Concept and Types of Annuity

An annuity is a sequence of equal payments made at equal intervals of time. The rent paid every month, a fixed premium paid every year on a life-insurance policy, the equal instalments in which a loan is cleared, and the equal sum a firm sets aside every year to replace a machine are all annuities. What makes a stream of payments an annuity is not the size of the payment but the fact that the amount is the same every time and the gap between payments is equal. The Odisha CHSE +2 (Std-11) Business Mathematics and Statistics syllabus studies annuities as an application of compound interest — every annuity formula is built by applying compound interest to each individual payment and then adding the results.

Two pieces of vocabulary are used throughout this chapter:

• The equal payment itself is called the periodic payment, instalment or annuity (denoted $P$ or $R$).
• The gap between two successive payments is the payment period (a year, half-year, quarter or month). The rate of interest is always taken per payment period — an 8% per annum rate compounded half-yearly means $i = 4\\%$ per half-year, and $n$ counts the number of half-years, not the number of years.

Types of annuity. Annuities are classified along three independent lines:

1. By certainty of the term:

   • An annuity certain runs for a fixed, known number of periods (e.g. a loan repaid in exactly 60 monthly instalments). This is the type this chapter computes.
   • An annuity contingent runs until an uncertain event occurs (e.g. a pension that stops on the pensioner's death). Its term is not fixed in advance.

2. By the timing of the payment within each period:

   • In an ordinary annuity (also called an immediate annuity or annuity in arrears) each payment is made at the end of its period. A loan instalment paid at the end of every month is the standard example, and unless a problem says otherwise, an annuity is taken to be ordinary.
   • In an annuity due each payment is made at the beginning of its period. Rent paid in advance and insurance premiums paid at the start of the year are annuities due.

3. By when the payments begin:

   • An immediate annuity begins at the end of the first period.
   • A deferred annuity begins only after a stated number of idle periods (the period of deferment) have passed.
   • A perpetuity is an annuity that continues forever — it has no final payment (e.g. the income from an endowment fund that pays out only its interest).

Every calculation in this chapter answers one of just two questions about such a stream: what is it worth at the end (its future value or amount, §2), or what is it worth today (its present value, §3). Sinking funds (§5) are an application of future value; loan and overdraft repayment (§6) are an application of present value.