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Business Math17th EditionMary Hansen 3,684 solutions Lesson 10: AnnuitiesThis lesson discusses annuities in the context of the compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:
Definition of an AnnuityAn annuity is a series of equal cash flows, or payments, made at regular intervals (e.g., monthly or annually). The payments must be equal, and the interval between payments must be regular. The following compound interest functions in AH 505 involve annuities:
Ordinary AnnuityAn ordinary annuity is an annuity in which the cash flows, or payments, occur at the end of the period. An ordinary annuity of cash inflows of $100 per year for 5 years can be represented like this: The cash flows occur at the end of years 1 through 5. And the first cash flow occurs at the end of year 1. Most appraisal problems involve ordinary annuities; that is payments are assumed to occur at the end of the period. All of the formulas and factors in AH 505 pertain to ordinary annuities only. Annuity DueAn annuity due is an annuity in which the cash flows, or payments, occur at the beginning of the period. An annuity due is also called an annuity in arrears. An annuity due of cash inflows of $100 per year for 5 years can be represented like this: The cash flows occur at the beginning of years 1 through 5. And the first cash flow occurs at time 0 (now). Practical Applications Converting Annuity FactorsAs noted, most appraisal problems assume that payments occur at the end of the period (ordinary annuity). But if payments occur at the beginning of the period (annuity due), an ordinary annuity factor in AH 505 can be converted to its corresponding annuity due factor with a relatively simple calculation. Although financial calculators and spreadsheet software make it even easier to convert from an ordinary annuity to an annuity due, it is useful to understand how to "manually" convert the ordinary annuity factors in AH 505 to annuity due factors. Conversion of ordinary annuity factor to annuity due factor for FW$1/P or PW$1/P: To determine the Future Worth of $1 Per Period (FW$1/P) or Present Worth of $1 Per Period (PW$1/P) factor for an annuity due, refer to the corresponding factor in AH 505 for an ordinary annuity and multiply it by a factor of (1 + the periodic interest rate). The periodic rate will differ depending on the compounding interval in the problem. For example, with annual compounding, the periodic rate would be the same as the annual rate; with monthly compounding the periodic rate would be the annual rate divided by 12. Example 1: Conversion to annuity due factor for FW$1/P Solution:
Example 2: Conversion to annuity due factor for PW$1/P Solution:
Conversion of ordinary annuity factor to annuity due factor for SFF or PR: To determine the Sinking Fund Factor (SFF) or Periodic Repayment (PR) Factor for an annuity due, refer to the corresponding factor in AH 505 for an ordinary annuity and divide it by a factor of (1+ the periodic interest rate). Be sure to divide, not multiply, when converting the SFF and PR factors. Note: the periodic rate will differ depending on the compounding interval in the problem. Example
3: Conversion to annuity due for SFF Solution: Example 4: Conversion to annuity due for PR Solution: What do you call the annuity paid in the end of the period?If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuityimmediate, or ordinary annuity. Mortgage payments are annuityimmediate, interest is earned before being paid.
What type of annuity is being referred to wherein the payments are due after the end of each period?Ordinary annuities: An ordinary annuity makes (or requires) payments at the end of each period. For example, bonds generally pay interest at the end of every six months.
