Time Value of Moey ad Ivestmet Aalysis Explaatios ad Spreadsheet Applicatios for Agricultural ad Agribusiess Firms Part I. by Bruce J. Sherrick Paul N. Elliger David A. Lis V 1.2, September 2000 The Ceter for Farm ad Rural Busiess Fiace Departmet of Agricultural ad Cosumer Ecoomics ad Departmet of Fiace Uiversity of Illiois, Urbaa-Champaig
Time Value of Moey ad Ivestmet Aalysis: Table of Cotets Part I. Itroductio... 1 Basic Cocepts ad Termiology... 3 Categories of Time Value of Moey Problems... 4 1. Sigle-Paymet Compoud Amout (SPCA)... 4 2. Uiform Series Compoud Amout (USCA)... 5 3. Sikig Fud Deposit (SFD)... 5 4. Sigle-Paymet Preset Value (SPPV)... 6 5. Uiform-Series Preset Value (USPV)... 6 6. Capital Recovery (CV)... 7 Coceptualizatio ad Solutio of Time Value of Moey Problems... 8 Notatio Summary... 8 Other covetios commoly employed i this booklet ad i other TVM materials... 9 Derivatio of Formulas used for each category of problem... 9 Impact of Compoudig Frequecy... 13 Summary of Materials i Part I... 15 Summary of TVM Formulas... 16
TIME VALUE OF MONEY AND INVESTMENT ANALYSIS INTRODUCTION This documet cotais explaatios ad illustratios of commo Time Value of Moey problems facig agricultural ad agribusiess firms. The accompayig spreadsheet files cotai applicatios that mirror each sectio of the booklet, ad provide the tools to do real-time computatios ad illustratios of the ideas preseted i the text. Take together, they provide the capacity to aalyze ad solve a wide array of real-world ivestmet problems. Time Value of Moey problems refer to situatios ivolvig the exchage of somethig of value (moey) at differet poits i time. I a basic sese, all ivestmets ivolve the exchage of moey at oe poit i time for the rights to the future cash flows associated with that ivestmet. Expressig all of the values that are exchaged i terms of a commo medium of exchage, or moey, allows differet sets of products or services to be compared i terms of a sigle stadard of value (e.g., dollars). However, the passage of time betwee the outflows ad iflows i a typical ivestmet situatio results i differet curret values associated with cash flows that occur at differet poits i time. Thus, it is ot possible to assess a ivestmet simply by addig up the total cash iflows ad outflows ad determiig if they are positive or egative without first cosiderig whe the cash flows occur. There are four primary reasos why a dollar to be received i the future is worth less tha a dollar to be received immediately. The first ad most obvious reaso is the presece of positive rates of iflatio which reduce the purchasig power of dollars through time. Secodly, a dollar today is worth more today tha i the future because of the opportuity cost of lost earigs -- that is, it could have bee ivested ad eared a retur betwee today ad a poit i time i the future. Thirdly, all future values are i some sese oly promises, ad cotai some ucertaity about their occurrece. As a result of the risk of default or operformace of a ivestmet, a dollar i had today is worth more tha a expected dollar i the future. Fially, huma prefereces typically ivolve impatiece, or the preferece to cosume goods ad services ow rather tha i the future. 1
Iterest rates represet the price paid to use moey for some period of time. Iterest rates are positive to compesate leders (savers) for foregoig the use of moey for some iterval of time. The iterest rate must offset the collective effects of the four reasos cited above for preferrig a dollar today to a dollar i the future. The iterest rate per period alog with the other iformatio about the sizes ad timigs of cash flows permit meaigful ivestmet aalyses to be coducted. Ufortuately, typical ivestmet decisios are much more complicated tha simply calculatig the expected cash flows ad iterest rates ivolved. Icluded i real-world aalyses may be ivestmet optios with differet legth lives, differet sized ivestmets, differet fiacig terms, differig tax implicatios, ad the overall feasibility of makig the iitial ivestmet. I respose, the materials i this documet ad the accompayig spreadsheets were developed to assist i placig each of these issues ito a cotext that permits meaigful comparisos across differig ivestmet situatios. I each case, the cash flows associated with a ivestmet are coverted to similar terms ad the coverted to their equivalet values at a commo poit i time usig tools ad techiques that collectively comprise the cocepts kow as the Time Value of Moey. The materials i this documet are orgaized ito three sectios. The first sectio discusses the coceptual uderpiigs of time value of moey techiques alog with the resultig mathematical expressios, ad provides coveiet summary of the formulas that are used to solve may time value of moey problems. The secod sectio discusses iformatioal eeds, alterative approaches to ivestmet aalysis, ad commo problems ecoutered i real world aalyses of time value of moey problems. The third sectio cotais a collectio of idividual chapters devoted to descriptios of the spreadsheet applicatios for use i coductig meaigful ivestmet aalyses. I total, we hope this package is useful for learig ad applyig time value of moey cocepts to make better fiacial decisios. Farm Aalysis Solutio Tools 2
Time Value of Moey ad Ivestmet Aalysis Part I: BASIC CONCEPTS AND TERMS Time value of moey problems arise i may differet forms ad situatios. Thus, it is importat to establish some commo cocepts ad termiology to permit accurate characterizatio of their features. Amog the most importat characteristics of time value problems are: (i) the directio i time that cash flows are coverted to equivalet values, (ii) whether there is a sigle cash flow, or a series of cash flows, ad (iii) the decisio variable or ukow value of the problem. The first feature to establish ivolves the directio i time toward which cash flows are coverted. Compoudig refers to situatios where a curret value is beig coverted to its equivalet future value for compariso to aother future value. Discoutig ivolves movig back through time, or the coversio of a cash flow to be received i the future ito its equivalet curret value. The secod importat feature to establish is whether the cash flow type is a sigle paymet at some poit i time or a series of paymets through time. Periodic paymet, or series problems, ca be solved as a collectio of sigle-paymet problems, but fortuately there are more coveiet solutio techiques for series problems tha solvig a set of sigle paymet procedures. A further distictio of series problems that ca be made is whether it ivolves a series with fixed paymet size (commoly referred to as uiform series problems) or whether the series paymet size is growig or decliig through time. Fially, the decisio variable or item whose value is beig sought i the problem must be established. The collectio of time value of moey techiques ca be used to solve for preset values, future values, the paymet size i a series, the iterest rate or yield, or the legth of time ivolved i a decisio. Although there are may variats of time value of moey problems, they ca early all be placed ito oe of six categories. The variats of each category of problem permit solutios for differet decisio variables, but each ivolves the same basic formulas. The followig pages idetify 3
ad describe the six basic categories of time value of moey problems. Brief examples of each type of problem are provided to help develop idetificatio skills for real world applicatios. There are also a series of example situatios described i a compaio documet that ca be used as a self-test for those iterested i hoig their skills at classifyig ad solvig TVM problems. Categories of Time Value of Moey Problems The six basic types of time value of moey problems are described below. These six ca also be described i terms of the elemets itroduced earlier to characterize problems: (i) the directio i time that cash flows are coverted to equivalet values, (ii) whether the cash flow is a sigle value or a repeated series, ad (iii) the decisio variable or ukow value of the problem. Of the six basic types the first three categories ivolve compoudig, or the coversio of curret ad series paymets to future values. Categories four through six ivolve discoutig, or fidig curret values associated with future cash flows. Categories oe ad four apply to sigle paymet problems ad differ oly by whether the future or preset value is beig sought. Categories two ad five are used to address series paymets rather tha sigle paymet situatios ad differ oly by whether the future or preset value is beig sought. Categories three ad six are employed whe the size of the paymet i a series is beig sought whe the total value of the series of paymets is already kow at some poit i time. Thus, they differ from two ad five respectively oly by which item is the ukow or decisio variable i the aalysis. I each case, oce the appropriate category is idetified for the solutio of a problem, the associated formula ca be rearraged to solve for differet variats of the problem. These six problem types are described more fully below with example situatios i which they would each apply. 1. Sigle-Paymet Compoud Amout (SPCA) This category refers to problems that ivolve a kow sigle iitial outlay ivested at a specified iterest rate ad compouded at a regular basis. It is used whe oe eeds to kow the value to which the origial sigle pricipal or ivestmet will grow by the ed of a specified time period. A savigs deposit accout that pays iterest represets a SPCA problem whe oe desires to kow how much 4
a iitial deposit will grow to by the ed of a specific time period. Aother example would be to fid the value of a savigs bod payig a kow iterest rate, at some poit i time i the future. Variats of the formula used to solve this problem ca be used to solve for (i) the legth of time eeded for a ivestmet to double i value at a kow iterest rate, ad (ii) the yield o a ivestmet that doubled i valued over a kow iterval of time. 2. Uiform Series Compoud Amout (USCA) This category of problem ivolves kow periodic paymets ivested at a regular itervals ito a iterest bearig accout or iterest payig ivestmet that permits iterest to be reivested ito the project. It is used to solve for the future value that this uiform series of paymets of deposits grows ito at compoud iterest, whe cotiued for the specified legth of time. This cocept is complicated by the fact that each succeedig deposit ears iterest for oe less period tha the precedig deposit. Examples of this applicatio iclude solvig for the size of a retiremet accout expected if regular mothly deposits are made ito a iterest payig ivestmet accout. Aother example would be to solve for the value of a savigs accout for college expeses at the time a child turs 18, if aual deposits are made to their accout. Life isurace policies cash value computatios utilize the formula associated with this problem as well. Variats of this problem iclude solvig for: (i) at what poit i time will a accout be worth some amout if regular deposits of kow size are made ito the accout with a kow iterest rate ad regular compoudig, ad (ii) what rate of retur is eeded if kow periodic paymets are made ito a accout ad a kow future amout (e.g., eough to retire) is eeded. 3. Sikig Fud Deposit (SFD) A third variatio of the compoudig problem occurs whe the desire is to make regular uiform deposits that will geerate a predetermied amout by the ed of a give period. The compoud iterest rate ad the umber of deposits to be made are kow, but the size of the ecessary deposits is ukow. For example, oe might decide whe a child is bor to make mothly deposits 5
toward a college educatio. If the goal is to have $30,000 at the ed of 18 years, for example, the formula associated with this category ca be used to solve for the size of the required mothly deposits eeded to meet that goal. Aother example that ofte arises is to fid the savigs amout eeded each period i a iterest earig accout to be able to purchase somethig of kow value i the future. SFD calculatios ca also be used to fid the size of the diversio i icome eeded to be able to retire a balloo paymet o a loa whe it comes due i the future. Variats of this problem iclude situatios such as solvig for the legth of time oe would eed to work util retiremet if regular deposits are made util retiremet of a kow size ito a accout payig a kow iterest rate, ad if a kow retiremet accout threshold must be reached. Or, the formula associated with the SFD problem ca be used to solve for the required yield eeded for a series of kow ivestmet cotributios to grow to a give size i a specified iterval of time. 4. Sigle-Paymet Preset Value (SPPV) Sigle-paymet preset value problems ivolve calculatios solvig for the discouted value of a future sigle paymet that results i a equivalet value i exchage today (preset). Solvig for the preset value of a kow future paymet is the iverse of the problem of solvig for the future value of a kow preset value. The oly differece betwee SPCA ad SPPV problems is the directio i time toward which moey is beig coverted. I the SPPV problem, time takes o a egative value that is the problem is used to trasfer a future kow value backwards i time to the preset. Examples iclude calculatios of the price to pay today for a pure discout bod, or the value today of a promise that someoe makes to pay you a kow amout at some poit i time i the future. Variats of this problem, like SPCA problems, ivolve solvig for the iterest rate or time factors eeded to covert a future value to its kow equivalet preset value uder differet circumstaces. 5. Uiform-Series Preset Value (USPV) I this category, a series of paymets of equal size is to be received at differet poits of time i the future, ad the preset value of the total series of paymets is beig sought. Although this type 6
problem is coceptually equivalet to a series of SPPV problems, the formula ivolved is much simpler if the paymets ca be expressed as a series. For example, if oe were etitled to receive fixed paymets at the ed of each year for five years, the there are really five SPPV problems with the sum of the results beig equal to the USPV. Examples iclude calculatio of the curret value of a set of scheduled retiremet paymets, a series of sales receipts, or other situatios i which there is a series of future cash iflows. Traditioal ivestmet theory asserts that the value of a ivestmet today is equal to the discouted sum of all future cash flows. That statemet of equivalece betwee future cash flows ad preset value is the most geeral applicatio of the formula associated with this category of time value of moey problems. Variats of this problem iclude (i) calculatio of factorig rates, of the implicit cost of borrowig if oe were to sell a set of receivables for a kow curret amout, or, (ii) fidig the legth of time eeded to retire a obligatio if the periodic maximum paymets ad iterest rate are kow. 6. Capital Recovery (CR) A problem closely related to the USPV is the capital recovery problem, also kow as the loa amortizatio paymet problem. I this case the preset value is kow, (the origial loa balace which must be repaid) ad the iterest rate o upaid remaiig pricipal is kow. I questio is the size of equal paymets (coverig both iterest ad pricipal) which must be made each time period to exactly retire the etire remaiig pricipal with the last paymet. Typical ledig situatios provide the bulk of the examples of this problem with variats that are aalogous to those i the USPV case. It should be oted that the differece betwee the USPV case ad the CR case is whether the preset value (e.g., iitial loa amout) or size of paymet is the ukow. Commo variats i practice iclude (i) fidig the maximum size loa that ca be borrowed with a kow icome stream or debt repaymet capacity, or (ii) fidig the legth of time over which a loa must be amortized for the loa paymets to be of a give acceptable size. 7
Coceptualizatio ad Solutio of Time Value of Moey Problems The two ecessary phases of solvig time value of moey problems are: 1) correctly idetifyig which type of problem exists ad what factor is ukow; ad 2) correctly applyig the appropriate mathematical calculatios to fid the aswer. Classificatio of problems is a skill that ca be developed by carefully reducig the problem ito its kow ad ukow values, ad the rulig out approaches that do ot apply. It is extremely helpful to draw a timelie associated with the cashflows of the problem to assist i problem classificatio ad descriptio. Oce classified, the mathematical formulas cotaied below ad i the spreadsheets supplied with this text ca be used to fid the actual values. These values are ofte ecessary i makig maagemet or ivestmet decisios. It is useful to first provide stadardized otatio that ca be used i solvig each of the problems. A summary of the otatio used is provided below. Notatio Summary: P t = Paymet of size P at time t. Paymets may differ through time. V o = preset value, or sometimes PV, value at time 0. V t = geeral otatio for total value at time t. = a period of time (could be moth, half-year, year, etc) also a poit i time, with the fial period i time ofte give as N. t = time idex, especially commo i cotiuously compouded problems, fial poit i time is ofte give as T. A = auity, or simply the periodic paymet amout (always a costat amout) m = umber of iterest rate compoudigs per period of time. r = iterest rate per period of time. exp, e, or e = base of the atural logarithm. 8
Other covetios commoly employed i this bookelt ad i other TVM materials: The curret time period, or preset, is always time 0. Discrete time problems usually use for itervals of time, ad by covetio, the paymets flows occur at the ed of the time iterval uless otherwise idicated. Thus, a paymet P 1 is a paymet that occurs at the ed of the first period. Cotiuous time problems usually use t to represet a poit i time. Loa amouts are special cases of V o ad are sometimes writte as L Bods ad ivestmets payig fixed coupos represet special cases of P t ad are sometimes writte as C t to represet coupo paymet Derivatio of formulas used for each category of problem. This sectio provides the mathematical relatioships ad algorithms that are associated with each of the six categories above. I additio, it cotais the cotiuous-time formulas that are ofte used uder cotiuous compoudig, or as simplificatios of the discrete time versios. All six of the formulas ca be derived from the same basic priciples, ad therefore the relatioships amog the formulas should be apparet after workig through this sectio. At the ed of this sectio, each of the formulas is restated i summary form o a sigle page iteded as oe page pullout referece, ad thus, this sectio ca be skipped without loss of cotiuity or applicability. To begi, cosider a iitial pricipal deposit, V 0, deposited i a iterest bearig accout that pays (compouds) a fixed iterest rate, r, aually. After oe period, the iterest earigs of P 1 = r*v 0 could be withdraw leavig the origial balace i place to ear iterest for the ext period. After the secod period, the same iterest-oly paymet could be made from this accout. I fact, the iterest could be withdraw each period i perpetuity ad ever disturb the origial pricipal balace i the accout. Thus, i exchage for the iitial pricipal, the ivestor ca receive a perpetual series of paymets equal to r*v 0 which are all also equal to P 1 sice all iterest withdrawals would be equal. The relatioship r*v 0 - P 1 is kow as the fudametal capitalizatio formula ad is usually rearraged ad writte as: 9
P1 [1] V, 0 = r This relatioship ca be depicted graphically as show below: Next, cosider the same deposit, but ow assume the iterest earigs are ever withdraw, but istead are left i the accout to accumulate iterest as well i all future periods. After oe period, the accout will be worth the iitial pricipal plus iterest earigs or V 0 + r*v 0. This equatio ca be rewritte as: V 0 (1+r) 1. That amout, if left udisturbed for the secod period will be worth its iitial value at the begiig of the period or V 0 (1+r) 1 plus r*v 0 (1+r) 1 i iterest earigs. This amout ca be rewritte as V 0 (1+r)(1+r) or equivaletly V 0 (1+r) 2. Each successive period will result i ed of that period s value equal to its begiig period value times (1+r). Thus, after periods, a iitial deposit of V 0 will grow to: [2] V ( + r ) = V 0 1 which ca be graphically depicted as: SPCA The value (1+r) is the SPCA iterest rate factor which, whe multiplied by ay size iitial deposit gives its future value after periods at iterest rate per period of r, compouded oce per period. Oce the preset value ad future value are liked through the iterest rate ad time relatioship, equatio [2] ca be rearraged to solve for V 0 i terms of V givig: 10
V [3] V, 0 = ( 1+ r) which is ofte rewritte as V 0 = V (1+r) -. The egative expoet i this versio highlights the idea of movig backward i time to get back to a preset value. The relatioship ca be graphically depicted as: SPPV Combiig these relatioships permits the valuatio of series of paymets, although the algebra is a bit more ivolved. To begi, cosider the perpetual series of paymets of size P 1 begiig at the ed of the first period ad lastig forever (labeled Series I. i the figure below). From the fudametal capitalizatio formula, it is kow that its curret value is simply P 1 /r. Now cosider a secod series of paymets (labeled Series II. i the figure below), but this time the first paymet is received at time period +1, ad at the ed of every period thereafter. At the future poit i time, that series is worth P +1 /r. If you begi with series I ad subtract series II, what remais is a series of paymets arrivig at the ed of each of the first periods ito the future ad the zero thereafter -- or a uiform series lastig periods. Graphically, 11
Usig the formula for SPPV, series II has a curret value of (P +1 /r)(1+r) -. Sice the paymets are of equal size at all poits i time, the subscripts ca all be writte as 1 ad each paymet referred to as P 1. Thus, the formula for a uiform series preset value ca be writte as V 0 = (Series I preset value) mius (Series II preset value) or: [4] V 0 P1 P1 1 = r r 1+ r Factorig out P, multiplyig through both parts o the right had side by (1+r) results i the typical expressio for the USPV relatioship: ( 1 + r) 1 [5] V0 = P1. r( 1 + r) Equatio [5] ca be rearraged to fid the size of paymet per period for periods at iterest rate r that has preset value equal to V 0. The result is the capital recovery formula (CR), or as more commoly called, the loa amortizatio formula. Rearragig eq. [5] for P 1 gives: [6] P V r r ( 1+ ) 1 = 0, ( 1+ r) 1 which is used to fid the size of equal periodic paymet at the ed of each period for periods at iterest rate r that will retire a iitial loa pricipal of size V 0. The uiform series compoud amout (USCA) differs from the USPV formula by solvig for the future value rather tha the preset value of a series of paymets of kow size for a kow umber of periods ad a kow iterest rate. Give the SPCA formula that liks preset to future values, the USCA formula ca be foud from the USPV by simply compoudig the preset value from the USPV equatio to the ed of the time horizo. Algebraically, multiply both sides of eq. [4] by (1+r) to get: 12
( 1+ r) 1 [7] V0 ( 1+ r) = P1 ( 1 r), r( 1+ r) + ad recogizig that V 0 (1+r) = V, ad the rearragig the expressio results i the formula for the USCA: ( 1 + r) 1 [8] V = P1. r The sikig fud deposit formula (SFD) ca easily be foud from the above formula [8] by rearragig for the size of periodic paymet, P 1 that results i a future value of V after periodic paymets at a kow iterest rate. Doig so results i the followig: [9] P = V 1 r ( 1 + r) 1 Impact of Compoudig Frequecy Iterest rates are typically stated i aual form. However, may times the paymet frequecy or compoudig iterval differs from the aual rate. For example, a savigs accout may have a associated aual iterest rate of 8% but have iterest earigs that are credited to the accout o a quarterly basis. The result is that part of the iterest is available earlier ad itself ears iterest over the remaider of the ivestmet period. Wheever the frequecy of compoudig ad the iterest rate time iterval differ, adjustmets must be made to the formulas above to accout for the more or less frequet compoudig. Fortuately, the adjustmet is very simple ivolvig oly the iterest rate per period, or r; the legth of time, or ; ad the frequecy of compoudig per period, or m. I each case, the iterest rate is divided by m ad the umber of time periods is multiplied by m. Whe m = 1, it ca be omitted from the equatios above. Whe it differs from 1, it has the effect of covertig the iterest rate per year (or other iterval) to a iterest rate per compoudig period ad the effect of covertig the 13
umber of years (or other iterval) to the umber of total paymet periods. Some examples will help illustrate. Suppose you deposited $100 ito a accout payig 8% iterest per year ad left it udisturbed for 10 years. Usig the stadard SPCA formula, the termial value could be foud as V = V 0 (1+r) or i this case 100(1.08) 10 = $215.89. If istead, the iterest eared were credited quarterly, the m=4 ad the resultig formula is: V = V 0 (1+r/m) *m or i this case 100(1.02) 40 = $220.80. The icreased value with more frequet compoudig simply represets the additioal iterest eared through time o the iterest paymets that were credited to the accout earlier. If the iterest were credited mothly i this case, the fial value would have bee $221.96. Uder daily compoudig, the resultig calculatio is $100(1+.08/365) 365*10 = $222.53. Naturally, more frequet compoudig results i higher values i future value calculatios. Ad, it is atural to ask what the limitig value that frequecy of compoudig implies about the future value. Ad so, with o further plays o words, the atural logarithm is itroduced with its implicatios to the time value of moey. Note that V = V 0 (1+r/m) *m ca be rewritte as: [10] V = V 0 1 + 1 m r m *( r* ) r ad recallig that h lim 1+ 1 = h h e, where e is the base of the atural logarithm eq.[10] ca be evaluated at the limit of the frequecy of compoudig as m approaches ifiity or: m r* r r* lim V 1 + V e [11] m m 0 1 = 0, r 14
because the term i the curly braces has a limitig value of e, or the base of the atural logarithm. Hece, uder cotiuous compoudig, the $100 deposit at 8% aual iterest for 10 years would be worth $100e.08*10 = $222.55. This formula ca be easily rearraged for the preset value that a future kow value represets. I this case, the fial expressio i eq. [11] ca be rewritte as: [12] V 0 = V e -r*. Summary of Materials i Part I The above materials are meat to help the user to (i) uderstad reasos to use time value of moey approaches i problems ivolvig cash flows through time, (ii) uderstad the categories of problems that require applicatio of time value of moey cocepts, (iii) be able to classify problems ito the appropriate category for solutio, (iv) uderstad the backgroud ad derivatio of the formulas used to solve TVM problems. To provide a simple ad compact referece, the major formulas are repeated below i summary form with descriptios idetifyig their applicatio purposes. Each of the formulas described o the followig page is icluded i the accompayig spreadsheet with utilities to make their use simple ad direct, thus avoidig may of the calculatio errors that ca occur whe workig through the formulas by had with a calculator. 15
Summary of TVM Formulas 1. (SPCA) Sigle paymet compouded future amout. (Ukow value is future amout, kow values are: iterest rate per period, iitial pricipal ad the umber of periods) V = V 0 (1+r) Ad uder cotiuous compoudig, V = V 0 e rt 2. (USCA) Uiform series compoud amout. (Ukow value is future amout, kow values are: iterest rate per period, periodic paymets, ad the umber of periods). V ( 1+ r) 1 = P1 r 3. (SFD) Sikig fud deposit. (Ukow value is size of periodic paymet, kow values are: iterest rate per period, future value, ad the umber of periods). P 1 = V r ( 1 + r) 1 4. (SPPV) Sigle paymet preset value. (Ukow value is preset value, kow values are: iterest rate per period, future value, ad the umber of periods). V 0 = V (1+r) - Ad uder cotiuous discoutig V 0 = V e -rt 5. (USPV) Uiform series preset value. (Ukow value is preset value, kow values are: iterest rate per period, periodic paymets, ad the umber of periods). V ( 1 + r) 1 = P r( 1 + r) 0 1 6. (CR) Capital recovery or the loa paymet problem. (Ukow value is the size of paymets, kow values are: iterest rate per period, iitial pricipal value, ad the umber of periods). P V r r ( 1 + ) 1 = 0 ( 1 + r) 1 7. Fudametal capitalizatio formula: (Ukow value is preset value, kow values are: iterest rate per period, periodic perpetual paymets). V o = P 1 /r 16