You deposit $300 each year for 15 years at 6%. How much will you have at the end of that time? Annuities Using Tables to Solve Future Value of Annuity Problems Practice Excercise  HYPERLINK "javascript:openWin('/common/table2.pdf','550','420');" \o "Display Future Value of Annuity Factors Table" Future Value of Annuity Factors The following excercise should aid in using tables to solve annuity problems. Please answer the questions below and proceed by clicking the Check My Answers button located at the bottom of the page. Top of Form 1. You invest $2,000 in IRA's each year for 5 years. If interest on these IRA's is 4%, how much will you have at the end of those 5 years?   HTMLCONTROL Forms.HTML:Option.1  $10,000   HTMLCONTROL Forms.HTML:Option.1  $10,832.60   HTMLCONTROL Forms.HTML:Option.1  $8,903.60 2. If you deposit $4,500 each year into an account paying 8% interest, how much will you have at the end of 3 years?   HTMLCONTROL Forms.HTML:Option.1  $13,500   HTMLCONTROL Forms.HTML:Option.1  $14,608.80   HTMLCONTROL Forms.HTML:Option.1  $11,596.95 Bottom of Form Present Value Using Tables to Solve Present Value Problems  HYPERLINK "javascript:popUp('Present%20Value:%20Reciprocal%20of%20future%20value.');" \o "Present Value: Reciprocal of future value." Present value is simply the reciprocal of compound interest. Another way to think of present value is to adopt a stance out on the time line in the future and look back toward time 0 to see what was the beginning amount. Present Value = P0 = Pn / (1+I)n    HYPERLINK "javascript:openWin('/common/table3.pdf','550','420');" \o "Display Present Value Factors Table" Time Value of Money Table 3: Present Value Factors Table 3 shows present value factors. Note that they are all less than one. Therefore, when multiplying a future value by these factors, the future value is discounted down to present value. The table is used in much the same way as the other time value of money tables. To find the present value of a future amount, locate the appropriate number of years and the appropriate interest rate, take the resulting factor and multiply it times the future value. An example illustrates the process. How much would you have to deposit now to have $15,000 in 8 years if interest is 7%? Present Value Using Tables to Solve Present Value Problems Practice Excercise  HYPERLINK "javascript:openWin('/common/table3.pdf','550','420');" \o "Display Present Value Factors Table" Present Value Factors The following excercise should aid in using tables to solve present value problems. Please answer the questions below and proceed by clicking the Check My Answers button located at the bottom of the page. Top of Form 1. If you want to have $10,000 in 3 years and you can earn 8%, how much would you have to deposit today?   HTMLCONTROL Forms.HTML:Option.1  $7938.00   HTMLCONTROL Forms.HTML:Option.1  $25,771   HTMLCONTROL Forms.HTML:Option.1  $12,597 2. If you think you can sell an asset for $25,000 in five years and you think the appropriate discount rate is 5%, how much would you be will to pay for the asset today?   HTMLCONTROL Forms.HTML:Option.1  $25,000   HTMLCONTROL Forms.HTML:Option.1  $19,587.50   HTMLCONTROL Forms.HTML:Option.1  Cannot Tell From This Information Bottom of Form Present Value Using Tables to Solve Present Value Problems Practice Excercise Results Top of Form 1. If you want to have $10,000 in 3 years and you can earn 8%, how much would you have to deposit today?   HTMLCONTROL Forms.HTML:Option.1  $7938.00   HTMLCONTROL Forms.HTML:Option.1  $25,771   HTMLCONTROL Forms.HTML:Option.1  $12,597 2. If you think you can sell an asset for $25,000 in five years and you think the appropriate discount rate is 5%, how much would you be will to pay for the asset today?   HTMLCONTROL Forms.HTML:Option.1  $25,000   HTMLCONTROL Forms.HTML:Option.1  $19,587.50   HTMLCONTROL Forms.HTML:Option.1  Cannot Tell From This Information Bottom of Form Present Value of an Annuity Using Tables to Solve Present Value of an Annuity Problems To find the present value of an annuity, use Table 4. Find the appropriate factor and multiply it times the amount of the annuity to find the present value of the annuity.   HYPERLINK "javascript:openWin('/common/table4.pdf','550','420');" \o "Display Present Value Of Annuity Factors Table" Time Value of Money Table 4: Present Value Of Annuity Factors An example illustrates the process. Find the present value of a 4-year, $3,000 per year annuity at 6%. Intrayear Compounding If a cash flow is compounded more frequently than annually, then  HYPERLINK "javascript:popUp('Intrayear%20Compounding:%20More%20frequent%20compounding%20than%20annual.');" \o "Intrayear Compounding: More frequent compounding than annual." intrayear compounding is being used. To adjust for intrayear compounding, an  HYPERLINK "javascript:popUp('Interest%20Rate%20per%20Compounding%20Period:%20Annual%20rate%20divided%20by%20time%20per%20year%20compounding.');" \o "Interest Rate per Compounding Period: Annual rate divided by time per year compounding." interest rate per compounding period must be found as well as the total  HYPERLINK "javascript:popUp('Number%20of%20Compounding%20Periods:%20Number%20of%20years%20times%20number%20of%20times%20per%20year%20compounding.');" \o "Number of Compounding Periods: Number of years times number of times per year compounding." number of compounding periods. The interest rate per compounding period is found by taking the annual rate and dividing it by the number of times per year the cash flows are compounded. The total number of compounding periods is found by multiplying the number of years by the number of times per year cash flows are compounded. For instance, suppose someone were to invest $5,000 at 8% interest, compounded semiannually, and hold it for five years. The interest rate per compounding period would be 4%, ( 8% / 2 ). The number of compounding periods would be 10, ( 5 2 ). To solve, find the future value of a single sum looking up 4% and 10 periods in the Future Value table. FV = PV(FVIF) FV = $5,000(1.480) FV = $7,400   HYPERLINK "javascript:openWin('/common/table1.pdf','550','420');" \o "Display Future Value Factors Table" Time Value of Money Table 1: Future Value Factors Annuities Due  HYPERLINK "javascript:popUp('Annuity%20Due:%20Beginning%20of%20year%20cash%20flows.');" \o "Annuity Due: Beginning of year cash flows." Annuities due are beginning-of-year annuities. To work annuities due, simply set up the problem the same way as would be done with an ordinary annuity, then multiply the resulting factor by (1+I). This is done whether the problem is present value or future value. To illustrate this: Find the future value of a three-year 6% annuity due or $4000. FVa = Annuity(FVIFa)(1+I) FVa = 4000(3.1836)(1.06) FVa = $13,498.46 Perpetuities  HYPERLINK "javascript:popUp('Perpetuity:%20Cash%20flow%20with%20no%20fixed%20time%20horizon.');" \o "Perpetuity: Cash flow with no fixed time horizon." Perpetuity is a cash flow without a fixed time horizon. For example if someone were promised that they would receive a cash flow of $400 per year until they died, that would be a perpetuity. To find the present value of a perpetuity, simply take the annual return in dollars and divide it by the appropriate discount rate. To illustrate this: If someone were promised a cash flow of $400 per year until they died and they could earn 6% on other investments of similar quality, in present value terms the perpetuity would be worth $6,666.67. ( $400 / .06 = $6,666.67 ) Using a Financial Calculator Financial calculators may be used to solve time value of money problems. To use a financial calculator, it is necessary to understand the owner's manual. Generally, the following steps should be followed. 1. Read the problem thoroughly. 2. Make sure what is being asked in the problem. 3. Clear the calculator. 4. Input the known value. 5. Input the number of compounding periods per year. 6. Input the annual interest rate. 7. Input the total number of compounding periods. 8. Request the unknown. One thing to keep in mind is that cash outflows should carry a minus sign. The following three examples illustrate the use of a financial calculator in solving time value of money problems. The Hewlett Packard 10B calculator is used in these examples.Using a Financial Calculator Example 1: How much will $15,000 be worth in five years if interest is 8% compounded quarterly? 1. Read the problem thoroughly. 2. Make sure what is being asked in the problem. What is the future value of the $15,000? 3. Clear the calculator. Gold. Clear All. 4. Input the known value. 15000 +/ PV 5. Input the number of compounding periods per year. 4 Gold P/YR 6. Input the annual interest rate. 8 I/YR 7. Input the total number of compounding periods. 20 N 8. Request the unknown. FV  Answer: $22,289.21 Using a Financial Calculator Example 2: How much would you have in four years if you deposit $40,000 at the beginning of each year for four years and interest is 10% compounded annually? 1. Read the problem thoroughly. 2. Make sure what is being asked in the problem. What is the future value of a $40,000 four-year annuity due? 3. Clear the calculator. Gold. Clear All. 4. Input the known value. 40,000 +/ PMT 5. Input the number of compounding periods per year. 1 Gold P/YR 6. Input the annual interest rate. 10 I/YR 7. Input the total number of compounding periods. Gold BEG/END 4 N 8. Request the unknown. FV  Answer: $204,204.00 Using a Financial Calculator Example 3: What would you have to deposit today to have $50,000 in eight years if you can earn 6% interest, compounded semiannually? 1. Read the problem thoroughly. 2. Make sure what is being asked in the problem. What is the present value of $50,000? 3. Clear the calculator. Gold. Clear All. 4. Input the known value. 50,000 FV 5. Input the number of compounding periods per year. 2 Gold P/YR 6. Input the annual interest rate. 6 I/YR 7. Input the total number of compounding periods. 16 N 8. Request the unknown. PV  Answer: $31,158.35 Additional Resources Time Value Using Microsoft Excel It is possible to use financial spreadsheet programs to solve time value of money problems. A Microsoft Excel template illustrates using a spreadsheet approach to solve time value of money problems.   HYPERLINK "http://www.studyfinance.com/templates/timevalue.xls" \o "Excel Template: Time Value" \t "_blank" Excel Template: Time Value  HYPERLINK "http://www.studyfinance.com/lessons/timevalue/index.mv?page=19&xls" \o "xls Document. What is an xls Document?" (xls) Time Value of Money Problem Bank Now that you have completed the Time Value of Money Overview, use the Problem Bank as practice.   HYPERLINK "http://www.studyfinance.com/lessons/timevalue/index.mv?problembank" \o "Time Value of Money Problem Bank" Time Value of Money Problem Bank Time Value of Money Problem Bank HYPERLINK "http://www.studyfinance.com/lessons/timevalue/index.mv?page=19" \o "Exit Time Value of Money Problem Bank" Exit Problem Bank You are planning to retire in twenty years. You'll live ten years after retirement. You want to be able to draw out of your savings at the rate of $10,000 per year. How much would you have to pay in equal annual deposits until retirement to meet your objectives? Assume interest remains at 9%. You can deposit $4000 per year into an account that pays 12% interest. If you deposit such amounts for 15 years and start drawing money out of the account in equal annual installments, how much could you draw out each year for 20 years? What is the value of a $100 perpetuity if interest is 7%? You deposit $13,000 at the beginning of every year for 10 years. If interest is being paid at 8%, how much will you have in 10 years? You are getting payments of $8000 at the beginning of every year and they are to last another five years. At 6%, what is the value of this annuity? How much would you have to deposit today to have $10,000 in five years at 6% interest compounded semiannually? Construct an amortization schedule for a 3-year loan of $20,000 if interest is 9%. If you get payments of $15,000 per year for the next ten years and interest is 4%, how much would that stream of income be worth in present value terms? Your company must deposit equal annual beginning of year payments into a sinking fund for an obligation of $800,000 which matures in 15 years. Assuming you can earn 4% interest on the sinking fund, how much must the payments be? If you deposit $45,000 into an account earning 4% interest compounded quarterly, how much would you have in 5 years? How much would you pay for an investment which will be worth $16,000 in three years? Assume interest is 5%. You have $100,000 to invest at 4% interest. If you wish to withdraw equal annual payments for 4 years, how much could you withdraw each year and leave $0 in the investment account? You are considering the purchase of two different insurance annuities. Annuity A will pay you $16,000 at the beginning of each year for 8 years. 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