of the relationship between time and the value of money.

Transcription

1 TIME AND THE VALUE OF MONEY Most agrbusess maagers are famlar wth the terms compoudg, dscoutg, auty, ad captalzato. That s, most agrbusess maagers have a tutve uderstadg that each term mples some relatoshp betwee the value of moey ad the passage of tme, but few would be able to accurately descrbe that relatoshp. For example, you probably recogze that the preset value of a $100 promssory ote payable oe year hece s somethg less tha the ote s face value. You also probably recogze that the preset value of a seres of twelve mothly reveues of $10 to be receved the future s somethg less tha the sum total of those reveues. However, are you able to provde a detaled explaato of the relatoshp betwee the value of moey ad the passage of tme? Probably ot-except to state your awareess of the exstece of such a relatoshp. Every agrbusess maager s faced wth makg decsos cocerg alteratve uses of avalable captal. A farm supply busess maager may have to decde whether to elarge hs curret product le or elarge hs facltes. A frut warehouse maager may have to decde whether to vest fuds ew, more effcet equpmet wth a mmedate retur or expaded C.A. facltes whch may ot geerate a et retur for several years. Or, a gra elevator maager may have to decde whether to sell some used gra trucks ow at a stated prce, or wat a few more years ad sell at some ukow prce. Regardless, all three maagers are faced wth a decso, whch requres a uderstadg of the relatoshp betwee tme ad the value of moey. Exactly what s ths tme-value relatoshp ad what should a agrbusess maager kow about t to make better decsos? Actually, there exsts o sgle tme-value relatoshp. Istead there are several, cludg: a) future value of a preset sum, b) preset value of a future sum, c) preset value of a aual reveue, d) preset value of a auty for a gve tme, ad e) future value of a auty for a gve tme. Every agrbusess maager should have at least a workg kowledge of these fve smple relatoshps. Through a seres of bref llustratos, t s hoped that ths paper wll provde such kowledge. of a Preset Sum Let s assume that you are the maager of a fertlzer supply frm, whch specalzes, aeral applcatos. At the preset tme your frm s major vestmet s arcraft wth a remag useful lfe of about fve years. I order to facltate the purchase of ew arcraft, you decde to place $20,000 a specal bak accout yeldg sx percet terest per aum. Your plas are to wthdraw the $20,000, plus terest fve years ad use ths amout to help buy the ew arcraft. Allowg the terest to accumulate over the etre perod, how much moey wll you be able to wthdraw fve years? Ths future value of a preset sum ca be determed by usg the followg formula: A= P(1 + ) 1 WASHINGTON STATE UNIVERSITY & U.S. DEPARTMENT OF AGRICULTURE COOPERATING

2 where: A = future value P = preset sum = terest rate (per coverso perod) = umber of coverso perods I our llustrato, we see that the preset sum ($20,000) s coverted (perod for whch terest s pad) fve tmes at a terest rate of sx percet. Moreover, the formula becomes: = $20,000 (1.06) 5 = $20,000 (1.3382) = $26,764 The factor (1.06) 5 ca be had calculated, but t ca be more easly obtaed from a compoud terest table foud the appedx of most mathematcs or statstcs books, e.g., accordg to the approprate table values for ad, (1.06) 5 equals As show above, $26,764 could be wthdraw from the specal accout at the ed of the fve-year perod. of a Future Sum Suppose a dary cooperatve purchases ts supples from a large federated supply cooperatve. Ths year, the dary cooperatve receved a certfcate of equty for $20,000, whch represets the cooperatve s share of the federato s eargs. At the preset, the federato s payg ts certfcates (revolvg ts equtes) fve years. Let s assume that you are the maager of the dary cooperatve ad that your board of drectors has asked you to select a represetatve dscout rate ad determe the preset value of the $20,000 certfcate. Selectg a dscout rate of sx percet, you could use the followg formula to calculate the preset value: P = A ( 1+ ) where: P = preset value A = future sum = dscout rate (per coverso perod) = umber of coverso perods I our llustrato, we see that the future sum of $20,000 s beg dscouted at sx percet for fve coverso perods ad our formula becomes: $20,000 P = ( 1.06) 5 $20,000 = (1.3382) The factor (1.06) 5 ca be take from the compoud terest table oted earler. Ofte, however, t wll be easer to solve the formula usg a dfferet table -- oe called the dscout table, showg the preset values for varous values of ad, ad also foud most mathematcs books. It ca be show that the followg mathematcal relatoshp exsts: 1 ( 1+ ) ( 1 ) = + Substtutg to our formula we fd that: P = A(1 + ) - = $20,000 (.7473) = $14,946 The preset value of the certfcate s $14,946 ad ca be expressed two ways: a) you have cocluded that you would be just as well off to receve $14,949 ow as to receve $20,000 fve years from ow, or b) you are cocludg that t would take $14,946 vested today at sx percet terest per aum to yeld $20,000 fve years. 2

3 Iterest or Dscout So far we have used the terms terest ad dscout. Oe should be able to dstgush betwee the two. I our frst llustrato, we started wth a preset sum ad moved to the future to determe the value of that sum for a future tme. I fact, whe movg from a earler tme to a future tme, terest s always volved. I our secod llustrato, the movemet was reversed,.e., we started wth a future sum ad determed the value of the sum at the preset tme. Therefore, whe movg from ay future tme to a earler tme, dscoutg s always volved. of Costat Aual Reveue Now let s assume that you are the maager of a frut warehouse. Your frm s stackg ts loose frut boxes o a oe-acre plot just to the rear of your warehouse. Had your frm ot owed ths plot of lad, t would probably have had to pay a aual ret of $200 for the prvlege of stackg ts boxes there. Cosderg the fact that owershp of ths acre allows your frm to avod ths retal fee, ths acre s earg your frm a equvalet of $200 per year. If you expect all frm assets to create a rate of retur of at least te percet, what s the captalzed (preset) value of the oe-acre storage area? To aswer ths questo, oe should use the followg formula: P = R where P = preset (or captalzed) value R = reveue (per coverso perod) = dscout rate (per coverso perod) I our llustrato, we see that: $200 P = = $2, I other words, accordg to your stated mmum rate of retur (dscout), the preset value of your storage area s $2,000. Moreover, ths ca be descrbed two ways: a) t s the preset value of a aual reveue of $200 for a defte tme, or b) t s the sum you would have to vest at te percet terest order to obta a aual retur of $200 for a defte umber of years. of a Auty for a Gve Tme Let s assume that as the maager of a truckg frm you are cosderg the purchase of a ew two-to truck. Accordg to your records, you ca expect the ew truck to produce aual et reveue of $1000 for ts useful lfe of te years, after whch o salvage value exsts. As our earler example, you have establshed a mmum requremet of a te percet rate of retur before you wll make ay ew vestmets. You ow wsh to determe the maxmum amout you could pay for the ew truck ad stll acheve your expected rate of retur. To solve ths problem, oe should use the followg formula: ( 1 ) 1 + P = R where: P = preset value R = reveue (per coverso perod; = terest rate (per coverso perod) = umber of coverso perods I our llustrato, the formula becomes: ( ) P = $1, = 1, = $6,145 3

4 Based o your expectatos ad your requred rate of retur, you could pay up to $6,145 for the ew truck ad stll have a satsfactory vestmet. Actually, ths aswer could be terpreted two ways: a) t s the preset value of te aual reveues of $1,000 each, wth reveues receved at the ed of each year ad wth a te percet dscout rate, ad b) t also represets the dollar amout, whch f vested at te percet terest, wll be completely exhausted after wthdrawals of $1,000 at the ed of each of te cosecutve years. of a Auty for a Gve Tme Let s assume that you have bee the maager of a cooperatve farm supply frm for twety years ad ow expect to retre fve years. Your board of drectors has just formed you that, apprecato of your may years of servce, they have decded to establsh a specal retremet fud for you. Each year, $2,000 s to be vested ths specal fud, whch wll receve a aual terest rate of te percet. At the tme of your retremet, fve years hece, the fud wll be gve to you as a gft. What wll be the total amout of moey receved at the tme of your retremet? To aswer ths questo, the followg formula should be used: R = ( 1 + ) 1 where: A= future value R= reveue (per coverso perod) = terest rate (per coverso) = umber of coverso perods I our llustrato, the formula becomes: A = $2, = $2, = $12,210 ( ) 5 The result of your frm vestg $2,000 each of fve cosecutve years at te percet terest, the reveues accumulatg alog wth terest, wll provde a retremet fud of $12,210. Rates ad Relatoshps I our llustratos, we used varyg terest ad dscout rates. What effect does the selected rate have o the partcular tme-value relatoshp? I fact, the effect may be postve or egatve, depedg o the partcular relatoshp to whch we are referrg. Table 1 llustrates the varyg effects. You wll otce Table 1 that creasg terest rates has a postve effect o the relatoshps cocered wth future values, ad creasg dscout rates has a egatve effect o the relatoshp s cocered wth preset values. Ths s, of course, completely logcal as a creased rate smply speeds up the compoudg or dscoutg process, the frst creasg the future value ad the latter decreasg the preset value. Tme ad Relatoshps I our llustratos, we also chose to use varyg tme perods. What effect does the selected tme perod legth have o the partcular tme-value relatoshp? Aga, the effect vares. I determg the preset 4

5 Table 1 The Effects of Rate Chages o Tme-Value Relatoshps Tme-Value Relatoshp If Iterest or (Dscout) Rate: The: Effect o Future or (Preset) Value of Preset Sum Postve of Future Sum () (Negatve) of Aual Reveue () (Negatve) of a Auty () (Negatve) of a Auty Postve value of costat aual reveue, tme perod legth has zero effect because the aual reveues are expected for a defte perod of tme. The other four tme-value relatoshps, however, are affected dfferetly by chages the tme perod legth selected. These varyg effects are show Table 2. (See top of ext page) You wll otce Table 2 that creasg tme perod legth has a postve effect o all relatoshps except for the preset value of a future sum. Aga, ths s completely logcal because as a gve sum s to be receved oly after a loger perod of tme, oe would expect ts preset value to decrease. Dverse Perods So far our dscusso, the coverso perod the llustratos has bee detcal wth the reveue perod,.e., occurrg oce at the ed of each year. For some agrbusess stuatos ths may ot be true, e.g., terest s ofte compouded semaually or mothly whle reveues (or paymets) occur aually, or reveues (or paymets) occur mothly whle terest s compouded quarterly. Whe terest (or dscoutg) ad reveues (or paymets) are based o detcal perods, our calculatos are relatvely smple, as show. Whe they are based o dverse perods, however, the calculatos become slghtly more complex. Two adjustmets must be made. Frst, the terest or dscout rate must be stated terms of a rate per coverso perod,.e., wth te percet terest compouded semaually, ths must be adjusted to fve percet per coverso perod. Secod, the tme perod volved must be stated terms of umber of coverso perods,.e., f terest s compouded semaually for fve years, the umber of coverso perods wll be te. The frst three relatoshps dscussed ths paper ca be adjusted to dverse perods usg 5

6 Table 2 The Effects of Tme Perod Legth Chages o Tme-Value Relatoshps Tme-Value Relatoshp If tme Perod Legth: The: Effect o Future or (Preset) Value of Preset Sum Postve of Future Sum (Negatve) of a Auty (Postve) of a Auty Postve the two-step coverso just descrbed above. The two auty relatoshps, however, requre a thrd coverso before they become adaptable to dverse perod. The auty formulae used earler cluded the term whch was defed as the umber of coverso perods. Uder dverse perods, the coverso perod must be cosdered as the base perod,.e., the perodc reveue (or paymet), the umber of paymets, ad the terest or dscout rate must be stated terms of the coverso perod. Now we must add a ew factor to our two auty formulae,.e., e s defed as the umber of perodc reveues (or paymets) each coverso perod. For example, f reveue s receved mothly ad dscoutg occurs semaually, ad the e s sx. R was defed earler as the reveue (paymet) per coverso perod. Uder dverse perods, where reveue ad coverso perods dffer, we see that R ow equals e multpled by the dollar reveue (or paymet). Whe determg the preset value of a auty uder dverse perods, we use the same formula show for detcal perods except that we sert a correcto factor whch we shall deote as as follows: ( 1 ) 1 + P = R where s selected from a coverso factor table for the approprate values of e ad. Usg our earler llustrato o the purchase of a ew two-to truck, suppose we ow chage the percet aual dscout rate to a sem-aual rate. The coverso perod s ow sx moths ad our formula becomes: 20 1 (1.05) P = $500 (.9756) = $500 (.9756).05 = $6,080 6

7 where: P = preset value R = reveue (per coverso perod) = $500 = ½ of $1,000 receved aually = correcto factor (table value for, e) = 5 percet dscout rate (per coverso perod) e = ½ = umber of reveues (per coverso perod) = 20 = umber coverso perods (durg te years) If the dscout rate had remaed at te percet aually, but the $1,000 aual reveue were chaged to $500 receved semaually, the we see that R = $1,000, = 10, e = 2, = 10, ad would be selected from the coverso table for the approprate values of ad e. If the future value of a auty s to be determed wth dverse perods, the correcto factor would aga be serted to our earler formula to form: A = R Note: ( 1 + ) 1 I workg wth autes, we defed R as the reveue (or paymet) per coverso perod. I workg wth terest or dscout rates, we referred to the future or preset value of a sum. The llustratos used were descrbed mostly as vestmets. Had we bee referrg to costs, rather tha reveues (or paymets), the calculatos would ot have chaged, for a sum ca be receved as a come, or pad as a cost. Summary Fve dfferet relatoshps betwee tme ad the value of moey have bee descrbed. Each relatoshp s desged to ft a dfferet stuato so far as t apples to preset or future value ad dscout or terest rates. Every agrbusess maager must make decsos cocerg the vestmet of avalable captal. Before such decsos ca be made tellgetly, the maager must recogze, uderstad ad apply umerous tme-value relatoshps. Ths paper presets several tme-value llustratos a attempt to assst the maager developg ths recogto, uderstadg, ad ablty. The effects of chages rates, tme perod legths, ad dverse perods o tme-value relatoshps were also cosdered. Those relatoshps descrbed were: a) future value of a preset sum, b) preset value of a future sum, c) preset value of a costat aual reveue, d) preset value of a auty, ad e) future value of a auty. For a more detaled dscusso of the cocepts oted ths paper, the reader s ecouraged to wrte to the Supertedet of Documets, U. S. Govermet Prtg Offce, Washgto, D.C for a copy of Agrcultural Hadbook No. 349, The Evaluato of Ivestmet Opportutes by Arthur J. Wabrath ad W. L. Gbso, Jr., February 1968, U.S.D.A. Ke D. Duft Exteso Marketg Ecoomst 7