MORE ON TVM, "SIX FUNCTIONS OF A DOLLAR", FINANCIAL MECHANICS. Copyright 2004, S. Malpezzi

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1 MORE ON VM, "SIX FUNCIONS OF A DOLLAR", FINANCIAL MECHANICS Copyrgh 2004, S. Malpezz I wan everyone o be very clear on boh he "rees" (our basc fnancal funcons) and he "fores" (he dea of he cash flow model). Le's look a he rees n hs handou. See also he Appendx n Shllng, pp. 653 ff. In he handou on presen values we learned how o: () Compue he fuure value of a one me depos (compoundng). (2) Compue he fuure value of a seres of perodc paymens (an annuy). (3) Compue he presen value of a one me fuure recep (dscounng). (4) Compue he presen value of a seres of fuure receps. We wll revew hese, and also how o: (5) Deermne how much mus be se asde perodcally o accumulae a ceran value (snkng fund). (6) Deermne he consan perodc paymen necessary o repay (amorze) a loan. Pay parcular aenon o how each of hese "funcons of a dollar" are relaed. We sared wh a famlar concep, "he mracle of compound neres," and buld hem up one by one. Revew of he Frs Four Funcons he key o undersandng all hs suff s smply hs: a dollar oday s worh more han a dollar omorrow. How much more? ha depends on your dscoun rae. For each of hese funcons you have several opons open o you: () Wre down he cash flows, fgure ou he weghs for each perod and add hem up. I prefer o do hs n a spreadshee when I can. (2) Shorcu formulas. hese can be useful, bu excessve focus on "wha's he formula" whou undersandng s dervaon can obscure wha s acually gong on. (3) Punch daa no your fnancal calculaor. Fas and easy, bu as wh (2) make sure you undersand wha goes on behnd he scenes. (4) Fnancal ables. hese are obsolee, and we no longer each hem. If you re neresed, see Shllng s appendx. (5) Spreadshee funcons. hs s wha you ll probably do mos ofen f you become a real esae professonal. We llusrae

2 2 hese mehods n class, bu canno rely on hem for exams (for obvous reasons). We go back and forh among mehods. Mehod yelds superor undersandng. Funcon : Fuure Value of a Dollar oday Suppose we know he presen value (a dollar oday s worh a dollar oday), and we wan o fnd he fuure value. We showed n he PV handou ha he general formula for he fuure value A of an amoun A 0 nvesed oday a rae o me s: Fuure Value FV A = A 0 *(+) hs calculaon s easy o carry ou wh any calculaor ha calculaes exponens, or usng he exponen operaor ("^") n a spreadshee. Funcon 2: Fuure Value of a Seres of Perodc Paymens By smply exendng he formula above o many paymens, A, FV = A (+) + A 2 (+) A In he specal case where each paymen A s he same, namely A: FV = A = (+ ) - (+ ) - = A Agan, one can use hs formula, 2 or he fnancal calculaor; one can use he =FV funcon n a spreadshee. FV reurns he fuure value of an nvesmen based on perodc, consan paymens and a consan neres rae: FV(rae,nper,pm,pv,ype) Rae s he neres rae per perod. In my handou I wroe A as he amoun n me (he fuure value) and A 0 as he amoun n me 0 (oday, he presen value). Somemes you wll see hs wren dfferenly, e.g. F or FV for fuure value and P or PV for presen value. Ge used o seeng dfferen noaon from dfferen sources. If you rely oo much on one se of noaon you wll be unable o handle fuure work, snce (for beer or for worse) everyone uses dfferen symbols. 2 Mah fans! Sep by sep dervaon of all he mporan formulas, ncludng he properes of nfne seres whch we use o derve hem, can be found n Sanley Grossman, Calculus (Academc Press, 977), Chaper 6, "Exponenals and Logarhms," especally he secons on applcaons o economcs and fnance.

3 Nper s he oal number of paymen perods n an annuy. Pm s he paymen made each perod; canno change over he lfe of he annuy. If pm s omed, you mus nclude he pv argumen. Pv s he presen value, or he lump-sum amoun ha a seres of fuure paymens s worh rgh now. If pv s omed, s assumed o be 0 (zero), and you mus nclude he pm argumen. ype s he number 0 or and ndcaes when paymens are due. If ype s omed, s assumed o be 0. Se ype equal o 0 f paymens are due a he end of he perod, and f due a he begnnng of he perod. (End of perod dscounng s he norm n real esae). Funcon 3: Presen Value of a One-me Recep 3 urnng he noon of compoundng on s head, f we wan o know how much an amoun receved a me s worh oday (me 0), we showed ha by solvng Funcon for A 0 we ge PV A 0 = A /(+) whch s he presen value of fuure amoun A. Calculang usng he formula s sraghforward, n a calculaor or n Excel. Bu be careful; he =PV funcon n Excel does no reurn he presen value of a one me recep; raher =PV(,n,A) reurns he presen value of a seres of equal receps A (see Funcon 4, below). Of course o calculae he presen value of a sngle gven cash flow you may also use he Appendx able, Column 4, "Presen Value of." Funcon 4(a): Presen Value of a Seres of Fuure Receps o represen PV more generally, we make use of he followng noaon, where he Greek leer Sgma means "sum" (add over all perods, from he nal perod 0 o he fnal perod ) PV A0 = =0 A (+ ) Spreadshees and fnancal calculaors wll calculae he presen values of any se of cash flows you ener n. Be careful f enerng a se of flows n a calculaor; n an exam (as n a real world problem) 's a good dea o ener he daa and calculae he PV wce, as 's easy o mspunch one of a seres of cash flows. Excel's =PV funcon calculaes he ne presen value of a seres of equal cash flows: =PV(,n,A). Excel's =NPV funcon calculaes he ne presen value of a seres of uneven (dfferen) cash flows: =NPV(, range conanng he cash flows). A few qurks: Excel (and mos oher spreadshees) assume ha he cash flow n he frs cell n he range s perod, he second cell perod 2, ec. hs mples ha f you have a perod 0 (undscouned) cash flow you mus calculae as per he followng example:

4 4 =C2+NPV(.,D2:F2), where he perod 0 cash flow s n cell C2 and cells D2, E2 and F2 conan perods, 2 and 3 respecvely, and hese are o be dscouned a 0 percen. Funcon 4(b): Presen Value of A Seres of Equal Paymens When each amoun A s he same, hen we have he presen value of an annuy (see Funcon 2 for he fuure value of same). If we can compue he fuure value we can ceranly dscoun back o he presen,.e. by compung: -(+ ) PV = A -n Snce presen values are addve, Funcon 4(b) can be handy even when some flows are dfferen; one can compue he presen value n pars, f one s careful o keep rack o avod double counng. Personally, I prefer o use a fnancal calculaor and/or a spreadshee whou he shorcu o lessen he chance of error. Alernavely you can use he =PV funcon n Excel. Funcon 4(c): Presen Value of a Perpeuy A shorcu: usng calculus can be shown ha, n he lm, f each and every A s he same (le's say A), as approaches nfny hs formula becomes: A0 = A whch s easy and requres no able or fancy calculaor. wo New Funcons Funcon 5: Snkng Fund Accumulaon How much wll have o be se asde each perod n an accoun payng gven neres o accumulae a parcular fuure value? hs would be very useful o you f you ever have o save for a parcular purpose (e.g. o pay your chldren's college uon!). In apprasal we wll use hs o compue he so-called "recapure premum" requred for a "boom up" cap rae (of whch more laer). From Funcon 2, we know how o compue he fuure value of a seres of paymens: FV = A (+) + A 2 (+) A bu hs s a smple problem where () we assume each A s he same (equal paymens), call he amoun A, and (2) we know he fuure value nsead of he

5 5 paymens. he soluon s sraghforward: FV = A * [(+) + (+) ] by rearrangng erms we ge A = FV * S where S s he "snkng fund facor" we need,.e. S = (+ ) +(+ ) - = (+ ) - S can also be found usng a fnancal calculaor. One can also fnd he requred paymen drecly usng he =PM funcon n Excel, usng he form =PM(,n,0,FV) where and n are as before; 0 s enered as he hrd elemen because hs elemen s he presen value of he expresson (whch s zero n hs case), and FV s he desred fuure value. Funcon Sx: he Consan Perodc Paymen Requred o Amorze a Loan Wha are he paymens requred o fully amorze (pay off prncpal as well as neres) a parcular loan amoun, gven a consan neres rae and a fxed erm? Rehnk he problem as follows. Each perod he lender receves an equal paymen, whch he or she dscouns o he presen (see funcon 4). he neres rae s, n hs case, he rae a whch we dscoun. Our problem hen becomes urnng PV on s head: Gven PV and an neres rae a whch o dscoun, wha seres of equal annual paymens s equvalen? Example. Wha annual paymens are requred o amorze an 8 percen, 3 year, $5,000 loan? 5000 = A/.08 + A/ A/.08 3 = A [/.08 + / /.08 3 ] = A * So A = 5000/2.577 = 5000 *.3880 = $940.7 Generalzaon. Wh a bunch of manpulaon, s no oo hard o solve for he paymen A whch yelds a presen value PV, A= PV * = (+ = PV * - ) (+ ) where s he neres rae and s he erm of he loan. One can use he formula; or any busness calculaor has such a funcon bul n drecly. In Excel use he =PM funcon: =PM(,n,PV).

6 6 ps on Applyng hese Sx Funcons One of he mos common msakes s o confuse monhly and annual amouns n calculaons or ables. I's easy enough o calculae a monhly morgage paymen and mulply by 2 o ge annual paymens on a monhly morgage; bu 's also easy enough o forge (I sll do so occasonally). A second common msake s o forge o add posve cash flows and subrac negave ones. Sounds hard o forge, bu you'd be surprsed. Anoher common msake s o forge o remember ha spreadshee formulas assume dscounng of he frs cash flow when n many problems he frs flow s undscouned (see above, dscusson of =NPV). Sudens ofen nally confuse he =PM, =PV and =NPV funcons n Excel and/or he equvalen funcons n her fnancal calculaors. Spend a lle me praccng wh hem f you are usng spreadshees. ha suggess one hng he appendx ables are good for: see f you can replcae numbers n he Appendx ables, e.g. he paymen needed o amorze a $ loan. hs s good pracce wheher you use spreadshees, fnancal calculaors, or formulas. Morgage Balance Wha s he ousandng balance on a morgage? Answerng hs queson s easy f we realze ha hs s jus he presen value of he remanng sream of neres paymens, dscouned a he rae of neres. For example, suppose we had a $00, year fxed rae morgage a 2 percen (monhly paymens). Monhly paymens would be $,028.6 (check). Wha s he ousandng balance afer 5 years? snce here are 300 MB= 028.6* 300 = = 028.6* = $97, (.0 ) paymens lef afer 5 years. Afer 5 years, he balance has declned by only $2, here are oher ways o calculae he balance. Somemes I prefer o calculae he neres and paymen of each morgage paymen separaely and explcly, f I don' have oo many o do. Excel has funcons =CUMIPM and =CUMPRINC whch compue he poron of a paymen ha s neres and prncpal, respecvely. Prned amorzaon schedules are also wdely avalable.