Financial Mathematics of Ordinary Simple Annuity

Definitions

An annuity is a sequence of periodic payments, usually equal, made at equal intervals of time. Premium on insurance, mortgage payments, interest payment on bonds and debentures, payments of rent, payments of hire purchase and dividends are just few examples of annuities.

The time between successive payments is called payment period. The time from the beginning of the first payment period to the end of the last payment period is called the term of an annuity. When the term of an annuity is fixed, i.e. the dates of the first and the last payment is fixed, the annuity is called an annuity certain. When the term of an annuity depends on some uncertain event, the annuity is called contingent annuity. Bond interest payments form an annuity certain: life-insurance premiums form a contingent annuity( they cease with the death of the insured). Unless otherwise specified the word annuity will refer to an annuity certain.

When the payments are made at end of each payment period, the annuity is called ordinary annuity. Examples include loan repayments and interest payment on bonds and debentures. When the payments are made at the beginning of each payment period is called annuity due. Insurance premium are a good example of this type of annuity.

When the payment period and interest rate coincide, the annuity is called a simple annuity, otherwise it is a general annuity.

The accumulated value of an annuity at the equivalent dated value of a set of payments due at the end of the term. Similarly, the discounted value of an annuity is defined as the equivalent dated value of a set of payments due at the beginning of the term. In this article I will use the following notations as follows:

R = the periodic payment of the annuity

n = the number of interest period during the term of the annuity

i = the interest rate per payment period

S = the accumulated value of an annuity

A= Present value of an annuity.

Accumulated value of an ordinary simple annuity

The accumulated value “S” of an ordinary simple annuity is defined as the amount due at the end of the term equivalent to the dated values of the payments comprising the annuity with the date of the last payments as the focal date. As ordinary simple annuity is shown on the time diagram below with the interest period ( which equals the payment period) as the unit of measure.