THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS


 Maurice McKinney
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1 VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely possible o ge he firm ino "rouble" hrough poor financing decisions or improper managemen of working capial, he value of he firm is principally deermined by he prospecs for is invesmens. Invesmens by he firm ake wo forms: (i) inernallygeneraed projecs which, if underaken, creae new asses; and (ii) he acquisiion of exernal alreadyexising asses from oher firms by eiher direc purchase of he asses or he acquisiion of he whole firm by merger, consolidaion, or akeover. Mergers and acquisiions are imporan opics for financial managemen and will be discussed in Secion XV. However, wih he excepion of a few specialized firms, he primary funcion of he business firm is o find and underake profiable new projecs, and i is his form of invesmen which is he opic of his secion. The capial budgeing problem is how o selec hose physical invesmens or projecs so as o maximize he value of he firm. Much of he formal apparaus has already been developed in Secions II, VI, and o some exen in Secion V. However, o pu hese ools in a more specific framework, we examine he various radiional capial budgeing mehods used o evaluae projecs. Before proceeding, we begin wih some definiions: A projec is defined by he series of ne cash flows i generaes a he end of each period, {X(1),X(2),...,X(N)}. These flows {X()} can be eiher posiive or negaive. If X() is posiive, hen he projec provides a ne flow of cash ino he firm a he end of period, and if X() is negaive, hen i causes a new flow of cash ou of he firm. Since mos projecs require an iniial ouflow, i is a common convenion o denoe his flow by " I0" where I0 is he (posiive) ouflow or iniial invesmen in he projec. For symmery, we will also denoe I0 by "X(0)", he ne cash flow a he end of he "zeroh" period (or he beginning of he firs period). X() = [Revenues Coss Depreciaion] (1 ax rae) + Depreciaion Invesmen (in he projec) 134
2 = Aferax Operaing Profis ne new invesmen (in he projec) Rober C. Meron Le k denoe he cos of capial o he firm (measured in percen per period) where he cos of capial is he (exernal) rae of reurn required by invesors for providing funds o he firm and i reflecs all he marke opporuniies available o invesors. In a world of cerainy (which is he formal seing for his secion), he cos of capial is simply he marke rae of ineres, r. However, we follow radiion of using "k" raher han "r" o include he possibiliy in a quasiuncerainy sense (made rigorous in Secion XIV) ha differen risk projecs will have differen required reurns (and in paricular, required raes differen from he riskless ineres rae). Following he pracice of Secion II, o simplify he analysis, i is assumed ha he explici opporuniy cos o invesors for invesing in he firm, k, is consan over ime. If k were changing over ime, hen in an analogous fashion o R() in Secions II and V, we could define K() by "[1 + k] " appears. [1+ K() ] [1+k()], and use "[1 + K()] " everywhere in he formulas when j=1 Independen Projecs are projec such ha he firm can decide o do boh or eiher one or neiher. (Noe: his definiion has no implicaions of saisical independence among projecs.) Muually Exclusive Projecs are projecs such ha he firm can only do one or he oher, bu no boh. Tradiional Mehods of Projec Selecion I. PayBack Mehod 135
3 Finance Theory If I0 is he iniial invesmen, hen he payback period is ha value of T such ha. I 0 = T X I.e., i is he minimum lengh of ime unil he ne cash flows sum o he value of he iniial =1 invesmen. The payback mehod says rank all (independen) projecs from he shores o he longes and hen ake (inves in) all projecs wih a payback period less han or equal o some given ime, T *. When choosing among muually exclusive projecs, selec he one wih he smaller payback period. II. Presen Value Mehod (Review Secion II) The (ne) presen value of a projec is N N X() X() PV =  I + =. 0 =1 (1+k ) =0 (1+k ) As described in Secion II, he presen value rule says rank all (independen) projecs from he highes o he lowes, and hen ake all invesmens wih posiive (or as a maer of indifference, zero) presen value. When choosing among muually exclusive projecs, selec he one wih he larges presen value. Noe: If he cos of capial were changing over ime, hen he presen value of he projec will be. N X() PV =  I 0 + and he mehod is sill applicable [1+ K() ] =1 III. Inernal Rae of Reurn Mehod (Review Secion V on YieldoMauriy) The inernal rae of reurn for a projec, i, is ha discoun rae such ha he presen value of he projec (compued a ha rae) is zero. I.e., i is he soluion o 136
4 Rober C. Meron N X() 0 =  I 0 +. =1 [1+ i ] I is called an inernal rae because, unlike k (he cos of capial), which is an (exernal) marke (opporuniy cos) rae, i depends only on he naure of he imeflow paerns of he projec and is compleely unrelaed o any marke rae. The inernal rae of reurn rule says rank all (independen) projecs from he highes o he lowes, and hen ake all invesmens whose inernal rae of reurn is greaer han some specified rae i * (usually aken o be he cos of capial, i.e., i * = k). When choosing among muuallyexclusive projecs, selec he one wih he larges inernal rae of reurn. IV. Profiabiliy Index Mehod Profiabil iy Index T X() / I =1 (1+ k ) = PI = 0 Mehod: Rank all (independen) projecs form he highes o he lowes and ake all invesmens wih profiabiliy index greaer han one. When choosing among muually exclusive projecs, selec he one wih he larges profiabiliy index. Evaluaion of hese Mehods: Problems wih Payback 1. Neglecs he ime value of money (no discouning) 2. Neglecs all flows beyond he payback period (implici "infinie" discouning) Therefore, misses fuure negaive or posiive flows. A relaed mehod someimes used is he "Modified" Payback Mehod. The modified payback period is defined as he minimum T such ha I 0 = T =1 X() (1+k ) 137
5 Finance Theory Presen Value In perfec capial markes and cerainy, he value of he firm is equal o he presen value of all is fuure flows discouned a he (markedeermined) cos of capial. Hence, he presen value rule maximizes he value of he firm. I is someimes called a conservaive rule because he firm always has available invesmens which will earn k: namely, i can buy is own sock. Thus, he firm should never ake negaive PV projecs. In uncerainy, he rule can be modified according o he "riskadjused" mehod o be discussed laer: Namely, PV =  I 0 + N =1 α X () (1+r ) where α = a cerainy equivalen and X () is he expeced cash flow. In general, presen value is he mos appropriae of hese four radiional echniques. Inernal Rae of Reurn While he presen value mehod assumes ha he flows can be reinvesed a he cos of capial (which is always possible), he inernal rae of reurn assumes ha he flows can be reinvesed a he inernal rae of reurn i. Technical Problems ha Can Arise wih Inernal Rae of Reurn 1. There may be eiher more han one value of i or no value of i which makes he presen value of he projec zero. 2. If he cos of capial is varying over ime, he "cuoff" rule of aking only projecs wih i = k is no welldefined. Example: Presen Value vs. Inernal Rae of Reurn Assume k =
6 Rober C. Meron Presen Value Projec A (Muually Exclusive of B) Year End Ne Cash Flow Discoun 1 Presen Value (1+k) 01,000, ,000, ,150, ,998, ,307, ,999, ,157, ,001,192 Presen Value of A = Sum of PV = 0 Projec B (Muually Exclusive of A) Year End Ne Cash Flow Discoun 1 (1+k) Presen Value 01,000, ,000, ,210, ,055, ,433, ,114, ,224, ,057,605 Presen Value of B = Sum of PV = 4,548 So, by he Presen Value Mehod, A is preferred o B. Example (coninued) Inernal Rae of Reurn Projec A: Le x = 1 + i 139
7 Finance Theory 3,150,000 3,307,500 1,157,630 PV =  1,000, = x x x or find he roos of he cubic equaion: x x x = 0 3 roos: ( x1,x2, x3 ) are x1= 1+i1= 1.05 x2= 1+i2= 1.05 so i A=.05 x3= 1+i3= 1.05 Projec B: Le x = 1 + i 3,210,000 3,433,800 1,224,080 PV =  1,000, = x x x or find he roos of he cubic equaion: 3 2 x x x = 0 ( x1,x2, x3 ) are x1= 1+i1= roos: x2 = 1+i2= 1.07 x3= 1+i3= 1.10 B B B So here are hree inernal raes of reurn i1 =.04; i2 =.07;i3 =
8 Rober C. Meron Example (coninued) "Swiching Poins" a k =.05 ake A over B a k =.08 ake B over A 141
9 Finance Theory As noed in Secion VII, his is no a paradox because differen ineres raes imply differen "worlds" wih differen alernaives. Thus, which echnology o use (e.g., wood bridge versus a seel bridge) rarely can be answered wih knowledge of he echnology only. The swiching problem (or mulipleroos problem) occurs when here is more han one posiive roo which makes he presen value equal o zero. One can use he following rule o check o see wheher n n1 n2 more han one such roo can occur: if x + a1 x + a2 x an = 0, hen (Descare's rule of signs) he number of posiive roos eiher is equal o he number of variaions of signs of he ai's or is less han his number of variaions by an even ineger. In he example, boh projecs had hree sign changes, and hence, eiher hree or one posiive roos. I should be noed ha from he ables in his example, boh he payback and he modified payback mehods would have picked Projec B over Projec A. More on Presen Value versus Inernal Rae of Reurn If X(0) = I0 < 0 and all X() 0, for = 1,2,... for all he projecs being considered and if he projecs are independen, hen he Presen Value Rule and Inernal Rae of Reurn Rule will lead o he same answer wih respec o which projecs will be aken. To see his, noe ha a plo of presen value versus cos of capial will look like: 142
10 Rober C. Meron Hence, if i > k, hen he presen value will be posiive. However, even in he case of a single posiive roo, he rankings of projecs by he wo mehods can be differen. Hence, danger lurks for evaluaing muually exclusive projecs using i or in using i in he case of capial raioning as he following example illusraes. Example: Suppose ha you have $1000 and you can purchase eiher Projec A or Projec B. Given ha he only invesmen alernaive available in fuure years for any money received will be o suff i in a maress or bury i in a coffee can (i.e., k = 0), which should you ake? Projec A: Pay $1000 oday (i.e., I0 = 1000) and you receive no paymens unil he end of fifeen years when you will receive $4,177 (i.e., x(1) = x(2) = x(3) =... = x(14) = 0 and x(15) = 4177). Projec B: Pay $1000 oday (i.e., I0 = 1000) and you receive $214 a he end of each year for fifeen years (i.e., x(1) = x(2) = x(3) =... = x(14) = x(15) = 214). We know by Descare's rule of signs ha boh bonds have only one posiive roo. Hence, he inernal rae of reurn for boh is unique. Using he presen value ables and he formula for an annuiy, he inernal rae of reurn on A is ia =.10 and on B is ib =.20. Clearly, on a IRR basis, B is preferred o A. Wha abou presen value? A k = 0, PV PV B A 4177 = (1+0 ) = = 3, = (15x214) (1+0 ) = 2210 Clearly, by he Presen Value Rule, A is preferred o B. Which is "more" correc? Fis, noe ha since all inerim paymens canno be invesed o earn a posiive reurn, i is easy o compue how much money we will have a he end of fifeen years from each projec: for Projec A, we have $4177 and for Projec B, we will have only $3210. Since hey boh cos he same, which do 15 and 143
11 Finance Theory you prefer? Furher, since we know he final amouns, we can compue an acual average compound reurn per year for boh. I.e., rue reurn per year from A is 10%, and (1+ R A ) = has he soluion RA =.10. So, he (1+ RB ) = has he soluion RB = 8.2%. So, he 1000 rue reurn per year from B is 8.2% NOT 20%. Hence, Presen Value is a beer ranker. Noe: he inernal reurn, ib, is a number and need no bear a close relaionship o he acual reurns earned. E.g., 20% versus 8.2%. In bond evaluaion, yieldomauriy is jus an inernal rae of reurn calculaion, and herefore, as noed in Secion V, he same warnings apply o comparing alernaive bond invesmens by yieldomauriy even when he bonds have he same mauriy dae. Imperfecions and Capial Budgeing If he firm is a "perfec compeior" for capial (i.e., he firm's cos of capial is unaffeced by he scale of is invesmens) and capial markes are "reasonably" perfec, hen he correc capial budgeing decision rule is presen value. However, in he face of cerain imperfecions, his decision rule may require modificaion. Capial Raioning: an examinaion of all he decision rules given shows ha each assumes ha here is no budge consrain for profiable invesmens. I.e., each period, he firm looks over all available projec proposals and selecs all projecs wih posiive presen value. This done, hen a budge is esablished o deermine how much capial is needed (and from which sources i will be raised) o carry ou he program. If he esimaes of he cos of capial and he cash flows are accurae, hen here should be lile problem in raising he necessary (addiional) funds in he capial marke. Furher, his procedure is opimal relaive o he (efficiency) crierion of maximizing marke value. Noe: he procedure o be described is conrary o he one an individual consumer would follow in allocaing his income (and wealh) over various consumpion goods a differen poins in ime. 144
12 Rober C. Meron However, in cerain siuaions, here may be a (predeermined) absolue limi o he amoun ha can be invesed by he firm in any one period. This siuaion is called capial raioning. I may occur for he firm in counries where here are no (or poorlyorganized) capial markes; or for divisions of firms where (incorrecly deermined) decenralizaion rules dicae a fixed budge for each division prior o he examinaion of he projecs available; i is no an infrequen case in he public secor where resources are a imes allocaed (prior o specific knowledge of projecs) on he basis of "las year's" allocaion (of I0)". Under capial raioning, i is someimes suggesed ha he Profiabiliy Index (or "Benefi/Cos" raio) is a beer rule han presen value. While i is rue ha he profiabiliy index gives he mos Presen value per dollar of iniial invesmen which is highly suggesive of wha one should do in a consrained siuaion, i does no reflec fuure budgeary consrains. Thus, a plan may saisfy he curren budge consrain, bu violae all fuure consrains. The bes echnique in his siuaion is o maximize presen value subjec o he budge consrain in each year using mahemaical programming echniques. While such a procedure is no opimal relaive o (unconsrained) maximizing of marke value, i does produce a feasible program. Moreover, he "shadow prices" or dual variables will give an explici esimae of he marginal coss of he raioning. These values can ofen be used o argue for he eliminaion of he consrains, paricularly if he coss are high. Always, ask yourself: why he consrain? How much is i cosing? Is i raional? Rising Cos of Capial. I is ypically assumed ha he cos of capial is a consan funcion of he amoun of invesmen, in each period. However, if k depends on he scale, hen programming echniques mus be employed. 145
13 Finance Theory Applicaion of Presen Value: The Replacemen Problem The produc decisions are already made and he decision is o choose beween wo alernaive machines o produce he produc. Technical change is negleced and he opimal horizon for he produc run is given. I. Replacemen ime for each machine is known. Same produc for T years: which machine? Life of machine A is T1 years. Life of machine B is T2 years. 146
14 Rober C. Meron Machine A coss IA and has operaing coss per year (2T 1  T) years (o go), of S. C,C,..., C 1 2 T1 A A A and has salvage value wih 1 2 T 2 Machine B coss IB and has operaing coss per year C B,C B,...,C B. Assume replace each machine wih he same machine. The presen value of he coss of machine A over is life is P A = I A + T1 =1 A C (1+ k ) For machine B PB = I B + T 2 =1 B C (1+ k ) If we choose machine A, hen i mus be replaced a ime T1. A ha ime, he presen value of coss will be I T  T1 C S A A TT =1 (1+k ) (1+k ) + , 1 because i is no used for is full life If we choose machine B, hen i mus be replaced a ime T2. A ha ime, he presen value of coss will be PB again because i is used for is full life. To decide which machine o use, we compare he presen values of coss for he enire produc life, which are, oday, and P P I T T1 ' 1 CA S A = A + T A + T T ( 1+ k) = 1 ( 1+ k) ( 1+ k)
15 Finance Theory ' 1 1 PB = PB + P T B = PB 1 +, 2 T 2 ( 1+ k) ( 1+ k) and choose he smaller one beween P and P. ' A ' B A common siuaion is when he lengh of he produc run is anicipaed o run indefiniely ino he fuure (formally, T = ). Hence, if a machine has life of lengh n, we anicipae replacing he machine every n years and making an "infinie" number of replacemens. If P is he presen value of coss of he machine over one cycle, i.e., P= I n C +, hen, as above, he presen value of coss over he produc life will be =1 (1+k ) ' P P P P P = P K ( 1+ k) ( 1+ k) ( 1+ k) ( 1+ k) n 2n 3n 4n ' = P n 1 n 2 n 3 n 4 ( 1+ k) ( 1+ k) ( 1+ k) ( 1+ k) K or ' j P P X X = where j= 0 1 ( 1+ k ) n N 1 N j X 1 1 From (II.14), we have ha X = = as N for X < 1. So subsiuing for X X X, we have j=0 n ' 1 ( 1 = k ) P = P = P n 1 1 ( 1+ k ) 1 n ( 1 k + ) ' So o deermine which machine o choose, ake he one wih he smaller P. 148
16 Rober C. Meron An alernaive represenaion (due o Lewellen) is o conver he presen value calculaions ino a consan annual cos (flow) comparison. This approach would be useful for comparison beween he choice beween buying and mainaining he machine or rening (or leasing) he machine wih a service conrac from anoher firm, i.e., wha is he (maximum) consan annual paymen ha you would be willing o pay a he end of each year for rening he machine and having i serviced? Clearly, his represens an annuiy paymen problem. The presen value of he annuiy is P; he (maximum) rae o be paid is k; find he annual paymens implied: From Secion II, we have ha he formula for a Nyear annuiy is where y = he annual paymen. 1 y(1  (1+ k ) AN = k Corresponding o he perpeuiy (N = ), we have y A = k N ) so he annual flow (cos) will be y = k kp = P 1 1 (1+k ) n Again, we can selec beween wo machines by choosing he one wih he smaller y, and if a leasing conrac is available for less han y, hen ake i. II. Opimal Replacemen Time In he previous analysis, i was assumed ha he machines' replacemen imes were known and ha hey corresponded o heir physical lenghs of life. Rarely is his he case. Normally, he decision o replace a machine is an economic one. As before, assume ha we will always be replacing old machines wih new machines and ha replacemen goes on indefiniely. 149
17 Finance Theory Le I = iniial invesmen and C1,C2,C3,...,CT be he annual operaing coss up o he end of he physical life of he machine, T; le S1,S2,S3,...,ST1,ST (= 0) be he salvage value of he machine a he end of each year. From he assumpions of he problem, an opimal replacemen ime soluion will be he same for all ime, i.e., i will never be opimal o replace every wo years for a while and hen swich o replacing every hree years, ec. Le τ = lengh of ime beween replacemens (0 < τ T). To "conver" he problem o he ype of he previous secion, consider ha each of T differen replacemen sraegies defines a "differen" machine (which i does in he economic sense). Thus, le P(τ) = presen value of coss for one cycle for he machine replaced every τ years. Then τ ( ) C Sτ P τ = I + . τ (1 + k) (1+ k) = 1 Le P(τ) be he presen value of coss of he machine (replaced every τ years) over he produc life. Then 1 1 P ( τ) =  P ( ). 1 ( 1+k) τ τ The opimal replacemen ime, τ *, will he be τ such ha * P( τ ) P( τ ) for all possible τ = 1,2,3,...,T We can also wrie he condiions in erms of equalized annual coss by: y = kp ( τ ) and hen τ * τ is such ha y * τ y τ for all possibleτ. Noe: τ * will depend on k, he srucure of he operaing coss, and he salvage values, and hence will be differen for differen cos of capial, ec. 150
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