last installment is paid in the beginning of nth year, hence its present value is given as ( ) i If P denotes the present value of annuity due, then P = a + ... + + = a ... < F = a T SSSSSSSS V WWWWWWWW = a > H G P = a ( + i ) (v) Perpetual Annuity Perpetual annuity is an annuity whose payment continuous for ever. As such the amount of perpetuity is undefined as the amount increases without any limit as time passes on.
We know that the present value P of immediate annuity is given by - - P = i ) i < F Now as per the definition of perpetual annuity as n → ∞, we know that ( ) i n " since + i > . Here P = i a [ - ] P = i NOTE In all the above formulae the period is of one year. Now if the payment is made more than once in a year then ‘ i ’ is replaced by k i and n is replaced by nk , where k is the number of payments in a year. Example .
A person pays ` , per annum for years at the rate of % per year. Find the amount of an ordinary annuity [( . ) = . ].
Here a = , , n = and i = = . Amount of ordinary annuity ( A ) = i a [( + i ) n - ] . 64000 [( + . ) – ] = , , [( .
) – ] = , , [ . – ] = , , [ . ] = ×23184 ∴ A = ` , , Example . What amount should be deposited annually in an ordinary annuity scheme, so that after years, a person receives ` , , if the interest rate is % [( .
To find : a Now A = i a [( + i ) n – ]