# Time Value of Money Module

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1 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.

3 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, /360). However, f the term of ths note s 5 months, the smple nterest s,500 (0, /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, / , nd quarter 0, / , rd quarter 0, / , th quarter 0, / , th quarter, / , Compound nterest on 0,000 at 2% compounded quarterly for 5 quarters, * 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

4 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, 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? 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, by December 3, 20. Ths s the future value. The total nterest of 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 ) (Col. 2 Col. 3) 2008, , , , , , , , 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

5 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 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 as follows:, , 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, , 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, 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) Step 2 f,000( ), 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

6 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) f n, 4% (.4) f n 2, 2% (.02) f n 2, 4% (.4) f n 3, 2% (.02) f n 3, 4% (.4) f n 4, 2% (.02) f n 4, 4% (.4) f n 40, 2% (.02) f n 40, 4% (.4) 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% 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, ; then, to arrve at the answer of,262.48, the calculaton s: f,000( ), 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, Factor for f n 28,? ,000.00

7 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, s the future value on ths date Quarterly Interest Rate Unknown The factor of 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 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, s equal to (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 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, when t earns 6% compound nterest per year for four years, then t follows that, to be receved four years from now s worth,000 now at tme perod zero; that s,,000 s the present value of, 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.

8 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, wll be receved on ths date 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, 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, 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, to be receved four years from now wth nterest of 6% compounded annually can be calculated n two steps: Step p n 4, 6% (.06) 4 Step 2 p,262.48( ),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

9 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, 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 Then the future value of, s multpled by ths present value of factor to obtan the present value amount of,000, as follows: p,262.48( ),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, , (Factor for p n 0,? ) 3, Factor for p n 0,? , The factor of 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 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, s n the 0% column. Thus the annual rate s 0%. If the factor of 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, s the known present value 0 annual tme perods 0,000 s the known future value to be pad Annual Interest Rate Unknown

10 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, annual cash flows of,000 each,000, Interest Rate Is 6% Compounded Annually, 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.

11 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, (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 = , = 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% Step 2 F 0,000( ) 4, Ths two-step approach s used to solve the problem when factors are not avalable.

12 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 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 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, 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, 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 )

13 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, ,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 Interest Rate Is 2% Compounded Annually 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, Interest Rate Is 6% Compounded Annually 20

14 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% Step 2 Subtract wthout nterest from the value obtaned n step. ( ) Step 3 Ths s the converted future value factor for F dn 4, 6% 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, 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,

15 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, 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 :, ,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) * *The value of s slghtly smaller than the factor for P 0n = 4, = 6% of n Table 4 dscussed later n ths secton; ths s the result of roundng each of the four factors for P n,.

16 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, 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 =, 000 = 3, ( ) 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

17 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% = = 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 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 Ths factor s then multpled by,000 to determne the present value fgure of 3,465.: P 0,000( ) 3,465. Over the 4 perods, the annuty yelds nterest each perod as follows: Begnnng Cash Endng Perod Balance Interest Flow Balance 3, (,000) 2, , (,000), , (,000) (,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.

18 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, 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, , Remember that each of these payments of 7, 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, (7, ,000). For the year 2008 the nterest s,36.9 [2% (00,000 5,698.42)], and the retrement of prncpal s 6, (7,698.42,36.9). The last payment of 7, 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 Interest Rate Is 2% Compounded Annually 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, The nterest rate s 6% compounded annually. Example M-5 shows the facts of ths problem.

19 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, 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 =, = 3, 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, )

20 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% = = Step 2 P d,000( ) 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 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( ) 3,673.0 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% Step 2 Add wthout nterest to the value obtaned n step Step 3 Ths s the converted present value factor for P dn 4, 6% 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, 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 ( ) 3,673.0.

21 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 Interest Rate Is 8% Compounded Annually 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, The down payment of, 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.

22 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, , , 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, 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[( )( )] 2, 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% ( ) P 0k 3, 6% ( )

23 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, 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% ( ) P 0k 5, 2% ( ) Then the amount of each cash flow s 50,000 C 5, 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, 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[( )( )] 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.

24 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 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

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