1. Time Value of Money 3 2. Discounted Cash Flow Statistics and Market Returns Probabilities Key Formulas 109

Transcription

1 1. Time Value of Money 3 2. Discouned Cash Flow Saisics and Make Reuns Pobabiliies Key Fomulas 109 Candidae Noe: This is a lenghy Sudy Session ha, along wih Sudy Session 3, you should plan on eviewing moe han once, paiculaly if any o all of he opic aeas ae new o you. We sugges eviewing i wih a CFA Insiue-appoved financial calculao so ha you can wok hough he examples and become familia wih he calculaions. Sudy ime spen on his Sudy Session could benefi you gealy on exam day. The numeical examples shown in he Quaniaive Mehods Sudy Sessions ae displayed wih ounded numbes, bu he calculaions ae done wih geae pecision. You may find a small amoun of wha appeas o be ounding eo, bu his is because numbes in inemediae seps ae displayed wih some ounding. Howeve, exac figues ae used houghou he calculaions, and he final answes eflec his geae pecision. Noe ha any necessay key fomulas and appendices ae locaed a he end of Sudy Session 3. All ighs eseved. Waning: Copyigh violaions will be posecued. Any use of hese maeials wihou he expess wien consen of he publishe is a violaion of fedeal and/o inenaional copyigh laws. 2-1

2 Time Value of Money 1. Time Value of Money Leaning Objecives This summay includes a eview and an analysis of he pinciples se foh by CFA Insiue. Upon eview of his summay, you should be able o: Undesand inees aes as equied aes of eun, discoun aes, o oppouniy coss...pg. 4 Undesand a nominal inees ae as he sum of a eal isk-fee ae, expeced inflaion, and pemiums ha compensae invesos fo disinc ypes of isk...pg. 4 Calculae he fuue value (FV) of a single sum of money...pg. 6 Calculae he pesen value (PV) of a single sum of money...pg. 8 Calculae he FV of a egula annuiy...pg. 11 Calculae he FV of an annuiy due...pg. 12 Calculae he PV of a egula annuiy...pg. 14 Calculae he PV of an annuiy due...pg. 15 Calculae he PV of a pepeuiy...pg. 17 Calculae he PV of a seies of uneven cash flows...pg. 18 Calculae he FV of a seies of uneven cash flows...pg. 20 Calculae ime value of money poblems when compounding peiods ae no annual...pg. 21 Calculae he effecive annual ae when given he saed annual inees ae and he fequency of compounding...pg. 22 Using a ime line, illusae how o solve ime value of money poblems (such as mogages, cedi cads, savings fo college uiion, o eiemen)...pg

3 Discouned Cash Flow 2. Discouned Cash Flow Leaning Objecives This summay includes a eview and an analysis of he pinciples se foh by CFA Insiue. Upon eview of his summay, you should be able o: Calculae he ne pesen value (NPV) and inenal ae of eun (IRR) of a capial invesmen pojec...pg. 36 Discuss he use of he decision ule unde he ne pesen value and inenal ae of eun mehods...pg. 37 Discuss he poblems wih he inenal ae of eun mehod...pg. 37 Define, calculae, and inepe holding peiod euns (oal eun)...pg. 38 Diffeeniae beween money-weighed and ime-weighed aes of eun...pg. 39 Compue and appaise he pefomance of a pofolio s money-weighed and imeweighed aes of eun...pg. 39 Compue he holding peiod yield, effecive annual yield, money make yield, and bank discoun yield fo a U.S. Teasuy bill and ohe money make insumens...pg. 43 Conve among holding peiod yields, money make yields, and effecive annual yields...pg. 45 Compue bond equivalen yield (BEY)...pg

4 Sudy Session 2 Ne Pesen Value Leaning Objecive: Calculae he ne pesen value (NPV) and inenal ae of eun (IRR) of a capial invesmen pojec. Ne pesen value is simply he esimaed value of a pojec based on he cash flows i is expeced o poduce. If he esimaed value is posiive, hen acceping he pojec will incease he value of he fim and, hus, incease shaeholdes wealh. The NPV mehod elies on discouned cash flow (DCF) echniques and does no suffe fom he shocomings of he egula and discouned payback mehods. The seps in he NPV mehod ae: 1. Deemine each cash inflow and ouflow and discoun hem a he pojec s cos of capial o find he pesen value (PV) of each. 2. Sum hese discouned cash flows o poduce he pojec s NPV. 3. If he NPV is posiive, he pojec should be acceped. If he NPV is negaive, he pojec should be ejeced. The equaion fo he NPV is: CF CF CF NPV = CF ( 1 + k ) ( 1 + k ) ( 1 + k ) Whee CF is he expeced ne cash flow fo peiod and k is he cos of capial. (Cash ouflows ae eaed as negaive numbes and inflows ae eaed as posiive numbes.) Assume ha a pojec s cash flows consis of an iniial invesmen of $5,000, and annual cash inflows of $1,000, $2,000, $3,000, and $4,000. The NPV is compued as follows: 1,000 2,000 3,000 4,000 NPV s = -5, = $2, ( 1.12) ( 1.12) ( 1.12) ( 1.12) If a pojec has a posiive NPV, hen i should be acceped. An NPV of $2,164 means ha if pojec X is undeaken, he fim s value (and heefoe shaeholdes wealh) will incease by $2,164 as soon as he pojec is acceped. If he NPV is negaive, hen he pojec should be ejeced. Inenal Rae of Reun The IRR is he discoun ae ha equaes he PV of he pojec s expeced cash inflows o he PV of he pojec s coss. I is he ae ha foces he NPV o equal zeo. In he following equaion, if CF 0 CF n ae known, he IRR is he ae ha makes he PV of all fuue cash flows equal CF 0, hus poducing an NPV of zeo. n n 2-36 Sudy Guide fo he Level I 2015 CFA Exam - Reading Highlighs

5 Discouned Cash Flow CF CF CFn NPV = CF = ( 1 + IRR) ( 1 + IRR ) ( 1 + IRR) The IRR can easily be found using any financial calculao. I can also be solved hough a ialand-eo pocess by eneing a discoun ae ino he fomula and seeing if he equaion solves o zeo. If i does no, a diffeen ae is eneed unil he ae foces he equaion o equal zeo. The IRR fo he pojec in he pevious secion is 27.3%. If a pojec s IRR is geae han he fim s cos of capial, hen he pojec should be acceped. The IRR is he expeced aveage annual ae of eun on he pojec. If he IRR exceeds he cos of he funds used o finance he pojec, a suplus emains afe paying fo he capial. This suplus accues o he sockholdes, heeby inceasing shaeholdes wealh. Using Ne Pesen Value and Inenal Rae of Reun Leaning Objecive: Discuss he use of he decision ule unde he ne pesen value and inenal ae of eun mehods. The NPV and IRR cieia always lead o he same accep/ejec decision if independen pojecs ae being consideed. Fo muually exclusive pojecs, NPV is he coec mehod. IRR should no be used as he selecion mehod when choosing beween muually exclusive pojecs because IRR and NPV can give conflicing esuls. One pojec can have he highe IRR while he ohe pojec has he highe NPV. Two condiions can cause NPV and IRR o yield conflicing ankings: 1. When he cos of one pojec is lage han he ohe. 2. When he iming of cash flows fom he wo pojecs diffes so ha mos of he cash flows fom one pojec come in he ealy yeas while mos of he cash flows fom he ohe pojec come in he lae yeas. The NPV mehod is heefoe he pefeed mehod fo evaluaing muually exclusive pojecs, especially fo hose pojecs ha diffe in cos and/o iming. Addiional Poblems wih Inenal Rae of Reun Leaning Objecive: Discuss he poblems wih he inenal ae of eun mehod. n If a pojec has abnomal cash flows, he IRR mehod may no be usable. Nomal cash flows occu if one o moe cash ouflows ae followed by a seies of cash inflows (and heefoe hee is only one change in he signs of he cash flows). If a pojec equies cash ouflows afe cash inflows have occued, hen he pojec has abnomal cash flows. (Abnomal cash flows have moe han one sign change. A negaive cash flow in he middle of a pojec would poduce hee sign changes in he cash flow seam. A negaive cash flow a he end of a pojec would poduce wo sign changes in he cash flow seam.) 2-37

6 Sudy Session 2 Fo example, he following seies of cash flows is nomal: -$1,000, $500, $600, $700, and $800. Thee is only one change in sign (fom -$1,000 o $500). Howeve, he following seies of cash flows is no nomal: -$1,000, $500, -$600, $700, and $800. Thee ae hee changes in sign (fom -$1,000 o $500, fom $500 o -$600, and fom -$600 o $700). If he IRR equaion is solved fo a pojec wih abnomal cash flows, i is possible o obain moe han one value of IRR, meaning ha muliple IRRs occu. (In fac, hee will be one IRR pe sign change in he cash flow seam.) In such cases, he IRR mehod should no be used o evaluae o ank he pojec. The IRR mehod also assumes ha he pojec s cash flows ae einvesed a he IRR ove he pojec s life. When a pojec has a high IRR, i may no be ealisic o assume ha he cash flows fom he pojec can be einvesed a ha same high ae of eun. Holding Peiod Reun Leaning Objecive: Define, calculae, and inepe holding peiod euns (oal eun). The peiod duing which an inveso owns an invesmen is called he holding peiod. The eun fo he peiod duing which an inveso owns an invesmen is called he holding peiod eun. Holding peiod eun (HPR) is given by he following fomula: Whee: HPR = holding peiod eun fo ime 1 o P P + D 1 HPR = P 1 P = pice of he invesmen a ime P 1 = pice of he invesmen a ime 1 D = any cash disibuion (such as a dividend) a ime Fo example, suppose an inveso puchases an asse fo $300 and ges back $330 a he end of he yea. The holding peiod is one yea and he holding peiod eun is ($330 $300)/$300 = 0.10 o 10%. The holding peiod eun is also efeed o as he oal eun. This is, as saed above, simply he pice appeciaion (P P 1 ) plus income (D ), divided by he cos of he invesmen. This is he oal eun of he invesmen. Noe ha his eun is no necessaily annualized, bu ahe is he oal eun of he invesmen ove he holding peiod. Only unde he special case of he holding peiod being pecisely one yea, is he oal eun equal o he annualized eun Sudy Guide fo he Level I 2015 CFA Exam - Reading Highlighs

7 Discouned Cash Flow Leaning Objecive: Diffeeniae beween money-weighed and ime-weighed aes of eun. Money-Weighed Reun Money-weighed (MW) eun is he inenal ae of eun on a pofolio, aking accoun of all cash flows. I ies o measue eun on aveage invesmen duing he peiod. The bes way o compue a money-weighed eun is o use he inenal ae of eun (IRR) mehod. Time-Weighed Reun Time-weighed (TW) eun emoves he effecs of he iming and amoun of cash flows in and ou of he pofolio. I eflecs he compound ae of gowh of one uni of cuency invesed ove a saed measuemen peiod. Leaning Objecive: Compue and appaise he pefomance of a pofolio s money-weighed and ime-weighed aes of eun. Money-weighed eun is he inenal ae of eun (IRR), which is he R p in he equaion: Whee: FV PMT PV = + (1 + R ) (1 + R ) p p R p = FV = PV = PMT = money-weighed eun on pofolio P value of pofolio P a he end of he peiod value of pofolio P a he beginning of he peiod cash flow a ime Example An invesmen of $1,000 pays $40 a he end of each yea fo 5 yeas, plus $1,100 a he end of he five yeas. Wha is he money-weighed ae of eun? Soluion The money-weighed ae of eun will be he ae R ha saisfies he equaion: 5 $1,100 $40 $1,000 = + (1 + R ) (1 + R ) p =1 p 2-39

8 Sudy Session 2 We can use a financial calculao s bond funcion o solve his. Remembe o se paymens pe yea equal o one if necessay (ahe han he bond convenion of semiannual paymens): [N] = 5 [PV] = [PMT] = 40 [FV] = 1100 [2 nd ] [I/Y] = 1 [ENTER] [CE/C] [CPT] [I/Y] You should ge I/Y = 5.78, which means 5.78%. Money-weighed (MW) eun is fo he peiod and can be annualized by compounding. Fo example, if we compued he quaely MW eun and we need he annual MW eun, we can compue i as follows: Whee: R annual = (1 + R q1 )(1 + R q2 )(1 + R q3 )(1 + R q4 ) 1 R annual = R q = annual MW eun MW eun fo quae Time-Weighed Reun Because open-ended muual fund manages canno diecly conol he cash inflows o hei pofolio, hee is inees in a eun measue ha emoves he effecs of he iming and amoun of cash flows in and ou of he pofolio. Time-weighed eun compuaion divides he measuemen peiod ino sub-peiods. The fis sub-peiod sas a he beginning of he measuemen peiod and ends wih he dae of he fis cash flow. The second sub-peiod sas he day afe he fis sub-peiod and ends on he dae of he second cash flow and so foh. The las sub-peiod begins he day afe he las cash flow and ends a he end of he measuemen peiod. TW eun equies valuaion of he pofolio each ime a cash flow occus. The sub-peiod eun is equal o he change in pofolio value duing he peiod, divided by he oal amoun invesed fo he peiod: Reun fo peiod = Ending value Amoun invesed Amoun invesed All sub-peiod euns ae hen compounded o aive a he eun fo he measuemen peiod: R w = (1 + R s1 )(1 + R s2 )(1 + R s3 ) (1 + R sk ) Sudy Guide fo he Level I 2015 CFA Exam - Reading Highlighs

9 Discouned Cash Flow Whee: k = R sk = oal numbe of sub peiods pofolio eun duing sub-peiod k To obain he ime-weighed aveage eun ove a peiod of yeas, ake he geomeic mean of he oal ime-weighed eun fo he sudy peiod. Example You ae Sewa Muha, an invesmen manage wih Clone Em Equiy Panes. You fund s pefomance is shown below (figues in millions). Peiod: Sa Value $1,000 $1,300 $1,800 $2,500 $2,500 New Invesmens $100 $300 $250 $500 ($1,000) Amoun Invesed $1,100 $1,600 $2,050 $3,000 $1,500 End Value $1,300 $1,800 $2,500 $2,500 $1,800 Assuming new invesmens ae acceped only a he beginning of each peiod, find he imeweighed aveage eun fo he enie peiod. Soluion The eun fo peiod 1 is found by: $1,300 $1,100 Reun fo peiod = = 18.18% $1,100 Similaly, he euns fo all peiods can be found: Peiod Reun 18.18% 12.50% 21.95% % 20.00% To ge he oveall ime-weighed aveage eun, we use he equaion: R w = (1 + R s1 )(1 + R s2 )(1 + R s3 ) (1 + R sk ) 1 R w = (1.1818)(1.1250)(1.2195)(0.8333)(1.2000) 1 = = 62.14% 2-41

10 Sudy Session 2 By adding one o his value, and aking he fifh oo of i, we can ge he aveage annual imeweighed eun: = 10.15% Noe: This will diffe fom he dolla-weighed eun. The dolla-weighed eun will equie use of a speadshee o a financial calculao s CF and IRR funcion. (The bond funcion used ealie was useful fo a simple example.) To compue dolla-weighed eun, we mus compae he ne inflow and ouflow fo each peiod, and find he inees ae which equaes hem ove ime. The beginning pofolio value should be consideed an invesmen (ouflow), and he ending pofolio value should be consideed a eun (inflow). Fo he example above, ou ne inflows and ouflows fo each peiod will be: Time Ne Inflow 0-1, , ,800 Noice he designaion of he cash flow iming. The saing value in he fis peiod is denoed as ime = 0. Also, he ne inflow of $1,000 occus a he beginning of he 5 h yea, while he final pofolio value of $1,800 is a he end of he 5 h yea. Fo he Texas Insumens BA II Plus, make he following enies: [CF] [2 nd ] [CE/C] his cleas all enies in he cash flow shee 1100 [+/-] [ENTER] [] 300 [+/-] [ENTER] [][] 250 [+/-] [ENTER] [][] 500 [+/-] [ENTER] [][] 1000 [ENTER] [][] 1800 [ENTER] [IRR] [CPT] You should ge 7.57, which in his conex means 7.57%. Thus, he dolla-weighed eun is much lowe. The ime-weighed eun shows ha he dollaweighed eun is negaively affeced by he cash flows in and ou of he fund Sudy Guide fo he Level I 2015 CFA Exam - Reading Highlighs

11 Discouned Cash Flow Leaning Objecive: Compue he holding peiod yield, effecive annual yield, money make yield, and bank discoun yield fo a U.S. Teasuy bill and ohe money make insumens. Holding Peiod Yield (oal eun fo he peiod, no annualized): + HPY = P P D P Whee P 1 is he mauiy value, P 0 is he puchase pice, and D 1 is he dividend a he end of he holding peiod. Fo a T-bill, which pays no explici dividend D 1, HPY is simply: 1 0 HPY = P P P Effecive Annual Yield (365-day compound-annualized HPY): EAY = (1 + HPY) 1 Money Make Yield o CD-Equivalen Yield (360-day simple-annualized HPY): MM D P BD = = HPY = = BD P0 P0 360 BD The D in he fomula above efes o he T-bill discoun, P 1 P 0. Bank Discoun Yield (use aihmeic annualizaion wih face value F as denominao): BD D = F 360 Noe ha T-bills ae discoun insumens - hey pay back he face amoun upon mauiy, bu make no ohe paymens. The inees eaned on T-bills comes fom he fac ha hey sell a a discoun, so he diffeence beween he discoun and he mauiy value is he inees. Commecial pape and bankes accepances also use his convenion fo pices and yields. Howeve, negoiable CDs - which ae also money make insumens - acually pay inees on op of a pincipal amoun. 2-43

12 Sudy Session 2 The elaionships among money make insumens and how hey ean inees is summaized in he following able: Wha You Inves Wha You Ge A Mauiy T-Bills, Commecial Pape, Bankes Accepances Face Discoun Face Negoiable CDs Face Face + Inees Example A U.S. Teasuy bill, wih a face value (o pa value) of $100,000 and 160 days unil mauiy, is selling fo $97,000. Calculae is holding peiod yield, effecive annual yield, money make yield, and bank discoun yield. Soluion Fo a T-bill, which pays no explici dividend D 1, HPY is simply: 100,000 97,000 So hee, HPY = = %. 97,000 Effecive Annual Yield: 1 0 HPY = P P P EAY = (1 + HPY) 1 = ( %) 1 = % Money Make Yield (CD-equivalen Yield): MM D 360 3, = = = % P 97, Bank Discoun Yield (use aihmeic annualizaion wih face value F as denominao): BD D 360 $3, = = = 6.75% F $100, Noe: The pice of sho-em discoun insumens such as Teasuy bills ae quoed using bank discoun yields, so we mus conve fom bank discoun yield o pice Sudy Guide fo he Level I 2015 CFA Exam - Reading Highlighs

13 Discouned Cash Flow Example Thee is a $100,000 T-bill, wih 160 days o mauiy, quoed a Wha is he pesen puchase pice fo his T-bill? Soluion A quoe of means ha he bank discoun yield (o discoun yield) is % = 6.75%. Wih he discoun yield, we solve fo he dolla discoun, D, as follows: 160 D = F = 6.75% 100,000 = 3,000 BD Wih he dolla discoun calculaed, he puchase pice fo he T-bill is is face value minus he dolla discoun, F D = $100,000 $3,000 = $97,000. Leaning Objecive: Conve among holding peiod yields, money make yields, and effecive annual yields. To conve among he hee vesions of sho em yields, you need o know ha all hose yields come fom hee pieces of basic infomaion: he pesen value (PV), he fuue value o face value (FV), and he numbe of days in beween. The Holding Peiod Yield is he oal eun fo he specific peiod. I is no annualized: + HPY = P P D P The Effecive Annual Yield is he holding peiod eun annualized using 365-day compounding: EAY = (1 + HPY) 1 The Money Make Yield o CD-Equivalen Yield is he holding peiod yield annualized using 360-day convenion wih no compounding: MM D = = HPY P 0 So he key is o emembe ha Holding Peiod Yield is fo he holding peiod only, no necessaily fo he yea. Effecive Annual Yield is annualized HPY using 365-day compounding. Money Make (CD) Yield is annualized HPY using 360-day convenion wih no compounding. 2-45

14 Sudy Session 2 Example Suppose a savings insumen poduces a holding peiod eun of 5% fo 180 days. Find he effecive annual yield and he CD-equivalen yield. Soluion EAY = ( ) 1 = 10.40% MM 360 = 0.05 = 10.00% 180 Example A 270-day CD poduces a CD-equivalen of 7.25%. Find he holding peiod yield and he effecive annual yield. Soluion Theefoe: MM 360 = = HPY ; HPY = % EAY = ( ) 1 = % Example An effecive annual yield fo a 90-day secuiy is 5.50%. Find he holding peiod yield and he money make equivalen yield. Soluion = (1 + HPY) 1 ln(1.055) = ( ) ln 1 + HPY = ln ( 1 + HPY ) ( ) ln 1 + HPY e = e = 1 + HPY 2-46 Sudy Guide fo he Level I 2015 CFA Exam - Reading Highlighs

15 Discouned Cash Flow HPY = % The money make equivalen yield can now be found: MM 360 = = % 90 Leaning Objecive: Compue bond equivalen yield (BEY). Peiodic bond yields fo boh saigh and zeo-coupon bonds ae convenionally compued based on semiannual peiods, as U.S. bonds ypically make wo coupon paymens pe yea. Fo example, a zeo-coupon bond selling a $ wih a mauiy of en yeas will maue in weny 6-monh peiods. To ge he peiodic yield fo ha bond,, we can solve he equaion: o ge = 3%. $ = $1, (1 + ) This yield of 3% is, of couse, an inenal ae of eun wih semiannual compounding. In bond make eminology, i is he semiannual yield o mauiy. How do we annualize i? The convenion is o double i and call he esul he bond s yield o mauiy. In ou example, he semiannual yield o mauiy is 3%; he bond s yield o mauiy is 6%. The yield o mauiy calculaed in his way, which ignoes compounding fo he six monhs, is a bond-equivalen yield. Annualizing a semiannual yield by doubling i is puing he yield on a bond-equivalen basis. This yield o mauiy (on a bond-equivalen basis) is he acual yield quoed in makes. I is he one we use in alking o a bond deale. All ighs eseved. Waning: Copyigh violaions will be posecued. Any use of hese maeials wihou he expess wien consen of he publishe is a violaion of fedeal and/o inenaional copyigh laws. 2-47