1. The Time Value of Money

Transcription

1 Corporate Face [ ]. The Tme Value of Moey. Compoudg ad Dscoutg Captalzato (compoudg, fdg future values) s a process of movg a value forward tme. It yelds the future value gve the relevat compoudg rate (retur rate, terest rate, growth rate). Actualzato (dscoutg, fdg preset values) s the reverse process. Whe we compute preset values, we move backward tme. Dscoutg yelds the preset value of a future value gve the relevat dscoutg rate (decle rate, terest rate, reducto rate). Compoudg (or dscoutg) s also a process of covertg flow varables (such as sales, expedtures, cash flows, all tems show come statemet or cash flow statemet)) to a stock varable (value, a balace sheet tem) two steps:. Idetfyg the cash flows provded by the asset. Compoudg or dscoutg these cash flows at the approprate growth or dscoutg rate. Table. Future Value ad Preset Value Factors Factor Formula Method of Calculato Future value of a sgle sum, FVF, ( + ) ( + ) Preset value of a sgle sum, PVF, (+ ) (+ ) Future value of a ordary auty, ( + ) t FVFA, Future value of a auty due ( + ) (+ ) Preset value of a auty, PVFA, t ( + Preset value of a auty due ( + ) t ( + t ) ) (+ ) ( + ) ( + ) ( ) + ( + ) ( ) +

2 Corporate Face [ ]. A sgle cash flow The future value factor (growth factor) s the future value of $ at terest rate for perods. The future value of a sgle sum years s determed by () FVF +, ( ) The preset value factor (dscoutg factor) s the preset value of $ receved or pad at the ed of the th year wth the rate. Dscoutg s smply the reverse of compoudg. The preset value factor s equal to () PVF, ( + ) The future value (FV) s equal to the preset value (PV) multpled by the future value factor (3) FV PV( + ) The preset value (PV) s equal to the future value multpled by the preset value factor (4) PV FV ( + ) The rate (compoudg or growth rate, dscoutg rate) s foud as FV (5) PV (6) Number of perods s gve by FV l PV l( + ) Dscrete Compoudg To reflect the frequecy of compoudg perods, two adjustmets are requred. Frst, the terest rate s coverted to a per-perod rate by dvdg the aual rate by the umber of compoudg perods a year. Secod, the umber of years s multpled by the umber of perods that occur each year, m. The calculato of future value usg dscrete compoudg s () FV PV + m m Cotuous Compoudg If the compoudg perod approaches to zero, the future value s () FV lm PV + PVe m m where e s Euler s costat, whch s approxmately,7. m

3 Corporate Face [ ].3 Equal cash flows A auty s a seres of level uterrupted cash flows occurrg at regular tervals. Ordary auty (or deferred auty) s a sequece of uterrupted, equal cash flows wth paymets (recepts) occurrg at the ed of each perod. Auty due s a sequece of uterrupted, equal cash flows wth paymets (recepts) occurrg at the begg of each perod. The future value factor for a auty determes the future value (at the ed of perod t) of the sum of captalzed cash flows of $ receved or pad at the ed of each perod (begg from t) for a specfed umber of perods. It represets the sum of a seres of future value sgle-sum factors: (7) (8) FVFA, ( + ) - + ( + ) ( + ) ( + ) + The future value of auty PMT (paymet) at the ed of perods s foud as ( + ) FV PMT The future value of auty due wth the paymets occurrg at the begg of each perod ca be calculated as ( + ) (9) FV PMT ( + ) The preset value factor for a auty s the sum of a seres of preset value factors for a sgle sum. It determes the preset value (at the ed of perod t0) of the sum of a seres of dscouted cash flows of $ receved or pad at the ed of each perod (begg from t) for a specfed umber of perods. Thus, (0) PVFA, ( + ) + ( + ) ( + ) ( + ) ( + ) The preset value of a ordary auty PMT (paymet) receved or pad at the ed of perods s foud as () ( + ) PV PMT ( + ) The preset value of a auty due wth the paymets occurrg at the begg of each perod ca be calculated as ( + ) () PV PMT ( + ) ( + ) There are four varables ths equato. If you kow the values of ay three, you ca fd the value of the forth. If you kow the preset value, the rate ad the umber of perods, the paymet (ordary auty) s (3) ( + ) PMT PV ( + ) 3

4 Corporate Face [ ] If you kow the future value, the rate ad the umber of perods, the paymet (ordary auty) s (4) PMT FV ( + ) A perpetuty s a seres of equal paymets that cotue forever (to fty). The preset value of a perpetuty receved or pad at the ed of each perod ca be foud wth a formula: (5) or (6) (+ ) (+ ) (+ ) (+ ) PV lm PMT lm PMT (+ ) (+ ) (+ ) PV PMT The preset value of a perpetuty receved or pad at the begg of each perod ca be foud as follows: (7) PV PMT ( + ).4 Growg Cash Flows The geeral equato used to fd the future value of a -perod growg auty at a costat rate g s show below: (8) (9) FV PMT ( + )... + PMT ( + g) - + PMT ( + g)( + ) ( + ) + PMT ( + g) + PMT ( + g), ( + ) 3 ( + ) ( + g) PMT g The preset value of a -perod auty growg at a costat rate g s gve by PMT PV g + g + A growg perpetuty s oe whch the cash flows crease each perod by costat rate, that cash flow ay perod s (+g) tmes larger tha the prevous cash flow. Ths process results equato +... (0) PMT + g PMT PV lm g + g 4

5 Corporate Face [ ].5 Ueve Cash Flow Streams I practce most vestmet or facal decsos volve ueve or ocostat cash flows. PV The preset value of ueve cash flows s foud as the sum of the preset values of the dvdual cash flows: CF CF CF () PV (+ ) (+ ) (+ ) NPV I practce postve cash flows (cash flows) ad egatve cash flows (cash outflows) are usually dstgushed. The Net Preset Value (NPV) of a stream of cash flows s the dfferece betwee the preset value of the flows ad the preset value of the outflows, that s (3) NPV PV flows - Pv outflows The equato for the NPV s usually preseted as follows: NCF NCF NCF () NPV ( + ) ( + ) ( + ) Cash flows may be postve or egatve. The rate s the same for all perods (tradtoal approach). It s also possble to use dfferet dscoutg rates for each perod (o-arbtrage approach). The NPV s the calculated as NCF NCF NCF (3) NPV (+ ) (+ ) (+ ) IRR Iteral rate of retur (IRR) s the dscoutg rate that makes NPV0, or, equvaletly, the rate that makes the preset value of flows equal to the preset value of outflows. Fdg IRR meas solvg the followg equato: NCF NCF NCF (4) ( + IRR) ( + IRR) ( + IRR) Ths equato s a polyomal of the th degree. If cash flows occur over more tha two perods, the IRR caot be solved drectly, ad therefore the tral-ad error method or terpolato method becomes ecessary. 5

6 Corporate Face [ ] UAS The uform aual seres (UAS) s a auty paymet that s equvalet preset value terms to a rregular cash flow patter that occurs over the same tme perod. The UAS s used specal stuatos captal budgetg. The UAS s a auty paymet (PMT) foud by solvg the equato: PMT PMT PMT ( ) ( ) (5) NPV... PMT PMT (+ ) (+ ) (+ ) (+ ) (6) Questos: Thus, the uform aual seres (PMT) s calculated as NPV NPV(+ ) PMT (+ ) (+ ) (+ ). What s the compoudg process? What s dscoutg?. What s the dfferece betwee smple terest ad compoud terest? 3. What s the dfferece betwee a ordary auty ad auty due? 4. What s the dfferece betwee a auty ad a perpetuty? 5. Iterpret the NPV. 6. Iterpret the IRR. 7. Expla the UAS. 6