Dr. Maddah ENMG 400 Engineering Economy 06/24/09. Chapter 2 Factors: How time and interest affect money

Size: px

Start display at page:

Download "Dr. Maddah ENMG 400 Engineering Economy 06/24/09. Chapter 2 Factors: How time and interest affect money"

Aubrey Gregory
5 years ago
Views:

1 Dr Maddah ENM 400 Egieerig Ecoomy 06/4/09 Chapter Factors: How time ad iterest affect moey Sigle Paymet Factors Recall that P dollars ow are equivalet to F dollars after time periods at a iterest rate of i per time period, where F P( i) Rewrite this as F P ( F / P, i, ), where ( F / P, i, ) ( i) is the F/P factor I additio, this implies that F P F ( P / F, i, ), ( i) where ( P / F, i, ) ( i) is the P/F factor Tables ad Spreadsheets The P/F ad F/P factors, as well as other factors, are tabulated at the ed of your textbook You may use these tables or the formulas that we will derive Spreadsheets (ie, Excel) has built-i fuctio for factors (or you ca easily build your ow fuctios) for calculatig the factors Excel is very suited for practical iterest calculatios Here ad elsewhere, whe the type of iterest is ot specified, assume it s compoud iterest

2 Uiform Series Factors Suppose oe will pay A dollars every time period for period startig with the ed of period (see figure below) 0 - A A A A P=? F=? The, this series of cash flows is ow equivalet to j A A A A A i P i ( i) ( i) i j0 i i i Upo simplificatio, ( i) A P A A ( P / A, i, ) i( i) i ( i) ( /,, ) [( ) P A i i ]/[ i( i) ] is the P/A factor I additio, i( i) A P P ( A/ P, i, ) ( i), ( /,, ) ( ) A P i i i /[( i) ] is the A/P factor Fially, to fid the future amout, F, equivalet to the uiform series of cash

3 ( i) F P( i) A A( F / A, i, ), i i( i) F i( i) i A P F ( i) ( i) ( i) ( i) F ( A/ F, i, ), ( F / A, i, ) [( i) ]/ i is the F/A factor ( A/ F, i, ) i /[( i) ] is the A/F factor Ruig amortizatio Amortizatio is the process of substitutig a curret paymet P for periodic paymets of A per period (eg car or home loa) Oe ca view each amortizatio paymet (A) as composed of two parts: (i) iterest o ruig (outstadig) balace ad (ii) partial repaymet of pricipal This procedure is equivalet to re-amortizig the ruig balace every period over the remaiig time horizo This is cosistet with accoutig practice Eg, cosider a loa of $,000 issued o Ja, 006, to be paid back i equal mothly paymets over 5 years at a iterest rate of % per moth The mothly paymet is 60 (00)( 00) A 000 $4 60 ( 00) 3

4 The, the outstadig balace o Feb, 006 is $,000 mius the mothly paymet ($4) plus the mothly iterest (00,000=$0), which gives $98776 The (ruig) amortizatio of the $98776 at -Mar-006 over the remaiig 59 moths is 59 (00)( 00) A $4 59 ( 00) The (ruig) amortizatio of the $97539 at -Apr-006 over the remaiig 58 moths is also $4, ad so o Date Previous balace Iterest Paymet Received New Balace -Ja-06 $,000 -Feb-06 $,000 $000 $4 $ Mar-06 $98776 $988 $4 $ Apr-06 $97539 $975 $4 $9690 -May-06 $9690 $963 $4 $9508 -Dec-0 $4383 $044 $4 $0 -Ja- $0 $0 $4 $000 4

5 Arithmetic radiet Factors 0 3 Base = A time i I some cases cash flows icrease by a fixed amout i each time period startig with period Startig with a cash flow of A at the ed of period, the cash flows icrease by the gradiet,, i each period That is, the cash flows are A, A +, A +,, A +, i periods,,, The, at time 0, this series of cash flows is equivalet to P P P, A where P A is the equivalet at time zero of the series with ( i) uiform cash flows A per time period, PA A i ( i ) P is the equivalet at time zero of the arithmetic cash flow series with gradiet (ie, the series havig,,, ( ) cash flows at the ed of periods, 3,, 5

6 P is evaluated as follows P ( ) ( ) 3 ( i) ( i) ( i) ( i) ( ) j ( i) [ i ( ) i]/( i) j i j i i i ( i) ( i) ( i) ( P /, i, ) i i( i) i i( i) ( i) ( i) i i( i) ( i) ( P /, i, ) is the "P/ factor" I the above we have used the fact, that for j ad m itegers m j m j ( y) ( y my) /( y) j/( y) y Fially, w defie A/ ad F/ factors eometric radiet Factors Suppose ow that i a series of a cash flows the amouts icrease (or decrease) by a fixed amout (+g), i each time period startig with period A (+g) A A (+g) A (+g) 6

7 0 3 At time 0, this geometric gradiet series is equivalet to P g A A ( g) A ( g) A g i ( i) ( i) ( i) j0 i It follows that j P g [( g) /( i)] A, if i g i g A, if i g i Summary of termiology F/P factor: Compoud Amout Factor P/F factor: Preset Worth Factor P/A factor : Uiform-Series Preset Worth Factor A/P factor: Capital Recovery Factor A/F factor: Sikig Fud Factor F/A factor: Uiform-Series Compoud Amout Factor P/ factor: Arithmetic gradiet preset worth factor A/ factor: Arithmetic gradiet uiform-series factor 7

Single-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P)

Single-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P) Sigle-Paymet Factors (P/F, F/P) Example: Ivest $1000 for 3 years at 5% iterest. F =? i =.05 $1000 F 1 = 1000 + (1000)(.05) = 1000(1+.05) F 2 = F 1 + F 1 i = F 1 (1+ = 1000(1+.05)(1+.05) = 1000(1+.05) 2