Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

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1 Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig preset values from aual amouts. Calculatig future ad preset values from gradiet amouts. Calculatig preset value of a future perpetual amouts. Calculatig deferred auities. 2 Learig Objectives Uit Upo completio of this uit you should uderstad: Calculatig future values from aual amouts. Calculatig preset values from aual amouts. Calculatig future ad preset values from gradiet amouts. Calculatig preset value of a future perpetual amouts. Calculatig deferred auities. 3

2 Itroductio The prior chapter covered sigle paymet fuctios where a cash iflow occurred at oe poit i time ad a cash outflow occurred at a secod poit i time. May fiacial trasactios have elemets that occur at multiple poits i time. These ca iclude: Equal aual cash flow. Liear gradiet cash flow. No liear gradiet cash flow. Mixed aual cash flow. 4 Itroductio This chapter covers three types of multiple paymet situatios: Equal aual amouts (A) equal dollar amouts flow ito or out of a ivestmet or project each year. Liear gradiet amouts (G) dollar amouts flowig ito or out of a ivestmet or project icrease/decrease each year by a costat amout (liear). Mixed aual amouts differig dollar amout flow ito ad/or out of a ivestmet or project each year. 5 Equal Aual Amouts Assumptios for equal aual amout aalysis iclude: Cash flow occurs at the ed of each year. All cash flows are equal ad occur each year. Note that most iterest table such as those i appedix B of the Bowma text are based o ed of year trasactios. Iterest table are available that use the begiig or middle of time periods. If usig a time value of moey fuctio o your calculator, check the maual to determie if the time basis is ed of period (year) or some other basis. 6 2

3 Equal Aual Amouts Notatios used for time value of moey calculatio Future Value (oe time occurrece) F Preset Value (oe time occurrece) P Equal Aual Amout A Cash flow diagrams represet aual amouts as equal legth lies as illustrated i Figure 5 : 7 Future Value Calculatios Future value for a equal aual amout is determied by the followig equatio: ( i ) + F = A i Where: F = Future Value ($) A = Aual Amout ($) = Time (years) i = Iterest (% per year) 8 Future Value Calculatios Solutio methods for fidig future values: Use the F/A colum o a Iterest Factors table (Bowma text appedix B, page 580). Notatio F/A is iterpreted as Fid F give A. Notatio (F/A,, i) is iterpreted as Fid F give A for years at i iterest rate. Use the Excel fuctio FV(rate, per, pmt, pv, type) Use the formula ad calculator. Note that the cash outflows are etered as a egative umber. 9 3

4 Future Value/Aual Example Your pla is to save $00 at the ed of each year at 8% iterest. What will be the size of the accout i 0 years? F = A ( F/A,0,8% ) = $00 ( 4.487) F = $, FValue/Aual Example (cotiued) Solutio usig Excel : Note that the aual amout was etered as a egative umber which idicates a cash outflow. FValue/Aual Example (cotiued) Solutio usig formula: F = Future Value =? A = Aual Amout = $00 = Time = 0 years i = Iterest = 8% per year The slight differece betwee this amout ad the amout determied by the factor from the tables is due to roudig. ( + i) F = A i 0 ( ) = $ = $ = $00 [ ] F = $,

5 Future Value Calculatios Equal aual amouts for a future value is determied by the followig equatio: Note that this formula is the iverse of the formula to fid F give A. i A = F + i ( ) Where: F = Future Value ($) A = Aual Amout ($) = Time (years) i = Iterest (% per year) 3 Future Value Calculatios Solutio methods for fidig aual amouts: Use the A/F colum o a Iterest Factors table (Bowma text appedix B, page 580). Notatio A/F is iterpreted as Fid A give F. Notatio (A/F,, i) is iterpreted as Fid F give A for years at i iterest rate. Use the Excel fuctio PMT(rate, per, pv, fv, type) Use the formula ad calculator. Note that the cash outflows are etered as a egative umber. 4 Aual/Future Value Example Your goal is to save $7,500 for a car dow paymet i 4 years by ivestig part of your ed of year bous. How much to you eed to save aually at 4% iterest? A = F ( A/F,4,4% ) = $7,500 ( ) A = $,766.25/year 5 5

6 Aual/FValue Example (cotiued) Solutio usig Excel : Note that the fuctio returs a egative umber which idicates a cash outflow. 6 Aual/FValue Example (cotiued) Solutio usig the formula: F = Future Value = $7,500 A = Aual Amout =? = Time = 4 years i = Iterest = 4% per year The slight differece betwee this amout ad the amout determied by the factor from the tables is due to roudig. i A = F ( + i) 0.04 = $7,500 4 ( 0.04) = $7, = $7,500 A = $,766.7 [ ] 7 Ed Uit Material Additioal Readig Fiacial Fuctios: Go to Uit 2 Preset Value Amouts 8 6

7 Chapter 5 Uit 2 Preset Value Amouts IET 350 Egieerig Ecoomics Learig Objectives Uit 2 Upo completio of this uit you should uderstad: Calculatig future values from aual amouts. Calculatig preset values from aual amouts. Calculatig future ad preset values from gradiet amouts. Calculatig preset value of a future perpetual amouts. Calculatig deferred auities. 20 Preset Value Calculatios Preset value for a equal aual amout is determied by the followig equatio: P ( + i ) i( + i) + A = Where: P = Preset Value ($) A = Aual Amout ($) = Time (years) i = Iterest (% per year) 2 7

8 Preset Value Calculatios Solutio methods for fidig preset values: Use the P/A colum o a Iterest Factors table (Bowma text appedix B, page 580). Notatio P/A is iterpreted as Fid P give A. Notatio (P/A,, i) is iterpreted as Fid P give A for years at i iterest rate. Use the Excel fuctio PV(rate, per,pmt, fv, type). Use the formula ad calculator. Note that the cash outflows are etered as a egative umber. 22 Preset Value/Aual Example You scheduled to receive $5,000 at the ed of the ext 7 years. If the curret iterest rate is 6%, what is the equivalet amout today? P = A ( P/A,7,6% ) = $5,000 ( ) P = $83, PValue/Aual Example (cotiued) Solutio usig Excel : Note that the fuctio returs a egative umber which idicates a cash outflow. 24 8

9 PValue/Aual Example (cotiued) Solutio usig the formula: P = Preset Value =? A = Aual Amout = $5,000 = Time = 7 years i = Iterest = 6% per year ( + i) P = A i( + i) 7 ( ) = $5, ( 0.06) = $5,000 (0.06) (.50363) = $5,000 P = $83, [ ] 25 Preset Value Calculatios Equal aual amouts for a preset value is determied by the followig equatio: Note that this formula is the iverse of the formula to fid P give A. A = P i ( + i ) ( + i) Where: P = Preset Value ($) A = Aual Amout ($) = Time (years) i = Iterest (% per year) 26 Preset Value Calculatios Solutio methods for fidig preset values: Use the A/P colum o a Iterest Factors table (Bowma text appedix B, page 580). Notatio A/P is iterpreted as Fid A give P. Notatio (A/P,, i) is iterpreted as Fid A give P for years at i iterest rate. Use the Excel fuctio PMT(rate, per, pv, fv, type). Use the formula ad calculator. Note that the cash outflows are etered as a egative umber. 27 9

10 Aual/Preset Value Example You $5,000 ivest i a accout that returs 6% aual iterest. How much ca you withdraw each semester (twice/year) over the ext 4 years for books ad supplies? A = P ( A/P,8,3% ) = $5,000 ( 0.425) A = $72.50/semester Time periods other tha a year ca be used, however, the tabulated iterest rate is a yearly rate so it must be adjusted to match the umber of periods/ year 6% per year/2 periods per year = 3% per period. Also the total umber of periods is used 4 yrs x 2 periods/yr = 8 periods. 28 Aual/PValue Example (cotiued) Solutio usig Excel : Remember that i ad must be adjusted for time periods other tha yearly. 29 Aual/PValue Example (cotiued) Solutio usig the formula (see page 87 Bowma text): P = Preset Value = $5,000 A = Aual Amout =? = Time = 4 years M = #Periods/year = 2 r = Aual Iterest = 6% m r r A + m m = P m m r + m ( + ) 2 2 = $5, ( + ) = $5, = $5,000 [ ] A = $72.28/semester m 30 0

11 Fidig Ukow or i Values Occasioally a egieerig ecoomic aalysis will occur whe the umber of years () or the iterest rate (i) is ukow. Like sigle paymet py calculatios, if three of the four factors are kow, we ca solve for the ukow. Future value factors F, A, i,. Preset value factors P, A, i,. 3 Ukow i ad Calculatios Solutio methods for fidig iterest or time period values: Iterpolate usig the appropriate colum o a Iterest Factors table (Bowma text appedix B, page 580). Use the Excel fuctios : RATE(per, pmt, pv, fv, type, guess) returs the iterest rate per period for a cash flow. NPER(rate, pmt, pv, fv, type) returs the umber of periods for a cash flow with a costat iterest rate. Rearrage the appropriate formula ad solve with your calculator. Note that the cash outflows are etered as a egative umber. 32 Ukow Iterest Example You have $5,000 to ivest i a accout ad would like to withdraw $,550 per year for the ext four years.. What iterest rate will be required to meet the eeds? You will eed to ivest the $5,000 at 9.2% aual iterest rate. 33

12 Ukow Time Example You pla to ivest $,250 per year i a security with a 4.75% aual retur rate. How may years before the accout grows to $2,500? Time Required 8 years 4 moths 5 days 34 Ed Uit 2 Material Go to Uit 3 Gradiet Amouts 35 Chapter 5 Uit 3 Gradiet Amouts IET 350 Egieerig Ecoomics 2

13 Learig Objectives Uit 3 Upo completio of this uit you should uderstad: Calculatig future values from aual amouts. Calculatig preset values from aual amouts. Calculatig future ad preset values from gradiet amouts. Calculatig preset value of a future perpetual amouts. Calculatig deferred auities. 37 Gradiet Amouts Ulike equal aual amouts, gradiet amouts icrease or decrease each time period. Types: Liear chage i cash flow is by a equal amout for each time period. Gradiet factors tabulated i the iterest tables or determied by formula. No liear chage is cash flow varies betwee time periods. No liear gradiet fuctios must be calculated with a series of P/A or A/P for each time period. 38 Gradiet Amouts Assumptios for liear gradiet amout aalysis iclude: Cash flow occurs at the ed of each year. Chage i cash flow year to year is at a costat rate. The amout of chage is desigated G Iitial cash flow is desigated A 39 3

14 Gradiet Calculatios Solutio methods for fidig preset values: Use the A/G colum o a Iterest Factors table (Bowma text appedix B, page 580). Future or preset values ca the be determied usig the aual amout (A): Notatio A/G is iterpreted as Fid A give G. Notatio (A/G,, i) is iterpreted as Fid A give P for years at i iterest rate. Use the Excel fuctio XNPV(rate, values, dates) Use the formula ad calculator. Note that the cash outflows are etered as a egative umber. 40 Gradiet Calculatios Equal aual amouts for a liear gradiet values is determied by the followig equatio: Whe the chage is icreasig betwee time periods, the gradiet is added (+) to the iitial value ad subtracted ( ) whe decreasig. A = A ± G i + i ( ) Where: A = Aual Amout ($) A = Iitial Cash Flow($) G = Gradiet Amout ($) = Time (years) i = Iterest (% per year) 4 Gradiet Example Your st year s salary is $45,000. Your cotract states that your raise will be $5,000/year i years 2 through 6. What is the preset value of the cotract at 5% iterest? Cash Flow Diagram: 42 4

15 Gradiet Example (cotiued) Solutio method: Fid aual value (A) of the gradiet (G). Covert the aual amout (A) ito the preset value (P). A = A + G ( A/G,6,5% ) = $45,000 + $5,000 ( ) A = $56,789.50/year P = A(P/A,6,5%) = $56, (5.0757) P = $288, Gradiet Example (cotiued) Solutio usig Excel. You must create a schedule of amouts with a date. The schedule must start at time = 0 (today). No liear gradiets ca be solved with this method. 44 Gradiet Example (cotiued) Solutio usig the formulas: P = Preset value =? A = Aual Amout =? A = Iitial Amout = $45,000 G = Gradiet = $5,000 = Time = 6 years i = Iterest = 5% per year A = A ± G i ( i) + 6 = $45,000 + $5, ( 0.05) + 6 = $45,000 + $5, = $45,000 + $5,000 [ ] A = $56,789.6/year ( + i) P = A i( + i) 6 ( ) = $56, ( 0.05) + = $56,789.6 [ ] 45 P = $288,

16 Ed Uit 3 Material Go to Uit 4 Perpetual Amouts ad Deferred Auities 46 Chapter 5 Uit 4 Perpetual Amouts ad Deferred Auities IET 350 Egieerig Ecoomics Learig Objectives Uit 4 Upo completio of this uit you should uderstad: Calculatig future values from aual amouts. Calculatig preset values from aual amouts. Calculatig future ad preset values from gradiet amouts. Calculatig preset value of a future perpetual amouts. Calculatig deferred auities. 48 6

17 Perpetual Gradiet Amouts Perpetual gradiet amouts icrease or decrease each time period. Assumptios: Cash flow occurs at the ed of each year with chage at a costat rate. The amout of chage is desigated G Project life is cosidered ifiite: 49 Perpetual Gradiet Amouts Preset value for project with a ifiite life ad a gradiet icrease i cash flow is determied by the followig equatio: P = P + P + P Total 0 Aual A G P Total = P i i Gradiet Where: P 0 = Iitial Project Cost ($) A = Iitial Aual Amout ($) G = Gradiet Amout ($) i = Iterest (% per year) 50 Perpetual Gradiet Example A warehouse was costructed at a iitial cost of $500,000 ad is expected to last forever. First year maiteace cost is $5,000 ad is expected to icrease at a costat $500/year. If the firm uses a 7.5% iterest rate, what is the total preset value of the project? P = Preset value =? P 0 = $500,000 A = Maiteace = $5000/yr G = Gradiet = $500/yr i = 7.5% per year P P Total Total A G = P i i $5000 $500 = $500, = $500,000 + $66, $ = $655,

18 Deferred Auities A auity is equivalet to a aual amout. A deferred auity is a set of aual cash flows that will occur i the future (deferred) rather tha immediately. Figure 5 8 from the Bowma text illustrates a ivestmet made a t 0 (preset time) with aual disbursemet begiig i the 4 th year. 52 Deferred Auities Solutio steps for deferred auities: Determie the future value (F) of the iitial ivestmet at the begiig time poit of the deferred auity. Set the future value (F) () to the deferred preset value (P ). Determie the aual amout (A) usig the deferred preset value (P ). 53 Deferred Auity Example For your 8 th birthday your rich ucle gave you $25,000 for college which you ivested i a deferred auity at 4%. You begi drawig o the auity at age 8 for 4 years. How much will you receive each year? F = P ( F/P,0,4% ) = $25,000(.480) F = $37,000 A = P (A/P, 4,4%) withp = F = $37,000(0.2755) A = $0,93.50/year 54 8

19 Ed Chapter 5 Material Studet Study Guide Chapter 5 Homework Assigmet Problem Set