C h a p t e r TIME VALUE O MONEY 6. TIME VALUE O MONEY The idividual s preferece for possessio of give amout of cash ow, rather tha the same amout at some future time, is called Time preferece for moey. Three reasos may be advaced to accout for the idividual s time preferece for moey. They are :. Ucertaity: As a idividual is ot certai about future cash receipts, he prefers receivig cash ow.. Preferece for preset cosumptio : Most people have subjective preferece for preset cosumptio over future cosumptio of goods ad service either because of urgecy of their preset wats of because of the risk of ot beig i the positio to ejoy future cosumptio that may be caused by illess or death or because or iflatio. A moey is a meas by which idividual acquire most goods ad service they may prefer to have moey ow. 3. Ivestmet opportuities : Most idividual prefer preset cash to future cash because of the available ivestmet opportuities to which they ca put preset cash to ear additioal cash. or Example, a idividual who is offered Rs. 00 ow or Rs. 00 oe year from ow, would prefer Rs. 00 ow. If he put the sum of Rs 00 o the bak @ 5 % iterest rate, his total cash oe year from ow will be Rs. 05. Thus if he wishes to icrease to cash resources, the opportuity to ear iterest would lead him to prefer Rs. 00 ow, ot Rs. 00 after oe year. I case of busiess firm as well as idividuals, the justificatio for time preferece for moey lies simply i the availability of ivestmet of opportuities. The time preferece of moey is geerally expressed as iterest rate + risk premium if ay exist. The time preferece of moey ca be broadly classified as follows: () Compoud value or future value Compoud value of lump sum. Compoud value of a auity () Preset value Preset value of lump sum Preset value of a auity (3) Value of a auity due Compoud value of a auity due Preset value of a auity due 6.. COMPOUND VALUE (UTURE VALUE) O MONEY The iterest (i) that is paid o pricipal (p) as well as o ay iterest eared but ot with draw durig earlier period is called compoud iterest. The process of fidig the future value of a paymet or services of paymet (or receipt) whe applyig the cocept of compoud iterest is called compoudig. The compoudig may be categorized as compoud value of lump sum ad compoud value of auity.
Time Value of Moey Compoud value of lump sum Let us assumes Rs. 00 is placed at savig accout at 5% iterest rate. The followig grow o pricipal is obtaied. Amout deposited Rs. 00 @ 5% per aum At the ed of first year: 00 + 00 x 0.05 05 At the ed of secod year: 05 + 05 x 0.05 0.5 At the ed of third year: 0.5 + 0.5 x 0.05 5.76 Let i represeted the iterest rate ad P is the priciple, the we get: P + Pi P ( + i) Where, compoud value at ed of first year Similarly, + x i ( + i) P ( + i) ( + i) P ( + i) Similarly, 3 + x i ( + i) 3 P ( + i) ( + i) P ( + i) 3 Similarly, P ( + i) ------------ () The term ( + i) is compoud value factor (CV) of a lump sum of Re, ad it always has value greater tha for positive iterest rate i.e. i idicatig that CV icreases as i ad icreases. Solved problem id the future value (lump sum) at the ed of seve year where Rs. 50 is deposited @ 7% iterest. Here, P 50 7 I 0.07 7 P ( + i) 50 ( + 0.07) 7 Rs. 007 (b) Compoud value of auity A auity is the fixed paymet (or receipts) each year for specific umber of years. Example, if you ret a flat ad promise to make a series of paymet over a agreed period, you have created a auity. The equal istallmet loas from the house fiacig compaies or employees are commo example of auities. Let us assume, Re is deposited i a savig accout at the ed of each year for four years @ 6% iterest. i.e. Re deposited at ed of first year Re deposited at ed of secod year Re deposited at ed of third year Re deposited at ed of fourth year This implies that Re deposited at the ed of first year will grow for three years. Similarly Re deposited at the ed of secod year will grow for two years. Similarly Re deposited at the ed of third year will grow for oe year. Ad Re deposited at the ed of fourth year will ot yield ay iterest. Now, Compoud value of Re deposited i fourth year is oly Re, But deposited i first year is ( + 0.06).06, deposited i secod year is ( + 0.06).,deposited i third year is ( + 0.06) 3.9.Now, the aggregate compoud value of Re deposited at the ed of every year for four years would be :(a + b + c + d) ( +.06 +. +.9).375 This ca be show that A ( + i) 3 + A ( + i) + A ( + i) + A Where is the future value at ed of fourth year. 95
A Text Book of Operatioal Research ad ood Plat Maagemet A auity i iterest rate A [( + i) 3 + ( + i) + ( + i) + ] We ca exteds above equatio for period ad rewrite the equatio as (+ i ) - A. () i Where A costat period flow of cash ad term withi bracket are compoud value factor for a auity (CVA) Solved problem Determied compoud value if Rs. 00 are deposited at the ed of each year for four years @ 0% iterest. Here, (+ 0.) - 00 00 x.6 6. 0. 6.. PRESENT VALUE O MONEY It has bee so far, how compoudig techiques ca be used for the adjustig the time value of moey. It icreases ivestor s aalytical power to compare cash flows that are separated by more tha oe period, give his iterest rate period. The compoudig techiques similarly could be used for the preset value of future cash iflow or cash outflow. The preset value of a future cash iflow or outflow is the amout of curret cash that is equivalet to the future cash iflow or outflow. The process of determiig the preset value of future paymet (receipts) or series of future paymets (receipts) is called discoutig. The compoud iterest rate used for discoutig cash flows is called the discoutig rate. (a) Preset value of a lump sum I dealig with compoud value of lump sum, we kow that there is o differece betwee Re ow ad Re ( + i) (where i is iterest rate) oe year from ow, or Re ( + i) after two years, or Re ( + i) after years. This was give by P ( + i). However, we ca determied the preset value of lump sum P by arragig the equatio as P.. (3) Where ad i are provided. Let us cosider a example, with 0% iterest rate Re at ed of first year will be P ( + i) ( + 0.). However, for obtaiig Re at ed of first year, the amout that is eeded is give by: P x. or P 0.909 i.e. if 0.909 is deposited it will become Re at the ed of first year or ext year. This implies that. preset value of Re (if iterest rate is 0%) to be received after oe year is equivalet to Re 0.909. Stated differetly Re 0.909 deposited ow will grow to Re after oe year. The preset value of Re iflow at the ed of two years ca also be worked out easily. A amout P deposited ow would grow to P ( + i) after two years. There for if expected to be received after two years is Re, the followig amout should be deposited ow. P ( + i) or P.0 0.86 i.e. 0.86 is the preset value of Re at 0% iterest rate to be received after two years. The ivestors will be idifferet betwee Re after two years ad Re 0.86 ow. At 0% iterest rate Re 0.86 ow will grow to Re after two years. Similarly the preset value of Re to be received after three years will be 3 P ( + i) 3 P 3 / ( + i) 3 /(+0.) 3 0.75 Thus, the formula 96
Time Value of Moey P.. () ca be used for determiig the preset value of for year with iterest i. The term is the preset value factor (PV) ad it is always less tha for positive i idicatig that future amout has a smaller preset value. (b) Preset value of auity The preset value of auity is the reserve of future value of auity. To fid the preset value of auity, we will have to fid out the preset value of the amout every year ad will have to aggregate all the preset values to get the total preset value of the auity. or example, a ivestor who has determied his iterest rate 0% may have opportuity to receive a auity of Re for four years. The preset value of Re received after oe year is give by P Similarly, after two years P After three years P 3 3 After four years P ( + i) (.) 3 (.) (.) 0.75 0.683 0.909. 0.86() Thus, the total preset value of auity of Re is P 0.909 + 0.86 + 0.75 + 0.683 3.69 Thus, the preset value of auity of Re for four year at 0% iterest rate is Rs. 3.69. This ca be determied by formula - P A (5) i Solved problem What is the preset value of 50000 to be received after 5 years if the iterest rate is 9%? Here, 50000 i 0.09 5 By usig formula P 50000 (+ 0.09) 5 Rs. 376 Solved problem 5 Determie the preset value of auity, if the perso receives auity of Rs.5000 for years, cosiderig the rate of iterest as 0%. Here, I 0. A 50000 97
A Text Book of Operatioal Research ad ood Plat Maagemet - - Now, P A ( + 0.) i 50000 5893.7 0. 6..3 VALUE O AN ANNUITY DUE The cocept of compoud value ad preset value of auity discussed earlier are based o the assumptio that the series of paymet are made at the ed of the year. I practice paymet could be made at the begiig of the year. Example, whe you buy a refrigerator o istallmets sales, the dealer requires you to make the first paymet immediately i.e., o begiig of the first period ad subsequet istallmet o begiig of each period. A series of fixed paymet startig at the begiig of each year for a specified umber of years is called a auity due. a b Compoud value of a auity due Recall the compoud value of a auity, whe Re deposits are made at the ed of year, the compoud value at ed of four year is.375. However, the Re deposited i the begiig of each year of through the year four will ear iterest respectively for years, 3 years, years ad year. Thus, the compoud value of this series is A ( + i) + A ( + i) 3 + A ( + i) + A ( + i) (.06) + (.06) 3 + (.06) + (.06).637 This ca be give by formula - A ( + i) (6) i Preset value of a auity due We have calculated that preset value of auity Re for four years whe paymet are made at the ed of year to be 3.69 (0% iterest rate). However, if the preset value of this auity if each paymet is made at the begiig of the year is give by : or first year, preset value (at the begiig of the year) is or secod year, preset value (at the begiig of the year) is /. 0.909 Similarly, usig P 0.86 ad (.) 3, the preset value for third year (begiig), fourth year are (.) 0.75, Thus, the preset value of a auity due is equal to + 0.99 + 0.86 0.75-3.87 the formula is give by P A ( + i) i. (7) 6..3 SUMMARY O ORMULAS Compoud value (a) Compoud value of lump sum, P ( + i) (b) Compoud value of auity, ( + i) - A i Preset value (a) Preset value of lump sum, P 98
Time Value of Moey (b) Preset value of auity, P - ( + i) A i Value of auity due (a) Compoud (future) value of auity due, - A ( + i) i - (b) Preset value of auity due, P A ( + i) i Note these values ca be obtaied from table too. This table is give o ay fiacial maagemet book. The abbreviatio used are : P Preset value A Auity uture value i iterest rate, discoutig rate, opportuity cost, required rate of retur, miimum rate of retur, desired rate of retur. Solved Problem 6 Calculate the preset value of Rs. 600 assume 5% time preferece rate. (a) received oe year from ow. (b) received at the ed of five years. (c) received at the ed of fiftee years. Solutio. (a) received oe year form ow i.e. preset value of 600 at the ed of oe year P 600 57. ( + i) ( + 0.05) (b) preset value of 600 received at the ed of five years. P 600 70. ( + 0.05) 5 (c) received at ed of fiftee years i.e. preset value of 600 at ed of fiftee years. P 600 ( + 0.05) 5 88.6 Solved Problem 7 Determie the preset value of Rs. 700 each paid at the ed of each of ext six years. Assume 8% rate of iterest Solutio. - - (+ 0.08) 6 Here, preset value of auity is give by P A i 700 336.0 0.08 Solved Problem 8 Assume 0% discout rate. Compute the preset value of Rs.00, Rs.900, Rs.500 ad Rs.700 received at the ed of oe through four year. Solutio. The data are give as ueve cash flow. The preset value for ueve cash flow are computed as 00 900 500 700 P + + + ( + 0.) ( + 0.) ( + 0.) 3 ( + 0.) 99
A Text Book of Operatioal Research ad ood Plat Maagemet 000 + 73.80 + 6.97 + 78.0 338.88 Solved Problem 9 Below is give the iformatio related to the cash flows of three project X, Y, Z. the iitial cost of each project is Rs.55000 ad the firm s cost of capital is 0%. Calculate the total preset value of each of three projects. Project X: Cash flows Rs.0000 per year ad life 5 years Project Y: Year 3 5 Cash flow 30000 0000 50000 60000 0000 Project Z: Cash flow Rs.5000 per year from to year ad 30000 at the ed of 5 th year Solutio: (a) Project X cash flow 0000 per year for 5 years. Here, 0000 cash flow is cotiued for 5 years, so the preset value of auity is determied for 5 years. - - P (+i) (+ 0.) 5 A i 0000 563.7 0. (b) Project Y Year Cash flow Preset value 30000 30000 / ( + 0.) 77.77 0000 0000 / ( + 0.) 33057.85 3 50000 50000 / ( + 0.) 3 37565.7 60000 60000 / ( + 0.) 0980.80 5 0000 0000 / ( + 0.) 5 836.85 Total 6373.96 Here, preset value of lump sum, P (c) Project Z Here, cash flow of 5000 is cotiued for year thus, preset value of auity is determied for years ad preset value of lump sum of 5 th year is determied as follows: Year Cash flow Preset value - 5000 - ( + 0.) 63.95 5000 0. 5 30000 30000 867.63969 ( + 0.) 5 Total 67.57 6. CAPITAL BUDGETING AND NATURE O INVESTMENT DECISION The ivestmet decisio of the firm is geerally called the capital expediture decisio or capital budgetig. A capital budgetig decisio may be defied as the firm s decisio to ivest its curret fuds most efficietly i the log-term assets i aticipatio of a expected flow of beefits over a series of years. The log-term assets are those that affect the firm s operatio beyod oe-year period. The firm s ivestmet decisios would geerally iclude expasio, acquisitio, moderizatio ad replacemet of log-term assets. Similarly, sale of divisio or busiess (divestmet) is also aalyzed as ivestmet decisio. It is importat to ote that ivestmet i logterm assets ivariably requires fuds to be tied up i the curret assets such as ivetories ad receivables. The followig are the features of ivestmet decisios: (a) (b) The exchage of curret fuds for future beefits. The fuds are ivested i log-term assets. 00
Time Value of Moey (c) The future beefits will occur to the firm over a series of years. The expediture ad beefits of ivestmets should be measured i cash. I the ivestmet aalysis, it is cash flow that is importat, ot the accoutig profit. It may also be poited that the ivestmet decisio affects the firm s value. The firm value will icrease if the ivestmets are profitable ad add to the shareholder s wealth. Thus, ivestmet should be evaluated o the basis of criterio that is compatible with the objective of the shareholder s wealth maximizatio. A ivestmet will add to the shareholder s wealth if it gives retur greater tha the curret iterest rate or opportuity rate. The eed of the soud capital expediture policy arises due to the followig:. The amout ivolved i capital expediture is usually comparatively heavy tha the reveue expediture. So, it should be see that amout ivested i capital expediture gives the best possible results.. Ivestmet i capital expediture is for a loger period, so risk i such type of expediture is heavy. Capital ivestmet affects future profit for a umber of years. So, the future profitability of the cocer is liked with capital ivestmet decisios. 3. Heavy losses are likely to occur due to obsolescece of fixed assets o accout of rapid techological chages. This fact should be take ito cosideratio.. The progressive maagemet should always be o the look out for ways of improvig the profitability of the compay by wise ivestmet of available capital fuds. 6.. TYPES O INVESTMENT DECISIONS There are may ways to classify ivestmets. Oe classificatio is as follows: a. Expasio of existig busiess b. Replacemet ad moderizatio Aother useful way to classified ivestmet is as follows: a. Mutually exclusive ivestmets b. Idepedet ivestmet c. Cotiget ivestmet Mutually exclusive ivestmets: It serves the same purpose ad competes with each other. If oe ivestmet is uder take, others will have to be excluded e.g. a compay may acquire either automatic plat or semi automatic plat. He caot ivest i two plats for the same productio. Idepedet ivestmets: They serve differet purpose ad do ot compete with each other. E.g. A compay may ivest o ew mixig machie ad packig machie. These two are idepedet decisio. Cotiget ivestmets: These are depedet projects, the choice of oe ecessitates that oe or more ivestmet should also be uder take e.g. A compay decides to ivest i sugar plat i remote, it has to ivest i electricity, road etc. 6.. STEPS IN CAPITAL BUDGETING The capital budgetig geerally cosists of the followig steps. (i) Project geeratio The geeratio of project meas the search for ivestmet alterative. As we kow capital budgetig is the ivestmet i the fixed log-term assets, which geerate the retur or yield the retur for the period of time. The project might be either ivestmet i ew assets or replacemet of existig assets or moderizatio of the existig facilities etc. These ivestmets might be mutually exclusive oe or idepedet oe or cotiget ivestmets. 0
A Text Book of Operatioal Research ad ood Plat Maagemet (ii) Project evaluatio It is the secod ad most crucial step i the project evaluatio. Amog lot of alteratives or for sigle opportuity, which project is to be take or whether the project is profitable, oe or ot the aswer is obtaied by evaluatig the project uder various criterio. The overall ivestmet is evaluated i terms of cash flow, which is also called as the icremetal-after tax-cash flows. The cash flow approach to measure beefits is superior to the accoutig system of profit basis because it: (a) Avoids ambiguities of the accoutig profit cocept. (b) Measure the total beefits (c) Take cogizace of the time value of moey. The accoutig profit is useful for performace measure, but cash flow is useful as decisio criterio. The measure differece betwee the cash flow ad the accoutig profit approaches relate to the treatmet of depreciatio, which is igored as a item of cost i computatio uder the cash flow approach. The methods of evaluatio we deal o the successive headigs. (iii) Project selectio Oce the cash flow of the alteratives are aalyzed or evaluated usig differece evaluatio criterio, which icludes moder approach ad traditioal approach. Those project which has the highest preset value or higher cash flow or higher iteral rate of retur are selectig. To summarize, the best project is selected. The selected project might be mutually exclusive, idepedet or cotiget ivestmet. (iv) Project implemetatio: The selected projects are the implemeted. The implemetatio should be carried out i a very systematic maer. A capital budget, which iclude the source time ad amout of capital should be prepared ad ivestmet should be doe accordig to the plaed budget. (v) Project moitorig ad cotrol: The completed ad remaiig work of the project should be moitored a evaluated with pla i frequet iterval so that ay deviatio could be corrected o time. This esures that the project will be completed o time with the budgeted ivestmet figure. 6..3 PROJECT EVALUATION OR INVESTMENT EVALUATION CRITERION Three steps are ivolved i the evaluatio of the ivestmet. a. Estimatio of cash flows. b. Estimatio of required rate of retur c. Applicatio of the decisio rule for makig a choice. The first two steps are out of our scope. However, we have discussed it i precedig topics. Our discussio will be cofied to the merit ad demerit of various ivestmet criterios. The ivestmet decisio rule may be referred to as capital budgetig techiques or ivestmets criterio. A soud appraisal techique should be used to measure the ecoomic worth of a ivestmet project. The essetial property of a soud techique is that it should maximize the shareholder s wealth. The followig character should be possessed by the soud ivestmet evaluatio criterio. a. It should be a measure of the project s profitability by cosiderig all cash flows. b. It should provide a meas of distiguishig betwee acceptable ad uacceptable projects. c. It should provide a rakig of the projects i order of their ecoomic desirability. d. It should recogize the fact that bigger cash flows are preferable to smaller oes ad early cash flows are preferable to later oes. e. It should help to choose amog mutually exclusive projects that project which maximizes the shareholder s wealth. f. It should be a criterio, which is applicable to ay coceivable ivestmet project idepedet of others. A umber of evaluatio techiques or capital budgetig techiques are used i practice. They may be grouped i the followig two categories. Discouted cash flow (DC) criteria (moder method) (a) Net preset value (NPV) (b) Iteral rate retur (IRR) 0
Time Value of Moey (c) Profitability idex No-discouted cash flow criteria (Traditioal method) (a) Pay back period (PB) (b) Accoutig rate of retur (ARR) We will show i the successive topics that the NPV criterio is the most valid techique of evaluatig ivestmet project. It is geerally cosistet with the objective of maximizig shareholder s wealth. Payback period The payback period (PB) is the most popular ad widely recogized traditioal methods of evaluatig ivestmet proposals. It is defied as the umber of years required to recover the origial cash flow (cash outlay) ivested i a project. If the project geerates costat aual cash iflows, the payback period is give by Net cash outlay (ivestmet) Iitial ivestmet PBP (8) Net cash iflow (aual) Aual cash iflow Solved example 0 ) Compute PBP from the iformatio furished below. Iitial ivestmet Rs. 50,000 Life of the project 5 years Cash flow after tax Rs. 3,000 per year Solutio: PBP Iitial ivestmet Cash flow after Tax 50000 3.86 years 3000 However, if the cash iflow is uequal, the pay back period ca be foud out by addig up the cash iflows util the total is equal to the iitial cash outlays. Solved example Compute PBP from the iformatio furished below. Iitial ivestmet Rs. 80,000 Life of project 5 years Cash iflow Solutio : Year 3 5 CAT Rs.,000,000 30,000 5,000 0,000 Here, we fid that the cash iflow is ueve so that formula () could ot be used. Calculatio of PBP of the project Year CAT Cum. CAT Remarks 000 000 000 5000 3 30000 75000 Miimum period Recoverable balace is (80000 75000) 5000 5000 00000 5 0000 0000 03
A Text Book of Operatioal Research ad ood Plat Maagemet Amout to be recovered Now PBP Miimum period + CAT of maximum year Acceptace rule: The payback period should be less tha or equal to stadard PBP. or mutually exclusive project, those project is selected which has less PBP. I the above projects, project B is selected. Merits of PBP It is simple to uderstad ad easy to calculate. The cost of aalysis of PBP is very less due to simplicity. It gives isight ito liquidity of the project. Demerits of PBP It fails to take accout of cash iflows eared after the payback period. It is ot appropriate method of measurig the profitability of a ivestmet project as it does t cosider the etire cash flows yielded by the project. It fails to cosider the patter of cash iflows i.e. magitude ad timig of the cash iflows. It does ot cosider the time value of moey, thus it does ot maximize the market value of the firm s share. Share value does ot deped o the payback period of the ivestmet projects. Accoutig rate of retur The accoutig rate of retur (ARR) is determied by dividig the average after tax profit (average et icome after tax) by average ivestmet. The after tax profit is obtaied from the fiacial statemet, so this seeks the help of the fiacial accoutig statemet. ARR Average et icome after tax 00. (9) Averageivestmet Here, Average et icome after tax Total et icome Life of the project Average ivestmet Cost of ivestmet for costructio depreciatio Begiig book value of ivestmet + edig book value of ivestmet Book value value after deductig depreciatio. Solved example A Compay is cosiderig two proposals-m & N, each costig Rs.3, 00,000.00 to the compay. The expected life of both the proposals is years. The compay has to pay 0% tax o the earig. Depreciatio is charged o straight-lie basis. The before tax cash flow of the two proposals are estimated to be the follows. Year Proposal M Proposal N 5000 50000 5000 35000 3 5000 0000 5000 5000 Solutio: (a) Aual depreciatio Cost of proposal Expected life or M depreciatio Rs.300000 / Rs.75000 / year or N depreciatio Rs.300000 / Rs.75000 / year (b) Calculatio of average earig after tax Project M Year 3 CBT (Cash flow before tax) 5000 5000 5000 5000 Less depreciatio 75000 75000 75000 75000 0
Time Value of Moey Earig before tax (EBT) 50000 50000 50000 50000 Less tax 0% 0000 0000 0000 0000 Earig after tax 30000 30000 30000 30000 Average et icome after tax 30000 + 30000 + 30000 + 30000 30000 Project N Year 3 CBT 50000 35000 000 5000 Less depreciatio 75000 75000 75000 75000 EBT 75000 60000 5000 50000 Less tax 30000 000 8000 0000 EAT 5000 36000 7000 30000 5000 + 36000 + 7000 + 30000 Average et icome after tax 3500 30000 or project M, ARR 00 0% 50000 3500 or project N, ARR 00 3% 50000 Acceptace rule: The project N is selected as it has higher ARR. As accept or reject criterio, this method will accept all those projects whose ARR is higher tha the miimum rate established by maagemet ad reject those projects which have ARR less tha miimum rate. This method would rak a project as umber oe of it has highest ARR ad lowest rak would be assiged to the project with lower ARR. Merits of ARR method: Simple to uderstad ad use. It ca be easily calculated from accoutig data. It icorporates the etire stream of icome i calculatig the project s profitability. Demerits of ARR method: It uses the accoutig profits ot cash iflows i appraisig the projects. The averagig of icome igores the time value of moey. Net preset value (NPV) NPV is oe of the moder methods of evaluatig the ivestmet proposals. It is oe of the discouted cash flow (DC) techiques explicitly recogizig the time value of moey. It correctly postulates that cash flows arisig at differet time periods differ i value ad are comparable oly whe their equivalets preset values are foud out. NPV methods are as follows: a. Appropriate rate of iterest should be selected to discout the forecasted cash flows. This appropriate rate of iterest is the firm s opportuity cost of capital, which is equal to the miimum rate of retur expected by ivestors to be eared by the firm o ivestmet of equivalet risk. b. Secod the preset value of the ivestmet cash flow should be computed usig opportuity cost of capital of the discoutig rate. c. The et preset value should be foud out by subtractig the iitial cash out lay from the preset value of cash iflows. The firm should accept the project if NPV is positive. Net preset value Total preset value Net cash outlay, where, Or NPV TPV NCO.. (0) Solved Example 3 Below is give the iformatio related to the cash flows of there projects x, y, z. The iitial cost of each project is Rs. 55000 ad the firm cost of capital is 0%. Calculate the preset value of each three project ad decide the best project accordig to NPV. 05
A Text Book of Operatioal Research ad ood Plat Maagemet Solutio: Project Year 3 5 X Cash low 0000 0000 0000 0000 0000 Y Cash low 30000 0000 50000.60000 0000 Z Cash low 5000 5000 5000 5000.30000 Step I : calculatio of total preset value. Project X : Here eve cash flow of 0000 per year for years so preset value of auity @ 0% cash flow determied as. + ( i) P A ( + 0.) 5 0000 i 563.07 0. Project Y : The cash flow is differet i all 5 years. So the preset value of each year should be determied by usig formula of preset value of lump sum i.e. P ( + i) Year Cash P Preset value flow ( + i ) 3 5 30000 0000 50000 60000 0000 30000/(.) 0000/(.) 50000/(.) 3 60000/(.) 0000/(.) 5 77.7 33057.85 37565.7 0980.8 836.8 Total preset value 6373.96 Project Z : Here, eve cash flow of 5000 for years ad 30000 for 5 th year has occurred so for cumulative - year preset value of auity is determie ad too 5 th year preset value of lump sum is determied. Year Cash flow Preset value formula Preset value - 5000 5 30000 63.9 5000 (.) 0. 30000 867.6 (.) 5 Total preset value 67.5 Step II: determiatio of NPV Project TPV - NCO NPV Remarks X 563. -55000-3368.6 Reject Y 6373.9-55000 873.9 Accept ad select Z 67.5-55000 67.3 Accept Here, project X is rejected due to egative NPV. However amog project Y & Z, project Y is selected due to higher NPV tha the later oe. Acceptace Rule of NPV: The project is accepted if NPV positive ad rejected if NPV is egative. The market value of the firm s share would rise if project with positive ad higher NPV are selected. This will be so, because the +ve NPV will be displayed by those projects which would geerate cash iflows at a rate higher tha the miimum required 06
Time Value of Moey by the ivestors. The NPV method ca be used to select betwee mutually exclusive projects, the oe with the higher NPV should be selected. Merit of NPV method a. It recogizes the time value of moey, a rupee received today is worth tha a rupee received tomorrow. b. It uses all cash flows occurrig over the etire life of projects i calculatig its worth c. The discoutig process facilitates measurig cash flows ad discoutig rate are kow Demerit of NPV method a. It is quite difficult to obtai the estimate of cash flows due to ucertaity. b. It is quite difficult to measure the discout rate. Note: NPV is regarded as the most beeficial evaluatio criterio. If o demarcatio is doe about the criterio, NPV should be used if possible. Profitability Idex Yet aother time adjusted method of evaluatig the ivestmet proposal is the beefit- cost (B/C) ratio or profitability idex (PI).PI is the ratio of the preset value of cash iflows at the required rate of retur to the iitial cash outflows of the ivestmet which is called the et cash outlay or NCO TPV PI.. () NCO Takig the solved example 3, PI is calculated as Project TPV NCO PI Remarks X 563. 55 563./550000.98 Reject Y 6373.9 5500 6373.9/55000.06 Accept ad Select Z 67.5 55000 67.5/55000.0 Accept Acceptace rule The accept or reject rule, usig PI is to accept project if its pl is greater tha oe,sice the project will have +ve et preset value. The higher PI is selected. Evaluatio The merit & demerit of PI are same as NPV. Iteral Rate of Retur Iteral rate of retur (IRR) is aother discouted cash flow techique which takes accout of the magitude ad timig of cash flows. Iteral ratio of retur (IRR) ca be defied as that rate which equates the preset value of cash iflow with the preset value of cash outflow of a ivestmet. It is the rate at which the NPV of the ivestmet is zero. It is called iteral rate because it depeds solely o the outlay ad proceed associated i the ivestmet ad ot a ay rate determied outside the ivestmet. The followig formula will be used to determie IRR. TPV LR - NCO IRR LR + (HR LR) () TPV LR -TPV HR Where LR lower discoutig rate HR higher discoutig rate TPV LR total preset value at lower rate TPV HR total preset value at higher rate NCO et cash outlay. 07
A Text Book of Operatioal Research ad ood Plat Maagemet Acceptace rule The accept or reject rule usig IRR method is to accept the project if its IRR is higher tha the opportuity cost of capital. The cost of capital may also be expressed as required rate of retur or cut off rate or hurdle rate. They shall be rejected if its iteral rate of retur is lower tha the opportuity cost of capital. Merit Like NPV method the IRR method recogize the time value of moey ad cosider all cash flows occurrig over the etire life of projects to calculate its rate of retur. It geerally gives the same acceptace rule as NPV method. IRR ca be determied by fidig the discout rate where NPV is zero. or this, trial ad error approach could be utilized. If the cash outlay (preset value of cash outlay) is higher tha the preset value of cash iflow i.e. NPV is egative, lower rate should be tried. If cash outlay is less tha the PV of cash iflow NPV is positive, the higher rate should be tried. This will be further illustrated from the solved example. Solved example A project costs Rs. 0000 ad is expected to geerate cash iflows of Rs. 0000, 35000,35000, 30000, ad 0000 respectively eve its 5 year life. Calculate the iteral ratio rate of retur. The preset value at % discout rate ad 0%discoutrate for 5 years is give below Preset value factor (PV) Discout 3 5 rate % 0.893 0.797 0.7 0.636 0.567 6% 0.86 0.73 0.6 0.55 0.76 Solutio : Year Cash Iflow (Rs) PV (%) PV@% PV@6% PV@6% 0000 0.893 3570 0.86 380 35000 0.797 7895 0.73 7005 3 30000 0.7 360 0.6 930 35000 0.636 60 0.55 930 5 0000 0.567 680 0.76 9050 Total preset value 995 8075 Less cost of project 0000 0000 Net preset value + 9,95-95 Now we have Positive NPV of Rs 995 at % (lower rate) ad egative of Rs 95 at Rs. 6% (higher rate) ad the TPV beig Rs 995 at % ad Rs 8075 at 6%., IRR is thus give by : IRR LR + TPV LR - NCO 995-0000 (HR LR) % + (6% % ) 5.35% TPV -TPV 995-8075 LR HR Solved example 5 Project cost 6000 ad is expected to geerate cash iflow of 8000, 7000 ad 6000 at the ed of each year for ext 3 years. The preset value factor for the discout rate 5%,6%.0% for 3 year is give below: Preset value factor Discout st yr d yr 3 rd yr rate 6% 0.86 0.703 0.6 0% 0.833 0.69 0.57 5% 0.870 0.756 0.658 Solutio: Calculatio of NPV at differet rates. 08
Time Value of Moey Year Cash flows PV@6% PV PV@0% PV PV@5% PV 8000 0.86 6896 0.833 666 0.870 6960 7000 0.73 50 0.69 858 0.756 59 3 6000 0.6 386 0.579 37 0.658 398 Total Preset Value 593 996 600 Less Cost of Project 6000 0000 6000 NPV - 57-00 + 000 Hece, we come to kow that IRR lies betwee 5 to 6% or ca be i 5 to 0%.Calculatig IRR by formula : TPV LR - NCO IRR LR + (HR LR) TPV LR -TPV HR 600-6000 or 5 % to 6 %, we have IRR 5 % + (6% 5% ) 5.77% 600-593 600-6000 or 5 % to 0 %, we have, IRR 5 % + (0% 5% ) 5.99% 600-996 rom the above we came to kow that the lower the differece betwee higher rate ad lower rate the more accurate result will be obtaied. Solved example 6 rom the followig statemet calculate NPV ad IRR. Year Cash low D (0%) D (0%) - 700000.798 3.70 5 600000 0.76 0.6 6 930000 0.0 0.56 Solutio Calculatio of NPV Year Cash low D (6%) PV D (0%) PV - 700000.798 958600 3.70 9000 5 600000 0.76 85600 0.6 37600 6 930000 0.0 38300 0.56 550 TPV 65500 360 NCO 3000000 3000000 NPV - 37500 60 Now, TPV LR - NCO IRR LR + (HR LR) TPV LR -TPV HR or 6 % to 0 %, we have 360-3000000 0 % + (6% 0% ).% 360-65500 09