EC 2 Polika Lectue 11 Egieeig Ecoomics Pat II Auities & Sikig uds D. R. Polika
Recall 1. Daw pope cash-flow diagam (yealy, quately, etc. compoudig) 2. Do ot add values uless they ae located at the same poit o the time-lie 3. Make sue you fial aswe is ouded to eaest $0.01 (o sigificat digits) - keep all sigificat digits the calculato holds i ay itemediate steps i the calculatio 4. Make sue aswe is easoable 5. The compoud iteest elatios ae = C P C = 1+ N
Auities Auities ivolve a seies of equal paymets at egula itevals - examples iclude istallmet paymets, motgage paymets, etiemet beefits, sikig fuds, etc auities
Auities A A A A 0 1 2 3 4 = iteest ate yea = A + (1+)A + (1+) 2 A + (1+) 3 A = A [1 + (1+) + (1+) 2 + (1+) 3 ] = A (1 + u + u 2 + u 3 ) u=(1+) (1) u=a(u + u 2 + u 3 + u 4 ) multiply (1) by u (2) u - = A (u 4-1) subtact (1) fom (2) u 4-1 = A u-1 = A (1+) 4-1
Sikig uds AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 0 1 yea 2 3 utue woth: = A + (1+i)A + (1+i) 2 A + (1+i) 3 A +... + (1+i) N-1 A whee i = : iteest ate : compoudig peiod
Sikig ud AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA utue woth =? 0 1 yea 2 3 i = = A + (1+i)A + (1+i) 2 A + (1+i) 3 A +... + (1+i) N-1 A multiply each side of the equatio by (1+i) (1+i) = (1+i)A + (1+i) 2 A + (1+i) 3 A +... + (1+i) N-1 A + (1+i) N A Subtact the fist equatio fom the secod (sum of a geometic seies) (1+i) - = i = (1+i) N A - A = A [(1+i) N -1] = A [(1+ ) N -1] = A (C - 1) A( C 1 = A( C 1) = )
Sikig uds AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 0 1 yea 2 3 = A (C - 1) o = S A whee S = (C - 1) sikig fud facto C = 1 + N
Sikig uds Sikig fuds ae ofte used to accumulate sufficiet moey to eplace wo-out o obsolete equipmet. Example: How much moey would be accumulated by a sikig fud to puchase ew equipmet if $90 is deposited at the ed of each moth fo 3 yeas at a ate of etu of 10 pecet (values ae compouded mothly)? AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A = 90: = 12 N = 3: = 0.10 0 1 yea 2 3 0.1 C = (1 + ) 12(3) = 1.348 1818 S = C - 1 = 41.781 821 12 0.1/12 = S A = $3 760.36
Example Istallmet Loa Example: You wat to puchase a ew ca ad you ca affod to sped $200 pe moth fo 3 yeas compouded mothly. You have a $3 000 dow paymet. The iteest ate is 4%. What pice ca ca you affod (today)? P 1 P 2 = $3 000 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A = 200: = 12 N = 3: = 0.04 0 1 yea 2 3 C 1 0.04 12 C = (1 + 0.04 12 ) 12(3) = 1.127272 S = = 38.18157 S P 1 = A = $6 774.15 C puchase pice = 6 774.15 + 3 000 = $9 774.15 Recall: =P*C =S*A
Aalysis of Alteatives Aalysis of Alteatives: Egiees must make compaiso of alteative pojects. Defiitios: ist cost - this is the iitial cost of the puchase ad icludes items such as feight, sales tax, ad istallatio Life - this efes to the umbe of yeas of sevice the use expects fom the item o popety Salvage - this is the et sum to be ealized fom the disposal of a item o popety afte sevice. It omally icludes ecoveed value less emoval costs, feight out, etc. Alteatives must be compaed fo the same legth of time, ad eplacemets due to shot-life expectacies must be cosideed. The method of compaisos iclude: compaiso of peset woth called capitalized cost compaiso of uifom costs called aveage aual cost (o a coespodig uifom peiod of time such as aveage mothly cost) compaiso of futue woths
Example Coside the puchase of two compute-aided-desig (CAD) systems. Assume the aual iteest ate is 12% compouded yealy. System 1 System 2 Iitial cost (P 1 ) $100 000 $65 000 Maiteace ad opeatig cost ( )$4 000 / yea $8 000 / yea Salvage afte 5 yeas (P 3 ) $18 000 $5 000 P 1 0 1 2 3 4 yea P 3 5 = 1: = 0.12: N 1 = 5: N 2 = 5: N 3 = 0
Example Sikig fud System 1 System 2 Iitial cost (P 1 ) $100 000 $65 000 Maiteace ad opeatig cost ( )$4 000 / yea $8 000 / yea Salvage afte 5 yeas (P 3 ) $18 000 $5 000 P 1 0 1 2 3 4 yea P 3 5 = 1: = 0.12: N 1 = 5: N 2 = 5: N 3 = 0 = 1 + 2 + 3 = P 1 C 1 + S 2 + P 3 C 3 C 1 = (1 + 0.12) 5 = C 2 = 1.7623417 : C 3 = 1: S 2 = = 6.3528475 C 2-1 0.12
Example optio 1 = 1 + 2 + 3 = P 1 C 1 + S 2 + P 3 C 3 = 100 000(1.7623417) + 4 000(6.3528475) + 18 000 = $219,645.56 optio 2 (lowe cost optio) = 1 + 2 + 3 = P 1 C 1 + S 2 + P 3 C 3 = 65 000(1.7623417) + 8 000(6.3528475) + 5 000 = $170,374.99
Example optio 1 = 1 + 2 + 3 = P 1 C 1 + S 2 + P 3 C 3 = 100 000(1.7623417) + 4 000(6.3528475) + 18 000 = $219,645.56 P = C = $124 632.79 A = = $34,574.35 y -1 S optio 2 (lowe cost optio) = 1 + 2 + 3 = P 1 C 1 + S 2 + P 3 C 3 = 65 000(1.7623417) + 8 000(6.3528475) + 5 000 = $ 170,374.99 P = C = $96,675.34 A = S = $26,818.68 y -1
Auity Summay Give A,, N, ; solve fo, the sikig fud Give A,, N, ; solve fo P, the etiemet pla Give P,, N, ; solve fo A, the motgage, istallmet paymets N C = 1 + ; = CP C 1 S= ; = SA
Solutios of the sikig fud equatio fo N 1 = SA ; = CP; S= C ; C= 1+ Elimiate, S, C fom these equatios: N Summay give,,p,a solve fo N CP C 1 SA= CP; S= = ; C = 1+ A C = P A 1 = 1+ 1 N N P 1 = 1+ A N Solutio fo N if,, P, A ae kow N = l 1 P A l 1+
Summay Solutios of the sikig fud equatio fo = SA = CP S C 1 ; ; = ; C= 1 + Elimiate, S, C fom these equatios: N Solutio fo if N,, P, A ae kow, o diect solutio ca be foud. 1 SA CP S CP C = ; = = ; A 1 C = = 1+ P 1 A N C = + 1 N 1 = 1+ P A N N P 1 1+ = 0 A
Egieeig Ecoomics = C P compoud iteest fomula: used fo sigle paymets = S A sikig fud fomula: used fo uifom paymets C (1 + ) N compoud iteest facto Help (C - 1) S sikig fud facto = futue woth: P = peset woth: A = uifom amout = ate of etu: = compoudig time pe yea: N = yeas fo the poject All calculatios i egieeig ecoomics (at least fo ENG102) ca be doe by combiatios of these fou elatios! MAKE SURE YOU HAVE THEM ON YOUR REERENCE SHEET ad kow how to use them!!!