Arithmetic gradient.ppt
- 1. EGR 312 – Spring ‘05 1 Arithmetic 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 $175 G = $25 Base = $100 0 1 2 3 4 $2000 $1500 $1000 $500 G = -$500 Base = $2000 0 1 2 3 4
- 2. EGR 312 – Spring ‘05 2 Arithmetic Gradient Factors (P/G, A/G) Equivalent cash flows: + Note: the gradient series by convention starts in year 2. $100 $125 $150 $175 G = $25 Base = $100 0 1 2 3 4 0 1 2 3 4 $100 0 1 2 3 4 $25 $50 $75
- 3. EGR 312 – Spring ‘05 3 Arithmetic 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 nG P n n n i n i i i i G P ) 1 ( ) 1 ( 1 ) 1 ( n n n i n i i i i ) 1 ( ) 1 ( 1 ) 1 ( 1
- 4. Problems 1. The maintenance on a machine is expected to be $155 at the end of the first year, and increasing $35 each year for the following seven years. What present of money would need to be set aside now to pay the maintenance for the eight year period ? Assume 6% of interest. EGR 312 – Spring ‘05 4
- 5. 2. The tuition fee in a school are expect to inflate at the rate of 8%per year and the tuition at year 1 is P60,000. The parents are then given a financial planning by the bank that says they must deposit an amount of money today that will cover for their child’s whole 4-year education. If the amount deposited has an interest rate of 5% per year compounded annually, how much money must be deposited to the bank today. EGR 312 – Spring ‘05 5
- 6. EGR 312 – Spring ‘05 6 Arithmetic 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) $155 $125 $150 $175 0 1 2 3 4 0 1 2 3 4 $155 0 1 2 3 4 $35 $70 $105
- 7. EGR 312 – Spring ‘05 7 Arithmetic 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 $500 0 1 2 3 4 0 1 2 3 4 $2000 0 1 2 3 4 $500 $1000 $1500
- 8. EGR 312 – Spring ‘05 8 Arithmetic 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 nG 0 1 2 3 … n A 1 ) 1 ( 1 n i n i
- 9. EGR 312 – Spring ‘05 9 Geometric Gradient Factors (Pg /A) A Geometric gradient is when the periodic payment is increasing (decreasing) by a constant percentage: A1 = $100, g = 0.1 A2 = $100(1+g) A3 = $100(1+g)2 An = $100(1+g)n-1 $100 $110 $121 $133 0 1 2 3 4
- 10. EGR 312 – Spring ‘05 10 Geometric Gradient Factors (Pg /A) To find the Present Worth, Pg , for a geometric gradient cash flow G: Pg $100 $110 $121 $133 0 1 2 3 4 i g i n A P i g g i i g A P g n g 1 1 1 1 1 1
- 11. EGR 312 – Spring ‘05 11 Determining Unknown Interest Rate To 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 tables (and interpolation) 3) Spreadsheet (Excel) a) =IRR(first cell: last cell) b) =RATE(number_years,A,P,F)
- 12. EGR 312 – Spring ‘05 12 Determining Unknown Interest Rate Example: 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% annually
- 13. EGR 312 – Spring ‘05 13 Determining Unknown Interest Rate Example: 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. 48 48 ) 1 ( 1 i) (1 $658.25 $25000 i i 48 48 ) 1 ( 1 i) (1 37.9795 i i
- 14. EGR 312 – Spring ‘05 14 Determining 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 tables 3) Spreadsheet (Excel) a) =NPER(i%,A,P,F)
- 15. EGR 312 – Spring ‘05 15 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) 0.2 = (P/F,5%,n) n ~ 33 periods