CE 314 Engineering Economy. Interest Formulas


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1 METHODS OF COMPUTING INTEREST CE 314 Engineering Economy Interest Formulas 1) SIMPLE INTEREST  Interest is computed using the principal only. Only applicable to bonds and savings accounts. 2) COMPOUND INTEREST  Interest is calculated on the principal plus the total amount of interest accrued in previous periods. "Interest on top of Interest" Example: An individual borrows $18,000 at an interest rate of 7% per year to be paid back in a lump sum payment at the end of 4 years. Compute the total amount of interest charged over the 4year period using the simple interest and compound interest formulas. Compute the total amount owed after 4 years using simple and compound interest. Using simple interest: Interest = Principal (number of periods) (interest rate) I = P(n)(i) I = 18,000 (4)(0.07) = $5,040 And the Amount owed = Principal + Interest accrued F = P + I F = 18, ,040 = $23,040 Using compound interest: Year Interest Charge Accrued Amount 1 18,000 (0.07) = $1,260 18, ,260 =$19, ,260 (0.07) = $1, , , = $20, ,608.20(0.07) = $1, , , = $22, ,050.77(0.07) = $1, , , = $23, Total Interest charged = $23, $18,000 = $5, (11% increase)
2 Interest Formulas: Symbols P  F  A  G  g  A 1  n  i  t  Present value, value of money at the present (time = 0); $'s Future value, value of money at some time in the future; $'s Uniform Series, a series of consecutive, equal, end of time period amounts of money; $'s/ month, $'s/ year, etc. Constant arithmeticgradient, periodbyperiod linear increase or decrease in cash flow; $ s/month, $ s/year, etc. Geometric gradient, periodbyperiod constant increase or decrease in cash flow; $ s/month, $ s/year, etc. First payment in a geometric gradient (time =1), $ s Number of interest periods; months, years, etc. Interest rate or rate of return per period; percent per month, percent per year, etc. time, stated in periods; months, years, etc. Derivation of the relationship between a future amount and a present amount: (Single Payment Formulas) Previous example: P = $18,000 i = 7% per year n = 4 F 4 =? F 1 = 18, ,000 (0.07) = $19,260 F 1 = P + P(i) = P (1 + i) F 2 = F 1 + F 1 (i) = P (1 + i) + [P (1 + i)](i) = P (1 + i) [1 + i] = P (1 + i) 2 F 3 = P (1 + i) 2 + P (1 + i) 2 (i) = P (1 + i) 2 (1 + i) = P (1 + i) 3 In general, F = P (1 + i) n The term (1 + i) n is called the single payment compound amount factor.
3 To compute a present amount from a future amount, solve for P: F = P (1 + i) n P = F / (1 + i) n The term 1 / (1 + i) n is called the single payment present worth factor. Derivation of the relationship between a uniform series and a future worth and a uniform series and a present worth: F = A 1 (1 + i) 4 + A 2 (1 + i) 3 + A 3 (1 + i) 2 + A 4 (1 + i) + A 5 But: A 1=A 2=A 3=A 4=A 5=A Equation 1: F= A [(1 + i) 4 + (1 + i) 3 + (1 + i) 2 + (1 + i) + 1] A [(1 + i) 4 + (1 + i) 3 + (1 + i) 2 + (1 + i) + 1]  F = 0 Now multiply each side by (1 + i): Equation 2: F(1 + i) = A [ (1 + i) 5 + (1 + i) 4 + (1 + i) 3 + (1 + i) 2 + (1 + i)] A [ (1 + i) 5 + (1 + i) 4 + (1 + i) 3 + (1 + i) 2 + (1 + i)]  F( 1 + i) = 0 A [ (1 + i) 5 + (1 + i) 4 + (1 + i) 3 + (1 + i) 2 + (1 + i)]  F  Fi = 0 Equation 2  Equation 1: A [ (1 + i) 5 + (1 + i) 4 + (1 + i) 3 + (1 + i) 2 + (1 + i)]  F  Fi = 0 A [(1 + i) 4 + (1 + i) 3 + (1 + i) 2 + (1 + i) + 1]  F = 0 A [ (1 + i) 51]  Fi = 0 A [ (1 + i) 51]  Fi = 0 Fi = A [ (1 + i) 51] F = A{ [ (1 + i) 51] / i}
4 In general, F = A{ [ (1 + i) n  1] / i} The term { [ (1 + i) n  1] / i} is called the uniform series compound amount factor. The term { i / [ (1 + i) n  1]} is called the sinking fund factor. A = F{ i / [ (1 + i) n  1]} Sinking fund is the annual amount invested by a company to finance a proposed expenditure. Derivation of the relationship between a uniform series and a present amount: A = F{ i / [ (1 + i) n  1]} and F = P (1 + i) n Substitute P (1 + i) n for F in equation 1: A = P (1 + i) n { i / [ (1 + i) n  1]} = P [i( + i) n / (1 + i) n  1] The term [i( + i) n / (1 + i) n  1] is called the capital recovery factor. Capital recovery refers to the amount of money required each year to offset an initial investment. To compute a present amount from a uniform series. Solve for P: A = P [i(1 + i) n / (1 + i) n  1] P = A {[(1 + i) n  1] / i( 1 + i) n } An arithmetic gradient is a cash flow series that either increases or decreases by a constant amount:
5 To compute a present amount from a linear gradient series use: The term in the brackets is called the arithmeticgradient series present worth factor. To compute an equivalent annual series from a linear gradient use: The term in the brackets is called the arithmeticgradient uniformseries factor. To compute a future amount from a linear gradient series use: The term in the brackets is called the arithmeticgradient series future worth factor. The general equations for calculating total present worth are P T = P A + P G and P T = P A  P G. The general equations for calculating the equivalent total annual series are A T = A A + A G and A T = A A  A G.
6 It is common for cash flow series, such as operating costs, construction costs, and revenues to increase or decrease from period to period by a constant percentage. The uniform rate of change defines a geometric gradient series of cash flows: To compute a present amount from a geometric gradient series use: Use only if g does not equal i. The term in the brackets is called the geometricgradientseries present worth factor. Use if i = g. To compute a future worth from a geometric gradient series use: F = A 1 [((1 + i) n  (1 + g) n )/(i  g)] use only if i does not equal g. The term [(1(1 + g) n (1 + i) n )/(i  g)] is called the geometricgradientseries future worth factor. F = na 1 (1 + i) n1 use if i = g.
7 Standard Notation: To compute a future amount given a present amount: F = P (F/P, i%, n) Looking for a F given a P To compute a present amount given a future amount: P = F (P/F, i%, n) Looking for a P given a F To compute a present amount given a geometricgradientseries: P = A 1 (P/A 1,g,i,n) Tables are available on pages in your textbook, which have factors computed for all of the formulas (excluding the geometricgradientseries) for different values of i and n.
8 Convention: The present value of a series cash flow is computed one period prior to the first series payment. The future value of a series cash flow is computed at the same time period as the last series payment.
9 The present value of a linear gradient series is computed by breaking the linear gradient into two parts: a uniform series cash flow and a conventional linear gradient series. The present value of a conventional linear gradient series is computed two periods prior to the first payment in the conventional linear gradient. The future value of a conventional linear gradient is computed at the same time period as the last payment in the conventional linear gradient.
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