LEARNING OBJECTIVES. 2.1 Derivation by Recursion: F/P factor. 2.1 Basic Derivations: F/P factor. 2.1 P/F factor discounting back in time

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1 LEARNING OBJECTIVES Developed By: Dr. Do Smith, P.E. Departmet of Idustrial Egieerig Texas A&M Uiversity College Statio, Texas Executive Summary Versio Chapter 2 Factors: How Time ad Iterest Affect Moey. F/P ad P/F factors 2. P/A ad A/P factors 3. Iterpolate for factor values 4. P/G ad A/G factors 5. Geometric gradiet 6. Calculate i 7. Calculate 8. Spreadsheets 9//200 9// Basic Derivatios: F/P factor F/P Factor To fid F give P P 0 To Fid F give P. Compoud forward i time F 2. Derivatio by Recursio: F/P factor F = P(+i) F 2 = F (+i)..but: F 2 = P(+i)(+i) = P(+i) 2 F 3 =F 2 (+i) =P(+i) 2 (+i) = P(+i) 3 I geeral: F = P(+i) F = P(F/P,i%,) 9// // Preset Worth Factor from F/P Sice F = P(+i) 2. P/F factor discoutig back i time Discoutig back from the future We solve for P i terms of F N P = F{ / (+i) } = F(+i) - Thus: P = F(P/F,i%,) where (P/F,i%,) = (+i) - Thus, the two factors are:. F = P(+i) fids the future worth of P; 2. P = F(+i) - fids the preset worth from F P. F P/F factor brigs a sigle future sum back to a specific poit i time. 9// //200 6

2 Sct 2. Sigle-Paymet Factors (F/P ad P/F) Objective: Derive factors to determie the preset or future worth of a cash flow Cash Flow Diagram basic format F 2.2 Example- F/P Aalysis Give: P= $,000; =3; i=0% What is the future value, F? F =?? i% / period P=$, i=0%/year P 0 P 0 = F /(+i) (P/F,i%,) factor: Excel: =PV(i%,,,F) F = P 0 (+i) (F/P,i%,) factor: Excel: =FV(i%,,,P) F 3 = $,000[F/P,0%,3] = $,000[.0] 3 = $,000[.330] = $, // // Example P/F Aalysis Give: F = $00,000, 9 years from ow. Fid: The preset worth of this amout ow if i =5%? i = 5%/yr F 9 = $00, Sct 2.2 Uiform-Series: Preset Worth Factor (P/A) ad Capital Recovery Factor(A/P) Cash flow profile for P/A factor i% per iterest period $A per iterest period -2 - Fid P P=?? P 0 = $00,000(P/F, 5%,9) = $00,000(/(.5) 9 ) = $00,000(0.2843) = $28,430 at time t = 0 9//200 9 Required: To fid P give A Cash flows are equal, uiterrupted ad flow at the ed of each iterest period 9//200 0 (P/A) Factor Derivatio Setup the followig: P = A ( ) ( ) ( ) i i i ( i) () Multiply by to obtai a secod equatio (+i) P = A i ( i) ( i) ( i) ( i) (2) Subtract () from (2) to yield i P = A (3) + + i ( + i) ( + i) (P/A) ad (A/P) Factor Formulas Simplify (3) to yield ( + i) P = A for i 0 i( + i) Solve (4) for A to get (A/P) factor i( + i) A = P ( + i) (5) (4) (P/A,i%,) factor Excel: =PV(i%,,A) (A/P,i%,) factor Excel: =PMT(i%,,P) 9//200 9//

3 ANSI Stadard Notatio for Iterest Factors Stadard otatio has bee adopted to represet the various iterest factors Cosists of two cash flow symbols, the iterest rate, ad the umber of time periods Geeral form: (X/Y,i%,) X represets what is ukow Y represets what is kow i ad represet iput parameters; ca be kow or ukow depedig upo the problem Notatio - cotiued Example: (F/P,6%,20) is read as: To fid F, give P whe the iterest rate is 6% ad the umber of time periods equals 20. I problem formulatio, the stadard otatio is ofte used i place of the closed-form equivalet relatios (factor) Tables at the back of the text provide tabulatios of commo values for i% ad 9// //200 4 Sct 2.3 Sikig Fud Factor ad Uiform Series Compoud Amout Factor (A/F ad F/A) Cash flow diagram for (A/F) factor i% per iterest period A=? per iterest period Fid A, give F Start with what has already bee developed i( + i) A = F ( + i) ( + i) i A = F ( + i) F = give (F/A) factor from (A/F) Give: i (A/F,i%,) factor A = F ( i) + Solve for F i terms of A to yield Excel: =PMT(i%,,,F) ( + i) (F/A,i%,) factor F = A i Excel: =FV(i%,,A) 9// //200 6 Sct 2.4 Iterpolatio i Iterest Tables Whe usig tabulated iterest tables oe might be forced to approximate a factor that is ot tabulated Ca apply liear iterpolatio to approximate See Table 2-4 Factors are oliear fuctios, hece liear iterpolatio will yield errors i the 2-4% rage Use a spreadsheet model to calculate the factor precisely 2.4 Iterpolatio of Factors Typical Format for Tabulated Iterest Tables 9// //

4 2.4 Basic Setup for Iterpolatio Work with the followig basic relatioships 2.3 Example 2.5 It is desired to kow the future worth of $,000,000 ivested at the ed of each year for 8 years, startig oe year from ow. The iterest rate is assumed to be 4% per year A = $,000,000/yr; = 8 yrs, i = 4%/yr F 8 =?? 9// // Example 2.5 Solutio: The cash flow diagram shows the aual paymets startig at the ed of year ad edig i the year the future worth is desired. Cash flows are idicated i $000 uits. The F value i 8 years is F = l000(f/a,4%,8) = 000( ) = $3, = millio 8 years from ow. 9//200 2 Sct 2.5 Arithmetic Gradiet Factors (P/G) ad (A/G) Cash flow profile Fid P, give gradiet cash flow G Base amout = A A +G A +2G A +(-2)G A +(-)G CF = A ± (-)G 9// Gradiet Example $400 $300 $200 $00 $500 $600 $700 Gradiet Compoets (-2)G (-3)G Fid P of gradiet series G 2G 0G Base amout = A / period (-)G Gradiets have two compoets:. The base amout ad the gradiet 2. The base amout (above) = $00/time period Preset worth poit is period to the left of the 0G cash flow For preset worth of the base amout, use the P/A factor (already kow) For preset worth of the gradiet series, use the P/G factor (to be derived) 9// //

5 Gradiet Decompositio As we kow, arithmetic gradiets are comprised of two compoets. Gradiet compoet 2. Base amout Whe workig with a cash flow cotaiig a gradiet, the (P/G) factor is oly for the gradiet compoet Apply the (P/A) factor to work o the base amout compoet P = PW(gradiet) + PW(base amout) Derivatio Summary for (P/G) Start with: P = G( P / F, i, 2) + 2 G( P / F, i,3) + 3 G( P / F, i, 4) [(-2)G](P/F,i,-)+[(-)G](P/F,i,) Multiply () by (+i) to create a secod equatio Subtract () from the secod equatio ad simplify Yields G ( + i) ( + i) i (P/G,i,) factor P= i ( ) i + i ( ) + i = 2 i + i No Excel relatio exists () 9// // Use of the (A/G) Factor 2.5 Gradiet Example A = G(A/G,i,) Fid A, give gradiet cash flow G G 2G (-2)G A A A... A A CF = (-)G Equivalet A of gradiet series (-)G 9// Cosider the followig cash flow $00 $200 $300 $ Preset Worth Poit is here! Ad the G amt. = $00/period $500 Fid the preset worth if i = 0%/yr; = 5 yrs 9// Gradiet Example- Base Auity First, The Base Auity of $00/period A = +$ Gradiet Example- Focus o the Gradiet Compoet $0 $00 $200 $300 $ PW(0%) of the base auity = $00(P/A,0%,5) PW Base = $00(3.7908)= $ Not Fiished: We eed the PW of the gradiet compoet ad the add that value to the $ amout We desire the PW of the Gradiet Compoet at t = 0 P G@t=0 = G( P/G,0%,5 ) = $00( P/G,0%,5 ) 9// //

6 2.5 Gradiet Example- The Set Up $0 $00 $200 $300 $ P G@t=0 = G(P/G,0%,5) = $00(P/G,0%,5) N P= G ( + i) N i ( ) N ( ) N i + i + i Could substitute =5, i=0% ad G = $00 ito the P/G closed form to get the value of the factor. 2.5 Gradiet Example- PW of the Gradiet Compoet P G@t=0 = G(P/G,0%,5) = $00(P/G,0%,5) P/G,0%,5) N P= G ( + i) N i ( ) N ( ) N i + i + i Sub. G=$00;i=0.0;= Calculatig or lookig up the P/G,0%,5 factor yields the followig: P t=0 = $00(6.868) = $686.8 for the gradiet PW 9// // Gradiet Example: Fial Result Sct 2.6 Geometric Gradiet Series Factor PW(0%) Base Auity = $ PW(0%) Gradiet Compoet = $686.8 Total PW(0%) = $ $686.8 Equals $ Note: The two sums occur at t =0 ad ca be added together cocept of equivalece 9// Geometric Gradiet Cash flow series that starts with a base amout A Icreases or decreases from period to period by a costat percetage amout This uiform rate of chage defies A GEOMETRIC GRADIENT Notatio: g = the costat rate of chage, i decimal form, by which future amouts icrease or decrease from oe time period to the ext 9// Typical Geometric Gradiet A Give A, i%, ad g% A (+g) A (+g) A (+g) - Required: Fid a factor (P/A,g%,i%,) that will covert future cash flows to a sigle preset worth value at time t = 0 9// Start with: Basic Derivatio: Geometric Gradiet A A ( + g) A ( + g) A ( + g) = ( + i) ( + i) ( + i) ( + i) Factor out A out ad re-write 2 ( + g) ( + g) ( + g) = A ( + i) ( + i) ( + i) ( + i) (2) Multiply by (+g)/(+i) to obtai Eq. (3 ) 2 ( + g ) ( + g ) ( + g ) ( + g ) ( + g ) = A ( + i) ( + i) ( + i ) ( + i ) ( + i ) ( + i ) Subtract Eq. (2 ) from Eq. (3 ) to yield + g ( + g ) P g A + + i = ( + i ) + i Solve for P g ad simplify to yield. 9// () (3) + g + i = A g i i g 6

7 Two Forms to Cosider + g + i = A g i i g Case: g = i To use the (P/A,g%,i%,) factor A is the startig cash flow A = ( + i) 9// P g Case: g = i There is NO base amout associated with a geometric gradiet The remaiig cash flows are geerated from the A startig value No tables available to tabulate this factor too may combiatios of i% ad g% to support tables 2.6 Geometric Gradiet: Example Assume maiteace costs for a particular activity will be $700 oe year from ow. Assume a aual icrease of maiteace costs of % per year over a 6-year time period. If the iterest rate is 8% per year, determie the preset worth of the future expeses at time t = 0. First, draw a cash flow diagram to represet the model. 9// Geometric Gradiet Example (+g) 2.6 Solutio g = +% per period; A = $700; i = 8%/yr $700 PW(8%) =?? $700(.) $700(.) 2 $700(.) 3 $700(.) 5 9// P = $700(P/A,%,8%,7) Need to calculate the P/A factor from the closed-form expressio for a geometric gradiet. From a spreadsheet we see: 303: Use "g" 667: use f-bar Geometric Gradiets "E" or g or f-bar = % i= 8% N= 7 P/A,g,i, factor is First Amt= $, P. Value = $, g + i = A g i i g 9// Geometric Gradiet ( -g ) Cosider the followig problem with a egative growth rate g. A = $000 $900 $80 $ P 0 =?? g = -0%/yr; i = 8%; = 4 We simply apply a g value = // Geometric Gradiet (-g value) Evaluate: For a egative g value = g + i P g = A g i i g 303: Use "g" 667: use f-bar Geometric Gradiets "E" or g or f-bar = -0% i= 8% N= 4 P/A,g,i, factor is First Amt= $, P. Value = $ 2, //

8 Sct 2.7 Determiatio of Ukow Iterest Rate Class of problems where the iterest rate, i%, is the ukow value For simple, sigle paymet problems (i.e., P ad F oly), solvig for i% give the other parameters is ot difficult For auity ad gradiet type problems, solvig for i% ca be tedious Trial ad error method Apply spreadsheet models The IRR Spreadsheet Fuctio Defie the total cash flow as a colum of values withi Excel Apply the IRR fuctio: =IRR(first_cell:last_cell, guess value) If the cash flow series is a A value the apply the RATE fuctio: =RATE(umber_years, A,P,F) See examples 2.2 ad 2.3 9// // Sct 2.8 Determiatio of Ukow Number of Years Class of problems where the umber of time periods (years) is the ukow I sigle paymet type problems, solvig for is straight forward I other types of cash flow profiles, solvig for requires trial ad error or spreadsheet I Excel, give A, P, ad/or F, ad i% values apply: =NPER(i%,A,P,F) to retur the value of Sct 2.9 Spreadsheet Applicatio Basic Sesitivity Aalysis Sesitivity Aalysis is a process of determiig what iput variables really matter i a give problem formulatio Sesitivity aalysis aids i evaluatig certai what-if scearios Spreadsheet modelig is the best approach to formulate sesitivity aalysis for a give problem 9// // Sesitivity Aalysis See Example 2.5 Illustrates a what-if situatio for receivig moey i three differet time periods Tabulates the associated rate of returs for the three situatios See Example 2.6 The evaluatio of o-sequetial cash flows. Foudatios: Overview. F/P ad P/F Factors 2. P/A ad A/P Factors 3. F/A ad A/F Factors 4. Iterpolate Factor Values 5. P/G ad A/G Factors 6. Geometric Gradiet 7. Calculate i 8. Calculate 9. Spreadsheets 9// //