an investment and earns interest. Next payment will earn interest for one period less and so on, the last payment will earn interest of one period. For example saving schemes, life insurance payments, etc. Deferred annuity: An annuity which is payable after the lapse of a number of periods is called deferred annuity.
T he derivation of the following formulae are given for better understanding and are exempted from examination. (i) Amount of immediate annuity (or) Ordinary annuity (or) Certain annuity Let ‘ a ’ be the ordinary annuity and i percent be the rate of interest per period. In ordinary annuity, the first installment is paid after the end of first period. Therefore it earns interest for ( n – ) period, second installment earns interest for ( n – ) periods and so on.
The last installment earns for ( n – n ) periods. (i.e) earns no interest. For ( n – ) periods, The amount of first annuity = a ( + i ) n - The amount of second annuity = a ( + i ) n - The amount of third annuity = a ( + i ) n - and so on. ` The total amount of annuity A for n period at i percent rate of interest is A = a ( + i ) n - + a ( + i ) n - +...
a ( + i )+ a = a [( + i ) n - +( + i ) n - +...+( + i )+ ] = a [ +( + i )+( + i ) +...+( + i ) n- ] = a [ + r + r +...+r n- ], where + = r = a [ r ], G.P with common ratio r > = a + - ; E A = i a [( + i ) n – ] (ii) Present Value of immediate annuity (or ordinary annuity) Let ‘ a ’ be the annual payment of an ordinary annuity, n be the number of years and - - Financial Mathematics i percent be the interest on one rupee per year and P