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 4-year 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 arithmetic-gradient, period-by-period linear increase or decrease in cash flow; $ s/month, $ s/year, etc. Geometric gradient, period-by-period 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) 5-1] - Fi = 0 A [ (1 + i) 5-1] - Fi = 0 Fi = A [ (1 + i) 5-1] F = A{ [ (1 + i) 5-1] / 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 arithmetic-gradient series present worth factor. To compute an equivalent annual series from a linear gradient use: The term in the brackets is called the arithmetic-gradient uniform-series factor. To compute a future amount from a linear gradient series use: The term in the brackets is called the arithmetic-gradient 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 geometric-gradient-series 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 geometric-gradient-series future worth factor. F = na 1 (1 + i) n-1 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 geometric-gradient-series: 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 geometric-gradient-series) 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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