Capital budgeting techniques

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Capital budgeting techniques"

Transcription

1 Capial budgeing echniques A reading prepared by Pamela Peerson Drake O U T L I N E 1. Inroducion 2. Evaluaion echniques 3. Comparing echniques 4. Capial budgeing in pracice 5. Summary 1. Inroducion The value of a firm oday is he presen value of all is fuure cash flows. These fuure cash flows come from asses are already in place and from fuure invesmen opporuniies. These fuure cash flows are discouned a a rae ha represens invesors' assessmens of he uncerainy ha hey will flow in he amouns and when expeced: CF Value of he firm = =1 (1+r) where CF is he cash flow in period and r is he required rae of reurn. The objecive of he financial manager is o maximize he value of he firm. In a corporaion, he shareholders are he residual owners of he firm, so decisions ha maximize he value of he firm also maximize shareholders' wealh. The financial manager makes decisions regarding long-lived asses; his process is referred o as capial budgeing. The capial budgeing decisions for a projec requires analysis of: is fuure cash flows, he degree of uncerainy associaed wih hese fuure cash flows, and he value of hese fuure cash flows considering heir uncerainy. We looked a how o esimae cash flows in a previous reading where we were concerned wih a projec's incremenal cash flows, comprising changes in operaing cash flows (change in revenues, expenses, and axes), and changes in invesmen cash flows (he firm's incremenal cash flows from he acquisiion and disposiion of he projec's asses). And we know he concep behind uncerainy: he more uncerain a fuure cash flow, he less i is worh oday. The degree of uncerainy, or risk, is refleced in a projec's cos of capial. The cos of capial is wha he firm mus pay for he funds o finance is invesmen. The cos of capial may be an explici cos (for example, he ineres paid on deb) or an implici cos (for example, he expeced price appreciaion of is shares of common sock). Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 1

2 In his reading, we focus on evaluaing he fuure cash flows. Given esimaes of incremenal cash flows for a projec and given a cos of capial ha reflecs he projec's risk, we look a alernaive echniques ha are used o selec projecs. For now all we need o undersand abou a projec's risk is ha we can incorporae risk in eiher of wo ways: (1) we can discoun fuure cash flows using a higher discoun rae, he greaer he cash flow's risk, or (2) we can require a higher annual reurn on a projec, he greaer he risk of is cash flows. 2. Evaluaion echniques Look a he incremenal cash flows for Projec X and Projec Y shown in Exhibi 1. Can you ell by looking a he cash flows for Invesmen A wheher or no i enhances wealh? Or, can you ell by jus looking a Invesmens A and B which one is beer? Perhaps wih some projecs you may hink you can pick ou which one is beer simply by gu feeling or eyeballing he cash flows. Bu why do i ha way when here are precise mehods o evaluae invesmens by heir cash flows? We mus firs deermine he cash flows from each invesmen and hen assess he uncerainy of all he cash flows in order o evaluae invesmen projecs and selec he invesmens ha maximize wealh. We look a six echniques ha are commonly used by firms o evaluaing invesmens in long-erm asses: 1. Payback period, 2. Discouned payback period, 3. Ne presen value, 4. Profiabiliy index, 5. Inernal rae of reurn, and 6. Modified inernal rae of reurn. Exhibi 1: Esimaed cash flows for Invesmens X and Y End of period cash flows Year Projec X Projec Y $1,000,000 -$1,000, $0 $325, $200,000 $325, $300,000 $325, $90 0,000 $325,000 We are ineresed in how well each echnique discriminaes among he differen projecs, seering us oward he projecs ha maximize owners' wealh. An evaluaion echnique should: Consider all he fuure incremenal cash flows from he projec; Consider he ime value of money; Consider he uncerainy associaed wih fuure cash flows, and Have an objecive crierion by which o selec a projec. Projecs seleced using a echnique ha saisfies all four crieria will, under mos general condiions, maximize owners' wealh. In addiion o judging wheher each echnique saisfies hese crieria, we will also look a which ones can be used in special siuaions, such as when a dollar limi is placed on he capial budge. A. Payback period The payback period for a projec is he ime from he iniial cash ouflow o inves in i unil he ime when is cash inflows add up o he iniial cash ouflow. In oher words, how long i akes o ge your Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 2

3 money back. The payback period is also referred o as he payoff period or he capial recovery period. If you inves $10,000 oday and are promised $5,000 one year from oday and $5,000 wo years from oday, he payback period is wo years -- i akes wo years o ge your $10,000 invesmen back. Suppose you are considering Invesmens X and Y, each requiring an invesmen of $1,000,000 oday (we're considering oday o be he las day of he year 2006) and promising cash flows a he end of each of he following years hrough How long does i ake o ge your $1,000,000 invesmen back? The payback period for Projec X is four years: Year Projec X Accumulaed cash flows $1,000, $0 -$1,000, , , , , , ,000 By he end of 2009, he full $1,000,000 is no paid back, bu by 2010 he accumulaed cash flow his (and exceeds) $1,000,000. Therefore, he payback period for Projec X is four years. The payback period for Projec Y is four years. I is no unil he end of 2010 ha he $1,000,000 original invesmen (and more) is paid back. We have assumed ha he cash flows are received a he end of he year. So we always arrive a a payback period in erms of a whole number of years. If we assume ha he cash flows are received, say, uniformly, such as monhly or weekly, hroughou he year, we arrive a a payback period in erms of years and fracions of years. 1 For example, assuming we receive cash flows uniformly hroughou he year, he payback period for Projec X is 3 years and 6.6 monhs (assuming $75,000 cash flow per monh). Our assumpion of end-of-period cash flows may be unrealisic, bu i is convenien o use his assumpion o demonsrae how o use he various evaluaion echniques. We will coninue o use his end-of-period assumpion hroughou he coverage of capial budgeing echniques. Is Projec X or Y more aracive? A shorer payback period is beer han a longer payback period. Ye here is no clear-cu rule for how shor is beer. If we assume ha all cash flows occur a he end of he year, Projec X provides he same payback as Projec Y. Therefore, we do no know in his paricular case wheher quicker is beer. In addiion o having no well-defined decision crieria, payback period analysis favors invesmens wih "fron-loaded" cash flows: an invesmen looks beer in erms of he payback period he sooner is cash flows are received no maer wha is laer cash flows look like. Payback period analysis is a ype of "break-even" measure. I ends o provide a measure of he economic life of he invesmen in erms of is payback period. The more likely he life exceeds he payback period, he more aracive he invesmen. The economic life beyond he payback period is referred o as he pospayback duraion. If pos-payback duraion is zero, he invesmen is worhless, no maer how shor he payback. This is because he sum of he fuure cash flows is no greaer han he iniial invesmen oulay. And since hese fuure cash flows are really worh less oday han in he fuure, a zero pos-payback duraion means ha he presen value of he fuure cash flows is less han he projec's iniial invesmen. 1 Bu hen we would have a challenge applying he mehods ha apply he ime value of money, so for simpliciy sake we assume end-of-period cash flows in illusraing he capial budgeing echniques. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 3

4 The payback mehod should only be used as a coarse iniial screen of invesmen projecs. Bu i can be a useful indicaor of some hings. Because a dollar of cash flow in he early years is worh more han a dollar of cash flow in laer years, he payback period mehod provides a simple, ye crude measure of he liquidiy of he invesmen. The payback period also offers some indicaion on he risk of he invesmen. In indusries where equipmen becomes obsolee rapidly or where here are very compeiive condiions, invesmens wih earlier payback are more valuable. Tha's because cash flows farher ino he fuure are more uncerain and herefore have lower presen value. In he personal compuer indusry, for example, he fierce compeiion and rapidly changing echnology requires invesmen in projecs ha have a payback of less han one year since here is no expecaion of projec benefis beyond one year. Because he payback mehod doesn' ell us he paricular payback period ha maximizes wealh, we canno use i as he primary screening device for invesmen in long-lived asses. B. Discouned payback period The discouned payback period is he ime needed o pay back he original invesmen in erms of discouned fuure cash flows. Each cash flow is discouned back o he beginning of he invesmen a a rae ha reflecs boh he ime value of money and he uncerainy of he fuure cash flows. This rae is he cos of capial -- he reurn required by he suppliers of capial (crediors and owners) o compensae hem for ime value of money and he risk associaed wih he invesmen. The more uncerain he fuure cash flows, he greaer he cos of capial. The cos of capial, he required rae of reurn, and he discoun rae We discoun an uncerain fuure cash flow o he presen a some rae ha reflecs he degree of uncerainy associaed wih his fuure cash flow. The more uncerain, he less he cash flow is worh oday -- his means ha a higher discoun rae is used o ranslae i ino a value oday. This discoun rae is a rae ha reflecs he opporuniy cos of funds. In he case of a corporaion, we consider he opporuniy cos of funds for he suppliers of capial (he crediors and owners). We refer o his opporuniy cos as he cos of capial. The cos of capial comprises he required rae of reurn (RRR) (ha is, he reurn suppliers of capial demand on heir invesmen) and he cos of raising new capial if he firm canno generae he needed capial inernally (ha is, from reaining earnings). The cos of capial and he required rae of reurn are he same concep, bu from differen perspecive. Therefore, we will use he erms inerchangeably in our sudy of capial budgeing. Calculaing he discouned payback period Reurning o Projecs X and Y, suppose ha each has a cos of capial of 10 percen. The firs sep in deermining he discouned payback period is o discoun each year's cash flow o he beginning of he invesmen (he end of he year 2006) a he cos of capial: Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 4

5 Projec X Projec Y Accumulaed discouned Accumulaed discouned Year Cash flows cash flows Cash flows cash flows $1,000, $1,000, $1,000, $1,000, $0.00 -$1,000, $295, $704, $165, $834, $268, $435, $225, $609, $244, $191, $614, $5, $221, $30, How long does i ake for each invesmen's discouned cash flows o pay back is $1,000,000 invesmen? The discouned payback period for boh X and Y is four years. Discouned payback decision rule I appears ha he shorer he payback period, he beer, wheher using discouned or nondiscouned cash flows. Bu how shor is beer? We don' know. All we know is ha an invesmen "breaks-even" in erms of discouned cash flows a he discouned payback period -- he poin in ime when he accumulaed discouned cash flows equal he amoun of he invesmen. Using he lengh of he payback as a basis for selecing invesmens, Projecs X and Y canno be disinguished. Bu we've ignored some valuable cash flows for boh invesmens, hose beyond wha is necessary for recovering he iniial cash ouflow. C. Ne presen value If offered an invesmen ha coss $5,000 oday and promises o pay you $7,000 wo years from oday and if your opporuniy cos for projecs of similar risk is 10 percen, would you make his invesmen? To deermine wheher or no his is a good invesmen you need o compare your $5,000 invesmen wih he $7,000 cash flow you expec in wo years. Because you deermine ha a discoun rae of 10 percen reflecs he degree of uncerainy associaed wih he $7,000 expeced in wo years, oday i is worh: Presen value of $7,000 o be received in 2 years = $7,000 ( ) 2 = $5, By invesing $5,000, oday you are geing in reurn, a promise of a cash flow in he fuure ha is worh $5, oday. You increase your wealh by $ when you make his invesmen. Anoher way of saing his is ha he presen value of he $7,000 cash inflow is $5,785.12, which is more han he $5,000, oday's cash ouflow o make he invesmen. When we subrac oday's cash ouflow o make an invesmen from he presen value of he cash inflow from he invesmen, he difference is he increase or decrease in our wealh referred o as he ne presen value. The ne presen value (NPV) is he presen value of all expeced cash flows. Ne presen value = Presen value of all expeced cash flows. The word "ne" in his erm indicaes ha all cash flows -- boh posiive and negaive -- are considered. Ofen he changes in operaing cash flows are inflows and he invesmen cash flows are ouflows. Therefore we end o refer o he ne presen value as he difference beween he presen value of he cash inflows and he presen value of he cash ouflows. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 5

6 We can represen he ne presen value using summaion noaion, where indicaes any paricular period, CF represens he cash flow a he end of period, i represens he cos of capial, and N he number of periods comprising he economic life of he invesmen: N CF NPV = presen value presen value = of cash inflows of cash ouflows = 1(1 + r) Cash inflows are posiive values of CF and cash ouflows are negaive values of CF. For any given period, we collec all he cash flows (posiive and negaive) and ne hem ogeher. To make hings a bi easier o rack, le s jus refer o cash flows as inflows or ouflows, and no specifically idenify hem as operaing or invesmen cash flows. Take anoher look a Projecs X. Using a 10 percen cos of capial, he presen values of inflows are: Projec X Discouned cash Year Cash flow flow $1,000,000 -$1,000, $0 $ , , , , , , L NPV = +$5, This NPV ell us ha if we inves in X, we expec o increase he value of he firm by $5, Calculaed in a similar manner, he ne presen value of Projec Y is $30, We can use a financial calculaor o solve for he NPV as well, inpuing he cash flows in order, making sure ha he $0 cash flow for year 2007 is included in he lis of cash flows. TI-83/84 {0,200000,300000,900000} STO lisname NPV(10, ,lisname) HP10B /- CF j 0 CF j CF j CF j CF j 10 i/yr NPV We can also use Microsof Excel o solve for he ne presen value. The Excel spreadshee enries for he daa would be: A B 1 Year Projec X $1,000, $ $200, $300, $900,000 and he ne presen value requires he use of he NPV uncion: =NPV(.1,B3:B6)+B2 Ne Presen Value Decision Rule A posiive ne presen value means ha he invesmen increases he value of he firm -- he reurn is more ha sufficien o compensae for he required reurn of he invesmen. A negaive ne presen value means ha he invesmen decreases he value of he firm -- he reurn is less han he cos of capial. A zero ne presen value means ha he reurn jus equals he reurn required by owners o Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 6

7 compensae hem for he degree of uncerainy of he invesmen's fuure cash flows and he ime value of money. Therefore, if... his means ha... and you... NPV > $0 NPV < $0 NPV = $0 he invesmen is expeced o increase shareholder wealh he invesmen is expeced o decrease shareholder wealh he invesmen is expeced no o change shareholder wealh should accep he projec. should rejec he projec. should be indifferen beween acceping or rejecing he projec Projec X is expeced o increase he value of he firm by $5,395.81, whereas Projec Y is expeced o increases add $30, in value. If hese are independen invesmens, boh should be aken on because boh increase he value of he firm. If X and Y are muually exclusive, such ha he only choice is eiher X or Y, hen Y is preferred since i has he greaer NPV. Projecs are said o be muually exclusive if acceping one precludes he accepance of he oher. D. Profiabiliy index The profiabiliy index uses some of he same informaion we used for he ne presen value, bu i is saed in erms of an index. Whereas he ne presen value is: The profiabiliy index, PI is: N CF NPV = presen value presen value = of cash inflows of cash ouflows = 1(1 + r) N CIF presen value of cash inflows = 1(1 + r) PI = = presen value N COF of cash ouflows = 1(1 + r) where CIF and COF are cash inflows and cash ouflows, respecively. Projec X Year Cash flow Discouned cash flow 2007 $0 $ , , , , , , N CIF = +$1,005, L = 1(1 + r) Therefore, he profiabiliy index is: $1, 005, PIX = = $1,000,000 The index value is greaer han one, which means ha he invesmen produces more in erms of benefis han coss. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 7

8 The decision rule for he profiabiliy index is herefore depends on he PI relaive o 1.0: if... his means ha... and you... PI > 1.0 PI < 1.0 PI = 1.0 he invesmen is expeced o increase shareholder wealh he invesmen is expeced o decrease shareholder wealh he invesmen is expeced no o change shareholder wealh should accep he projec. should rejec he projec. should be indifferen beween acceping or rejecing he projec There is no direc soluion for PI on your calculaor; wha you need o do is calculae he presen value of all he cash inflows and hen divide his value by he presen value of he cash ouflows. In he case of Projec X, here is only one cash ou flow and i is already in presen value erms (i.e., i occurs a he end of 2006). E. Inernal rae of reurn TI-83/84 {0,200000,300000,900000} STO lisname NPV(10,0,lisname) ENTER ENTER HP10B 0 +/- CF j 0 CF j CF j CF j CF j 10 i/yr NPV Suppose you are offered an invesmen opporuniy ha requires you o pu up $50,000 and has expeced cash inflows of $28, afer one year and $28, afer wo years. We can evaluae his opporuniy using a ime line, as shown in Exhibi 1. Exhibi 4 Time line of invesmen opporuniy $50,000 $28, $28, The reurn on his invesmen is he discoun rae ha causes he presen values of he $28, cash inflows o equal he presen value of he $50,000 cash ouflow, calculaed as: $28, $28, $50, 000 = + (1 + IRR) 1 (1 + IRR) 2 Anoher way o look a his is o consider he invesmen's cash flows discouned a he IRR of 10 percen. The NPV of his projec if he discoun rae is 10 percen (he IRR in his example), is zero: $28, $28, $50, 000 = + ( ) 1 ( ) 2 An invesmen's inernal rae of reurn (IRR) is he discoun rae ha makes he presen value of all expeced fuure cash flows equal o zero. We can represen he IRR as he rae ha solves: The IRR for X is he discoun rae ha solves: N CF $0 = = 1(1 + IRR) Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 8

9 Using a calculaor or a compuer, we ge he more precise answer of percen per year. $0 $200, 000 $300, 000 $900, 000 $1,000,000 = (1 + IRR) 1 (1 + IRR) 2 (1 + IRR) 3 (1 + IRR) 4 Looking back a he invesmen profiles of Projecs X and Y, you'll noice ha each profile crosses he horizonal axis (where NPV = $0) a he discoun rae ha corresponds o he invesmen's inernal rae of reurn. This is no coincidence: by definiion, he IRR is he discoun rae ha causes he projec's NPV o equal zero. Inernal rae of reurn decision rule TI-83/84 {0,200000,300000,900000} STO lisname IRR( ,lisname) HP10B /- CF j 0 CF j /- CF j /- CF j /- CF j IRR The inernal rae of reurn is a yield -- wha we earn, on average, per year. How do we use i o decide which invesmen, if any, o choose? Le's revisi Invesmens A and B and he IRRs we jus calculaed for each. If, for similar risk invesmens, owners earn 10 percen per year, hen boh A and B are aracive. They boh yield more han he rae owners require for he level of risk of hese wo invesmens: Invesmen IRR Cos of capial X % 10% Y % 10% The decision rule for he inernal rae of reurn is o inves in a projec if i provides a reurn greaer han he cos of capial. The cos of capial, in he conex of he IRR, is a hurdle rae -- he minimum accepable rae of reurn. For independen projecs and siuaions in which here is no capial raioning, hen if... his means ha... and you... IRR > cos of capial IRR < cos of capial IRR = cos of capial he invesmen is expeced o increase shareholder wealh he invesmen is expeced o decrease shareholder wealh he invesmen is expeced no o change shareholder wealh should accep he projec. should rejec he projec. should be indifferen beween acceping or rejecing he projec The IRR and muually exclusive projecs Wha if we were forced o choose beween projecs X and Y because hey are muually exclusive? Projec Y has a higher IRR han Projec X -- so a firs glance we migh wan o accep Projec Y. Wha abou he NPV of X and Y? Wha does he NPV ell us o do? If we use he higher IRR, i ells us o go wih Y. If we use he higher NPV if he cos of capial is 5 percen, we go wih X. Which is correc? Choosing he projec wih he higher ne presen value is consisen wih maximizing owners wealh. Why? Because if he cos of capial is 10 percen, we would calculae differen NPVs and come o a differen conclusion, as you can see from he invesmen profiles in Exhibi 3. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 9

10 When evaluaing muually exclusive projecs, he one wih he highes IRR may no be he one wih he bes NPV. The IRR may give a differen decision han NPV when evaluaing muually exclusive projecs because of he reinvesmen assumpion: NPV assumes cash flows reinvesed a he cos of capial. IRR assumes cash flows reinvesed a he inernal rae of reurn. This reinvesmen assumpion may cause differen decisions in choosing among muually exclusive projecs when: he iming of he cash flows is differen among he projecs, here are scale differences (ha is, very differen cash flow amouns), or he projecs have differen useful lives. Wih respec o he role of he iming of cash flows in choosing beween wo projecs: Projec Y's cash flows are received sooner han X's. Par of he reurn on eiher is from he reinvesmen of is cash inflows. And in he case of Y, here is more reurn from he reinvesmen of cash inflows. The quesion is "Wha do you do wih he cash inflows when you ge hem?" We generally assume ha if you receive cash inflows, you'll reinves hose cash flows in oher asses. Wih respec o he reinvesmen rae assumpion in choosing beween hese projecs: Suppose we can reasonably expec o earn only he cos of capial on our invesmens. Then for projecs wih an IRR above he cos of capial we would be oversaing he reurn on he invesmen using he IRR. Boom line: If we evaluae projecs on he basis of heir IRR, i is possible ha we may selec one ha does no maximize value. Wih respec o he NPV mehod: if he bes we can do is reinves cash flows a he cos of capial, he NPV assumes reinvesmen a he more reasonable rae (he cos of capial). If he reinvesmen rae is assumed o be he projec's cos of capial, we would evaluae projecs on he basis of he NPV and selec he one ha maximizes owners' wealh. The IRR and capial raioning Wha if here is capial raioning? Suppose Invesmens A and B are independen projecs. Projecs are independen if ha he accepance of one does no preven he accepance of he oher. And suppose he capial budge is limied o $1,000,000. We are herefore forced o choose beween A or B. If we selec he one wih he highes IRR, we choose A. Bu A is expeced o increase wealh less han B. Ranking invesmens on he basis of heir IRRs may no maximize wealh. We saw his dilemma in he previous reading peraining o projecs X and Y when we looked a heir invesmen profiles. The discoun rae a which X's NPV is $0.00 is X's IRR = percen, where X's profile crosses he horizonal axis. Likewise, he discoun rae a which Y's NPV is $0.00 is B's IRR = percen. The discoun rae a which X's and Y's profiles cross is he cross-over rae, percen. For discoun raes less han percen, X has he higher NPV. For discoun raes greaer han percen, Y has he higher NPV. If Y is chosen because i has a higher IRR and if Y's cos of capial is less han percen, we have no chosen he projec ha produces he greaes value. The source of he problem in he case of capial raioning is ha he IRR is a percenage, no a dollar amoun. Because of his, we canno deermine how o disribue he capial budge o maximize wealh because he invesmen or group of invesmens producing he highes yield does no mean hey are he ones ha produce he greaes wealh. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 10

11 Muliple inernal raes of reurn The ypical projec usually involves only one large negaive cash flow iniially, followed by a series of fuure posiive flows. Bu ha's no always he case. Suppose you are involved in a projec ha uses environmenally sensiive chemicals. I may cos you a grea deal o dispose of hem. And ha will mean a negaive cash flow a he end of he projec. Suppose we are considering a projec ha has cash flows as follows: End of period Period cash flow 0 -$ Wha is his projec's IRR? One possible soluion is IRR = percen, ye anoher possible soluion is IRR = percen. Exhibi 4: The case of muliple IRRs Ne presen value $60 $40 $20 $0 -$20 We can see his graphically in Exhibi 4, where he NPV of hese cash flows are shown for discoun raes from 0 percen o 250 percen. Remember ha -$40 -$60 -$ % 191.5% he IRR is he discoun rae ha causes he NPV o be zero. In erms of his graph, his Discoun rae means ha he IRR is he discoun rae where he NPV is $0, he poin a which he presen value changes sign -- from posiive o negaive or from negaive o posiive. In he case of his projec, he presen value changes from negaive o posiive a percen and from posiive o negaive a 250 percen. 0% 20% 40% 60% 80% 100% 120% 140% Boom line: We can use he inernal rae of reurn mehod if he sign of he cash flows change more han once during he projec s life. F. Modified inernal rae of reurn The inernal rae of reurn mehod assumes ha cash flows are reinvesed a he invesmen s inernal rae of reurn. Consider Projec X. The IRR is percen. If we ake each of he cash inflows from Projec X and reinves hem a percen, we will have $1,472, a he end of 2010: 160% 180% 200% 220% 240% Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 11

12 Number of periods earning Fuure value of cash flow a reurn reinvesed a % 3 $ , , , $1,473, The $1,473, is referred o as he projec s erminal value. 2 The erminal value is how much he company has from his invesmen if all proceeds are reinvesed a he IRR. So wha is he reurn on his projec? Using he erminal value as he fuure value and he invesmen as he presen value, FV = $1,473, PV = $1,000, N = 4 years $1, 473, i = 4 = % $1, 000, In oher words, by invesing $1,000,000 a he end of 2006 and receiving $1,473, produces an average annual reurn of percen, which is he projec s inernal rae of reurn. The modified inernal rae of reurn is he reurn on he projec assuming reinvesmen of he cash flows a a specified rae. Consider Projec X if he reinvesmen rae is 5 percen: Number of periods earning a reurn The modified inernal rae of reurn is percen: FV = $1,435,500 PV = $1,000,000 N = 4 years Fuure value of cash flow reinvesed a 5% 3 $ , , , $1,435, $1, 435,500 i = 4 = % $1, 000, For example, he 2008 cash flow of $200,000 is reinvesed a percen for wo periods (ha is, for 2009 and 2010), or $200,000 ( ) 2 = $242, Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 12

13 Exhibi 5 MIRRs for Projec X and Projec Y MIRR 12% 10% 8% 6% 4% 2% 0% 0% 1% 2% 3% 4% 5% 6% Projec X Reinvesmen rae Projec Y 7% 8% 9% 10% 11% The MIRR is herefore a funcion of boh he reinvesmen rae and he paern of cash flows, wih higher he reinvesmen raes leading o greaer MIRRs. You can see his in Exhibi 5, where he MIRR of boh Projec X and Projec Y is ploed for differen reinvesmen raes. Projec Y s MIRR is more sensiive o he reinvesmen rae because more of is cash flows are received sooner, relaive o Projec X s cash flows. If we wish o represen his echnique in a formula, MIRR= N =1 N N CIF (1+i) =1 COF (1+i) where he CIF are he cash inflows and he COF are he cash ouflows. In he previous example, he presen value of he cash ouflows is equal o he $1,000,000 iniial cash oulay, whereas he fuure value of he cash inflows is $1,435,500. If... his means ha... and you... N- MIRR > cos of capial MIRR < cos of capial MIRR = cos of capial he invesmen is expeced o reurn more han required he invesmen is expeced o reurn less han required he invesmen is expeced o reurn wha is required should accep he projec. should rejec he projec. are indifferen beween acceping or rejecing he projec G. Scale differences Scale differences -- differences in he amoun of he cash flows -- beween projecs can lead o conflicing invesmen decisions among he discouned cash flow echniques. Consider wo projecs, Projec Big and Projec Lile, ha each have a cos of capial of 5 percen per year wih he following cash flows: Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 13

14 End of period Projec Big Projec Lile 0 -$1,000,000 -$ , , , Applying he discouned cash flow echniques o each projec, Muually exclusive projecs Technique Projec Big Projec Lile NPV $89,299 $ PI IRR % % MIRR % % If Big and Lile are muually exclusive projecs, which projec should a firm prefer? If he firm goes sricly by he PI, IRR, or MIRR crieria, i would choose Projec Lile. Bu is his he beer projec? Projec Big provides more value -- $89,299 versus $0.18. The echniques ha ignore he scale of he invesmen -- PI, IRR, and MIRR -- may lead o an incorrec decision. Capial raioning If he firm is subjec o capial raioning -- say a limi of $1,000, and Big and Lile are independen projecs, which projec should he firm choose? The firm can only choose one -- spend $1 or $1,000,000, bu no $1,000,001. If you go sricly by he PI, IRR, or MIRR crieria, he firm would choose Projec Lile. Bu is his he beer projec? Again, he echniques ha ignore he scale of he invesmen -- PI, IRR, and MIRR -- leading o an incorrec decision. H. The invesmen profile We may wan o see how sensiive is our decision o accep a projec o changes in our cos of capial. We can see his sensiiviy in how a projec's ne presen value changes as he discoun rae changes by looking a a projec's invesmen profile, also referred o as he ne presen value profile. The invesmen profile is a graphical depicion of he relaion beween he ne presen value of a projec and he discoun rae: he profile shows he ne presen value of a projec for each discoun rae, wihin some range. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 14

15 The ne presen value profile for he wo projecs is shown in Exhibi 2 for discoun raes from 0 percen o 20 percen. To help you ge he idea behind his Exhibi 2: The invesmen profiles of Projecs X and Y graph, we've idenified he $500,000 NPV's of his projec for discoun raes of 5 percen and $400, percen. You should be able $300,000 o see ha he NPV is posiive % for discoun raes from 0 $200,000 Ne percen o percen, and presen $100,000 negaive for discoun raes value higher han percen. $0 The percen is he -$100,000 inernal rae of reurn; ha is, he discoun rae a which he -$200,000 ne presen value is equal o -$300,000 $0. Therefore, Projec X increases owners' wealh if he cos of capial on his projec is Required rae of reurn less han percen and decreases owners' wealh if he cos of capial on his projec is greaer han percen. Le's impose X's NPV profile on he NPV profile of Projec Y, as shown in he graph in Exhibi 3. If X and Y are muually exclusive projecs -- we inves in only one or neiher projec -- his graph clearly shows ha he projec we inves in depends on he discoun rae. For higher discoun raes, B's NPV falls faser han A's. This is because mos of B's presen value is aribued o he large cash flows four and five years ino he fuure. The presen value of he more disan cash flows is more sensiive o changes in he discoun rae han is he presen value of cash flows nearer he presen. 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Exhibi 3: Invesmen profiles of Invesmens X and Y Ne presen value $500,000 $400,000 $300,000 $200,000 $100,000 $0 -$100,000 Projec X Projec Y 7.495% If he discoun rae is less han percen, X adds more values han Y. If he discoun rae is more han percen bu less han percen, Y increases wealh more han X. If he discoun rae is greaer han percen, we should inves in neiher projec because boh would decrease wealh. is $88, $200,000 -$300,000 0% 2% 4% 6% 8% 10% 12% 14% Required rae of reurn 16% 18% 20% The percen is he cross-over discoun rae which produces idenical NPV's for he wo projecs. If he discoun rae is percen, he ne presen value of boh invesmens 3 The precise cross-over rae is percen, a which he NPV for boh projecs is $88,659. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 15

16 Example 1: The invesmen profile Problem Consider a projec ha has he following expeced cash flows: End of year Cash flow $1,000, , , , ,000 Draw his projec s invesmen profile for discoun raes from 0 percen o 20 percen. Soluion Sep 1: Calculae he NPV if he discoun rae = 0%. You calculae his by simply adding up all cash flows (boh posiive and negaive. In his example, his is $300,000. Sep 2: Calculae he IRR. In his case, his is 19.95% Sep 3: Calculae he NPV for some discoun rae beween 0% and he IRR. Sep 4: Mark he resul from Seps 1, 2 and 3 on he graph and connec he poins. $350,000 $300,000 $250,000 $200,000 NPV $150,000 $100,000 $50,000 $0 -$50,000 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% Discoun rae Solving for he cross-over rae For Projecs X and Y, he cross-over rae is he rae ha causes he ne presen value of he wo invesmens o be equal. Basically, his boils down o a simple approach: calculae he differences in he cash flows and hen solve for he inernal rae of reurn of hese differences. Year Projec X Projec Y Difference $1,000,000 -$1,000,000 $ $0 $325,000 -$325, $200,000 $325,000 -$125, $300,000 $325,000 -$25, $900,000 $325,000 $575,000 Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 16

17 The inernal rae of reurn of hese differences is he cross-over rae. Does i maer which projec s cash flows you deduc from he TI-83/84 HP10B oher? No a all jus be { , ,-25000,575000} STO lisname 0 +/- CF j consisen each period. IRR(0,lisname) Boom line: The cross-over rae is he decision poin beween wo muually exclusive projecs. Example Cross-over raes Problem Consider wo projecs, P & Q, wih he following ses of cash flows: End of P Q period 0 -$10 -$ Wha is he cross-over rae for hese wo projecs invesmen profiles? Soluion End of P Q Difference period 0 -$10 -$20 +$ Cross-over rae is he IRR of he differences, or 7.52 percen /- CF j /- CF j /- CF j CF j IRR 3. Comparing echniques If we are dealing wih muually exclusive projecs, he NPV mehod leads us o inves in projecs ha maximize wealh, ha is, capial budgeing decisions consisen wih owners' wealh maximizaion. If we are dealing wih a limi on he capial budge, he NPV and PI mehods lead us o inves in he se of projecs ha maximize wealh. The advanages and disadvanages of each of he echniques for evaluaing invesmens are summarized in Table 1. We see in his able ha he discouned cash flow echniques are preferred o he non-discouned cash flow echniques. The discouned cash flow echniques -- NPV, PI, IRR, MIRR -- are preferable since hey consider (1) all cash flows, (2) he ime value of money, and (3) he risk of fuure cash flows. The discouned cash flow echniques are also useful because we can apply objecive decision crieria -- crieria we can acually use ha ells us when a projec increases wealh and when i does no. We also see in his able ha no all of he discouned cash flow echniques are righ for every siuaion. There are quesions we need o ask when evaluaing an invesmen and he answers will deermine which echnique is he one o use for ha invesmen: Are he projecs muually exclusive or independen? Are he projecs subjec o capial raioning? Are he projecs of he same risk? Are he projecs of he same scale of invesmen? Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 17

18 Here are some simple rules: 1. If projecs are independen and no subjec o capial raioning, we can evaluae hem and deermine he ones ha maximize wealh based on any of he discouned cash flow echniques. 2. If he projecs are muually exclusive, have he same invesmen oulay, and have he same risk, we mus use only he NPV or he MIRR echniques o deermine he projecs ha maximize wealh. 3. If projecs are muually exclusive and are of differen risks or are of differen scales, NPV is preferred over MIRR. If he capial budge is limied, we can use eiher he NPV or he PI. We mus be careful, however, no o selec projecs simply on he basis of heir NPV or PI (ha is, ranking on NPV and selecing he highes NPV projecs), bu raher how we can maximize he NPV of he oal capial budge. In oher words, which se of capial projecs will maximize owners wealh? Try i! Capial budgeing echniques Suppose an invesmen requires an iniial oulay of $5 million and has expeced cash flows of $1 million, $3.5 million and $2 million for he firs hree years, respecively. Wha is his projec s: 1. Payback period? 2. Discouned payback period using a 10 percen required rae of reurn? 3. Ne presen value using a 10 percen required rae of reurn? 4. Inernal rae of reurn? 5. Modified inernal rae of reurn using 5 percen reinvesmen rae? 4. Capial budgeing echniques in pracice Among he evaluaion echniques in his chaper, he one we can be sure abou is he ne presen value mehod. NPV will seer us oward he projec ha maximizes wealh in he mos general circumsances. Bu wha evaluaion echnique do financial decision makers really use? We learn abou wha goes on in pracice by anecdoal evidence and hrough surveys. We see ha: here is an increased use of more sophisicaed capial budgeing echniques; mos financial managers use more han one echnique o evaluae he same projecs, wih a discouned cash flow echnique (NPV, IRR, PI) used as a primary mehod and payback period used as a secondary mehod; and he mos commonly used is he inernal rae of reurn mehod, hough he ne presen value mehod is gaining accepance. IRR is popular mos likely because i is a measure of yield and herefore easy o undersand. Moreover, since NPV is expressed in dollars -- he expeced incremen in he value of he firm and financial managers are accusomed o dealing wih yields, hey may be more comforable dealing wih he IRR han he NPV. The populariy of he IRR mehod is roublesome since i may lead o decisions abou projecs ha are no in he bes ineres of owners in cerain circumsances. However, he NPV mehod is becoming more widely acceped and, in ime, may replace he IRR as he more popular mehod. And is he use of payback period roublesome? No necessarily. The payback period is generally used as a screening device, eliminaing hose projecs ha canno even break-even. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 18

19 Furher, he payback period can be viewed as a measure of a yield. If he fuure cash flows are he same amoun each period and if hese fuure cash flows can be assumed o be received each period forever -- essenially, a perpeuiy -- hen 1/payback period is a rough guide o a yield on he invesmen. Suppose you inves $100 oday and expec $20 each period, forever. The payback period is 5 years. The inverse, 1/5= 20 percen per year, is he yield on he invesmen. Now le's urn his relaion around and creae a payback period rule. Suppose we wan a 10 percen per year reurn on our invesmen. This means ha he payback period should be less han or equal o 10 years. So while he payback period may seem o be a rough guide, here is some raionale behind i. Use of he simpler echniques, such as payback period, does no mean ha a firm has unsophisicaed capial budgeing. Remember ha evaluaing he cash flows is only one aspec of he process: cash flows mus firs be esimaed, cash flows are evaluaed using NPV, PI, IRR, MIRR or a payback mehod; and projec risk mus be assessed o deermine he cos of capial. 5. Summary The payback period and he discouned payback period mehods give us an idea of he ime i akes o recover he iniial invesmen in a projec. Boh of hese mehods are disappoining because hey do no necessarily consider all cash flows from a projec. Furher, here is no objecive crieria ha we can use o judge a projec, excep for he simple crierion ha he projec mus pay back. The ne presen value mehod and he profiabiliy index consider all of he cash flows from a projec and involve discouning, which incorporaes he ime value of money and risk. The ne presen value mehod produces an amoun ha is he expeced added value from invesing in a projec. The profiabiliy index, on he oher hand, produces an indexed value ha is useful in ranking projecs. The inernal rae of reurn is he yield on he invesmen. I is he discoun rae ha causes he ne presen value o be equal o zero. IRR is hazardous o use when selecing among muually exclusive projecs or when here is a limi on capial spending. The modified inernal rae of reurn is a yield on he invesmen, assuming ha cash inflows are reinvesed a some rae oher han he inernal rae of reurn. This mehod overcomes he problems associaed wih unrealisic reinvesmen rae assumpions inheren wih he inernal rae of reurn mehod. However, MIRR is hazardous o use when selecing among muually exclusive projecs or when here is a limi on capial spending. Each echnique we look a offers some advanages and disadvanages. The discouned flow echniques -- NPV, PI, IRR, and MIRR -- are superior o he non-discouned cash flow echniques -- he payback period and he discouned payback period. To evaluae muually exclusive projecs or projecs subjec o capial raioning, we have o be careful abou he echnique we use. The ne presen value mehod is consisen wih owners' wealh maximizaion wheher we have muually exclusive projecs or capial raioning. Looking a capial budgeing in pracice, we see ha firms do use he discouned cash flow echniques, wih IRR he mos widely used. Over ime, however, we see a growing use of he ne presen value echnique. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 19

20 6. Try i! Soluions Capial budgeing echniques 1. Payback period? The sum of he cash flows a he end of wo years is $4.5 million The sum a he end of hree years is $6.5 million Payback = Three years 2. Discouned payback period using a 10 percen required rae of reurn? 4 The sum of he discouned cash flows a he end of hree years is: $ = $ Discouned payback period = Three years. 3. Ne presen value using a 10 percen required rae of reurn? Presen value of inflows = $ million (we know his from he discouned payback period calculaion). Presen value of ouflows = $5 million NPV = $ = $ million 4. Inernal rae of reurn? We know ha he IRR mus be greaer han 10 percen because he NPV is posiive when he discoun rae is 10 percen. IRR = percen 5. Modified inernal rae of reurn using a 5 percen reinvesmen rae? Terminal value = $1 (1.05) 2 + $3.5 (1.05) + $2 = $ = $ million TV = FV = $6.7775; N = 3; PV = $5; Solve for i MIRR = percen 4 Why no check for discouned payback afer wo years? Because if i does no payback in wo years using undiscouned cash flows, i does no payback in erms of discouned cash flows. Capial budgeing echniques, a reading prepared by Pamela Peerson Drake 20

Topic Overview. Learning Objectives. Capital Budgeting Steps: WHAT IS CAPITAL BUDGETING?

Topic Overview. Learning Objectives. Capital Budgeting Steps: WHAT IS CAPITAL BUDGETING? Chaper 10: THE BASICS OF CAPITAL BUDGETING Should we build his plan? Topic Overview Projec Types Capial Budgeing Decision Crieria Payback Period Discouned Payback Period Ne Presen Value () Inernal Rae

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

Chapter 1.6 Financial Management

Chapter 1.6 Financial Management Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

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

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

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

Lecture III: Finish Discounted Value Formulation

Lecture III: Finish Discounted Value Formulation 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

More information

The Grantor Retained Annuity Trust (GRAT)

The Grantor Retained Annuity Trust (GRAT) WEALTH ADVISORY Esae Planning Sraegies for closely-held, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

SKF Documented Solutions

SKF Documented Solutions SKF Documened Soluions Real world savings and we can prove i! How much can SKF save you? Le s do he numbers. The SKF Documened Soluions Program SKF is probably no he firs of your supplier parners o alk

More information

Diagnostic Examination

Diagnostic Examination Diagnosic Examinaion TOPIC XV: ENGINEERING ECONOMICS TIME LIMIT: 45 MINUTES 1. Approximaely how many years will i ake o double an invesmen a a 6% effecive annual rae? (A) 10 yr (B) 12 yr (C) 15 yr (D)

More information

Understanding Sequential Circuit Timing

Understanding Sequential Circuit Timing ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

More information

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity .6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

More information

1. The graph shows the variation with time t of the velocity v of an object.

1. The graph shows the variation with time t of the velocity v of an object. 1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

More information

LEASING VERSUSBUYING

LEASING VERSUSBUYING LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

Market Analysis and Models of Investment. Product Development and Whole Life Cycle Costing

Market Analysis and Models of Investment. Product Development and Whole Life Cycle Costing The Universiy of Liverpool School of Archiecure and Building Engineering WINDS PROJECT COURSE SYNTHESIS SECTION 3 UNIT 11 Marke Analysis and Models of Invesmen. Produc Developmen and Whole Life Cycle Cosing

More information

One dictionary: Native language - English/English - native language or English - English

One dictionary: Native language - English/English - native language or English - English Faculy of Social Sciences School of Business Corporae Finance Examinaion December 03 English Dae: Monday 09 December, 03 Time: 4 hours/ 9:00-3:00 Toal number of pages including he cover page: 5 Toal number

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

Fair games, and the Martingale (or "Random walk") model of stock prices

Fair games, and the Martingale (or Random walk) model of stock prices Economics 236 Spring 2000 Professor Craine Problem Se 2: Fair games, and he Maringale (or "Random walk") model of sock prices Sephen F LeRoy, 989. Efficien Capial Markes and Maringales, J of Economic Lieraure,27,

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

Capital Budgeting and Initial Cash Outlay (ICO) Uncertainty

Capital Budgeting and Initial Cash Outlay (ICO) Uncertainty Financial Decisions, Summer 006, Aricle Capial Budgeing and Iniial Cash Oulay (ICO) Uncerainy Michael C. Ehrhard and John M. Wachowicz, Jr. * * The Paul and Beverly Casagna Professor of Finance and Professor

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

Chapter 6 Interest Rates and Bond Valuation

Chapter 6 Interest Rates and Bond Valuation Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Long-erm deb-loosely, bonds wih a mauriy of one year or more Shor-erm deb-less han a year o mauriy, also called unfunded deb Bond-sricly

More information

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were

More information

Chapter 2 Kinematics in One Dimension

Chapter 2 Kinematics in One Dimension Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

Performance Center Overview. Performance Center Overview 1

Performance Center Overview. Performance Center Overview 1 Performance Cener Overview Performance Cener Overview 1 ODJFS Performance Cener ce Cener New Performance Cener Model Performance Cener Projec Meeings Performance Cener Execuive Meeings Performance Cener

More information

TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED

TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific half-life. Sudens will develop mahemaical

More information

13. a. If the one-year discount factor is.905, what is the one-year interest rate?

13. a. If the one-year discount factor is.905, what is the one-year interest rate? CHAPTER 3: Pracice quesions 3. a. If he one-year discoun facor is.905, wha is he one-year ineres rae? = DF = + r 0.905 r = 0.050 = 0.50% b. If he wo-year ineres rae is 0.5 percen, wha is he wo-year discoun

More information

Newton's second law in action

Newton's second law in action Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

More information

Equities: Positions and Portfolio Returns

Equities: Positions and Portfolio Returns Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi

More information

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More information

Derivatives. Forwards and Futures. Forward. Futures. Options. Initial Cost

Derivatives. Forwards and Futures. Forward. Futures. Options. Initial Cost Derivaives Forwards and Fuures A derivaive securiy is a securiy whose value depends on he values of oher more basic underlying variables. Forward The mos common derivaive securiies are forward, fuures

More information

Benefit-Cost Analysis

Benefit-Cost Analysis Slide 1 Benefi-Cos Analysis Sco Pearson Sanford Universiy Sco Pearson is Professor Emerius of Agriculural Economics a he Food Research Insiue, Sanford Universiy. He has paricipaed in projecs ha combined

More information

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005 FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

More information

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion

More information

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed. Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

More information

Valuation Beyond NPV

Valuation Beyond NPV FIN 673 Alernaive Valuaion Approaches Professor Rober B.H. Hauswald Kogod School of Business, AU Valuaion Beyond NPV Corporae Finance revolves around hree fundamenal quesions: wha long-erm invesmens should

More information

Chapter Four: Methodology

Chapter Four: Methodology Chaper Four: Mehodology 1 Assessmen of isk Managemen Sraegy Comparing Is Cos of isks 1.1 Inroducion If we wan o choose a appropriae risk managemen sraegy, no only we should idenify he influence ha risks

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Cannibalization and Product Life Cycle Management

Cannibalization and Product Life Cycle Management Middle-Eas Journal of Scienific Research 19 (8): 1080-1084, 2014 ISSN 1990-9233 IDOSI Publicaions, 2014 DOI: 10.5829/idosi.mejsr.2014.19.8.11868 Cannibalizaion and Produc Life Cycle Managemen Ali Farrukh

More information

1. Explain why the theory of purchasing power parity is often referred to as the law of one price.

1. Explain why the theory of purchasing power parity is often referred to as the law of one price. Chaper Review Quesions. xplain why he heory of purchasing power pariy is ofen referred o as he law of one price. urchasing ower ariy () is referred o as he law of one price because he deerminaion of he

More information

Glenn P. Jenkins Queen s University, Kingston, Canada and Eastern Mediterranean University, North Cyprus

Glenn P. Jenkins Queen s University, Kingston, Canada and Eastern Mediterranean University, North Cyprus COST-BENEFIT ANALYSIS FOR INVESTMENT DECISIONS, CHAPTER 3: THE FINANCIAL APPRAISAL OF PROJECTS Glenn P. Jenkins Queen s Universiy, Kingson, Canada and Easern Medierranean Universiy, Norh Cyprus Developmen

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

More information

Depreciation and Corporate Taxes

Depreciation and Corporate Taxes 205 Depreciaion and Corporae Taxes Chris Hendrickson Carnegie Mellon Universiy Tung Au Carnegie Mellon Universiy 205.1 Depreciaion as Tax Deducion 205.2 Tax Laws and Tax Planning 205.3 Decision Crieria

More information

Chapter 4: Exponential and Logarithmic Functions

Chapter 4: Exponential and Logarithmic Functions Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 2010 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

More information

Follow links Class Use and other Permissions. For more information, send to:

Follow links Class Use and other Permissions. For more information, send  to: COPYRIGHT NOTICE: David A. Kendrick, P. Ruben Mercado, and Hans M. Amman: Compuaional Economics is published by Princeon Universiy Press and copyrighed, 2006, by Princeon Universiy Press. All righs reserved.

More information

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

More information

A Further Examination of Insurance Pricing and Underwriting Cycles

A Further Examination of Insurance Pricing and Underwriting Cycles A Furher Examinaion of Insurance ricing and Underwriing Cycles AFIR Conference, Sepember 2005, Zurich, Swizerland Chris K. Madsen, GE Insurance Soluions, Copenhagen, Denmark Svend Haasrup, GE Insurance

More information

Permutations and Combinations

Permutations and Combinations Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

Part 1: White Noise and Moving Average Models

Part 1: White Noise and Moving Average Models Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

More information

WHAT ARE OPTION CONTRACTS?

WHAT ARE OPTION CONTRACTS? WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be

More information

Chapter 9 Bond Prices and Yield

Chapter 9 Bond Prices and Yield Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value

More information

Acceleration Lab Teacher s Guide

Acceleration Lab Teacher s Guide Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

More information

Level I Study Handbook Sample

Level I Study Handbook Sample Level I Sudy Handbook Sample The UpperMark Sudy Handbooks for Level I are comprised of 3 Volumes, each covering abou 10 Topics from he CAIA curriculum. This is a sample of one of he Topic chapers. You

More information

Convexity. Concepts and Buzzwords. Dollar Convexity Convexity. Curvature, Taylor series, Barbell, Bullet. Convexity 1

Convexity. Concepts and Buzzwords. Dollar Convexity Convexity. Curvature, Taylor series, Barbell, Bullet. Convexity 1 Deb Insrumens and Markes Professor Carpener Convexiy Conceps and Buzzwords Dollar Convexiy Convexiy Curvaure, Taylor series, Barbell, Bulle Convexiy Deb Insrumens and Markes Professor Carpener Readings

More information

Chapter 7: Estimating the Variance of an Estimate s Probability Distribution

Chapter 7: Estimating the Variance of an Estimate s Probability Distribution Chaper 7: Esimaing he Variance of an Esimae s Probabiliy Disribuion Chaper 7 Ouline Review o Clin s Assignmen o General Properies of he Ordinary Leas Squares (OLS) Esimaion Procedure o Imporance of he

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

Chapter 8 Student Lecture Notes 8-1

Chapter 8 Student Lecture Notes 8-1 Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop

More information

Supply Chain Management Using Simulation Optimization By Miheer Kulkarni

Supply Chain Management Using Simulation Optimization By Miheer Kulkarni Supply Chain Managemen Using Simulaion Opimizaion By Miheer Kulkarni This problem was inspired by he paper by Jung, Blau, Pekny, Reklaii and Eversdyk which deals wih supply chain managemen for he chemical

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

FIN 472 Fixed-Income Securities Approximating Price Changes: From Duration to Convexity Professor Robert B.H. Hauswald Kogod School of Business, AU

FIN 472 Fixed-Income Securities Approximating Price Changes: From Duration to Convexity Professor Robert B.H. Hauswald Kogod School of Business, AU FIN 47 Fixed-Income Securiies Approximaing rice Changes: From Duraion o Convexiy rofessor Rober B.H. Hauswald Kogod School of Business, AU Bond rice Volailiy Consider only IR as a risk facor Longer M means

More information

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 2013 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

More information

Usefulness of the Forward Curve in Forecasting Oil Prices

Usefulness of the Forward Curve in Forecasting Oil Prices Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,

More information

Rotational Inertia of a Point Mass

Rotational Inertia of a Point Mass Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

More information

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009 ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

More information

4. The Poisson Distribution

4. The Poisson Distribution Virual Laboraories > 13. The Poisson Process > 1 2 3 4 5 6 7 4. The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy

More information

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly

More information

Vector Autoregressions (VARs): Operational Perspectives

Vector Autoregressions (VARs): Operational Perspectives Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

The yield curve, and spot and forward interest rates Moorad Choudhry

The yield curve, and spot and forward interest rates Moorad Choudhry he yield curve, and spo and forward ineres raes Moorad Choudhry In his primer we consider he zero-coupon or spo ineres rae and he forward rae. We also look a he yield curve. Invesors consider a bond yield

More information

Can Individual Investors Use Technical Trading Rules to Beat the Asian Markets?

Can Individual Investors Use Technical Trading Rules to Beat the Asian Markets? Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weak-form of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien

More information

Relationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**

Relationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith** Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia

More information

AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 2007 Scoring Guidelines AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

More information

Research Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis

Research Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis Inroducion o Hypohesis Tesing Research Quesion Is he average body emperaure of healhy aduls 98.6 F? HT - 1 HT - 2 Scienific Mehod 1. Sae research hypoheses or quesions. µ = 98.6? 2. Gaher daa or evidence

More information

1. Fund types and population covered

1. Fund types and population covered Performance and financial overview of invesmen funds - France 1 March 016 The Banque de France draws up he following informaion for invesmen funds: 1 monhly saisics on fund ousandings and flows, and on

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information