Valuing Long-Lived Assets

Transcription

1 Valuing Long-Lived Asses Olive Tabalski, This chape explains how you can calculae he pesen value of cash flow. Some vey useful shocu mehods will be shown. These shocus povide a good oppouniy fo valuaing cash flows. Thee ae diffeen ypes of cash flows. Some cash flows un fo a specific peiod (annuiies). Ohes have no specific end dae (pepeuiies). Fuhemoe, he valuaion of gowing cash flows will be shown and how o valuae hem. Cash flow - Sho Inoducion Fis, we define he fomula o use fo an asse ha poduces a one yea cash flow. C PV = DF C = DF is he discoun faco fo yea one and is he oppouniy cos fo a one yea invesmen. In he fomula, is he oppouniy cos of a one yea invesmen. Assume ha a one yea U.S. Teasuy is 7%. The pesen value is: C 00 PV = = = $86,9.07 Use his pimay fomula o calculae cash flow in yea wo: C PV = DF C = ( ) Suppose you ge a cash flow of $00 in yea wo (C =00) and he inees ae on a wo yea easuy is 8% pe yea. This means ha a dolla you invesed in wo yea noes will gow o.08²=$.6 by he end of wo yeas. The PV of yea- cash flow equals: C 00 PV = = = $57,0 ( ) (.08) Valuing long lived asses; Some shocus

2 Valuing cash flow in seveal peiods When you wan o bundle pesen values hough seveal peiods you can add hem vey easily. $00 $00 0 yea 00/.07 = $86,9 00/.08² = $57,0 Toal=PV = $444, Pesen Value of an invesmen ( yeas) Now we can add pesen values o he oal pesen value of an invesmen. The ule is: C C PV = = = $444. ( ) Hence we can use he addiive ule fo exended seam of cash flows: C C C PV =... ( ) ( ) This fomula is known as he discouned cash flow (DCF). A shohand way o wie his fomula is: C PV = ( ) Valuing long lived asses; Some shocus

3 Calculaion of Pesen Values and Ne Pesen Values Now we wan o calculae he ne pesen value of a eal esae invesmen. We will puchase a popey fo $ million. In he fis yea we ean $00,000 and $50,000 in he second. A he end of he second yea he boad decides o sell he popey fo $950,000. The inees ae is 5%. Peiod () 0 Puchase pice Income Payoff 950 Toal C 0 = C = 00 C = 00 Example figues: housands $ NPV = C 0 C C ( ) NPV =, ,00 (.05) $ $ yea -$ /.05 = $ /.05² = $997 7 Toal=NPV = $9 97 Calculaion showing he ne pesen value of he invesmen In his case, he invesmen is pofiable due o he posiive NPV. Valuing long lived asses; Some shocus

4 Valuaion of Annuiies and Pepeuiies Pepeuiies Pepeuiy is a consan and neve ending seam of paymens. The consan paymen sas a a fixed dae. P esen value of pepeuiy = C Example: Pesen value pepeuiies A fame keeps an inheied gold donkey in his sable which poduces gold in wondous way. Afe all coss ae paid, he gold donkey poduces $,000 each yea. Anohe fanasic aibue of he gold donkey is is immoaliy. The fame knows ha a gold donkey of his kind has go o be high in value. The pice of he gold donkey would be infiniely high. So he ies o sell he gold donkey. The fame invies many offes, bu no offe was highe han $50,000. The fame asks a poenial buye why he didn offe moe han $50,000 fo he gold donkey. Wha is he eason? If he buye inves fo example $60,000 on bank accoun wih he cuen make inees ae of % pe annum, he will ge back $,00. Howeve he gold donkey poduces only $,000 pe annum. Thus i is moe favoable o pu $60,000 ino a bank accoun insead of buying a gold donkey. If he buye invess $40,000 in a bank accoun, hen he eceives $800 pe annum wih he cuen make inees ae of % pe annum. The gold donkey poduces, howeve,,000 pe annum. Thus, i is moe favoable o buy a gold donkey fo 40,000 insead of invesing he money in a bank accoun. Bu wha is he acual value of his fabulous gold donkey? We can use he following fomula: C P esen value of pepeuiy =,000 P esen value of pepeuiy = = $50, Valuing long lived asses; Some shocus 4

5 Annuiies An annuiy poduces a fixed sum each yea fo a specified numbe of yeas. Examples of annuiies ae an insallmen cedi ageemen o he equal-paymen house mogage. The geneal fomula o value annuiy ha pays $ a yea fo each of yeas saing in yea is: P esen value of annuiy = ( Example: Pesen value annuiies Volvo offes a special offe fo hei cusomes. When you buy a new Volvo you pay $6,000 a he end of each of he nex five yeas and no cash down. The inees ae is 7%. How much money does he ca eally cos? To solve he poblem we could use ou sandad discouned cash flow fomula (DCF). $6 000 $6 000 $6 000 $6 000 $ yea $6,000/.07 = $5 607 $6,000/.07 = $5 4 $6,000/.07 = $4 898 $6,000/.07 = $4 577 $6,000/.07 = $4 78 Toal=PV = $4 60 Calculaion showing he yea-by-yea pesen value of paymens Bu we have leaned a fomula fo handling his kind of poblem: P esen value of annuiy = ( PV = 6,000 = 6, = $4, (.07) You can find he annuiy facos a he end of you wokbook (Appendix A, Table ). Valuing long lived asses; Some shocus 5

6 Fomulay of pepeuiies and annuiies. Thee-yea annuiy Yea: 4 5 Pesen Value $ $ $. Pepeuiy A $ $ $ $ $. Pepeuiy B $ $ $ Impoan fomulas ( ( ) Pesen Value Annuiies Due When we use he sandad annuiy fomula, we assume ha he fis paymen was made a he end of yea one. Bu wha will happen when he fis paymen is a he beginning of yea one? An annuiy due is woh () imes he value of an odinay paymen. P esen value of annuiy due = ( ( ) ( Annual Paymens: Example We eceive a mogage loan. The pesen value of he loan is $00,000 and he bank equies us o epay i in equal annual insallmens ove he nex 0 yeas. PV = mogage paymen 0 yea annuiy faco = $00,000 Mogage paymen = $00,000 / 0 yea annuiy faco Suppose ha he inees ae is 4%... 0 yea annuiy faco = = (.4) Mogage paymen = $ 00,000 / 6.6 = $45,97 This example is an amoizing loan. This means ha pa of he egula paymen ae is used o pay fo educion and he ohe pa fo inees. Valuing long lived asses; Some shocus 6

7 Fuue value of an annuiy: Example When we wan o calculae he fuue value of a level seam of paymens, we have o calculae he pesen value and hen muliply by (). The fomula is ( Fuue value of annuiy = ( = ( In his example, we hink abou buying a new ca in 5 yeas. We know ha we can save $5,000 evey yea and we ean 8% inees on hese savings. So how much money do we have afe saving fo 5 yeas? As fis we use he well-known fomula fo deemining he pesen value of an annuiy. The pesen value is PV PV = $5,000 5annuiy faco = $5, = $9,965 Now we have o hink abou how much money we have if we inves $9,965 oday. Value of heend of yea 5 == $9, = $9,5 Afe 5 yeas we should be able o buy a nice ca. Moe Shocus: Gowing Pepeuiies and Annuiies Ofen we have o value cash flows ha have consan gowh. In his secion, you will see some shocus o value pepeuiies and annuiies. In he oveview, g is he gowh ae. The following oveview shows you he ule fo gowing pepeuiies and annuiies. Yea: Pesen Value. Gowing Theeyea annuiy $ $x(g) $x(g) g ( g )(. Gowing Pepeuiy A $ $x(g) $x(g) $x(g) $x(g) 4... g. Gowing Pepeuiy B $x(g) $x(g) 4... ( g )( ) Valuing long lived asses; Some shocus 7

8 Fomulay Compendium wih all elevan fomulas fo execises. Cash flow, $ Yea: Pesen Value Pepeuiy peiod annuiy... -peiod annuiy due... ( )( ) ( ( Gowing pepeuiy x(g) x() - x(g) - x(g)... -gowing annuiy x(g) x(g) - x(g) - g ( g) g g ( Valuing long lived asses; Some shocus 8