Time Value of Money Module

Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the present value of a sngle sum. 4 Compute and use the future value of an ordnary annuty. 5 Compute and use the future value of an annuty due. 6 Compute and use the present value of an ordnary annuty. 7 Compute and use the present value of an annuty due. 8 Compute and use the present value of a deferred ordnary annuty. 9 Explan the conceptual ssues regardng the use of present value n fnancal reportng.

M2 Tme Value of Money Module Suppose someone asked you, Would you rather have 00 today or 00 next year? Your answer should be, I d rather have 00 today. Ths reply nvolves consderng the tme value of money. The dfference n worth between the two amounts, the tme value of money, s nterest. Interest s the cost of the use of money over tme. It s an expense to the borrower and revenue to the lender. Therefore, t s a very mportant element n the decson makng related to the acquston and dsposal of many of the resources of a company. Interest concepts are nvolved n the development of many values that a company reports on ts fnancal statements. Also, managers need to understand the concept of nterest when makng decsons where cash pad or receved now must be compared wth amounts that wll be receved or pad n the future. The cash flows at varous dates, say some at three years from now, some at two years from now, and some at one year from now, cannot be added together to produce a relevant value. Future cash flows, before they can be added, must be converted to a common denomnator by beng restated to ther present values as of a specfc moment n tme (often referred to as tme perod zero). The dollars to be receved or pad three years from now have a smaller present value than those to be receved or pad two years or one year from now. The converson of these future value amounts to the present value common denomnator s known as dscountng and nvolves the removal of the nterest or dscount the tme value of money from those dollars that would be receved or pad three years, two years, or one year from now. Instead of restatng some of the cash nflows and outflows to ther present values at tme perod zero, a common denomnator s also acheved by statng them at a future value by addng the tme value of money (nterest) to these nflows and outflows. The future value of any seres of nflows or outflows s the sum of these perodc amounts plus the compound nterest calculated on the amounts. A company uses the present value or the future value n many stuatons, such as () for measurement and reportng of some of ts assets and labltes, snce many accountng pronouncements requre the use of present value concepts n a number of measurement and reportng ssues; and (2) when t accumulates nformaton for decson makng nvolvng, for example, property, plant, and equpment acqustons. We dscuss these concepts n ths Module and we apply them n varous chapters when we dscuss how a company records and reports () long-term notes payable and notes recevable when the nterest rate s not specfed or dffers from the market rate at the tme of the transacton, (2) assets acqured by the ssuance of long-term debt securtes that carry ether no stated rate of nterest, or a rate of nterest that s dfferent from the market rate at the tme of the transacton, (3) bonds payable and nvestments n bonds and the amortzaton of bond premums and dscounts n each case, (4) long-term leases, (5) varous aspects of employees post-employment benefts, and (6) mparment of noncurrent assets. Varous compound nterest technques are used n the measurement of the values (costs) of these and other types of transactons. Most compound nterest applcatons can be calculated by a longhand arthmetc process. However, qucker approaches and shortcuts to the solutons of the problems are avalable. In ths Module we llustrate the basc prncples of compound nterest n a way that leads to the development of tables used to resolve ssues ntroduced throughout ths book. Note that many of the calculatons are rounded. SIMPLE INTEREST AND COMPOUND INTEREST Smple nterest s nterest on the orgnal prncpal (amount orgnally receved or pad) regardless of the number of tme perods that have passed or the amount of nterest that has been pad or accrued n the past. Interest rates are usually stated as an

Future Value of a Sngle Sum at Compound Interest M3 annual rate, whch s adjusted for any other tme perod. Thus smple nterest s calculated by the followng equaton: Interest Prncpal Rate Tme where tme s ether a fracton of a year or a multple of years. If the term of a note s stated n days, say 90 days, the denomnator of the tme fracton n the precedng equaton s usually stated n terms of a commercal year of 360 days rather than a full year of 365 days. In ths practce the year s assumed to be a perod of 2 months of 30 days each. For example, the smple nterest on a 0,000, 90-day, 2% note gven to a company by Allen Sanders s 300 (0,000 0.2 90/360). However, f the term of ths note s 5 months, the smple nterest s,500 (0,000 0.2 5/2). Observe that smple nterest for more than one year s stll calculated on only the prncpal amount (n ths case 0,000). Compound nterest s the nterest that accrues on both the prncpal and the past unpad accrued nterest. Smple nterest of 2% for 5 months on the Allen Sanders note s,500. If, on the other hand, the 2% nterest s compounded quarterly for 5 months (5 quarters), the total compound nterest s,592.74, as we show n Example M-. Note that n the compound nterest computaton, the future accumulated amount (value) at the end of each quarter becomes the prncpal sum used to compute the nterest for the followng perod. Understand smple nterest and compound nterest. EXAMPLE M- Computaton of Quarterly Compounded Interest Value at Begnnng Compound Value at End of Perod of Quarter* Rate Tme Interest Quarter st quarter 0,000.00 0.2 /4 300.00 0,300.00 2nd quarter 0,300.00 0.2 /4 309.00 0,609.00 3rd quarter 0,609.00 0.2 /4 38.27 0,927.27 4th quarter 0,927.27 0.2 /4 327.82,255.09 5th quarter,255.09 0.2 /4 337.65,592.74 Compound nterest on 0,000 at 2% compounded quarterly for 5 quarters,592.74 * Ths value s the amount on whch nterest s calculated. To help solve the many busness ssues stated n the ntroductory secton of ths Module, accountants need to know the varous types of compound nterest computatons. Although there are many varatons, there are only four basc types:. Future value (amount) of a sngle sum at compound nterest 2. Present value of a sngle sum due n the future 3. Future value (amount) of an annuty, a seres of recepts or payments 4. Present value of an annuty, a seres of recepts or payments FUTURE VALUE OF A SINGLE SUM AT COMPOUND INTEREST As we stated prevously, the man objectve of ths Module s to explan shortcut methods to determne and apply the compound nterest technques. We wll use the followng

M4 Tme Value of Money Module step-by-step procedure, ntroducng the entre topc only wth the future value of a sngle sum at compound nterest:. We dagram the dea or concept. 2. We make the computaton usng a longhand calculaton. 3. We make the computaton usng formulas. 4. We dscuss the method of constructng and usng tables. 5. We llustrate the use of the tables to solve a compound nterest problem. 2 Compute and use the future value of a sngle sum. The Idea The future value of a sngle sum at compound nterest s the orgnal sum plus the compound nterest, stated as of a specfc future date. It s also often referred to as the future amount of a sngle sum. For example, suppose you nvest a sngle amount of,000 n a savngs account on December 3, 2007. What wll be the amount n the savngs account on December 3, 20 f nterest at 6% s compounded annually each year? We show the ssue graphcally n Example M-2. Most compound nterest calculatons can be made by applyng longhand arthmetc. We follow ths procedure here only to clarfy the varous shortcut devces used. EXAMPLE M-2 Dagram of Future Value of a Sngle Sum,000 s nvested on ths date How much wll be n the savngs account (the future value) on ths date? 2007 2008 2009 200 Interest Rate Is 6% Compounded Annually 20 The future value of,000 for four years at 6% a year can be calculated as we show n Example M-3. The sngle sum of,000 nvested on December 3, 2007 has grown to,262.48 by December 3, 20. Ths s the future value. The total nterest of 262.48 for the four years s referred to as compound nterest. EXAMPLE M-3 Calculaton of Future Value of Sngle Sum at Compound Interest () (2) (3) (4) Annual Future Value Value at Compound at End Begnnng of Interest of Year Year Year (Col. 2 0.06) (Col. 2 Col. 3) 2008,000.00 60.00,060.00 2009,060.00 63.60,23.60 200,23.60 67.42,9.02 20,9.02 7.46,262.48 A slght varaton of the longhand arthmetc approach s to determne what nvested on December 3, 2007 wll amount to by December 3, 20 f nterest at 6% s

Future Value of a Sngle Sum at Compound Interest M5 compounded annually. Then ths amount s multpled by the prncpal sum to fnd the future value. In ths case, amounts to.26248 n four years. Knowng ths fact, the value of,000 dfferent nvestments (or,000) at the end of four years can be calculated by multplyng the,000 by.26248 as follows:,000.26248,262.48. To avod a sgnfcant roundng error n the fnal results, when solvng ths problem, the ntermedate fgures should not be rounded to the nearest cent. Formula Approach Each amount n column 4 of Example M-3 s.06 tmes the correspondng amount n column 2. The fnal future value s therefore,000.06.06.06.06,262.48. Ths means that.06 has been used as a multpler four tmes; that s,.06 has been rased to the fourth power. The future value s therefore,000 multpled by.06 to the fourth power: Future Value,000(.06) 4,262.48 Thus the formula to compute the future value of a sngle sum at compound nterest s: f p( ) n where f future value of a sngle sum at compound nterest for n perods p prncpal sum (present value) nterest rate for each of the stated tme perods n number of tme perods It s mportant to understand that the nterest rate s the rate of nterest applcable for the partcular tme perod for whch nterest s compounded. For example, a stated annual rate of nterest of 2% s 2% per year f nterest s compounded annually 6% per one-half year f nterest s compounded semannually 3% per quarter f nterest s compounded quarterly % per month f nterest s compounded monthly In general, an nterest rate per perod () s the annual stated rate (sometmes called the nomnal rate) dvded by the number of compoundng tme perods n the year, and n s the number of tme perods n the year multpled by the number of years. The formula for the future value of s: f n, ( ) n where f n, s the future compound value of ( or of any other monetary unt) at nterest rate for n perods. Usng the precedng formula for the future value of, a short formula for the future compound value of any sngle amount at compound nterest s: f p(f n, ) The example of the future value of,000 nvested at 6% wth nterest compounded annually can now be calculated n two steps: Step f n 4, 6% (.06) 4.2624796 Step 2 f,000(.2624796),262.48 Recall that ths s exactly the same as the second approach, whch we used n the prevous arthmetc method. Table Approach To develop addtonal shortcuts to the soluton of the compound nterest ssue, tables for the future value of have been constructed. These tables smply nclude calculatons of

M6 Tme Value of Money Module the future values of at dfferent nterest rates and for dfferent tme perods. They can be constructed by usng the precedng formula wth the desred nterest rates and tme perods. For example, suppose that you need tables of the future value of at 2% and 4% for tme perods through 4 and for 40 years. The nformaton for these can be calculated as follows: f n, 2% (.02).020000 f n, 4% (.4).40000 f n 2, 2% (.02) 2.040400 f n 2, 4% (.4) 2.299600 f n 3, 2% (.02) 3.06208 f n 3, 4% (.4) 3.48544 f n 4, 2% (.02) 4.082432 f n 4, 4% (.4) 4.688960 f n 40, 2% (.02) 40 2.208040 f n 40, 4% (.4) 40 88.88354 Ths nformaton can then be accumulated n a partal table as we show n Example M-4. In ths knd of table the factors are shown wthout the use of the dollar sgn. Each factor s an amount for a certan tme perod and rate. We provde more complete tables at the end of ths Module. EXAMPLE M-4 Future Value of Table ( ) n Perods 2% 4%.020000.40000 2.040400.299600 3.06208.48544 4.082432.688960......... 40 2.208040 88.88354 Snce the factors n Example M-4 and n Table at the end of ths Module are based on the formula ( ) n, the table approach can be expressed as: f p(factor for f n, ) To calculate the future value that,000 wll accumulate to n four years at 6% compounded annually, t s necessary to look up the table factor for f n 4, 6%, namely,.262477; then, to arrve at the answer of,262.48, the calculaton s: f,000(.262477),262.48. Summary and Illustraton In addton to the straghtforward stuaton of calculatng the future value of a sngle sum at compound nterest, you can solve other knds of problems wth the future value of table. Example: Fndng an Unstated Interest Rate If,000 s nvested on December 3, 2007 to earn compound nterest and f the future value on December 3, 204 s 2,998.70, what s the quarterly nterest rate on the nvestment? We show the facts n Example M-5. Usng the table approach f p(factor for f n, ) and substtutng n the formula the amounts shown n Example M-5, the factor s determned as follows: 2,998.70,000(Factor for f n 28,? ) 2,998.70 Factor for f n 28,? 2.99870,000.00

Present Value of a Sngle Sum M7 EXAMPLE M-5 Dagram of Future Value of a Sngle Sum Interest Rate to Be Determned,000 s nvested on ths date 28 quarterly tme perods 2,998.70 s the future value on ths date 2007 2008 2009 200 20 202 Quarterly Interest Rate Unknown 203 204 The factor of 2.99870 s the future value of for 28 tme perods at an unknown nterest rate. Usng the future value of table (Table ) at the end of ths Module, you look down the perods (n) column untl you get to 28. Then you move horzontally on the n 28 lne to the column factor closest to 2.99870. If the value appears n the table, you can determne the nterest rate (shown at the top of the column) that produces ths value. In ths case, 2.99870 s equal to 2.998703 (rounded) located n the 4% column; thus the quarterly nterest rate s 4%. Ths s often referred to as beng a stated annual rate of 6%; you should understand, however, that a quarterly rate of 4% compounded four tmes yelds an effectve rate of more than 6%. If the factor of 2.99870 does not appear n the table, an nterpolaton procedure s requred to approxmate the quarterly nterest rate. Calculators and computer software that compute the nterest rate are wdely avalable. You can solve other problems by usng the future amount of tables. Keep n mnd, however, that most tables are ncomplete. At tmes t wll be necessary to construct tables for odd nterest rates and tme perods, or to use a calculator or computer software. PRESENT VALUE OF A SINGLE SUM For the remanng compound nterest technques, we focus on the shortcut approach. After we dscuss the dea, we state the formula and use factors derved from the formula. The Idea The present value s the prncpal that must be nvested at tme perod zero to produce the known future value. Also, dscountng s the process of convertng the future value to the present value. For example, f,000 s worth,262.48 when t earns 6% compound nterest per year for four years, then t follows that,262.48 to be receved four years from now s worth,000 now at tme perod zero; that s,,000 s the present value of,262.48 dscounted at 6% for four years. Example M-6 presents ths nformaton graphcally.. You can use the followng sx steps to determne an nterest rate by lnear nterpolaton: () Calculate the compound nterest factor as shown n the precedng example. (2) Look up n compound nterest tables the two nterest rates that yeld the next largest and the next smallest factors from the calculated factor determned n step. (3) Determne (a) the dfference between the two factors n step 2, and (b) the dfference between the calculated factor from step and the factor of the smaller nterest rate from step 2. (4) Fnd the dfference between the two nterest rates found n step 2. (5) Apporton the dfference n the nterest rates n step 4 by multplyng t by a fracton: The numerator s the dfference determned n step 3b and the denomnator s the dfference determned n step 3a. (6) The nterest rate s then the lower rate found n step 2 plus the apportoned dfference from step 5.

M8 Tme Value of Money Module EXAMPLE M-6 Dagram of Present Value of a Sngle Sum,000 (the present value) must be nvested on ths date,262.48 wll be receved on ths date 2007 2008 2009 200 Interest Rate Is 6% Compounded Annually 20 3 Compute and use the present value of a sngle sum. Shortcut Approaches Whle t s possble to calculate the present value of,262.48 to be receved at the end of four years dscounted at 6% by a longhand approach by reversng the process descrbed n the calculaton of the future value, we do not show ths approach here. Instead we focus on the development of shortcut approaches to fnd the present value of a sngle sum. Frst we present the formula, then we explan how to create and use factors. Formula Approach Snce the present value of a sngle future amount s the recprocal value of the future value of a sngle sum, the formula for ths calculaton s: p f ( ) n where p present value of any gven future value due n the future f future value nterest rate for each of the stated tme perods n number of tme perods In ths example the present value of,262.48 receved at the end of 4 years dscounted at 6% s,000, calculated as follows: p,262.48,000 (.06) 4 The formula for the present value of s: p n, ( ) n where p n, s the present value of ( or of any monetary unt) at nterest rate for n perods. It s now possble to express the formula for the present value of any gven future amount as: p f(p n, ) The example of the present value of,262.48 to be receved four years from now wth nterest of 6% compounded annually can be calculated n two steps: Step p n 4, 6% 0.792094 (.06) 4 Step 2 p,262.48(0.792094),000 Table Approach Usng the formula for p n,, tables have been constructed for any nterest rate and for any number of perods by smply substtutng n the formula the selected varous nterest

Present Value of a Sngle Sum M9 rates for the varous tme perods desred. Table 3 at the end of ths Module shows the factors for the present value of (p n, ). Snce the factors n Table 3 are based on the formula p n, /( ) n, the generalzed table approach can be stated as: p f(factor for p n, ) To calculate the present value of,262.48 to be receved at the end of four years, dscounted at 6%, look up the factor for p n 4, 6% n Table 3; t s 0.792094. Then the future value of,262.48 s multpled by ths present value of factor to obtan the present value amount of,000, as follows: p,262.48(0.792094),000. Summary and Illustraton In addton to calculatng the present value of a sngle sum usng compound nterest, you can solve other knds of problems wth the present value of table. Example: Fndng an Unstated Interest Rate Assumng that the present value of 0,000 to be pad at the end of 0 years s 3,855.43, what nterest rate compounded annually s used n the calculaton of the present value? Example M-7 shows the known facts. Snce both the present value and the future amount are known, ths problem can be solved n two dfferent ways: () by usng the method we descrbed n the future value secton, or (2) by usng the present value approach we descrbe here. Snce we dscussed the future value approach earler n ths Module, we use only the present value approach here to solve the problem. Usng the table approach p f(factor for p n, ) and substtutng n the formula the known amounts shown n Example M-7, the factor s determned as follows: 3,855.43 0,000.00 (Factor for p n 0,? ) 3,855.43 Factor for p n 0,? 0.385543 0,000.00 The factor of 0.385543 s the present value of for 0 perods at an unknown nterest rate. Usng the present value of table (Table 3), you look down the perods (n) column untl you get to 0. Then you move horzontally on the n 0 lne to the column factor closest to 0.385543. If the amount appears n the table, you can determne the nterest rate (shown at the top of the column) that produces ths amount. In ths case, 0.385543 s n the 0% column. Thus the annual rate s 0%. If the factor of 0.385543 does not appear n the table, an nterpolaton procedure s requred to approxmate the annual nterest rate (see footnote ). EXAMPLE M-7 Dagram of the Present Value of a Sngle Sum Interest Rate to Be Determned 3,855.43 s the known present value 0 annual tme perods 0,000 s the known future value to be pad 0 2 3 4 5 6 7 8 9 0 Annual Interest Rate Unknown

M0 Tme Value of Money Module MEASUREMENTS INVOLVING AN ANNUITY An annuty s a seres of equal cash flows (deposts, recepts, payments, or wthdrawals), sometmes referred to as rents, made at regular ntervals wth nterest compounded at a certan rate. The regular ntervals between the cash flows may be any tme perod for example, one year, a sx-month perod, one month, or even one day. In solvng measurement problems nvolvng the use of annutes, these four condtons must exst: () the perodc cash flows are equal n amount, (2) the tme perods between the cash flows are the same length, (3) the nterest rate s constant for each tme perod, and (4) the nterest s compounded at the end of each tme perod. FUTURE VALUE OF AN ORDINARY ANNUITY The future value of an ordnary annuty s determned mmedately after the last cash flow n the seres s made. For the frst example, assume that Debb Whtten wants to calculate the future value of four cash flows of,000, each wth nterest compounded annually at 6%, where the frst,000 cash flow occurs on December 3, 2007 and the last,000 occurs on December 3, 200. Example M-8 presents ths nformaton graphcally. EXAMPLE M-8 Dagram of Future Value of Ordnary Annuty,000 2007 4 annual cash flows of,000 each,000,000 2008 2009 Interest Rate Is 6% Compounded Annually,000 200 The future value of an ordnary annuty s determned mmedately after the last,000 cash flow occurs In drawng a tme lne such as that n Example M-8, some accountants prefer to add a begnnng tme segment to the left of the tme when the frst cash flow occurs. For example, they would draw the tme lne for the future amount of an ordnary annuty, as we show n Example M-9. Ths approach s acceptable f t s understood that the tme from January, 2007 to December 3, 2007 (whch s the perod of tme mmedately before the frst cash flow occurs) s not used to compute the future value of the ordnary annuty. It s smlar to statng a decmal as.4 or 0.4. The zero n front of the decmal may help someone to understand the ssue better, but does not change t. In the case of the future value of an ordnary annuty, however, placng the broken lne segment to the left of the frst cash flow may lead someone to thnk that the cash flows n an ordnary annuty must occur at the end of a gven year. That statement s not true; the cash flows can occur, for example, on March 5 of each year, or November 5 of each year. For the calculaton to be the future value of an ordnary annuty, the future value s determned mmedately after the last cash flow n the seres occurs. Because of the potental msnterpretaton of the nformaton, we prefer not to use the broken lne segment to the left of the frst cash flow n the tme lnes descrbng the future value of an ordnary annuty.

Future Value of an Ordnary Annuty M EXAMPLE M-9 Alternatve Dagram of Future Value of an Ordnary Annuty 0 4 annual cash flows of,000 each,000,000 The future value of an ordnary annuty s determned mmedately after the last,000 cash flow occurs,000,000 2007 (No nterest accrues) 2007 (Interest accrues) 2008 (Interest accrues) Interest Rate Is 6% Compounded Annually 2009 (Interest accrues) 200 Shortcut Approaches Formula Approach The formula for the future value of an ordnary annuty of any amount s: F 0 where F 0 future value of an ordnary annuty of a seres of cash flows of any amount C amount of each cash flow n number of cash flows (not the number of tme perods) nterest rate for each of the stated tme perods In the example, the future value of an ordnary annuty of four cash flows of,000 each at 4% compounded annually s as follows: F 0 n = C + 4 = 000 06., = 4, 374. 62 006. 4 Compute and use the future value of an ordnary annuty. The formula for the future value of an ordnary annuty wth cash flows of each s as follows: F 0n, = + where F 0n, s the future value of an ordnary annuty of n cash flows of each at nterest rate. Wth the precedng formula for F 0n, t s possble to express another formula for the future value of an ordnary annuty of cash flows of any sze n ths manner: F 0 C(F 0n, ) n In a two-step approach, the future value of an ordnary annuty of four cash flows of,000 each at 4% compounded annually s calculated as follows: (.06) Step F 0n 4, 6% 4 4.37462 0.06 Step 2 F 0,000(4.37462) 4,374.62 Ths two-step approach s used to solve the problem when factors are not avalable.

M2 Tme Value of Money Module Table Approach The formula for F 0n, can be used to construct a table of the future value of any seres of cash flows of each for any nterest rate. Here the number of cash flows of and the nterest rates are substtuted nto the formula ( ) n Table 2 at the end of ths Module shows the factors for F 0n,. Turnng to Table 2, observe the followng:. The numbers n the frst column (n) represent the number of cash flows. 2. The future values are always equal to or larger than the number of cash flows of. For example, the future value of four cash flows of each at 6% s 4.37466. Ths fgure comprses two elements: (a) the number of cash flows of each wthout any nterest, and (b) the compound nterest on the cash flows, wth the excepton of the compound nterest on the last cash flow n the seres, whch n the case of an ordnary annuty does not earn any nterest. Snce Table 2 shows the calculaton of F 0n, or ( ) n values, the generalzed table approach s as follows: F 0 C(Factor for F 0n, ) To calculate the future value of an ordnary annuty of 4 cash flows of,000 each at 6%, you must look up the F 0n 4, 6% factor n the future value of an ordnary annuty of table (Table 2); t s 4.37466. Then the amount of each cash flow, here,000, s multpled by the Table 2 factor to obtan the future value of 4,374.62: F 0,000(4.37466) 4,374.62 Summary and Illustraton You can solve several knds of problems usng a future value of an ordnary annuty of table, such as () calculatng the future value when the cash flows and nterest rate are known (the precedng problem); (2) calculatng the value of each cash flow where the number of cash flows, nterest rate, and future value are known; (3) calculatng the number of cash flows when the amount of each cash flow, the nterest rate, and the future value are known; and (4) calculatng an unknown nterest rate when the cash flows and the future value are known. To demonstrate the analyss used n the soluton of all these problems, we show tem (2) as follows. Example: Determnng the Amount of Each Cash Flow Needed to Accumulate a Fund to Retre Debt At the begnnng of 2007 the Rexson Company ssued 0-year bonds wth a face value of,000,000 due on December 3, 206. The company wll accumulate a fund to retre these bonds at maturty. It wll make annual deposts to the fund begnnng on December 3, 2007. How much must the company depost each year, assumng that the fund wll earn 2% nterest compounded annually? Example M-0 shows the facts of the problem. The future value and the compound nterest rate are known. The amount of each of the 0 deposts (cash flows) s the unknown factor. Startng wth the formula F 0 C(Factor for F 0 )

Future Value of an Annuty Due M3 and then shftng the elements and substtutng the known amount and applcable factor (from Table 2), the amount of each annual depost s 56,984.6, calculated as follows: C F 0 Factor for F 0n, F 0 Factor for F 0n 0, 2%,000,000 7.548735 56,984.6 The 0 annual deposts of 56,984.6, plus the compound nterest, wll accumulate to,000,000 by December 3, 206. EXAMPLE M-0 Future Value of an Ordnary Annuty Amount of Cash Flows to Be Determned 0 cash flows of an unknown amount,000,000 s needed n fund on ths date 2007 2008 2009 200 20 202 203 Interest Rate Is 2% Compounded Annually 204 205 206 FUTURE VALUE OF AN ANNUITY DUE The future value of an annuty due (F d ) s determned perod after the last cash flow n the seres. For example, assume that Ronald Jacobson deposts n a fund four payments of,000 each begnnng December 3, 2007, wth the last depost beng made on December 3, 200. How much wll be n the fund on December 3, 200, year after the fnal payment, f the fund earns nterest at 6% compounded annually? Example M- shows the facts of ths problem. EXAMPLE M- Dagram of Future Value of Annuty Due How much wll be n the fund on ths date, whch s perod after the last cash flow n the seres?,000,000,000,000 2007 2008 2009 200 Interest Rate Is 6% Compounded Annually 20

M4 Tme Value of Money Module 5 Compute and use the future value of an annuty due. Soluton Approach By observng the nformaton contaned n Examples M- and M-8, you can determne a quck way to compute the future value of an annuty due. 2 When only the future value of an ordnary annuty table s avalable, you can use the factors by completng the followng steps: Step In the ordnary annuty table (Table 2), look up the value of n cash flows at 6% or the value of 5 cash flows at 6%. 5.637093 Step 2 Subtract wthout nterest from the value obtaned n step. (.000000) Step 3 Ths s the converted future value factor for F dn 4, 6% 4.637093 Multply the amount of each cash flow, here,000, by the converted factor for F dn 4, 6% determned n step 2: F d,000(4.637093) 4,637.09 Tables of the future value of an annuty due of cash flows of each are avalable n some fnance books, but not n ths book. Therefore, these values must be calculated usng the tables for the future value of an ordnary annuty. As we showed prevously, the general rule s to use the future value of an ordnary annuty factor for n cash flows and subtract from the factor. (Note that we do nclude n ths Module a present value of an annuty due table, as we dscuss later.) PRESENT VALUE OF AN ANNUITY The present value of an annuty s the present value of a seres of equal cash flows that occur n the future. In other words, t s the amount that must be nvested now and, f left to earn compound nterest, wll provde for a recept or payment of a seres of equal cash flows at regular ntervals. Over tme, the present value balance s ncreased perodcally for nterest and s decreased perodcally for each recept or payment. Thus, the last cash flow n the seres exhausts the balance on depost. A company frequently uses the present value of an annuty concept to report many tems n ts fnancal statements, as we stated n the ntroducton to ths Module. Because of the mportance of the present value of an annuty, we wll dscuss the () present value of an ordnary annuty, (2) present value of an annuty due, and (3) present value of a deferred annuty. 6 Compute and use the present value of an ordnary annuty. PRESENT VALUE OF AN ORDINARY ANNUITY The present value of an ordnary annuty s determned perod before the frst cash flow n the seres s made. For example, assume that Kyle Vasby wants to calculate the present value on January, 2007 of four future wthdrawals (cash flows) of,000, wth the frst wthdrawal beng made on December 3, 2007, year after the determnaton of the present value. The applcable nterest rate s 6% compounded annually. Example M-2 shows ths nformaton graphcally. 2. An alternatve approach s to multply the future value of an ordnary annuty factor by plus the nterest rate. Thus, the future value n ths example would be computed as,000 (4.37466.06) 4,637.09.

Present Value of an Ordnary Annuty M5 EXAMPLE M-2 Dagram of Present Value of an Ordnary Annuty The present value of an ordnary annuty s determned one perod before the frst wthdrawal,000 4 wthdrawals of,000 each,000,000,000 2007 2007 2008 2009 Interest Rate Is 6% Compounded Annually 200 Solvng by Determnng the Present Value of a Seres of Sngle Sums The soluton to ths problem can be determned by usng the present value of a sngle sum. For nstance, the answer can be calculated n the followng two steps: () determne the present value of four ndvdual cash flows of each for one, two, three, and four years, as we show n Example M-3; and (2) multply the fnal results of the summaton by,000. Step The present value of four cash flows of for one, two, three, and four years dscounted at 6% s determned n Example M-3. Step 2 Now t s possble to determne the present value of the four cash flows of,000 each by multplyng the,000 by 3.46505:,000 3.46505 3,465. The present value on January, 2007 s 3,465.; or we can say that 3,465. must be nvested on January, 2007 to provde for four wthdrawals of,000 each startng on December 3, 2007, gven an nterest rate of 6%. EXAMPLE M-3 Present Value of Four Cash Flows of for One, Two, Three, and Four Years at 6% Present value of on January, 2007 (from Table 3) 0.943396 0.889996 0.83969 0.792094 3.46505* 2007 2007 2008 2009 200 *The value of 3.46505 s slghtly smaller than the factor for P 0n = 4, = 6% of 3.46506 n Table 4 dscussed later n ths secton; ths s the result of roundng each of the four factors for P n,.

M6 Tme Value of Money Module Shortcut Approaches Formula Approach Even though the precedng approach can be used, t s tme-consumng for calculatons nvolvng a large number of cash flows. The formula for the present value of an ordnary annuty of any amount s: P where P 0 present value of an ordnary annuty of a seres of cash flows of any amount C amount of each cash flow n number of cash flows (not the number of tme perods) nterest rate for each of the stated tme perods In the example, the present value of an ordnary annuty of four cash flows of,000 each at 6% compounded annually can be calculated as follows: P 0 Based on these calculatons and formula observe that:. The results are the same as those produced n the frst approach, 3,465.. 2. The formula s developed from the formulas for both the future value of (f) and the present value of (p): ( ) n f p 3. Thus the formula can be restated as follows: p P 0 C The formula for the present value of an ordnary annuty can be converted to that for a seres of cash flows of each as follows: P 0 = C - 4.06 =, 000 = 3, 465. 0.06 ( ) n 0n, = - - where P 0n, s the present value of an ordnary annuty of n cash flows of each at nterest rate. Ths formula can be expressed for the present value of an ordnary annuty of cash flows of any sze as: P 0 C(P 0n, ) + + ( ) n n

Present Value of an Ordnary Annuty M7 In a two-step approach the present value of four future wthdrawals (cash flows) of,000 each dscounted at 6% s recalculated as follows: Step P0 n=4,=6% = - 06. 006. 4 = 34. 65 Step 2 P 0,000(3.465) 3,465. Ths calculaton s exactly the same as that of the frst formula, except that the process s dvded nto two steps. The two-step approach s the one used when tables of the present value of an ordnary annuty of are avalable. Table Approach The formula for P 0n, can be used to construct a table of the present value of any seres of cash flows of each for any nterest rate. All that s necessary s to substtute n the formula the desred number of cash flows for the varous requred nterest rates. Table 4 at the end of the Module shows the factors for P 0n,. Turnng to Table 4, observe the followng:. The numbers n the frst column (n) represent the number of cash flows of each. In ths calculaton the number of cash flows and tme perods are equal. 2. The present value amounts are always smaller than the number of cash flows of. For example, the present value of three cash flows of at 2% s 2.883883. Snce Table 4 shows the precalculaton of P 0n, or - + n the generalzed table approach s as follows: P 0 C(Factor for P 0n, ) Thus, to calculate the present value on January, 2007 of four future wthdrawals (cash flows) of,000 dscounted at 6%, wth the frst cash flow beng wthdrawn on December 3, 2007, t s necessary to look up the P 0n 4, 6% value n the present value of an ordnary annuty of table (Table 4); t s 3.46506. Ths factor s then multpled by,000 to determne the present value fgure of 3,465.: P 0,000(3.46506) 3,465. Over the 4 perods, the annuty yelds nterest each perod as follows: Begnnng Cash Endng Perod Balance Interest Flow Balance 3,465. 207.9 (,000) 2,673.02 2 2,673.02 60.38 (,000),833.40 3,833.40 0.00 (,000) 943.40 4 943.40 56.60 (,000) 0 Summary and Illustraton You can solve several knds of problems by usng the present value of an ordnary annuty of table. We present one addtonal example: a problem nvolvng the calculaton of the perodc cash flows when the present value and nterest rate are known.

M8 Tme Value of Money Module Example: Determnng the Value of Perodc Cash Flows When the Present Value Is Known Suppose that on January, 2007 Rex Company borrows 00,000 to fnance a plant expanson project. It plans to pay ths amount back wth nterest at 2% n equal annual payments over a 0-year perod, wth the frst payment due on December 3, 2007. What s the amount of each payment? Example M-4 shows the facts of the problem. The present value and the compound nterest rate are known. The amount of each of the 0 cash flows s the unknown tem and s 7,698.42, calculated as follows: C P 0 Factor for P 0n, P 0 Factor for P 0n 0, 2% 00,000 5.605223 7,698.42 Remember that each of these payments of 7,698.42 ncludes () a payment of annual nterest, and (2) a retrement of debt prncpal. For example, the nterest for 2007 s 2,000 (2% 00,000). Thus the amount of the payment on prncpal s 5,698.42 (7,698.42 2,000). For the year 2008 the nterest s,36.9 [2% (00,000 5,698.42)], and the retrement of prncpal s 6,382.23 (7,698.42,36.9). The last payment of 7,698.42 on December 3, 206, wll be suffcent to retre the remanng prncpal and to pay the nterest for the tenth year. EXAMPLE M-4 Dagram of the Present Value of an Ordnary Annuty Amount of Each Cash Flow to Be Determned 00,000 s the known present value of the 0 unknown cash flows dscounted at 2% 0 cash flows of an unknown amount 2007 2007 2008 2009 200 20 202 203 Interest Rate Is 2% Compounded Annually 204 205 206 7 Compute and use the present value of an annuty due. PRESENT VALUE OF AN ANNUITY DUE The present value of an annuty due (P d ) s determned on the date of the frst cash flow n the seres. For example, assume that Barbara Lvngston wants to calculate the present value of an annuty on December 3, 2007, whch wll permt four annual future recepts of,000 each, the frst to be receved on December 3, 2007. The nterest rate s 6% compounded annually. Example M-5 shows the facts of ths problem.

Present Value of an Annuty Due M9 EXAMPLE M-5 Dagram of the Present Value of an Annuty Due The present value of an annuty due s determned on ths date, whch s the date of the frst cash recept,000,000 4 cash flows of,000 each,000,000 2007 2008 2009 Interest Rate Is 6% Compounded Annually 200 Shortcut Approaches Formula Approach The formula for the present value of an annuty due of any amount s: P where d = C - + n- + P d present value of an ordnary annuty of a seres of cash flows of any amount C amount of each cash flow n number of cash flows (not the number of tme perods) nterest rate for each of the stated tme perods In the example, the present value of an annuty due of four cash flows of,000 each at 6% compounded annually s calculated as follows: P d =, 000-06. 006. 3 + = 3, 673. 0 The formula for the present value of an annuty due wth cash flows of each s: P dn, = C - + n- + where P dn, s the present value of an annuty due of n cash flows of each at nterest rate. Wth the precedng formula for P dn, t s possble to express another formula for the future value of an ordnary annuty of cash flows of any sze as: P d C(P dn, )

M20 Tme Value of Money Module In a two-step approach the present value of an annuty due of four cash flows of,000 each at 6% compounded annually s calculated as follows: - 3 Step P dn=4,=6% = 06. + = 3. 67302 006. Step 2 P d,000(3.67302) 3,673.0 Ths two-step approach s used to solve the problem when factors are not avalable. Table Approach The formula for P dn, can be used to construct a table of the future value of any seres of cash flows of each for any nterest rate. Table 5 at the end of ths Module shows the factors for P dn,. Snce the factors n Table 5 are based on the formula for P dn, or n + + values, the generalzed table approach s as follows: P d C(Factor for P dn, ) To calculate the present value of an annuty due of four cash flows of,000 each at 6%, the P dn 4, 6% factor s found n the present value of an annuty due table (Table 5); t s 3.67302. Then the amount of each cash flow, here,000, s multpled by the Table 5 factor to obtan the present value of 3,673.0: P d,000(3.67302) 3,673.0 Alternatve Table Approach By observng the nformaton contaned n Examples M-5 and M-2, you can determne another way to compute the present value of an annuty due. 3 When only the present value of an ordnary annuty table s avalable, you can use the factors to determne the present value of an annuty due by completng the followng steps: Step In the ordnary annuty table (Table 4), look up the present value of n cash flows at 6%, or the value of three cash flows at 6%. 2.67302 Step 2 Add wthout nterest to the value obtaned n step..000000 Step 3 Ths s the converted present value factor for P dn 4, 6%. 3.67302 Multply the amount of each cash flow, here,000, by the converted factor for P dn 4, 6% determned n step 2: P d,000(3.67302) 3,673.0 3. An alternatve approach s to multply the present value of an ordnary annuty factor by plus the nterest rate, whch s consstent wth the formula: + n + Thus, the present value n ths example would be computed as,000 (3.46506.06) 3,673.0.

Present Value of a Deferred Ordnary Annuty M2 Thus, f the present value of an annuty due s calculated usng tables for the present value of an ordnary annuty, the general rule s to use present value of an ordnary annuty factor for n cash flows and add to the factor. Another Applcaton Besdes determnng the present value of an annuty due where the amount of each cash flow s known, you can solve other types of problems by usng the precedng approaches. Suppose, for example, that Katherne Sprull purchases on January, 2007 an tem that costs 0,000. She agrees to pay for ths tem n 0 equal annual nstallments, wth the frst nstallment on January, 2007 as a down payment. The equal nstallments nclude nterest at 8% on the unpad balance at the begnnng of each year. After the nterest s deducted, the balance of each payment reduces the prncpal of the debt. Ths problem nvolves the present value of an annuty due. It requres the determnaton of the amount of each of 0 cash flows that have a present value of 0,000 when dscounted at an annual rate of 8%. Example M-6 shows these facts graphcally. EXAMPLE M-6 Dagram of the Present Value of an Annuty Due Amount of Each Cash Flow to Be Determned 0,000 s the present value of the 0 payments of an unknown amount on ths date of the frst cash payment 0 cash flows of an unknown amount 2007 2008 2009 200 20 202 203 Interest Rate Is 8% Compounded Annually 204 205 206 The soluton to ths problem requres the rearrangement of the present value of an annuty due formula: C P 0 Factor for P dn, 0,000,379.90 7.246888 The down payment of,379.90 plus nne more payments of ths same amount wll retre the prncpal n nne years, plus pay nterest at 8% on the balance of the prncpal outstandng at the begnnng of each year. PRESENT VALUE OF A DEFERRED ORDINARY ANNUITY The present value of a deferred ordnary annuty (P deferred ) s determned on a date two or more perods before the frst cash flow n the seres. Suppose, for example, that Helen Swan buys an annuty on January, 2007 that yelds her four annual recepts of,000 each, wth the frst recept on January, 20. The nterest rate s 6% compounded annually. What s the cost of the annuty that s, what s the present value on January, 2007 of the four cash flows of,000 each to be receved on January, 20, 202, 203, and 204 dscounted at 6%? Example M-7 shows the facts of ths problem dagrammatcally. 8 Compute and use the present value of a deferred ordnary annuty.

M22 Tme Value of Money Module EXAMPLE M-7 Dagram of the Present Value of a Deferred Ordnary Annuty The present value of the deferred annuty s determned on ths date, whch s 2 or more perods before the frst cash recept 4 cash flows of,000 deferred 3 perods,000 20,000 202,000 203,000 204 2007 2008 2009 200 Interest Rate Is 6% Compounded Annually There are two ways to compute the present value of a deferred annuty. The frst method nvolves a combnaton of the present value of an ordnary annuty (P 0 ) and the present value of a sngle sum due n the future (p). For the stated problem t s necessary to determne frst the present value of an ordnary annuty of four cash flows of,000 each to fnd a sngle present value fgure dscounted to January, 200. Note that because the present value of an ordnary annuty table s used, the present value of the four cash flows s computed on January, 200, not January, 20. That sngle sum s dscounted for three more perods at 6% to arrve at the present value on January, 2007. Usng the factors of each, the present value s stated as follows: P deferred C[(P 0n, )(p k, )] where P 0n, present value of the ordnary annuty of the n cash flows of at the gven nterest rate p k, present value of the sngle sum of for k perods of deferment Substtutng approprate factors from Tables 4 and 3, respectvely, n ths formula, the followng soluton s obtaned: P deferred C[(P 0n 4, 6% )(p k 3, 6% )],000[(3.46506)(0.83969)] 2,909.37 An alternatve approach nvolves a combnaton of two ordnary annutes. For example, t s possble to calculate the present value of an ordnary annuty of n k cash flows of. From ths amount s subtracted the present value of the k (the perod of deferment, whch s 3 n ths example) cash flows of. Ths procedure removes the cash flows that were not avalable to be receved; yet the dscount factor for the three perods of deferments on the four cash flows that are to be receved remans n the calculated factor. Ths dfference s multpled by the value of each cash flow to determne the fnal present value of the deferred annuty. Example M-8 llustrates ths approach. In effect, the present value of an ordnary annuty of n k cash flows, mnus the present value of an ordnary annuty of the k cash flows, becomes a converted factor for the present value of a deferred annuty, as follows: P deferred C(Converted Factor for Present Value of Deferred Annuty of ) Usng the factors from Table 4, the converted factor for the deferred ordnary annuty stated n the precedng problem s determned as follows: P 0n k 7, 6% (5.58238) P 0k 3, 6% (2.67302) 2.909369

Present Value of a Deferred Ordnary Annuty M23 EXAMPLE M-8 Dagram of Converted Table Factor of Present Value of a Deferred Ordnary Annuty Start wth Less Equals P deferred Interest Rate Is 6% Compounded Annually The present value of the four cash flows of,000 each, deferred three perods, s 2,909.37, calculated as follows: P deferred,000(2.909369) 2,909.37 Note that the two methods produce the same present value fgure. Also, note that the perod of deferment s only three perods and not four because the present value of an ordnary annuty table s used (see Example M-8 n the second approach). Ths assumpton s requred f the problem s to be solved by the use of ordnary annuty factors rather than annuty due factors. Another Applcaton Besdes determnng the present value of a deferred annuty, other types of problems can be solved by usng the prevous approaches. For example, suppose that Davd Jones wants to nvest 50,000 on January, 2007 so that he may wthdraw 0 annual cash flows of equal amounts begnnng January, 203. If the fund earns 2% annual nterest over ts lfe, what wll be the amount of each of the 0 wthdrawals? Example M-9 shows the facts of ths problem. A smpler method that can be used to solve ths problem s a varaton of the second suggested soluton. Here, the value of C can be determned from the followng expresson of the present value of a deferred annuty formula: P C deferred Converted Factor for Present Value of Deferred Annuty of Usng Table 4, the converted factor for 0 cash flows of each, deferred 5 perods at 2%, s as follows: Converted Factor P 0n k 5, 2% (6.80864) P 0k 5, 2% (3.604776) 3.206088 Then the amount of each cash flow s 50,000 C 5,595.33 3.206088 The accuracy of the answer produced by the second approach can be tested usng the amount of each cash flow and the soluton from the frst approach. The present value of 0 cash flows of 5,595.33 deferred 5 perods and dscounted at 2% must be 50,000 f the frst soluton s correct. The proof can be calculated as follows: P deferred 5,595.33[(5.650223)(0.567427)] 50,000 A slght roundng-error dfference may occur wth ths method because the soluton requres the multplcaton of two factors, P 0n, and p k,, whch are rounded.

M24 Tme Value of Money Module EXAMPLE M-9 Dagram of the Present Value of a Deferred Annuty Amount of Each Cash Flow to Be Determned 50,000 s the known present value on ths date of the 0 cash flows deferred 5 perods 0 cash flows of an unknown amount deferred 5 perods 203 2022 2007 202 Interest Rate Is 2% Compounded Annually SUMMARY OF PRESENT AND FUTURE VALUE CALCULATIONS The present and future value calculatons dscussed n ths Module may be summarzed by the followng dagrams: of a sngle sum for 3 perods Present Value Future Value of a 3-payment ordnary annuty of a 3-payment annuty due 9 Explan the conceptual ssues regardng the use of present value n fnancal reportng. CONCEPTUAL EVALUATION OF PRESENT VALUE TECHNIQUES IN FINANCIAL REPORTING Accountng prncples have evolved wthout a unfyng objectve or ratonale for determnng when present value technques should and should not be used. Among the ssues are the use of present value for the ntal valuaton of assets and labltes, the amortzaton of those assets and labltes, and any subsequent revaluaton when nterest rates change. Present values are used n generally accepted accountng prncples for certan monetary tems. A monetary tem s money or a clam to money that s not affected by changes n the prces of specfc goods or servces. For example, a note payable s a monetary tem, whereas a warranty payable s a nonmonetary tem. Monetary tems for whch present values are used n generally accepted accountng prncples nclude bonds payable and bond nvestments, long-term notes payable and recevable, leases, and postretrement benefts (e.g., pensons). Present value s not used for tems such as deferred ncome taxes. Some accountants argue that present value should be used for nonmonetary tems such as property, plant, and equpment. However, accountng prncples have not been extended to the use of present value for these nonmonetary tems, except for the mparment of noncurrent assets. Therefore, present values are not used for warrantes, unearned revenue, compensated absences, or for nonmonetary assets. We dscuss each of these topcs n ths book. Most accountants would argue that the use of present value creates a relevant accountng measurement. For example, n the stuatons we dscussed earler, present value amounts are more relevant than, say, the total of the undscounted cash flows because they represent the equvalent current cash amount. However, the use of present value may create

Conceptual Evaluaton of Present Value Technques n Fnancal Reportng M25 measurements that are less relable (especally f used for nonmonetary tems) because the computaton requres:. The estmaton of the future cash flows, ncludng the tmng, amount, and rsk of those cash flows. 2. The estmaton of the nterest rate. Interest rates that could be used nclude the hstorcal rate, the current rate, the average expected rate, the weghted average cost of captal, or the ncremental borrowng rate. 4 3. The degree to whch the cash flows from the ndvdual assets may be added (and the labltes subtracted) to gve a measure of the value of the company. In 2000, the FASB ssued FASB Statement of Concepts No. 7, Usng Cash Flow Informaton and Present Value n Accountng Measurements. 5 The Statement provdes a framework for usng future cash flows as the bass for an accountng measurement of both assets and labltes. It provdes general prncples governng the use of present value, as well as the objectves of present value accountng measurements. The Statement does not address recognton ssues, and therefore does not address when far value should be based on present value, or when assets or labltes should be remeasured usng present value. It descrbes fve elements that together may be used to determne the value of varous assets and labltes:. An estmate of the future cash flow(s) and the tmng of those cash flows 2. Estmates about varatons n the amount or tmng of those cash flows 3. The rsk-free nterest rate 4. An ncrease n the nterest for any expected rsk 5. Other factors, ncludng a lack of lqudty and market mperfectons The methodology ntroduced n the Statement permts development of a far value usng cash flow nformaton even f uncertantes exst about the tmng and/or amount of the cash flows. Present value calculatons have typcally been based on a sngle set of cash flows and a sngle dscount rate. The Statement ntroduces the concept of expected cash flows when usng present value technques for accountng measurements. Expected cash flows are a probablty-weghted average of the range of possble estmated cash flow amounts and/or estmated tmng of cash flows. For example, n regard to dfferng expected amounts, a company may estmate that there s a 20% probablty that the cash flow n a gven year wll be,000, a 50% probablty that t wll be,200, and a 30% probablty that t wll be,400. The company would use the expected cash flow of,220 [(,000 0.20) (,200 0.50) (,400 0.30)] n ts present value calculatons. Or n regard to the tmng of ts cash flows, a company mght determne that t has a 30% probablty of recevng,000 n one year but a 70% probablty of recevng,000 n two years. The present value would be calculated as [(30% the present value of,000 n one year) (70% the present value of,000 n two years)]. The Statement also dscusses the use of present value to estmate the far value for a transacton between wllng partes or to develop entty-specfc measurements. The entty-specfc value (or value n use) s the value of an asset or lablty to a partcular entty. In other words, the measurement substtutes the entty s assumptons for those of the market place. The FASB concluded that an entty-specfc measurement mght be approprate n some stuatons. As wth all FASB Statements of Concepts, the conclusons do not create specfc GAAP, but wll provde gudance for the development of future FASB Statements of Fnancal Accountng Standards. The FASB has also ssued an Exposure Draft on far value measurements that would clarfy the use of present value technques to estmate far value. 6 4. R. Aggarwal and C. H. Gbson, Dscountng n Fnancal Accountng and Reportng (Morrstown, N.J.: Fnancal Executves Research Foundaton, 989), p. 45. 5. Usng Cash Flow Informaton and Present Value n Accountng Measurements, FASB Statement of Fnancal Accountng Concepts No. 7 (Norwalk, Conn.: FASB, 2000). 6. Far Value Measurements, FASB Proposed Statement of Fnancal Accountng Standards (Norwalk, Conn.: FASB, 2004).

M26 Tme Value of Money Module S UMMARY At the begnnng of the Module, we dentfed several objectves you would accomplsh after readng the Module. The objectves are lsted below, each followed by a bref summary of the key ponts n the Module dscusson.. Understand smple nterest and compound nterest. Smple nterest s nterest on the orgnal prncpal regardless of the number of tme perods that have passed or the amount of nterest that has been pad or accrued n the past. Compound nterest s the nterest that accrues on both the prncpal and the past unpad accrued nterest. 2. Compute and use the future value of a sngle sum. The future value of a sngle sum s the orgnal sum plus the compound nterest, stated as of a specfc future date. The future value may be computed usng a formula approach or a table approach (Table at the end of ths Module). 3. Compute and use the present value of a sngle sum. The present value s the prncpal that must be nvested at tme perod zero to produce the known future value. Dscountng s the process of convertng the future value to the present value. The present value may be computed usng a formula approach or a table approach (Table 3). 4. Compute and use the future value of an ordnary annuty. An annuty s a seres of equal cash flows made at regular ntervals wth nterest compounded at a certan rate. The future value of an ordnary annuty s determned mmedately after the last cash flow n the seres s made. The future value may be computed usng a formula approach or a table approach (Table 2). 5. Compute and use the future value of annuty due. The future value of an annuty due s determned one perod after the last cash flow n the seres. The general rule for determnng the future value of an annuty due factor s to take the future value of an ordnary annuty factor (Table 2) for n cash flows and subtract from the factor. 6. Compute and use the present value of an ordnary annuty. The present value of an ordnary annuty s determned one perod before the frst cash flow n the seres s made. The present value may be computed usng a formula approach or a table approach (Table 4). 7. Compute and use the present value of annuty due. The present value of an annuty due s determned on the date of the frst cash flow n the seres. The present value may be computed usng a formula approach or a table approach (Table 5). The general rule for determnng the present value of an annuty due factor s to take the present value of an ordnary annuty factor (Table 4) for n cash flows and add to the factor. 8. Compute and use the present value of a deferred ordnary annuty. The present value of a deferred ordnary annuty s determned on a date two or more perods before the frst cash flow n the seres. The general rule for determnng the present value of a deferred ordnary annuty factor s to take the factor for the present value of an ordnary annuty of n k cash flows and subtract the factor for the present value of an ordnary annuty of the k cash flows. 9. Explan the conceptual ssues regardng the use of present value n fnancal statements. The conceptual ssues nclude the lack of a unfyng objectve or ratonale for determnng when present value technques should and should not be used. These nclude the use of present value for the ntal valuaton of assets and labltes, the amortzaton of those assets and labltes, and any subsequent revaluaton when nterest rates change. Another ssue s that present value may create measurements that are less relable. In 2000, the FASB ssued FASB Statement of Concepts No. 7, Usng Cash Flow Informaton and Present Value n Accountng Measurements. The Statement dentfes fve elements that together may be used to determne the value of varous assets and labltes: () an estmate of the future cash flow(s) and the tmng of those cash flows, (2) estmates about varatons n the amount or tmng of those cash flows, (3) the rsk-free nterest rate, (4) an ncrease n the nterest for any expected rsk, and (5) other factors, ncludng a lack of lqudty and market mperfectons. The FASB has ssued an Exposure Draft that would clarfy the use of present value to estmate far value. Q UESTIONS QM- Defne nterest. Explan how the cost of nterest s smlar to the prce of any merchandse tem. QM-2 Dscuss the followng concepts of nterest: smple nterest, compound nterest, tme value of money, dscount. QM-3 Dstngush between the future value of and the future value of an ordnary annuty of. QM-4 What s the nterest rate per perod and the frequency of compoundng per year n each of the followng? a. 8% compounded semannually b. 6% compounded quarterly c. 5% compounded monthly QM-5 Dstngush between the future value of and the present value of and between the present value of and the present value of an ordnary annuty of. QM-6 Dstngush between the future value of an ordnary annuty and the future value of an annuty due. Draw a tme lne of each. QM-7 Dstngush between the present value of an annuty due and the present value of a deferred annuty. Draw a tme lne of each.

Multple Choce M27 QM-8 Explan how to solve each of the followng wthout tables (n each case use the quckest approach possble): a. The present value of 0,000 for four years at 0% compounded annually b. The present value of 5,000 for fve years at 0% [start wth nformaton developed n (a)] c. The future value of fve cash flows of an ordnary annuty of 3,000 each at 0% compound nterest QM-9 Potter wshes to depost a sum that at 2% nterest, compounded semannually, wll permt two wthdrawals: 40,000 at the end of 4 years and 50,000 at the end of 0 years. Analyze the problem to determne the requred depost, statng the procedure to follow and the tables to use n developng the soluton. QM-0 The followng factors are taken from the compound nterest tables for the same number of tme perods and/or cash flows for the same nterest rate: a. 8.37249 b. 50.980352 c. 6.265060 d. 7.4268 e. 0.22892 Identfy each of the fve compound nterest table factors wthout reference to the tables. Dscuss brefly. QM- Explan how to determne the converted table factor for any deferred annuty by usng the present value of an ordnary annuty table. QM-2 Samuel Ames owes 20,000 to a frend. He wants to know how much he would have to pay f he pad the debt n three annual nstallments at the end of each year, whch would nclude nterest at 4%. Draw a tme lne for the problem. Indcate what table to use. Look up the table value and place n a bref formula, but do not solve. QM-3 Startng wth the gven value for (.6) 0 4.4435, descrbe the fastest way to solve each of the followng: a. p n 0, 6% b. f n 20, 6% c. F 0n 0, 6% d. P 0n 0, 6% e. F 0n 20, 6% M ULTIPLE C HOICE ( AICPA Adapted) Select the best answer for each of the followng. Items through 4 requre use of present value tables. The followng are the present value factors of dscounted at 8% for one to fve perods. Each tem s based on 8% nterest compounded annually from day of depost to day of wthdrawal. Present Value of Dscounted at Perods 8% per Perod 0.926 2 0.857 3 0.794 4 0.735 5 0.68 MM- What amount should be deposted n a bank today to grow to,000 three years from today? a.,000 0.794 b.,000 0.926 3 c. (,000 0.926) (,000 0.857) (,000 0.794) d.,000 0.794 MM-2 What amount should an ndvdual have n hs bank account today, before wthdrawal, f he needs 2,000 each year for four years, wth the frst wthdrawal to be made today and each subsequent wthdrawal at one-year ntervals? (He s to have exactly a zero balance n hs bank account after the fourth wthdrawal.) a. 2,000 (2,000 0.926) (2,000 0.857) (2,000 0.794) b. 2,000 4 0.735 c. (2,000 0.926) (2,000 0.857) (2,000 0.794) (2,000 0.735) d. 2,000 4 0.926 MM-3 If an ndvdual put 3,000 n a savngs account today, what amount of cash wll be avalable two years from today? 3,000 a. 3,000 0.857 c. 0.857 3,000 b. 3,000 0.857 2 d. 0.926 2 MM-4 What s the present value today of 4,000 to be receved sx years from today? a. 4,000 0.926 6 b. 4,000 0.794 2 c. 4,000 0.68 0.926 d. Cannot be determned from the nformaton gven MM-5 On January, 2007 Kern Company sold a machne to Burns Company. Burns sgned a non-nterest-bearng note requrng payment of 30,000 annually for seven years. The frst payment was made on January, 2007. The prevalng rate of nterest for ths type of note at the date of ssuance was 0%. Informaton on present value factors s as follows: Present Value of Present Value Ordnary Annuty Perods of at 0% of at 0% 6 0.56 4.36 7 0.5 4.87

M28 Tme Value of Money Module Kern should record the sale n January 2007 at a. 07,00 c. 46,00 b. 30,800 d. 60,800 MM-6 On May, 2007, a company purchased a new machne that t does not have to pay for untl May, 2009. The total payment on May, 2009 wll nclude both prncpal and nterest. Assumng nterest at a 0% rate, the cost of the machne would be the total payment multpled by what tme value of money concept? a. Future value of annuty of b. Future value of c. Present value of annuty of d. Present value of MM-7 An offce equpment representatve has a machne for sale or lease. If you buy the machne, the cost s 7,596. If you lease the machne, you wll have to sgn a noncancellable lease and make fve payments of 2,000 each. The frst payment wll be pad on the frst day of the lease. At the tme of the last payment you wll receve ttle to the machne. The present value of an ordnary annuty of s as follows: Present Value Number of Perods 0% 2% 6% 0.909 0.893 0.862 2.736.690.605 3 2.487 2.402 2.246 4 3.70 3.037 2.798 5 3.79 3.605 3.274 The nterest rate mplct n ths lease s approxmately a. 0% c. Between 0% and 2% b. 2% d. 6% MM-8 An accountant wshes to fnd the present value of an annuty of payable at the begnnng of each perod at 0% for eght perods. He has only one present value table, whch shows the present value of an annuty of payable at the end of each perod. To compute the present value factor he needs, the accountant would use the present value factor n the 0% column for a. Seven perods b. Seven perods and add c. Eght perods d. Nne perods and subtract MM-9 On July, 2007, James Rago sgned an agreement to operate as a franchsee of Fast Foods, Inc., for an ntal franchse fee of 60,000. Of ths amount, 20,000 was pad when the agreement was sgned and the balance s payable n four equal annual payments of 0,000 begnnng July, 2008. The agreement provdes that the down payment s not refundable and no future servces are requred of the franchsor. Rago s credt ratng ndcates that he can borrow money at 4% for a loan of ths type. Informaton on present and future value factors s as follows: Present value of at 4% for four perods 0.59 Future value of at 4% for four perods.69 Present value of an ordnary annuty of at 4% for four perods 2.9 Rago should record the acquston cost of the franchse on July, 2007 at a. 43,600 c. 60,000 b. 49,00 d. 67,600 MM-0 For whch of the followng transactons would the use of the present value of an annuty due concept be approprate n calculatng the present value of the asset obtaned or lablty owed at the date of ncurrence? a. A captal lease s entered nto wth the ntal lease payment due one month subsequent to the sgnng of the lease agreement. b. A captal lease s entered nto wth the ntal lease payment due upon the sgnng of the lease agreement. c. A 0-year, 8% bond s ssued on January 2, wth nterest payable semannually on July and January yeldng 7%. d. A 0-year, 8% bond s ssued on January 2, wth nterest payable semannually on July and January yeldng 9%. E XERCISES EM- Future Value of an Investment and Compound Interest Usng the future value tables, solve the followng:. What s the value on January, 204 of 40,000 deposted on January, 2007 whch accumulates nterest at 2% compounded annually? 2. What s the value on January, 203 of 0,000 deposted on July, 2007 whch accumulates nterest at 6% compounded quarterly? 3. What s the compound nterest on an nvestment of 6,000 left on depost for fve years at 0% compounded annually? EM-2 Future Value of an Investment Hugh Colson deposted 20,000 n a specal savngs account that provdes for nterest at the annual rate of 2% compounded semannually f the depost s mantaned for four years. Calculate the balance of the savngs account at the end of the four-year perod.

Exercses M29 EM-3 Present Value of a Sum and Compound Dscount Usng the present value tables, solve the followng problems:. What s the present value on January, 2007 of 30,000 due on January, 202 and dscounted at 2% compounded annually? 2. What s the present value on July, 2007 of 8,000 due January, 202 and dscounted at 6% compounded quarterly? 3. What s the compound dscount on 8,000 due at the end of fve years at 0% compounded annually? EM-4 Future Value of Annuty Usng approprate tables, solve the followng future value of annuty problems:. What s the future value on December 3, 203 of seven cash flows of 0,000, wth the frst cash payment beng made on December 3, 2007 and nterest at 2% beng compounded annually? 2. What s the future value on December 3, 204 of seven cash flows of 0,000, wth the frst cash payment made on December 3, 2007 and nterest at 2% beng compounded annually? EM-5 Present Value of an Annuty Samuel Davd wants to make fve equal annual wthdrawals of 8,000 from a fund that wll earn nterest at 0% compounded annually. How much would Davd have to nvest on:. January, 2007 f the frst wthdrawal s made on January, 2008? 2. January, 2007 f the frst wthdrawal s made on January, 2007? EM-6 Amount of Each Cash Flow Sx equal annual contrbutons are made to a fund, wth the frst depost on December 3, 2007. Usng the future value tables, determne the equal contrbutons that, f nvested at 0% compounded annually, wll produce a fund of 30,000, assumng that ths sum s desred on December 3, 202. EM-7 Amount of an Annuty Begnnng December 3, 20, fve equal annual wthdrawals are to be made. Usng the approprate tables, determne the equal annual wthdrawals f 25,000 s nvested at an nterest of 2% compounded annually on. December 3, 200 2. December 3, 20 3. December 3, 2007 EM-8 Amount of Each Cash Flow R. Lee Rouse borrows 0,000 that s to be repad n 24 equal monthly nstallments payable at the end of each subsequent month wth nterest at the rate of 2% a month. Usng the approprate table, calculate the equal nstallments. EM-9 Amount of Each Cash Flow On January, 2007 Charles Jamson borrows 40,000 from hs father to open a busness. The son s the benefcary of a trust created by hs favorte aunt from whch he wll receve 25,000 on January, 207. He sgns an agreement to make ths amount payable to hs father and, further, to pay hs father equal annual amounts from January, 2008 to January, 206, nclusve, n retrement of the debt. Interest s 2%. What are the annual payments? EM-0 Amount of an Annuty Begnnng wth January, 2007, fve equal deposts are to be made n a fund. Usng the approprate tables, determne the equal deposts f nterest at 0% s compounded annually and f 200,000 must be n the fund on. January, 202 2. January, 203 EM- Seres of Compound Interest Technques The followng are several stuatons nvolvng compound nterest. Usng the approprate table, solve each of the followng:. Hope Dearborn nvests 40,000 on January, 2007 n a savngs account that earns nterest of 8% compounded semannually. What wll be the amount n the fund on December 3, 202?

M30 Tme Value of Money Module 2. Ben Johnson receves a bonus of 5,000 each year on December 3. He starts depostng hs bonus on December 3, 2007 n a savngs account that earns nterest of 2% compounded annually. What wll be the amount n the fund on December 3, 20 after he deposts hs bonus receved on that date? 3. Ron Sewert owes 30,000 on a non-nterest-bearng note due January, 207. He offers to pay the amount on January, 2007 provded that t s dscounted at 0% on a compound annual dscount bass. What would he have to pay on January, 2007 under ths assumpton? 4. June Stckney purchased an annuty on January, 2007 whch, at a 2% annual rate, would yeld 6,000 each June 30 and December 3 for the next sx years. What was the cost of the annuty to Stckney? 5. Fve equal annual contrbutons are to be made to a fund, the frst depost on December 3, 2007. Determne the equal contrbutons that, f nvested at 0% compounded annually, wll produce a fund of 30,000 on December 3, 202. 6. Begnnng on December 3, 2008, sx equal annual wthdrawals are to be made. Determne the equal annual wthdrawals f,000 s nvested at 0% nterest compounded annually on December 3, 2007. EM-2 Amount of an Annuty John Goodheart wshes to provde for sx annual wthdrawals of 3,000 each begnnng January, 207. He wshes to make 0 annual deposts begnnng January, 2007, wth the last depost to be made on January, 206. If the fund earns nterest compounded annually at 0%, how much s each of the 0 deposts? EM-3 Present Value of Leased Asset On January, 2007 Ashly Farms leased a hay baler from Agrco Tractor Company. Ashly was havng cash flow problems, so Agrco drew up the lease to allow Ashly to reestablsh tself. The lease requres Ashly to make 3,000 payments on January of each year for fve years begnnng n 2007. The nterest rate s 2%. Calculate the present value of the cost of the lease payments to Ashly on January, 2007. EM-4 Number of Cash Flows On July, 2007 Boston Company purchased a machne at a cost of 80,000. It pad 56,046.06 n cash and sgned a 0% note for the dfference. Ths note s to be pad off n annual nstallments of 5,000 each, payable each July, begnnng mmedately. The 5,000 ncludes a payment of nterest on the balance of the prncpal at the begnnng of each perod and a payment on the prncpal. Calculate the number of annual payments to be made by Boston Company. P ROBLEMS PM- Future Value of an Investment Usng the future value tables, solve the followng:. What s the future value on December 3, 20 of a depost of 35,000 made on December 3, 2007 assumng nterest of 0% compounded annually? 2. What s the future value on December 3, 20 of a depost of 0,000 made on December 3, 2007 assumng nterest of 6% compounded quarterly? 3. What s the future value on December 3, 20 of a depost of 25,000 made on December 3, 2007 assumng nterest of 2% compounded semannually? PM-2 Present Value Issues Usng the present value tables, solve the followng:. What s the present value on January, 2007 of 30,000 due on January, 20 and dscounted at 0% compounded annually? 2. What s the present value on January, 2007 of 40,000 due on January, 20 and dscounted at % compounded semannually? 3. What s the present value on January, 2007 of 50,000 due on January, 20 and dscounted at 6% compounded quarterly? PM-3 Future Value Issues Usng the future values tables, solve the followng:. What s the future value on December 3, 206 of 0 cash flows of 20,000 wth the frst cash payment made on December 3, 2007 and nterest at 0% beng compounded annually?

Problems M3 2. What s the future value on June 30, 207 of 20 cash flows of 5,000 wth the frst cash payment made on December 3, 2007 and the annual nterest rate of 0% beng compounded semannually? 3. What s the future value on December 3, 207 of 20 cash flows of 5,000 wth the frst cash payment made on December 3, 2007 and the annual nterest rate of 0% beng compounded semannually? PM-4 Amount of Each Cash Flow On December 3, 204 Mchael McDowell desres to have 60,000. He plans to make sx deposts n a fund to provde ths amount. Interest s compounded annually at 2%. Compute the equal annual amounts that McDowell must depost assumng that he makes the frst depost on. December 3, 2009 2. December 3, 2008 PM-5 Value of an Annuty John Joshua wants to make fve equal annual wthdrawals of 20,000 from a fund that wll earn nterest at 2% compounded annually. How much would Joshua have to nvest on January, 2007 f he makes the frst wthdrawal on. January, 2008? 2. January, 2007? 3. January, 202? PM-6 Value of an Annuty Ralph Benke wants to make eght equal semannual wthdrawals of 8,000 from a fund that wll earn nterest at % compounded semannually. How much would Benke have to nvest on:. January, 2007 f the frst wthdrawal s made on July, 2007? 2. July, 2007 f the frst wthdrawal s made on July, 2007? 3. January, 2007 f the frst wthdrawal s made on January, 200? PM-7 Varous Compound Interest Issues You are gven the followng stuatons:. Thomas Petry owes a debt of 7,000 from the purchase of a boat. The debt bears nterest of 2% payable annually. Petry wll pay the debt and nterest n fve annual nstallments begnnng n one year. Calculate the equal annual nstallments that wll pay off the debt and nterest at 2% on the unpad balance. 2. On January, 2007 John Cothran offers to buy Ruth House s used tractor and equpment for 4,000 payable n 2 equal semannual nstallments, whch are to nclude payment of 0% nterest on the unpad balance and payment of a porton of the prncpal, wth the frst nstallment to be made on January, 2007. Calculate the amount of each of these nstallments. 3. Nadne Love nvests n a 60,000 annuty at 2% compounded annually on March, 2007. The frst of 5 recepts from the annuty s payable to Love on March, 207, 0 years after the annuty s purchased and on the date Love expects to retre. Calculate the amount of each of the 5 equal annual recepts. Usng the approprate tables, solve each of the precedng stuatons. PM-8 Value of an Annuty Usng the approprate tables, solve each of the followng:. Begnnng December 3, 2008, fve equal wthdrawals are to be made. Determne the equal annual wthdrawals f 30,000 s nvested at 0% nterest compounded annually on December 3, 2007. 2. Ten payments of 3,000 are due at annual ntervals begnnng June 30, 2008. What amount wll be accepted n cancellaton of ths seres of payments on June 30, 2007 assumng a dscount rate of 4% compounded annually? 3. Ten payments of 2,000 are due at annual ntervals begnnng December 3, 2007. What amount wll be accepted n cancellaton of ths seres of payments on January, 2007 assumng a dscount rate of 2% compounded annually? PM-9 Amount of Each Cash Flow On January, 2007 Phlp Holdng nvests 40,000 n an annuty to provde eght equal semannual payments. Interest s 0%, compounded semannually. Compute the equal semannual amounts that Holdng wll receve, assumng that the frst wthdrawal s to be receved on. July, 2007 2. January, 2007 3. July, 200 4. January, 202

M32 Tme Value of Money Module PM-0 Number of Cash Flows The followng are two ndependent stuatons.. Houser wshes to accumulate a fund of 40,000 for the purchase of a house and lot. He plans to depost 4,000 semannually at the end of each sx months. Assumng nterest at 4% a year compounded semannually, how many deposts of 4,000 each wll be requred and what s the amount of the last depost? 2. On January, 2007 Joan Campbell borrows 20,000 from Susan Rone and agrees to repay ths amount n payments of 4,000 a year untl the debt s pad n full. Payments are to be of an equal amount and are to nclude nterest at 2% on the unpad balance of prncpal at the begnnng of each perod. Assumng that the frst payment s to be made on January, 2008, determne the number of payments of 4,000 each to be made and the amount of the fnal payment. Usng the approprate tables, solve each of the precedng stuatons. PM- Seral Installments; Amounts Applcable to Interest and Prncpal Ronald McDuffe purchases a new car at a cost of 4,400. He pays 3,000 down and ssues an nstallment note payable by whch he promses to pay the balance n 8 equal monthly nstallments, whch nclude nterest at an annual rate of 8% on the remanng unpad balance at the begnnng of each month, startng wth the frst month after the purchase.. Compute the equal nstallment payments. 2. Compute the nterest that wll be pad for each of the frst two perods. Indcate the amount of each payment that wll be a reducton of prncpal. PM-2 Determnng Loan Repayments Rockness needs 40,000 to pay off a loan due on December 3, 206. Hs plans ncluded the makng of 0 annual deposts begnnng on December 3, 2007 n accumulatng a fund to pay off the loan. Wthout makng a precse calculaton, Rockness made three annual deposts of 4,000 each on December 3, 2007, 2008, and 2009, whch have been earnng nterest at 0% compounded annually. What s the equal amount of each of the next seven deposts for the perod December 3, 200 to December 3, 206 to reach the fund objectve, assumng that the fund wll contnue to earn nterest at 0% compounded annually? PM-3 Purchase of Asset Wllam Thomas ntends to purchase a tractor on credt. Two local mplement dealers have offered hm the followng payment plans for dentcal tractors:. Redd Truck & Tractor s plan calls for fve annual payments of 0,350, wth the frst payment now and the remanng payments at the begnnng of each of the next four years. 2. Greene Farm Implements requres semannual payments of 5,750 at the end of each of the next 0 semannual perods, wth the frst payment to be n sx months. Determne whch of the precedng plans offers Thomas the lower present value. The applcable annual nterest rate s 0% for both alternatves. PM-4 Fund to Retre Bonds At the begnnng of 2007 Shankln Company ssued 0-year bonds wth a face value of,000,000 due on December 3, 206. The company wants to accumulate a fund to retre these bonds at maturty by makng annual deposts begnnng on December 3, 2007. How much must the company depost each year, assumng that the fund wll earn 2% nterest a year compounded annually? PM-5 Asset Purchase Prce BWP, Inc., s consderng the purchase of an asset. BWP s requred rate of return on new assets s 2%. The expected net cash nflows generated by the new asset are as follows: Years Amount Nature of the Cash Inflows 4 3,000 Net operatng revenues 5 9 2,500 Net operatng revenues 0 2,000 Net operatng revenues 0,000 Sale of asset Gven that the net cash nflows can be realzed, what s the maxmum amount BWP should be wllng to pay for the new asset? Assume that each cash nflow occurs at the end of the year. (Contrbuted by Norma C. Powell) PM-6 Acquston of Asset SuMar Company purchased a new pece of machnery by payng 2,000 down and agreeng to pay,000 at the end of each year for fve years. The approprate nterest rate s 8%.. What s the cost of the machnery?

Problems M33 2. Prepare the journal entry to record the purchase of the machnery. 3. Prepare a table that shows the nterest and endng balance of the lablty each year. (Contrbuted by Norma C. Powell) PM-7 Present Value Issues Nello Constructon Company has just purchased several major peces of road-buldng equpment. Snce the purchase prce s so large, the equpment company s gvng Nello an opton of choosng one of four dfferent payment plans:. 600,000 mmedately n cash. 2. 200,000 down payment now; 65,000 per year for 2 years, begnnng at the end of the current year. 3. 200,000 down payment now; 25,000 per year for 3 years begnnng at the end of the current year; 75,000 per year for years begnnng at the end of the fourth year after the purchase. 4. 80,000 now and at the begnnng of each of the next 3 years. You have been asked by the Nello Constructon Company to decde whch payment plan wll provde the smallest present value. The expected effectve nterest rate durng the future perods stated above s 2%. PM-8 AICPA Adapted Comprehensve Part a. Reproduced n the followng table are the frst three lnes from the 2% columns of each of several tables of mathematcal values. For each of the followng tems, you are to select from among these fragmentary tables the one from whch the amount requred can be obtaned most drectly (assumng that the complete table was avalable n each nstance): Perods Table A Table B Table C Table D Table E Table F 0.0000.0000 0.9804.0200.0200.0000 0.9804.0200 2 0.962 2.0604.0404 0.4950.946 0.550 3 3.26 0.3268 2.8839 0.3468. The amount to whch a sngle sum would accumulate at compound nterest by the end of a specfed perod (nterest compounded annually). 2. The amount that must be approprated at the end of each of a specfc number of years to provde for the accumulaton, at annually compounded nterest, of a certan sum. 3. The amount that must be deposted n a fund that wll earn nterest at a specfed rate, compounded annually, n order to make possble the wthdrawal of certan equal sums annually over a specfed perod startng one year from date of depost. 4. The amount of nterest that wll accumulate on a sngle depost by the end of a specfed perod (nterest compounded semannually). 5. The amount, net of compound dscount, that f pad now would settle a debt of larger amount due at a specfed future date. Part b. The followng tables of values at 0% nterest may be used as needed to answer the questons n ths part of the problem. Future Future Present Value Value of Present Value Value of of at Annuty of of Annuty of at Compound Compound at End of at End of Perods Interest Interest Each Perod Each Perod.00 0.909.0000 0.909............... 6.776 0.5645 7.756 4.3553 7.9487 0.532 9.4872 4.8684 8 2.436 0.4665.4359 5.3349 9 2.3579 0.424 3.5795 5.7590 0 2.5937 0.3855 5.9374 6.446 2.853 0.3505 8.532 6.495 2 3.384 0.386 2.3843 6.837 3 3.4523 0.2897 24.5227 7.034 4 3.7975 0.2633 27.9750 7.3667 5 4.772 0.2394 3.7725 7.606 6 4.5950 0.276 35.9497 7.8237. Your clent has made annual payments of 2,500 nto a fund at the close of each year for the past three years. The fund balance mmedately after the thrd payment totaled 8,275. He has asked you how many more 2,500 annual payments

M34 Tme Value of Money Module wll be requred to brng the fund to 22,500, assumng that the fund contnues to earn nterest at 0% compounded annually. Compute the number of full payments requred and the amount of the fnal payment f t does not requre the entre 2,500. Carefully label all computatons supportng your answer. 2. Your clent wshes to provde for the payment of an oblgaton of 200,000 due on July, 204. He plans to depost 20,000 n a specal fund each July for 7 years, startng July, 2008. He wshes to make an ntal depost on July, 2007 of an amount that, wth ts accumulated nterest, wll brng the fund up to 200,000 at the maturty of the oblgaton. He expects that the fund wll earn nterest at the rate of 0% compounded annually. Compute the amount to be deposted July, 2007. Carefully label all computatons supportng your answer. PM-9 Comprehensve The followng are three ndependent stuatons:. M. Herman has decded to set up a scholarshp fund for students. She s wllng to depost 5,000 n a trust fund at the end of each year for 0 years. She wants the trust fund to then pay annual scholarshps at the end of each year for 30 years. 2. Charles Jordy s plannng to save for hs retrement. He has decded that he can save 3,000 at the end of each year for the next 0 years, 5,000 at the end of each year for years through 20, and 0,000 at the end of each year for years 2 through 30. 3. Patrca Karpas has 200,000 n savngs on the day she retres. She ntends to spend 2,000 per month travelng around the world for the next two years, durng whch tme her savngs wll earn 8%, compounded monthly. For the next fve years, she ntends to spend 6,000 every sx months, durng whch tme her savngs wll earn 2%, compounded semannually. For the rest of her lfe expectancy of 5 years, she wants an annuty to cover her lvng costs. Durng ths perod her savngs wll earn 0% compounded annually. Assume that all payments occur at the end of each perod.. In Stuaton, how much wll the annual scholarshps be f the fund can earn 6%? 0%? 2. In Stuaton 2, (a) How much wll Jordy have at the end of 30 years f hs savngs can earn 0%? 6%? (b) If Jordy expects to lve for 20 years n retrement, how much can he spend each year f hs savngs earn 0%? 6%? (c) How much would Jordy need to nvest today to have the same amount avalable at the tme he retres as calculated n 2(a) at 0%? 6%? 3. In Stuaton 3, how much wll Karpas s annuty be? C ASES CM- Cost of Insurance Plans The Johnson Company s consderng three dfferent tme perods for an nsurance polcy on ts man offce buldng. The premums on a fre nsurance polcy coverng the buldng for the amount of 2,000,000 on a one-year, three-year, and fve-year bass are as follows: One year 4,480 Three years,200 Fve years 7,920 In each case the entre premum for the full term of the polcy s payable at the begnnng of the year n whch the polcy s purchased. Evaluate the annual cost of each nsurance plan for the nsured, assumng that money s worth 2% compounded annually. Whch plan do you recommend? State the savngs for the company. CM-2 Acquston of Equpment The manager of the Taylor Company has consulted you, the controller, as to whch of the followng plans you would recommend n acqurng the use of a pece of heavy equpment:. Purchase the equpment and pay mmedately a cash prce of 36,800. The servce lfe of the heavy equpment s estmated to be fve years, wth a resale value at the end of that tme of 5,500. 2. Lease the equpment at the rate of 9,00 per year for fve years, payable at the begnnng of each year. Assumng that the tme value of money s 2%, evaluate the two alternatves and ndcate whch plan you would recommend to the manager, statng the value of savngs to the company. CM-3 Effectve Interest n Varous Stuatons On March, 2007 the Whte Company purchased 400,000 worth of nventory on credt wth terms of /20, n/60. In the past, Whte has always followed the polcy of makng payment one month (30 days) after the goods are purchased. A new member of Whte s staff has ndcated that the company he prevously worked for never passed up ts cash dscounts, and he wonders f ths s not a sound polcy. It was ponted out, however, that f Whte were to pay the bll on March 20 rather than on March 30, t would have to borrow the necessary funds for the 0 extra days. Whte s borrowng terms wth a local bank were estmated to be at 4% (annual rate), wth a 5% compensatng balance (a requrement by the bank that Whte mantan an amount n ts account equal to 5% of the loan) for the term of the loan. Most members of Whte s staff felt that t made lttle sense to take out a 4% loan

Compound Interest Tables M35 wth a compensatng balance of 5% n order to save % on 400,000 by payng the account 0 days earler than planned.. In terms of smple effectve annual nterest cost, explan whether t would be to Whte s advantage to borrow the amount necessary to take the % dscount by payng the bll 0 days early. 2. It has also been ponted out to Whte that f t does not take advantage of the cash dscount, t should wat the entre 60-day perod to pay the full bll rather than pay wthn 30 days. Explan how your answer to Requrement would change f Whte undertook ths polcy. 3. Your answer to Requrement 2 ndcates that, n relaton to Requrement, t has become ether more desrable or less desrable to borrow n order to take advantage of the % cash dscount. a. If you sad more desrable, explan why. b. If you sad less desrable, make a smlar explanaton. CM-4 Future Value of Sngle Investment and Annuty Jane Dough was a teller n a large northeastern bank. She was sngle and approachng age 30, and she consdered herself an honest and uprght ctzen. After consderng what she mght do to buld a retrement plan for the future, she decded to embezzle,500,000. Subsequently she gave herself up to the authortes but dd not return the,500,000. She was tred, convcted, and sentenced to 20 years n prson. After completng her 20-year term, she returned the,500,000 that she had stolen. She then decded to take a world cruse. On the shp someone asked her how she had accumulated enough money to afford the trp. She repled, Do you know how much nterest,500,000 wll earn n 20 years f nvested at an annual rate of 6% compounded quarterly?. Determne the answer to Jane Dough s queston. The table factor for f n 40, 4% s 4.8002. 2. Evaluate Jane s retrement decson, assumng that she could have earned 2,000 each year for each of the 20 years she was n prson. Assume that,000 s requred each year to cover lvng expenses and that she could have nvested the remanng 0,000 at the end of each year to earn nterest at 6% compounded annually. CM-5 Value of a Note You have just been promoted to manager at a natonal CPA frm. On your frst job a new accountant approaches you wth the followng stuaton: He has dscovered that the presdent of the clent company has a brother who s both the major stockholder and the presdent of a local bank. Your clent has a 300,000, fve-year note payable to the bank at 4% nterest compounded annually. Snce the gong nterest rate s 6%, the accountant suggests that the note be recorded at ts present value usng ths gong rate. The presdent says that the effectve lablty s 300,000 and should be reported on the balance sheet at ths fgure. The note was ssued on January, 2007 and s due on January, 202.. Explan who s correct. 2. At what amount should the company have valued the note on January, 2007, assumng that the accountant s assessment s correct? CM-6 Future Value and Present Value Issues Jean Perry has a 25,000 whole-lfe nsurance polcy that she began many years ago. She s presently 55 years old. One of the benefts of the polcy s that Perry can borrow up to a gven amount at 2% nterest (2% below the current rate), wth the prncpal due two years after the loan s made. The polcy states that should Perry default on the prncpal payment, t wll smply be deducted from the amount gven her benefcary when she des. However, the nterest wll contnue to accrue as long as the note s not pad. Perry has just borrowed 5,000 on ths polcy to take a vacaton n Hawa. Assumng that a woman of Perry s health s expected to lve to be 72, explan whether t would be fnancally advantageous for Perry to repay the prncpal on the loan n two years. (Calculatons are not requred.) C OMPOUND I NTEREST T ABLES Table : Future Value of : f n, ( ) n Table 2: Future Value of an Ordnary Annuty of : F0 Table 3: Present Value of : p n, ( ) n Table 4: Present Value of an Ordnary Annuty of : P0 Table 5: Present Value of Annuty Due: Pd n, = n, = n, + = n + + n + n

M36 Tme Value of Money Module Table FUTURE VALUE OF : f n, ( ) n n.5% 4.0% 4.5% 5.0% 5.5% 6.0% 7.0%.05000.040000.045000.050000.055000.060000.070000 2.030225.08600.092025.02500.3025.23600.44900 3.045678.24864.466.57625.7424.906.225043 4.06364.69859.9259.25506.238825.262477.30796 5.077284.26653.24682.276282.306960.338226.402552 6.093443.26539.302260.340096.378843.4859.500730 7.09845.35932.360862.40700.454679.503630.60578 8.26493.368569.4220.477455.534687.593848.7886 9.43390.42332.486095.55328.69094.689479.838459 0.6054.480244.552969.628895.70844.790848.9675.77949.539454.622853.70339.802092.898299 2.04852 2.9568.60032.69588.795856.90207 2.0296 2.25292 3.23552.665074.77296.885649 2.005774 2.32928 2.409845 4.23756.73676.85945.979932 2.609 2.260904 2.578534 5.250232.800944.935282 2.078928 2.232476 2.396558 2.759032 6.268986.87298 2.022370 2.82875 2.355263 2.540352 2.95264 7.288020.947900 2.3377 2.29208 2.484802 2.692773 3.5885 8.30734 2.02587 2.208479 2.40669 2.62466 2.854339 3.379932 9.32695 2.06849 2.307860 2.526950 2.765647 3.025600 3.66528 20.346855 2.923 2.474 2.653298 2.97757 3.20735 3.869684 2.367058 2.278768 2.52024 2.785963 3.078234 3.399564 4.40562 22.387564 2.36999 2.633652 2.92526 3.247537 3.603537 4.430402 23.408377 2.46476 2.75266 3.07524 3.42652 3.89750 4.740530 24.429503 2.563304 2.87604 3.22500 3.64590 4.048935 5.072367 25.450945 2.665836 3.005434 3.386355 3.83392 4.2987 5.427433 26.47270 2.772470 3.40679 3.555673 4.02329 4.549383 5.807353 27.494800 2.883369 3.28200 3.733456 4.24440 4.822346 6.23868 28.57222 2.998703 3.429700 3.92029 4.477843 5.687 6.648838 29.53998 3.865 3.584036 4.636 4.72424 5.48388 7.4257 30.563080 3.243398 3.74538 4.32942 4.98395 5.74349 7.62255 n 8.0% 9.0% 0.0% 2.0% 4.0% 6.0% 8.0%.080000.090000.00000.20000.40000.60000.80000 2.66400.8800.20000.254400.299600.345600.392400 3.25972.295029.33000.404928.48544.560896.643032 4.360489.4582.46400.57359.688960.80639.938778 5.469328.538624.6050.762342.92545 2.00342 2.287758 6.586874.67700.7756.973823 2.94973 2.436396 2.699554 7.73824.828039.94877 2.2068 2.502269 2.826220 3.85474 8.850930.992563 2.43589 2.475963 2.852586 3.27845 3.758859 9.999005 2.7893 2.357948 2.773079 3.25949 3.80296 4.435454 0 2.58925 2.367364 2.593742 3.05848 3.70722 4.4435 5.233836 2.33639 2.580426 2.8537 3.478550 4.226232 5.7265 6.75926 2 2.5870 2.82665 3.38428 3.895976 4.87905 5.936027 7.287593 3 2.79624 3.065805 3.45227 4.363493 5.4924 6.88579 8.599359 4 2.93794 3.34727 3.797498 4.8872 6.26349 7.98758 0.47244 5 3.7269 3.642482 4.77248 5.473566 7.37938 9.26552.973748 6 3.425943 3.970306 4.594973 6.30394 8.37249 0.748004 4.29023 7 3.70008 4.327633 5.054470 6.86604 9.276464 2.467685 6.672247 8 3.99609 4.7720 5.55997 7.689966 0.57569 4.46254 9.67325 9 4.3570 5.466 6.5909 8.62762 2.055693 6.77657 23.24436 20 4.660957 5.6044 6.727500 9.646293 3.743490 9.460759 27.393035 2 5.033834 6.08808 7.400250 0.803848 5.667578 22.57448 32.32378 22 5.436540 6.658600 8.40275 2.0030 7.86039 26.86398 38.4206 23 5.87464 7.257874 8.954302 3.552347 20.36585 30.376222 45.007632 24 6.348 7.9083 9.849733 5.78629 23.22207 35.23647 53.09006 25 6.848475 8.62308 0.834706 7.000064 26.4696 40.874244 62.668627 26 7.396353 9.39958.9877 9.040072 30.66584 47.4423 73.948980 27 7.98806 0.245082 3.09994 2.32488 34.389906 55.000382 87.259797 28 8.62706.6740 4.420994 23.883866 39.204493 63.800444 02.966560 29 9.37275 2.7282 5.863093 26.749930 44.69322 74.00855 2.50054 30 0.062657 3.267678 7.449402 29.959922 50.95059 85.849877 43.370638

Compound Interest Tables M37 Table 2 FUTURE VALUE OF AN ORDINARY ANNUITY OF : F + n.5% 4.0% 4.5% 5.0% 5.5% 6.0% 7.0%.000000.000000.000000.000000.000000.000000.000000 2 2.05000 2.040000 2.045000 2.050000 2.055000 2.060000 2.070000 3 3.045225 3.2600 3.37025 3.52500 3.68025 3.83600 3.24900 4 4.090903 4.246464 4.2789 4.3025 4.342266 4.37466 4.439943 5 5.52267 5.46323 5.47070 5.52563 5.5809 5.637093 5.750739 6 6.22955 6.632975 6.76892 6.8093 6.88805 6.97539 7.5329 7 7.322994 7.898294 8.0952 8.42008 8.266894 8.393838 8.65402 8 8.432839 9.24226 9.38004 9.54909 9.72573 9.897468 0.259803 9 9.559332 0.582795 0.8024.026564.256260.4936.977989 0 0.702722 2.00607 2.288209 2.577893 2.875354 3.80795 3.86448.863262 3.48635 3.8479 4.206787 4.583498 4.97643 5.783599 2 3.042 5.025805 5.464032 5.9727 6.38559 6.86994 7.88845 3 4.236830 6.626838 7.5993 7.72983 8.286798 8.88238 20.40643 4 5.450382 8.299 8.93209 9.598632 20.292572 2.05066 22.550488 5 6.68238 20.023588 20.784054 2.578564 22.408663 23.275970 25.29022 6 7.932370 2.82453 22.79337 23.657492 24.6440 25.672528 27.888054 7 9.20355 23.69752 24.74707 25.840366 26.996403 28.22880 30.84027 8 20.489376 25.64543 26.855084 28.32385 29.48205 30.905653 33.999033 9 2.79676 27.67229 29.063562 30.539004 32.0267 33.759992 37.378965 20 23.23667 29.778079 3.37423 33.065954 34.86838 36.78559 40.995492 2 24.470522 3.969202 33.78337 35.79252 37.786076 39.992727 44.86577 22 25.837580 34.247970 36.303378 38.50524 40.86430 43.392290 49.005739 23 27.22544 36.67889 38.937030 4.430475 44.847 46.995828 53.4364 24 28.63352 39.082604 4.68996 44.50999 47.537998 50.85577 58.7667 25 30.063024 4.645908 44.56520 47.727099 5.52588 54.86452 63.249038 26 3.53969 44.3745 47.570645 5.3454 54.96598 59.56383 68.676470 27 32.986678 47.08424 50.7324 54.66926 58.98909 63.705766 74.483823 28 34.48479 49.967583 53.993333 58.402583 63.23350 68.5282 80.69769 29 35.99870 52.966286 57.423033 62.32272 67.7354 73.639798 87.346529 30 37.53868 56.084938 6.007070 66.438848 72.435478 79.05886 94.460786 n 8.0% 9.0% 0.0% 2.0% 4.0% 6.0% 8.0%.000000.000000.000000.000000.000000.000000.000000 2 2.080000 2.090000 2.00000 2.20000 2.40000 2.60000 2.80000 3 3.246400 3.27800 3.30000 3.374400 3.439600 3.505600 3.572400 4 4.5062 4.57329 4.64000 4.779328 4.9244 5.066496 5.25432 5 5.86660 5.9847 6.0500 6.352847 6.6004 6.87735 7.5420 6 7.335929 7.523335 7.7560 8.589 8.53559 8.977477 9.44968 7 8.922803 9.200435 9.4877 0.08902 0.73049.43873 2.4522 8 0.636628.028474.435888 2.299693 3.232760 4.240093 5.326996 9 2.487558 3.02036 3.579477 4.775656 6.085347 7.58508 9.085855 0 4.486562 5.92930 5.937425 7.548735 9.337295 2.32469 23.52309 6.645487 7.560293 8.5367 20.654583 23.04456 25.732904 28.75544 2 8.97726 20.40720 2.384284 24.3333 27.270749 30.85069 34.93070 3 2.495297 22.953385 24.52272 28.02909 32.088654 36.78696 42.28663 4 24.24920 26.0989 27.974983 32.392602 37.58065 43.67987 50.88022 5 27.524 29.36096 3.772482 37.27975 43.84244 5.659505 60.965266 6 30.324283 33.003399 35.949730 42.753280 50.980352 60.925026 72.93904 7 33.750226 36.973705 40.544703 48.883674 59.760 7.673030 87.068036 8 37.450244 4.30338 45.59973 55.74975 68.394066 84.4075 03.740283 9 4.446263 46.08458 5.59090 63.43968 78.969235 98.603230 23.43534 20 45.76964 5.6020 57.274999 72.052442 9.024928 5.379747 46.627970 2 50.42292 56.764530 64.002499 8.698736 04.76848 34.840506 74.02005 22 55.456755 62.873338 7.402749 92.502584 20.435996 57.44987 206.344785 23 60.893296 69.53939 79.543024 04.602894 38.297035 83.60385 244.486847 24 66.764759 76.78983 88.497327 8.5524 58.658620 23.977607 289.494479 25 73.05940 84.700896 98.347059 33.333870 8.870827 249.24024 342.603486 26 79.95445 93.323977 09.8765 50.333934 208.332743 290.088267 405.2723 27 87.350768 02.72335 2.099942 69.374007 238.499327 337.502390 479.22093 28 95.338830 2.96827 34.209936 90.698887 272.889233 392.502773 566.480890 29 03.965936 24.35356 48.630930 24.582754 32.093725 456.30326 669.447450 30 3.2832 36.307539 64.494023 24.332684 356.786847 530.373 790.94799 0n, = n

M38 Tme Value of Money Module Table 3 PRESENT VALUE OF : p n, ( ) n n.5% 4.0% 4.5% 5.0% 5.5% 6.0% 7.0% 0.985222 0.96538 0.956938 0.95238 0.947867 0.943396 0.934579 2 0.970662 0.924556 0.95730 0.907029 0.898452 0.889996 0.873439 3 0.95637 0.888996 0.876297 0.863838 0.8564 0.83969 0.86298 4 0.94284 0.854804 0.83856 0.822702 0.80727 0.792094 0.762895 5 0.928260 0.82927 0.80245 0.783526 0.76534 0.747258 0.72986 6 0.94542 0.79035 0.767896 0.74625 0.725246 0.70496 0.666342 7 0.90027 0.75998 0.734828 0.7068 0.687437 0.665057 0.622750 8 0.8877 0.730690 0.70385 0.676839 0.65599 0.62742 0.582009 9 0.874592 0.702587 0.672904 0.644609 0.67629 0.59898 0.543934 0 0.86667 0.675564 0.643928 0.6393 0.58543 0.558395 0.508349 0.848933 0.64958 0.6699 0.584679 0.5549 0.526788 0.475093 2 0.836387 0.624597 0.589664 0.556837 0.525982 0.496969 0.44402 3 0.824027 0.600574 0.564272 0.53032 0.49856 0.468839 0.44964 4 0.8849 0.577475 0.539973 0.505068 0.472569 0.44230 0.38787 5 0.799852 0.555265 0.56720 0.4807 0.447933 0.47265 0.362446 6 0.78803 0.533908 0.494469 0.4582 0.42458 0.393646 0.338735 7 0.776385 0.53373 0.47376 0.436297 0.402447 0.37364 0.36574 8 0.76492 0.493628 0.452800 0.4552 0.38466 0.350344 0.295864 9 0.753607 0.474642 0.433302 0.395734 0.36579 0.33053 0.276508 20 0.742470 0.456387 0.44643 0.376889 0.342729 0.3805 0.25849 2 0.73498 0.438834 0.396787 0.358942 0.324862 0.29455 0.2453 22 0.720688 0.42955 0.37970 0.34850 0.307926 0.277505 0.22573 23 0.70037 0.405726 0.363350 0.32557 0.29873 0.26797 0.20947 24 0.699544 0.3902 0.347703 0.30068 0.276657 0.246979 0.9747 25 0.689206 0.3757 0.33273 0.295303 0.262234 0.232999 0.84249 26 0.67902 0.360689 0.38402 0.2824 0.248563 0.2980 0.7295 27 0.668986 0.34687 0.30469 0.267848 0.235605 0.207368 0.60930 28 0.659099 0.333477 0.2957 0.255094 0.223322 0.95630 0.50402 29 0.649359 0.32065 0.27905 0.242946 0.2679 0.84557 0.40563 30 0.639762 0.30839 0.267000 0.23377 0.200644 0.740 0.3367 n 8.0% 9.0% 0.0% 2.0% 4.0% 6.0% 8.0% 0.925926 0.9743 0.90909 0.892857 0.87793 0.862069 0.847458 2 0.857339 0.84680 0.826446 0.79794 0.769468 0.74363 0.7884 3 0.793832 0.77283 0.7535 0.7780 0.674972 0.640658 0.60863 4 0.735030 0.708425 0.68303 0.63558 0.592080 0.55229 0.55789 5 0.680583 0.64993 0.62092 0.567427 0.59369 0.4763 0.43709 6 0.63070 0.596267 0.564474 0.50663 0.455587 0.40442 0.370432 7 0.583490 0.547034 0.5358 0.452349 0.399637 0.353830 0.33925 8 0.540269 0.50866 0.466507 0.403883 0.350559 0.305025 0.266038 9 0.500249 0.460428 0.424098 0.36060 0.307508 0.262953 0.225456 0 0.46393 0.4224 0.385543 0.32973 0.269744 0.226684 0.9064 0.428883 0.387533 0.350494 0.287476 0.23667 0.9547 0.699 2 0.3974 0.355535 0.3863 0.256675 0.207559 0.68463 0.37220 3 0.367698 0.32679 0.289664 0.22974 0.82069 0.45227 0.6288 4 0.34046 0.299246 0.26333 0.204620 0.5970 0.2595 0.098549 5 0.35242 0.274538 0.239392 0.82696 0.40096 0.07927 0.08356 6 0.29890 0.25870 0.27629 0.6322 0.22892 0.09304 0.070776 7 0.270269 0.23073 0.97845 0.45644 0.07800 0.080207 0.059980 8 0.250249 0.2994 0.79859 0.30040 0.09456 0.06944 0.050830 9 0.2372 0.94490 0.63508 0.607 0.082948 0.059607 0.043077 20 0.24548 0.7843 0.48644 0.03667 0.072762 0.05385 0.036506 2 0.98656 0.63698 0.353 0.092560 0.063826 0.044298 0.030937 22 0.8394 0.5082 0.22846 0.082643 0.055988 0.03888 0.02628 23 0.7035 0.3778 0.678 0.073788 0.0492 0.032920 0.02228 24 0.57699 0.26405 0.0526 0.065882 0.04308 0.028380 0.08829 25 0.4608 0.5968 0.092296 0.058823 0.037790 0.024465 0.05957 26 0.35202 0.06393 0.083905 0.05252 0.03349 0.0209 0.03523 27 0.2587 0.097608 0.076278 0.046894 0.029078 0.0882 0.0460 28 0.594 0.089548 0.069343 0.04869 0.025507 0.05674 0.00972 29 0.07328 0.08255 0.063039 0.037383 0.022375 0.0352 0.008230 30 0.099377 0.07537 0.057309 0.033378 0.09627 0.0648 0.006975

Compound Interest Tables M39 n + Table 4 PRESENT VALUE OF AN ORDINARY ANNUITY OF : P0 = n, n.5% 4.0% 4.5% 5.0% 5.5% 6.0% 7.0% 0.985222 0.96538 0.956938 0.95238 0.947867 0.943396 0.934579 2.955883.886095.872668.85940.846320.833393.80808 3 2.92200 2.77509 2.748964 2.723248 2.697933 2.67302 2.62436 4 3.854385 3.629895 3.587526 3.54595 3.50550 3.46506 3.3872 5 4.782645 4.45822 4.389977 4.329477 4.270284 4.22364 4.0097 6 5.69787 5.24237 5.57872 5.075692 4.995530 4.97324 4.766540 7 6.59824 6.002055 5.89270 5.786373 5.682967 5.58238 5.389289 8 7.485925 6.732745 6.595886 6.46323 6.334566 6.209794 5.97299 9 8.36057 7.435332 7.268790 7.07822 6.95295 6.80692 6.55232 0 9.22285 8.0896 7.9278 7.72735 7.537626 7.360087 7.023582 0.078 8.760477 8.52897 8.30644 8.092536 7.886875 7.498674 2 0.907505 9.385074 9.858 8.863252 8.6858 8.383844 7.942686 3.73532 9.985648 9.682852 9.393573 9.7079 8.852683 8.35765 4 2.543382 0.56323 0.222825 9.89864 9.589648 9.294984 8.745468 5 3.343233.8387 0.739546 0.379658 0.03758 9.72249 9.0794 6 4.3264.652296.23405 0.837770 0.46262 0.05895 9.446649 7 4.907649 2.65669.7079.274066 0.864609 0.477260 9.763223 8 5.67256 2.659297 2.59992.689587.246074 0.827603 0.059087 9 6.42668 3.33939 2.593294 2.08532.607654.586 0.335595 20 7.68639 3.590326 3.007936 2.46220.950382.46992 0.59404 2 7.90037 4.02960 3.404724 2.8253 2.275244.764077 0.835527 22 8.620824 4.455 3.784425 3.63003 2.58370 2.04582.06240 23 9.33086 4.856842 4.47775 3.488574 2.875042 2.303379.27287 24 20.030405 5.246963 4.495478 3.798642 3.5699 2.550358.469334 25 20.796 5.622080 4.828209 4.093945 3.43933 2.783356.653583 26 2.398632 5.982769 5.466 4.37585 3.662495 3.00366.825779 27 22.06767 6.329586 5.45303 4.643034 3.89800 3.20534.986709 28 22.72677 6.663063 5.742874 4.89827 4.2422 3.40664 2.37 29 23.376076 6.98375 6.02889 5.4074 4.3330 3.59072 2.277674 30 24.05838 7.292033 6.288889 5.37245 4.533745 3.76483 2.40904 n 8.0% 9.0% 0.0% 2.0% 4.0% 6.0% 8.0% 0.925926 0.9743 0.90909 0.892857 0.87793 0.862069 0.847458 2.783265.759.735537.69005.64666.605232.565642 3 2.577097 2.53295 2.486852 2.4083 2.32632 2.245890 2.74273 4 3.3227 3.239720 3.69865 3.037349 2.9372 2.7988 2.690062 5 3.99270 3.88965 3.790787 3.604776 3.43308 3.274294 3.277 6 4.622880 4.48599 4.35526 4.407 3.888668 3.684736 3.497603 7 5.206370 5.032953 4.86849 4.563757 4.288305 4.038565 3.8528 8 5.746639 5.53489 5.334926 4.967640 4.638864 4.34359 4.077566 9 6.246888 5.995247 5.759024 5.328250 4.946372 4.606544 4.303022 0 6.7008 6.47658 6.44567 5.650223 5.266 4.833227 4.494086 7.38964 6.8059 6.49506 5.937699 5.452733 5.028644 4.656005 2 7.536078 7.60725 6.83692 6.94374 5.660292 5.9707 4.793225 3 7.903776 7.486904 7.03356 6.423548 5.842362 5.342334 4.90953 4 8.244237 7.78650 7.366687 6.62868 6.002072 5.467529 5.008062 5 8.559479 8.060688 7.606080 6.80864 6.4268 5.575456 5.09578 6 8.85369 8.32558 7.823709 6.973986 6.265060 5.668497 5.62354 7 9.2638 8.54363 8.02553 7.9630 6.372859 5.748704 5.222334 8 9.37887 8.755625 8.2042 7.249670 6.467420 5.87848 5.27364 9 9.603599 8.9505 8.364920 7.365777 6.550369 5.877455 5.3624 20 9.8847 9.28546 8.53564 7.469444 6.6233 5.92884 5.352746 2 0.06803 9.292244 8.648694 7.562003 6.686957 5.97339 5.383683 22 0.200744 9.442425 8.77540 7.644646 6.742944 6.0326 5.40990 23 0.37059 9.580207 8.88328 7.78434 6.792056 6.044247 5.43220 24 0.528758 9.70662 8.984744 7.78436 6.83537 6.072627 5.450949 25 0.674776 9.822580 9.077040 7.84339 6.872927 6.097092 5.466906 26 0.809978 9.928972 9.60945 7.895660 6.906077 6.883 5.480429 27 0.93565 0.026580 9.237223 7.942554 6.93555 6.36364 5.49889 28.05078 0.628 9.306567 7.984423 6.960662 6.52038 5.5060 29.58406 0.98283 9.369606 8.02806 6.983037 6.65550 5.50983 30.257783 0.273654 9.42694 8.05584 7.002664 6.7798 5.56806

M40 Tme Value of Money Module n + Table 5 PRESENT VALUE OF ANNUITY DUE: Pd = + n, n.5% 4.0% 4.5% 5.0% 5.5% 6.0% 7.0%.000000.000000.000000.000000.000000.000000.000000 2.985222.96538.956938.95238.947867.943396.934579 3 2.955883 2.886095 2.872668 2.85940 2.846320 2.833393 2.80808 4 3.92200 3.77509 3.748964 3.723248 3.697933 3.67302 3.62436 5 4.854385 4.629895 4.587526 4.54595 4.50550 4.46506 4.3872 6 5.782645 5.45822 5.389977 5.329477 5.270284 5.22364 5.0097 7 6.69787 6.24237 6.57872 6.075692 5.995530 5.97324 5.766540 8 7.59824 7.002055 6.89270 6.786373 6.682967 6.58238 6.389289 9 8.485925 7.732745 7.595886 7.46323 7.334566 7.209794 6.97299 0 9.36057 8.435332 8.268790 8.07822 7.95295 7.80692 7.55232 0.22285 9.0896 8.9278 8.72735 8.537626 8.360087 8.023582 2.078 9.760477 9.52897 9.30644 9.092536 8.886875 8.498674 3.907505 0.385074 0.858 9.863252 9.6858 9.383844 8.942686 4 2.73532 0.985648 0.682852 0.393573 0.7079 9.852683 9.35765 5 3.543382.56323.222825 0.89864 0.589648 0.294984 9.745468 6 4.343233 2.8387.739546.379658.03758 0.72249 0.0794 7 5.3264 2.652296 2.23405.837770.46262.05895 0.446649 8 5.907649 3.65669 2.7079 2.274066.864609.477260 0.763223 9 6.67256 3.659297 3.59992 2.689587 2.246074.827603.059087 20 7.42668 4.33939 3.593294 3.08532 2.607654 2.586.335595 2 8.68639 4.590326 4.007936 3.46220 2.950382 2.46992.59404 22 8.90037 5.02960 4.404724 3.8253 3.275244 2.764077.835527 23 9.620824 5.455 4.784425 4.63003 3.58370 3.04582 2.06240 24 20.33086 5.856842 5.47775 4.488574 3.875042 3.303379 2.27287 25 2.030405 6.246963 5.495478 4.798642 4.5699 3.550358 2.469334 26 2.796 6.622080 5.828209 5.093945 4.43933 3.783356 2.653583 27 22.398632 6.982769 6.466 5.37585 4.662495 4.00366 2.825779 28 23.06767 7.329586 6.45303 5.643034 4.89800 4.20534 2.986709 29 23.72677 7.663063 6.742874 5.89827 5.2422 4.40664 3.37 30 24.376076 7.98375 7.02889 6.4074 5.3330 4.59072 3.277674 n 8.0% 9.0% 0.0% 2.0% 4.0% 6.0% 8.0%.000000.000000.000000.000000.000000.000000.000000 2.925926.9743.90909.892857.87793.862069.847458 3 2.783265 2.759 2.735537 2.69005 2.64666 2.605232 2.565642 4 3.577097 3.53295 3.486852 3.4083 3.32632 3.245890 3.74273 5 4.3227 4.239720 4.69865 4.037349 3.9372 3.7988 3.690062 6 4.99270 4.88965 4.790787 4.604776 4.43308 4.274294 4.277 7 5.622880 5.48599 5.35526 5.407 4.888668 4.684736 4.497603 8 6.206370 6.032953 5.86849 5.563757 5.288305 5.038565 4.8528 9 6.746639 6.53489 6.334926 5.967640 5.638864 5.34359 5.077566 0 7.246888 6.995247 6.759024 6.328250 5.946372 5.606544 5.303022 7.7008 7.47658 7.44567 6.650223 6.266 5.833227 5.494086 2 8.38964 7.8059 7.49506 6.937699 6.452733 6.028644 5.656005 3 8.536078 8.60725 7.83692 7.94374 6.660292 6.9707 5.793225 4 8.903776 8.486904 8.03356 7.423548 6.842362 6.342334 5.90953 5 9.244237 8.78650 8.366687 7.62868 7.002072 6.467529 6.008062 6 9.559479 9.060688-8.606080 7.80864 7.4268 6.575456 6.09578 7 9.85369 9.32558 8.823709 7.973986 7.265060 6.668497 6.62354 8 0.2638 9.54363 9.02553 8.9630 7.372859 6.748704 6.222334 9 0.37887 9.755625 9.2042 8.249670 7.467420 6.87848 6.27364 20 0.603599 9.9505 9.364920 8.365777 7.550369 6.877455 6.3624 2 0.8847 0.28546 9.53564 8.469444 7.6233 6.92884 6.352746 22.06803 0.292244 9.648694 8.562003 7.686957 6.97339 6.383683 23.200744 0.442425 9.77540 8.644646 7.742944 7.0326 6.40990 24.37059 0.580207 9.88328 8.78434 7.792056 7.044247 6.43220 25.528758 0.70662 9.984744 8.78436 7.83537 7.072627 6.450949 26.674776 0.822580 0.077040 8.84339 7.872927 7.097092 6.466906 27.809978 0.928972 0.60945 8.895660 7.906077 7.883 6.480429 28.93565.026580 0.237223 8.942554 7.93555 7.36364 6.49889 29 2.05078.628 0.306567 8.984423 7.960662 7.52038 6.5060 30 2.58406.98283 0.369606 9.02806 7.983037 7.65550 6.50983