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 facor? DF = = ( + r ) (.05) = 0.89 c. Given hese one- and wo-year discoun facors, calculae he wo-year annuiy facor. The one and wo-year discoun facors of (a) and (b) ell us ha he ineres rae for boh periods = 0,5%. Thus o calculae he wo-year annuiy facor: Annuiy _ facor = = =, 74 ( + ) 0,05 0,05( + 0,05) r r r An alernaive way o calculae he wo-year annuiy facor is: AF = DF + DF = 0,905+0,89=,74 d. If he PV of $0 a year for hree years is $4.65,wha is he hree-year annuiy facor? Using he forula of he Presen Value of annuiy: PVannuiy = C, where : r r( r) + Annuiy _ facor = r r( + r)
Hence he hree-year annuiy facor is: $4.65 = $0 [AF 3 ] AF 3 =.465 d. Fro your answers o (c), and (d), calculae he hree-year discoun facor. AF 3 = DF + DF + DF 3.465 = 0.905 + 0.89 + DF 3 DF 3 = 0.74 5. A achine coss $380,000 and is expeced o produce he following cash flows: Year 3 4 5 6 7 8 9 0 CF(000s) 50 57 75 80 85 9 9 80 68 50 If he cos of capial is percen, wha is he achine s NPV? Period 0 3 4 5 6 7 8 9 0 Cash Flow - 380 50 57 75 80 85 9 9 80 68 50 Discoun rae - % % % % % % % % % % Discoun facor - 0,89 0,80 0,7 0,64 0,57 0,5 0,45 0,40 0,36 0,3 DCF - 380,00 44,64 45,44 53,38 50,84 48,3 46,6 4,6 3,3 4,5 6,0 NPV 3,70 ***Reeber no o discoun he iniial invesen (ade in period 0 = oday). The above able is equivalen of doing: NPV = + C o 0 = 0 C (.) $50,000 $57,000 $75,000 $80,000 $85,000 = $380,000 + + + + + 3 4 5..... $9,000 $9,000 $80,000 $68,000 $50,000 + + + + + = $3,696.5 6 7 8 9 0.....
6. Mike Polanski is 30 years of age and his salary nex year will be $40,000. Mike forecass ha his salary will increase a a seady rae of 5% per annu unil his reireen a age 60. a. If he discoun rae is 8%, wha is he presen value of hese fuure salary payens? Le S = salary in year 30 30 S 40,000 (.05) PV = = = (.08) = (.08) 30 30 (40,000 /.05) 38,095.4 = = = (.08 /.05) = (.086) = 38,095.4 = $760,66.53 30 0.086 0.086 (.086) Alernaively we could have used he growing annuiy forula: ( + g ) ( ) T PV = C T r g r g r + T ( +.05) = 40, 000 = 760, 66.53 T.08.05.08.05 ( +.08) b. If Mike saves 5% of his salary each year and invess hese savings a an ineres rae of 8%, how uch will he have saved by age 60? PV(salaries) x 0.05 = $38,033.3 Fuure value = $38,033.3x (.08) 30 = $38,74.3 By saving 5% of his salary each year, by he age of 60 Mark will have $38,74.3. c. If Mike plans o spend hese savings in even aouns over he subsequen 0 years, how uch can he spend each year? By using he forula of he Presen Value of Annuiy: PV = C + r r ( r) 3
$ 38,74.3 = C 0.08 0.08 (.08) 0 C = $ 38,74.3 = $38,980.30 0.08 0 0.08 (.08) 9. (Tuorial) As winner of a breakfas cereal copeiion, you can choose one of he following prizes: a) $00,000 now. PV = 00,000 b) $80,000 a he end of five years. PV = 80,000 (.) 5 = 0,37 c) $,400 a year forever PV =,400/0. = 95,000 d) $9,000 for each of 0 years PV = 9,000 0. 0. (.) 0 07,354 e) $6,500 nex year and increasing hereafer by 5% a year forever. PV = 6,500/(0.-0.05) = 9,857. If he ineres rae is %, which is he os valuable prize? $9,000 for each of 0 years is he os valuable prize, since i has he highes PV.. Kangaroo Auos is offering free credi on a new $0,000 car. You pay $,000 down and hen $300 a onh for he nex 30 onhs. Turle Moors nex door does no offer free credi bu will give you $,000 off he lis price. If he rae of ineres is 0% a year, (abou 0.83 a onh) which copany is offering a beer deal? 4
Kangaroo Auos Based on he annuiy forula: PV = C + r r ( r) of he payens o Kangaroo Auos: we calculae he presen value $,000 + $300 = $8,938 30 0.0083 0.0083 (.0083) Turle Moors A car fro Turle Moors coss $9,000 cash ($0,000-$,000). Therefore, Kangaroo Auos offers he beer deal, i.e., he presen value of he payens of he Kangaroo Auos deal is lower han he one of Turle oors.. 8. Wha would you prefer? a. An invesen paying ineres of % copounded annually. b. An invesen paying ineres of.7 percen copounded seiannually. c. An invesen paying.5 percen copounded coninuously. Work ou he value of each of hese invesens afer, 5 and 0 years. Hin: For he ease of your calculaions use $,000 as he aoun of your invesen. Le A represen he invesen a percen, copounded annually. Le B represen he invesen a.7 percen, copounded seiannually. Le C represen he invesen a.5 percen, copounded coninuously. r And reeber ha an annually copounded rae is copued as: + Afer one year: FV A = $,000 ( + 0.) = $,0.00 FV B = $,000 ( + 0.0585) = $,0.4 FV C = $,000 e (0.5 ) = $,.87 Leaving aside he aoun of he invesen we can also copare he raes of reurn as follows: 5
r 0, A: + = + =, 00% B: r 0,7 + = + =, 04% C: r e = = 0,5 e,8% Afer five years: FV A = $,000 ( + 0.) 5 = $,76.34 FV B = $,000 ( + 0.0585) 0 = $,765.67 FV C = $,000 e (0.5 5) = $,777.3 Afer weny years: FV A = $,000 ( + 0.) 0 = $9,646.9 FV B = $,000 ( + 0.0585) 40 = $9,79.9 FV C = $,000 e (0.5 0) = $9,974.8 Through all he periods he preferred invesen is C because i reurns a higher fuure value of he invesen. CHAPTER 3: Challenge quesions 38. You own a pipeline which will generae a $ illion cash reurn over he coing year. The pipeline s operaing coss are negligible, and i is expeced o las for a very long ie. Unforunaely, he volue of oil shipped is declining, and cash flows are expeced o decline by 4% per year. The discoun rae is 0%. a) Wha is he presen value (PV) of he pipeline s cash flows if is cash flows are assued o las forever? PV = /(0.-(-0.04)) = 4.9 b) Wha is he PV of he cash flows if he pipeline is scrapped afer 0 years? PV = 0.4 0.4 ( 0.96) (.) 0 0 3.35 6