ENG 29 Egieerig Ecoomy Lecture 9, Sectios 6.1 6.8 6.1) Preset Worth (PW) 6.2) Aual Equivalet (AE) 6.3) Future Worth (FW) 6.4) Iteral Rate of Retur (IRR) 6.5) Payback Period (PP) 6.6) Capitalized Equivalet Amout (CE) 6.7) Capital Recovery with Retur (CR) 6.8) Project Balace ENG 29 Egieerig Ecoomy
6.1) Preset Worth (PW) ( i) = F ( P F, i,) + F ( P / F, i,1) + F ( P / F, i ) PW, / 1 + PW PW ( i) = F ( P / F, i, t) t= ( i) = F t ( 1+ i) t= t t F F 1 F 2 F 1 2 ENG 29 Egieerig Ecoomy 1
Example 1: Fid the preset worth for the followig cash-flow for differet iterest rates. i PW(i) 4 4 4 4 6 1 268 2 35 22-3 3-133 4-26 1 1 2 3 4-1 ENG 29 Egieerig Ecoomy 2
6.2) Aual Equivalet (AE) ( i) = PW ( i)( A/ P, i ) AE, F F 1 F 2 F AE = t= ( i) F ( 1+ i) t t i( 1+ i) ( 1+ i) 1 1 2 ENG 29 Egieerig Ecoomy 3
Example 2: Fid the aual equivalet for the followig cashflow for i = 1%. 9 9 9 4 4 4 9 1 2 3 4-2 -3-4 4 9 1 1 1 1 1 2 1 4 9 AE( 1) = 1 + + 2 ( 1.1) ( 1.1) AE ( 1) = 61. 93 ( A/ P,1,2) ENG 29 Egieerig Ecoomy 4
6.3) Future Worth (FW) FW ( i) = F ( F P, i, ) + F ( F / P, i, 1) + + F ( F / P,,) / 1 i FW ( i) = F ( F / P, i, t) t= t F F 1 F 2 F FW ( i) = F t ( 1+ i) t= t 1 2 If we have two alterative cash-flows, A ad B, The PW PW ( i) ( i) A B = AE AE ( i) ( i) A B = FW FW ( i) A ( i) B PW(i), AE(i), ad FW(i) provide cosistet bases for compariso ENG 29 Egieerig Ecoomy 5
6.4.a) Iteral Rate of Retur (IRR) Sigle IRR IRR is the iterest rate that causes the equivalet receipts of a cash-flow to equal the equivalet disbursemet of that cash-flow. = PW i * ( ) = F t ( 1+ i) t= Example 3: Fid the IRR of the followig cash-flow. y y Slope = x2 x1 y y1 Slope = x x1 x = 12.76% t ( 12) 39 = = 51 12 13 12 51 = x 13 2 1 ( ) x 13 = 12 51 Ed of Year F t -1, 1-8 2 5 3 5 4 5 5 1,2 i PW(i) 9 12 39 13-12 ENG 29 Egieerig Ecoomy 6
Note: Last cash-flow has oly a sigle IRR ad PW(i) > for i < i* PW(i) = for i = i* PW(i) < for i > i* PW(i) The three coditios that guaratees that a cash-flow has a sigle IRR, are as follow: F < (F is the first o-zero cash-flow is disbursemet) i* Oe chage i sig i the sequece F, F 1, F 2,, F PW() > ENG 29 Egieerig Ecoomy 7
Example 4 (Cash-flow with a sigle IRR): Which of the followig cash-flows has a sigle IRR? Ed of year A B C D E -1, -1, -1, -1, 1 5-5 -1, 4,7 2 4-5 -1, -1, -1, 3 3-5 5 3,6 4 2 1,5 5 2, 5 1 2, 1,5 ENG 29 Egieerig Ecoomy 8
6.4.b) Iteral Rate of Retur (IRR) Multiple IRR Followig cash-flow has multiple IRR. PW(i) i* i* i* i ENG 29 Egieerig Ecoomy 9
6.5.a) Payback Period (PP) Without Iterest Let F F t i = The the first cost of the ivestmet the et cash-flow i period t payback period t= F t ENG 29 Egieerig Ecoomy 1
Example 5 (Payback without Iterest) Ed of year A B C -1, -1, -7 1 5 2-3 2 3 3 5 3 2 5 5 4 2 1, 5 2 2, 6 2 4, PW() 6 7, Payback Period 3 3 3 ENG 29 Egieerig Ecoomy 11
6.5.a) Payback Period (PP) With Iterest Let F F t I The the first cost of the ivestmet the et cash-flow i period t payback period, also called the discouted payback period the iterest rate t= ( 1+ ) F t i t ENG 29 Egieerig Ecoomy 12
Example 6 (Payback With Iterest): Cosider alterative A (i example 5) with i = 15% t = PW(15) = -1 the preset worth up to t =. t = 1 PW(15) = -1+5(1+.15) -1 = -565 the preset worth up to t = 1. t = 2 PW(15) = -565+3(1+.15) -2 = -338 the preset worth up to t = 2. t = 3 PW(15) = -338+2(1+.15) -3 = -27 the preset worth up to t = 3. t = 4 PW(15) = -27+2(1+.15) -4 = -92.5 the preset worth up to t = 4. t = 5 PW(15) = -92.5+2(1+.15) -5 = 7 the preset worth up to t = 5. 7 the the payback period = 5 ENG 29 Egieerig Ecoomy 13
Example 7: Cosider the followig cash-flow where is P = - 1, A = 12, = 2, ad i = 9%. Fid the discouted payback period. ( P / A, i, ') P + A ( P / A, i, ' ) P A from table the payback period = 17 ENG 29 Egieerig Ecoomy 14
6.6) Capitalized Equivalet Amout (CE) A special case of the preset worth PW(i) bases of compariso, where the cash-flow is repeated for ever. The CE(i) = PW(i) where the cash-flow exteds forever ( = ) CE ( i) = A( P / A, i, ) CE ( i) = lim A ( 1+ i) i( 1+ i) 1 ( i) CE = A i ENG 29 Egieerig Ecoomy 15
Example 8: A foudatio is cosiderig a gift to a city to build a park ad to maitai it forever. Suppose i = 8% ad the aual maiteace cost is expected to be KD 16, per year for the first 15 years, icreasig to KD 25, per year after 15 years. What is the preset that assure cotiuig maiteace o the park? 16, CE +.8 CE 9,.8 ( 8) = ( P / F,8,15) 16,.8 9,.8 ( 8 ) = + (.3153) = 235, 471 ENG 29 Egieerig Ecoomy 16
6.7) Capital Recovery with Retur (CR) A special case of the Aual Equivalet AE(i) bases of compariso, which is equivalet to the iitial cost of the ivestmet ad the salvage value. Let The P first cost of the asset F estimated salvage value estimated service life i years CR(i) Capital Recovery with retur F CR P ( A/ P, i, ) F( A/ F, i ) CR ( i) = P, ( P + F )( A/ P, i ) Fi CR ( i) =, + ENG 29 Egieerig Ecoomy 17
Example 9: A asset with a first cost of 5, has a estimated service life of 5 years ad a estimated salvage value of 1,. For a iterest rate of 1%, fid the capital recovery with retur CR? 1, 5 5, ENG 29 Egieerig Ecoomy 18
6.8.a) Project Balace (PB): without iflatio is the time profile that measures the et equivalet amout of Diars tied up. t ( 1+ i) PB( i) t Ft PB ( i) + = 1 Example 9: cosider the followig cash-flow. Fid the Project Balace at each period of the project service life? 1, 2, 8, 6, 3, i=2% 1,
6.8.b) Project Balace (PB): with iflatio Let PB f i i the project balace at costat purchasig power at t = T aual iflatio rate market iterest rate iflatio-free rate The PB'( i) T ( 1+ i' ) ( 1+ i) T t T = Ft for T =,1,, t t= This will be i term of costat purchasig power w.r.p. to t =
Example 1: cosider the followig cash-flow. Fid the Project Balace i costat-dollar domai at each period of the project service life? 1, 2, 8, 6, 3, i=21% 1, f=1% ENG 29 Egieerig Ecoomy 21