Recap from Last Week Time Value of Moey We leared: $ cash today is preferred over $ a year from ow there is time value of moey i the form of willigess of baks, busiesses, ad people to pay iterest for its use So this rises the questio of how to compare various cash flows dispersed over the time horizo? EW EW X EW 3 3 4 5... EW A A A A A EW 3 4 5... Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
Recap from Last Week Ecoomic Equivalece We leared: So this rises the questio of how to compare various cash flows dispersed over the time horizo? Ecoomic equivalece exists betwee cash flows that have the same ecoomic effect ad therefore be traded for oe aother Ecoomic Equivalece refers to the fact that: ay cash flow - whether a sigle paymet or a series of paymets - ca be coverted to a equivalet cash flow at ay poit i time. Ad that equivalece depeds o the iterest rate To covert the cash flows - whether a sigle paymet or a series of paymets - to a equivalet cash flow at ay poit i time, we eed egieerig ecoomy factors This week we will focus o the derivatios of the most commoly used egieerig ecoomy factors that take time value of moey ito accout Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
Sigle-Paymet Factors (F/P ad P/F) Derivatio The most fudametal factor i egieerig ecoomy is the oe that determies the amout of moey F accumulated after periods from a sigle preset worth P, with iterest compouded oe time per period. P Ed of time period Time Note That F = P + Pi = P( + i) 3 4 5 Ivest amout P P F Ed of time period Ed of time period 3 4 5 Time Note That F = F + F i = F ( + i) = P( + i)( + i) = P( + i) F F Ivest amout P Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 3 / 46
Sigle-Paymet Factors (F/P ad P/F) Derivatio P Ed of time period Ed of time period 3 4 5 Time Note That F = P( + i) F Ivest amout P P Ed of time period F Ed of time period Ed of time period 3 4 5 Time Note That F 3 = F + F i = F ( + i) = P( + i) ( + i) = P( + i) 3 Ivest amout P F F F 3 By Iductio F = P( + i) Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 4 / 46
Sigle-Paymet Factors (F/P ad P/F) Derivatio The factor ( + i) is called the sigle paymet compoud amout factor, or simply F/P factor. This coversio factor yields the future amout F of a iitial amout P after years at iterest rate i, whe it is multiplied by P. P = give i = give... - - Time F =? Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 5 / 46
Sigle-Paymet Factors (F/P ad P/F) Derivatio Reverse the situatio to determie P value for stated amout F that occurs periods i the future (+i) is the sigle paymet preset worth factor or the P/F This factor yields the preset amout P of a give future amout F after years at iterest rate i, whe it is multiplied by F. P =? i = give... - - Time F = give Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 6 / 46
Sigle-Paymet Factors (F/P ad P/F) Stadard Notatio The two factors derived so far are for sigle paymets. A stadard otatio is developed for all factors. icludes: two cash flow symbols, iterest rate ad umber of periods geeral form is (X/Y, i, ) X is the cash flow that is sought Y is the cash flow that is give i is the iterest rate i percet is the time periods ivolved (F/P, 8%, ) represets the factor to calculate future value F accumulated i periods at 8% iterest rate for a iitial dollar ivested at time zero. P (F/P, 8%, ) gives you the future F value. Tables of factors are available from.5 to 5% iterest rates ad time periods to large values. For a give iterest rate, the factor value is foud at the itersectio of the factor ame ad. (P/F, 8%, ) = = =.463 (+.8).8 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 7 / 46
Sigle-Paymet Factors (F/P ad P/F) Stadard Notatio Example A idustrial egieer receives a bous of $,, ad decides to ivest the whole amout to the bak for years at 8% per each year. Fid the amout amout of moey he will get years later. By Formula F =?, P =,, i = 8 percet ad = F = P( + i) =, ( +.8) By Stadard Factor =, (.8) =, (4.66957) = $55, 93.5 F =?, P =,, i = 8% per year ad = F = P(F/P, 8%, ) =, (4.66) = $55, 93 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 8 / 46
Sigle-Paymet Factors (F/P ad P/F) Stadard Notatio I Class Work Forema family decided to start ivestig to their so Eric s college fud. They started it by depositig $, o Jauary, 8 to the Washigto Mutual Bak. They pla to deposit $, o Jue 3, ad $3, o Jue 3, 3. If they maage to ivest accordig to that pla, how much moey will they have o o Jauary, 8 if the iterest is take as 8% per each year. i =8% F=? 3 4 5 6 7 8 9 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 9 / 46
Sigle-Paymet Factors (F/P ad P/F) i =8% Stadard Notatio F=? 3 4 5 6 7 8 9 3 Ed-of-period covetio All cash flows occur at the ed of the iterest period. F=? i =8% 3 6 4 5 7 8 9 3 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
By Formula: Sigle-Paymet Factors (F/P ad P/F) Stadard Notatio F = 3(.8) 4 + (.8) 7 + (.8) = $9668.4 OR F = + (.8) 3 (.8) 3 + 3 (.8) 4 = $9668.4 OR By Stadard Notatio: F = 3(F/P, 8%, 4) + (F/P, 8%, 7) + (F/P, 8%, ) = 3(.365) + (.738) + (.589) = $9668 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
Uiform-Series Factors : (P/A ad A/P) Derivatio: P/A Factor The equivalet preset worth P of a uiform series A of ed-of-period cash flows ca be determied by cosiderig each A value as a future worth F ad calculatig its preset worth with the P/F factor. P =? i = give - -... Time A = give P = A ( + i) = A + A ( + i) + + A ( + i) ( + i) + ( + i) + + ( + i) + ( + i) + A ( + i) Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
Uiform-Series Factors : (P/A ad A/P) Derivatio: P/A Factor P P + i P + i P = A ip + i (+i) i(+i) = A ( + i) + ( + i) + + ( + i) + ( + i) = A ( + i) + ( + i) 3 + + ( + i) + ( + i) + ( + i) + ( + i) = A ( + i) + ( + i) P = A ( + i) i ( + i) = A i( + i) is the uiform series preset worth factor,p/a factor used to calculate the equivalet preset worth P i year for a uiform ed-of-period series of A values begiig at the ed of period ad extedig for periods. Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 3 / 46
Uiform-Series Factors : (P/A ad A/P) Derivatio: A/P Factor To reverse the situatio, the preset worth P is kow ad the equivalet uiform-series amout A is sought. The first A value occurs at the ed of period. The we have: i(+i) (+i) A i( + i) = P ( + i) is the capital recovery factor or A/P factor. It calculates the equivalet uiform aual worth A over years startig from the ed of year for a give P i year. CAUTION! These formulas are derived with the preset worth P ad the first uiform aual amout A oe period apart. Therefore, the preset worth P must always be located oe period prior to the first A. Stadard otatios for these two factors are: (P/A, i%, ) ad (A/P, i%, ). Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 4 / 46
Uiform-Series Factors : (A/F ad F/A) Derivatio: A/F Factor This time, the cash flow diagram we are dealig with is: F = give i = give - -... Time A =? We already kow that: A i( + i) = P ( + i) F i( + i) = ( + i) ( + i) i = F ( + i) Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 5 / 46
Uiform-Series Factors : (A/F ad F/A) Derivatio: A/F Factor i (+i) is the sikig fud factor or A/F factor, used to determie the aual series that is equivalet to a give future worth F. CAUTION! The uiform series A begis at the ed of period ad cotiues through the period of the give F. Therefore, the future worth F is at the same period as the last A. Obviously, (+i) i is the F/A factor. Whe multiplied by the give uiform aual amout A, it yields the future worth of the uiform series. The stadard otatios for these two factors are: (F/A, i%, ) ad (A/F, i%, ). Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 6 / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Defiitio A arithmetic gradiet is a cash flow that either icreases or decreases by a costat amout. The cash flow, chages by the same amout each period. The amout of decrease or the icrease is the gradiet or G. Therefore, it is composed of base amout ad gradiet part. i = give - -... Time Base Base Base+G Base+(-3)G Gradiet Base+(-)G Base+(-)G The cash flow i period (CF ) is give as: CF = base amout +( )G Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 7 / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) i = give - -... Separatio Time Base Base Base+G Base+(-3)G Gradiet Base+(-)G Base+(-)G i = give 3 -... - G G (-)G (-)G CAUTION:Covetioal Gradiet The gradiet begis betwee years ad. Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 8 / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Derivatio: P/G Factor P = G ( + i) + ( )G + G ( + i) ( + i) 3 + + ( )G ( + i) = G ( + i) + ( + i) 3 + + ( + i) + ( + i) P( + i) = G ( + i) + ( + i) + + ( + i) + ( + i) Pi = G ( + i) + ( + i) + + ( + i) + ( + i) Pi G ( + i) = G ( + i) + ( + i) + + ( + i) + ( + i) + ( + i) G ( + i) Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 9 / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Derivatio: P/G Factor (+i) i i(+i) (+i) P = G i ( + i) i( + i) ( + i) is the arithmetic-gradiet preset worth factor or P/G factor that coverts the arithmetic gradiet (without base amout) for years to the preset worth at year. The stadard otatio is (P/G, i%, ). i = give 3 - -... G Time P G (-3)G (-)G (-)G i = give - -... Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Derivatio: A/G Factor The equivalet uiform aual series A for a arithmetic gradiet G is foud by multiplyig (P/G, i%, ) by (A/P,i%,) I equatio form: A = G(P/G, i%, )(A/P, i%, ) = G(A/G, i%, ) A = G ( + i) i( + i) i i( + i) ( + i) ( + i) = G i ( + i) i (+i) is the arithmetic-gradiet uiform-series factor ad is idetified by (A/G, i%, ). This factor coverts: Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Derivatio: A/G Factor i = give 3 -... - G G Time (-3)G (-)G (-)G A... - - i = give A F/G factor (arithmetic-gradiet future worth factor)ca be foud by multiplyig P/G ad F/P factors. The resultig (F/G, i%, ) is: Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Derivatio: F/G Factor F = G(P/G, i%, )(F/P, i%, ) = G(F/G, i%, ) = G ( + i) i i( + i) ( + i) ( + i) ( ( + i) ) = G i i The total preset worth P T of a gradiet series must cosider the base ad the gradiet separately. Let P A be the preset worth of the base amout; uiform series amout A startig from the ed of period extedig through period. Let P G be the preset worth of a icreasig gradiet ad P G be the preset worth of a decreasig gradiet. Therefore: P T = P A + P G ad P T = P A P G for icreasig ad decreasig gradiet series, respectively. Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 3 / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Example Example Three couties i Florida agreed to pool tax resources already desigated for couty-maitaied bridge refurbishmet. At a recet meetig, the couty egieers estimated that a total of $5, will be deposited at the ed of ext year ito a accout for the repair of old bridges throughout the three-couty area. Further, they estimate the deposits will icrease by $, per year for oly 9 years thereafter, the cease. Determie the equivalet preset worth if couty fuds ear iterest at a rate of 5% per year. Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 4 / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Example P T =? i = 5% 3 4 5 6 7 8 9 P A =? i = 5% 3 4 5 6 7 8 9 5 6 7 8 9 3 4 P T = P A + P G, we have G = ad A = 5. 5 P G =? 5 5 5 5 5 5 5 5 5 i = 5% 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 5 / 46
Arithmetic Gradiet Factors : (P/G, A/G ad F/G) Example Solutio by Stadard Notatio P T = 5(P/A, 5%, ) + (P/G, 5%, ) = 5(7.77) + (3.65) = 76, 5 => $7, 6, 5 Solutio by Formulas ( + i) P T = 5 i( + i) + i (.5) = 5.5(.5) +.5 ( + i) i( + i) ( + i) (.5).5(.5) (.5) = 5(7.77) + (3.65) = 76, 5 => $7, 6, 5 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 6 / 46
Geometric Gradiet Series Factor Derivatio Cash flow series that icrease or decrease from period to period by costat percetage. g is the costat rate of chage i decimal form, by which amouts chage The series starts at the ed of year, with amout A, which is ot cosidered as a base amout i = give - -... A A (+g) A (+g) (-3) A (+g) (-) A (+g) (-) Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 7 / 46
Geometric Gradiet Series Factor Derivatio The total preset worth P g for the etire cash flow series ca be derived as: P g = A ( + i) ( + g) P g ( + i) + A ( + g) ( + i) = A ( + i) = A + g ( + i) ( + g) ( + g) + A ( + i) + + A ( + i) + ( + g) ( + i) P g ( + g) ( + i) ( + g) ( + g) + + + ( + i) ( + i) ( + g) ( + g) + + + ( + i) 3 ( + i) + ( + g) = A ( + i) + ( + i) P g (g i) = A ( + g) ( + i) ( + g) ( + i) + Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 8 / 46
Geometric Gradiet Series Factor Derivatio Therefore, whe i g; (+g) (+i) P g = A (i g) whe i = g; P g = A ( + i) + ( + i) + + ( + i) + ( + i) = A ( + i) These two factors, depedig o the case whether g = i or ot, will trasform the geometric-gradiet series startig at the ed of period with amout A that is chagig with costat rate g to the the preset worth P g at time. The stadard otatio for the factor is (P/A, g, i, ). Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 9 / 46
Geometric Gradiet Series Factor Derivatio P g i = give - -... A A (+g) A (+g) (-3) A (+g) (-) P g i = give - -... A (+g) (-) A (-g) (-) A (-g) (-) A (-g) (-3) A (-g) A Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 3 / 46
Shifted Series Cautios o Usig Derived Factors CAUTION o usig P/A factor: (P/A, i%, ) The preset worth P ad the first uiform aual amout A are oe period apart. Therefore, the preset worth P must always be located oe period prior to the first A. P =? i = give - -... Time A = give Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 3 / 46
Shifted Series Cautios o Usig Derived Factors CAUTION o usig F/A factor: (F/A, i%, ) The uiform series A begis at the ed of period ad cotiues through the period. Therefore, the future worth F is at the same period as the last A. F =? i = give - -... Time A =give Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 3 / 46
Shifted Series Cautios o Usig Derived Factors CAUTION o usig P/G factor: : (P/G, i%, ) The gradiet begis betwee years ad, cotiues through the period. Therefore, the preset worth of a arithmetic gradiet will always be located two periods before the gradiet starts. P=? i = give 3 -... - Time G G (-3)G (-)G (-)G Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 33 / 46
Shifted Series Cautios o Usig Derived Factors CAUTION o usig A/G factor: : (A/G, i%, ) The gradiet begis betwee years ad, cotiues through the period. Therefore, the equivalet aual series of a arithmetic gradiet will start from period ad cotiue through the period. i = give 3 - -... G G (-3)G (-)G Time (-)G A... - - i = give Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 34 / 46
Shifted Series Examples Most estimated cash flow series do ot fit exactly the series for which the factors are derived so far. Therefore, we eed to combie or shift the give series. Reumberig the cash flow diagram is a good practice. Example The egieerig compay just purchased ew CAD software for $5, ow ad aual paymets of $5 per year for 6 years, startig 3 years from ow for aual upgrades. What is the preset worth of the paymets if the iterest rate is 8% per year. P =? i = 8% 3 4 5 6 7 8 5 5 5 5 5 5 5 P = 5 + P A Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 35 / 46
Shifted Series Examples P =? 3 4 i = 8% 5 6 7 8 3 4 5 6 5 P' A 5 5 5 5 5 5 P = 5 + P A (P/F, 8%, ) = 5 + 5(P/A, 8%, 6)(P/F, 8%, ) = 5 + 5(4.69)(.8573) = $698.6 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 36 / 46
Shifted Series Examples Example The average ispectio cost o robotics maufacturig lie has bee tracked for 8 years. Cost averages were steady at $ per completed uit for the first 4 years, but have icreased cosistetly by $5 per uit for each of the last 4 years. Fid the preset worth equivalet of the cost averages assumig that the applicable iterest rate is 8%? P=? i = 8% 3 4 5 6 7 8 5 5 3 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 37 / 46
Shifted Series Examples P A =? i = 8% 3 4 5 6 7 8 A = P G =? 3 4 i = 8% 5 6 7 8 3 4 5 P' G G=5 5 5 P = P A + P G = P A + P G (P/F, 8%, 3) = (P/A, 8%, 8) + 5(P/G, 8%, 5)(P/F, 8%, 3) = (5.7466) + 5(7.374)(.7938) = $867.7 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 38 / 46
Shifted Series Examples Example Compute the equivalet aual series i years through 7 for the cash flow estimates give below whe applicable iterest rate is 8%. 3 4 5 6 7 5 5 5 i = 8% 7 9 3 A = 5 + A G i = 8% 3 4 5 6 7 3 4 5 4 G= 6 8 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 39 / 46
Ca we use the A/G factor? Shifted Series Examples A=(A\G, 8%, 5) 3 4 5 6 7 3 4 5 P G i = 8% P' G 4 G= i = 8% 6 8 3 4 5 6 7 G= A = 5 + P G (A/P, 8%, 7) 3 4 4 6 5 8 = 5 + P G (P/F, 8%, )(A/P, 8%, 7) = 5 + (P/G, 8%, 5)(P/F, 8%, )(A/P, 8%, 7) = 5 + (7.374)(.8573)(.97) = 74. Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 4 / 46
Shifted Series Examples The use of arithmetic gradiet factors is the same for icreasig ad decreasig gradiets, except that i the case of decreasig gradiets the followig are true: The base amout is equal to the largest amout, the amout at period. The gradiet amout is subtracted from the base amout. 3 G is used i the computatios. Example Compute the preset worth at year at i = 8% per year for the cash flow show below: 3 4 5 6 7 8 9 P=? i = 8% 4 4 44 46 48 5 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 4 / 46
Shifted Series Examples 3 4 5 6 7 8 9 i = 8% 3 4 5 6 P A P' A A=5 P G i = 8% G= P' 8 G 46 3 4 5 6 3 4 5 6 7 CAUTION o Sigs of Cash Flows 8 9 So far we have all the cash flows o oe side of the cash flow diagram. Therefore we do ot cosider the sigs of the cash flows. Wheever we have cash flows o both sides of the cash flow diagram, it is good practice to take the oes o the upside as (+) ad dowside (-). P P = P A P G (if you do ot cosider sigs) = P G P A (Cosiderig upside (+)&dowside ( )) Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 4 / 46
Shifted Series Examples P A = P A (P/F, 8%, 4) = 5(P/A, 8%, 6)(P/F, 8%, 4) = 5(4.69)(.735) = $6, 989.575 P G = P G (P/F, 8%, 4) = (P/G, 8%, 6)(P/F, 8%, 4) = (.533)(.735) = $, 546.778 P P = P A P G = 6, 989.575, 546.778 = $5, 44.3795 (Must idicate it is poitig dowward) = P G P A =, 546.778 6, 989.575 = $5, 44.3795 (No eed to idicate directio) Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 43 / 46
Shifted Series Examples I Class Work 3 Fid the preset worth ad equivalet aual series for the followig cash flow sequece: i = 8% 5 4 3 3 4 5 6 7 8 9 53 35 4 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 44 / 46
i = 8% Shifted Series Examples 5 4 3 3 4 5 6 7 8 9 53 35 4 P' A A =5 i = 8% 3 4 5 6 7 8 9 3 4 5 A = P A i = 8% 3 4 5 6 7 8 9 3 4 5 5 G 3 = G =5 5 4 P G P' G P = P GREEN + P BLUE P RED Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 45 / 46
Shifted Series Examples P GREEN = (P/F, 8%, ) + (P/F, 8%, ) = (.489) + (.397) = 86 P BLUE = P A (P/F, 8%, 5) P G (P/F, 8%, 5) = (P/F, 8%, 5)5(P/A, 8%, 5) (P/G, 8%, 5) = 5(3.997)(.686) (7.374)(.686) = 8569.5 P RED = P A + P G = (P/A, 8%, 5) + 5(P/G, 8%, 5) = (3.997) + 5(7.374) = 67.6 P = 86 + 8569.5 67.6 = 76. A = 76.(A/P, 8%, ) = 76.(.37) = 3.4 Dr.Serha Dura (METU) IE 347 Week Idustrial Egieerig Dept. 46 / 46