Arithmetic Gradient Factors (P/G, A/G)Slide 2Slide 3Slide 4Slide 5Slide 6Geometric Gradient Factors (Pg/A)Slide 8Determining Unknown Interest RateSlide 10Slide 11Determining Unknown Number of Periods (n)Slide 13Arithmetic Gradient Factors (P/G, A/G)Cash flows that increase or decrease by a constant amount are considered arithmetic gradient cash flows. The amount of increase (or decrease) is called the gradient.$100$125$150$175G = $25Base = $1000 1 2 3 4$2000$1500$1000$500G = -$500Base = $20000 1 2 3 4Arithmetic Gradient Factors (P/G, A/G)Equivalent cash flows:=> +Note: the gradient seriesby convention starts inyear 2.$100$125$150$175G = $25Base = $1000 1 2 3 4 0 1 2 3 4$1000 1 2 3 4$25$50$75Arithmetic Gradient Factors (P/G, A/G)To find P for a gradient cash flow that starts at the end of year 2 and end at year n:or P = G(P/G,i,n)where (P/G,i,n) = 0 1 2 3 … n$G$2G$(n-1)G$PnnniniiiiGP)1()1(1)1(nnniniiii )1()1(1)1(1Arithmetic Gradient Factors (P/G, A/G)To find P for the arithmetic gradient cash flow:=> +$P = $100(P/A,i,4) + $25(P/G,i,4) $100$125$150$1750 1 2 3 4 0 1 2 3 4$1000 1 2 3 4$25$50$75Arithmetic Gradient Factors (P/G, A/G)To find P for the declining arithmetic gradient cash flow:=> - $P = $2000(P/A,i,4) - $500(P/G,i,4)$2000$1500$1000$5000 1 2 3 4 0 1 2 3 4$20000 1 2 3 4$500$1000$1500Arithmetic Gradient Factors (P/G, A/G)To find the uniform annual series, A, for an arithmetic gradient cash flow G:-> $A = $G(P/G,i,n) (A/P,i,4) = $G(A/G,i,n)Where (A/G,i,n) = 0 1 2 3 … n$G$2G$(n-1)G0 1 2 3 … n$A1)1(1niniGeometric Gradient Factors (Pg/A)A Geometric gradient is when the periodic payment is increasing (decreasing) by a constant percentage:A1 = $100, g = .1A2 = $100(1+g)A3 = $100(1+g)2An = $100(1+g)n-1$100$110$121$1330 1 2 3 4Geometric Gradient Factors (Pg/A)To find the Present Worth, Pg, for a geometric gradient cash flow G: $Pg$100$110$121$1330 1 2 3 4iginAPiggiigAPgng111111Determining Unknown Interest RateTo find an unknown interest rate from a single-payment cash flow or uniform-series cash flow, the following methods can be used: 1) Use of Engineering Econ. Formulas.2) Use of factor tables3) Spreadsheet (Excel)a) =IRR(first cell: last cell)b) =RATE(number_years,A,P,F)Determining Unknown Interest RateExample: The list price for a vehicle is stated as $25,000. You are quoted a monthly payment of $658.25 per month for 4 years. What is the monthly interest rate? What interest rate would be quoted (yearly interest rate)?Using factor table:$25000 = $658.25(P/A,i,48) 37.974 = (P/A,i,48) i = 1% from table 4, pg 705 0r 12 % annuallyDetermining Unknown Interest RateExample: The list price for a vehicle is stated as $25,000. You are quoted a monthly payment of $658.25 per month for 4 years. What is the monthly interest rate? What interest rate would be quoted (yearly interest rate)?Using formula:Use Excel trial and error method to find i.4848)1(1i)(1 $658.25 $25000ii4848)1(1i)(1 37.9795iiDetermining Unknown Number of Periods (n)To find an unknown number of periods for a single-payment cash flow or uniform-series cash flow, the following methods can be used: 1) Use of Engineering Econ. Formulas.2) Use of factor tables3) Spreadsheet (Excel)a) =NPER(i%,A,P,F)Determining Unknown Number of Periods (n)Example: Find the number of periods required such that an invest of $1000 at 5% has a future worth of $5000.$P = $F(P/F,5%,n) $1000 = $5000(P/F,5%,n) .2 = (P/F,5%,n) n ~ 33
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