Lecture III: Finish Discounted Value Formulation


 Bryce Bruce Andrews
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1 Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal rae of reurn: The soluion of he inernal rae of reurn does no analyically exis. Mehods for finding he inernal rae of reurn are, hus, numerical search echniques or ieraive processes. a. Simple approach: umeric gradien (i.) Compue he PV of an invesmen a wo poins, or for wo ineres raes. {The soluion echnique works bes if one poin has a posiive PV and he oher a negaive PV}. (ii.) Compue he line beween he wo poins and find where he line equals zero. Assume i >i and PV >PV. PV PV i3 = i i i i or i i i PV PV 3 = i i PV i 3 i i i (iii.) Decide wheher he PV is close enough o zerosop if yes, go back o sep (ii) if no. {If he PV a i 3 is posiivereplace i, if i is negaive, replace i }.
2 AEB 645 Lecure III (iv.) Problems: The procedure does no have a good convergence {i may ake a long ime}. I may be difficul o find a good saring placeyou need wo ineres raes ha bound he zero (one posiive PV and anoher wih a negaive PV). (v.) Advanage: I is robusi will always work. b. ewon s Mehod PV( ik ) ik+ = ik PV i i i= ik (i.) For a reference o ewon s mehod see Buringon, R. S. Handbook of Mahemaical Tables and Formulas Fifh Ediion (ew York: McGrawHill Book Company, 973): 89. See also my lecure noes on ewon s Mehod from AEB hp:// /chuck/aeb6533.mahprogramming/l ecure9.pdf. PV i PV i = = 0 = ( + i) ( i) i = + = = CF CF CF ( + i) (ii.) Choose an iniial i. (iii.) Apply he formula o compue a new i. (iv.) Check o see if he PV is close o zero. If no reierae, if yes sop. (v.) This echnique is similar o he firs, excep an analyical slope is subsiued for he numerical slope.
3 AEB 645 Lecure III PV i i i 3 i (vi.) Commens: This mehod is relaively quick and is robus if he ne cash flows are smooh. Unforunaely, i is possible o sick he algorihm and i may break down if PV has muliple opimum. B. Why is i wrong o IRR.. Remember ha he IRR is simply he i ha reduces he PV o zero. I wan o show ha: a. Under cerain condiions IRR and PV yield he same resuls, i.e. in he case of convenional invesmens. b. I can show ha under cerain condiions IRR yields an inferior invesmen decision ha PV correcly idenifies. c. Thus, PV dominaes IRR as an invesmen crieria.. Assume ha Big Green manufacuring comes ou wih a lease sysem ha charges lease paymens for 3 years and hen requires he invesor o purchase he machine. This machine generaes revenue wih cash flows. The oal cash flows for he invesmen are presened in able. Table. Cash Flows for Big Green Machinery Invesmen Year Cash Flow ,000 3,50 45,000 The IRR for his invesmen is.704 and he PV a a discoun rae of 5% is ex, I wan o generae he unrecovered balance for 3
4 AEB 645 Lecure III he invesmen in each year. The unrecovered balance is defined like a savings accoun. Deparing from he invesmen example in able, consider an invesmen wih a normal cash flow paern as depiced in able. Table. Balance for Invesmen wih ormal Cash Flow Paerm Ineres Year Cash Flow on Balance Balance As presened in able, he year 0 cash flow for his invesmen is 5,000 (corresponding wih an iniial invesmen). A he end of year 0 (beginning of year ) his negaive cash flow would have accrued an ineres chare of 750. Adding he posiive cash flow in period (5,000) o he ineres charge and he unrecovered balance of 5,000 yields an unrecovered balance a he beginning of period of 0,750. Applying his echnique o cash flows in able, bu using wo ineres raes (5% as in PV and.704 as in he IRR). Table 3. Balance Using Two Ineres Raes. Year Cash Flow Balance.704 Balance , ,447,64,000 7,566,745 3,50 5,000 4,95 45, ,805 The higher he ending balloon paymen, he higher he IRR and he lower he PV also he more undesirable he invesmen. 3. There exis cerain legiimae invesmen quesions for which he IRR does no exis. a. For he IRR o exis boh posiive and negaive cash flows mus occur. However, several imporan and relevan invesmens may exis ha have all negaive or all posiive cash flows. b. Examples: (i.) All negaive cash flows: Suppose a farmer is evaluaing he purchase of wo racors. Also assume ha he is no currenly consrained in racor ime so ha he addiional revenue of eiher racor is zero. The farmer wans a echnique ha chooses he racor wih he leas cos hrough ime. 4
5 AEB 645 Lecure III (ii.) Table 4. Comparison of Tracor Coss Year Tracor I Tracor II 05,0000, , ,50 , ,500 ,000 PV a 5% 8,9504,800 o negaive cash flows and he exremely profiable invesmen. Table 5. o egaive Cash Flows and an Exremely Profiable Invesmen Year o egaive Cash Flows Exremely Profiable IRR PV ,69.75 C. Conclusions. IRR is defined as he discoun rae ha reduces he PV of an invesmen o zero.. Problems: a. I does no handle mixed invesmen flows appropriaely. b. I may no exi or may have several IRRs. 5
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