Chapter 10 Annuities
Lecture Outline
10.1 Future Value Annuities A. Ordinary Annuities 1. Formula Method For Calculating a Future Value of an ordinary annuity 2. Table Method for Calculating a Future Value of an ordinary annuity B. Annuity Due 1. Formula Method for Calculating a Future Value of an annuity due 2. Table Method for Calculating a Future Value of an annuity due 10.2 Present Value Annuity A. Formula Method for Calculating a Present Value Annuity B. Table Method for Calculating a Present Value Annuity 10.3 Sinking Funds A. Formula Method for Calculating a Sinking Fund Payment B. Table Method for Calculating a Sinking Fund Payment 10.4 Amortization of an Amount of Money A. Formula Method for Calculating the Amortization Payment B. Table Method for Calculating the Amortization Payment
Teaching Comments
A. General Suggestions 1. We recommend that the general analysis of an annuity problem as payments accruing to a lump sum, or lump sum generating equal regular payments be taught to the students as a first step in problem analysis. 2. Although the formulas are shown for annuities, sinking funds, and amortizations, the easier means of calculating these is to use the table method. However, some problems can only be done on the calculator since there are no tables included for the necessary interest rate. 3. Explain that annuities differ from the basic compound interest problem in that equal periodic payments are made into (or out of) a fund. Develop the formula and show the
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4.
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6.
students how to use the calculator. For most classes, show one or two examples and move to the table method of calculation. Use the figures 10.6 and 10.7 to explain the concept of present value annuity and show the formula for present value. Many students will not understand the negative exponent, so it may be easiest to proceed immediately to the table method of calculation. Otherwise, the negative exponent will need to be developed and the use of the calculator for computation developed. Explain that a sinking fund is an account designed to have a specified future value at a particular time. Develop the sinking fund formula from the ordinary annuity formula, and explain how to calculate the regular payment using the formula. Explain that amortization is the determination of the regular equal payment which will liquidate a principal plus its interest over a given number of compounding periods. Show the formula and proceed to calculations using tables. The fact that using amortization to determine loan repayments is a true “interest on the unpaid principal” method of loan liquidation will be developed in Chapter 11.
B.
Teaching Notes 10.1 Future Value Annuities a. An annuity is a special type of compound interest account. b. The annuity differs from the basic concept of compound interest studied in chapter 9 in that the principal is increased or decreased by equal regular payments either into or out of the account. c. A future value annuity is an application of compound interest in which regular deposits are made into a lump sum account according to the principal. d. An ordinary annuity is a future value annuity in which the equal regular payments are added to the principal at the end of each compounding period.
e.
f. g.
The future value of an ordinary annuity can be calculated by using the formula (1 + i) n − 1 S=p× , or by using Appendix B column C of the tables. i An annuity due is a future value annuity in which the equal regular payments are added to the principal at the beginning of each compounding period. The future value of an ordinary annuity can be calculated by using the formula (1 + i) n − 1 S=p× × (1 + i), or by using Appendix B column C of the tables. i
10.2 Present Value Annuities
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a.
b.
c.
A present value annuity is a form of compound interest account that is established with a single lump sum which periodically pays out of the account a specified number of equal regular payments that liquidate (reduce to $0) the account. Each compounding period the account increases in size by interest earned. and decreases in size by the payment made out of the account. 1 − (1 + i) − n The present value can be calculated by the formula A = p × , or by i using the table in Appendix B column E.
10.3 Sinking Funds a. An annuity sinking fund is an annuity into which regular equal periodic deposits are made so that a specific lump sum of money is accumulated at the end of a set time. b. The sinking fund formula provides a way to calculate the equal regular payments of an ordinary annuity so that the desired lump sum can be achieved in n payments. i c. The formula is p = S × , or the table in Appendix B column D can be (1 + i) n − 1 used. 10.4 Amortization of an Amount of Money a. Amortization of a lump sum of money is an application of the present value annuity. b. Amortization is a present value annuity in which we are trying to determine the value of the regular payment p which will liquidate a sum A. c. Amortization is the most common method used to pay off loans. d. The payment p for amortization can be determined by the formula i p=A× , or by using the tables in Appendix B column F. 1 − (1 + i) − n
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