An annuity is a series of payments required to be made or received over time at regular intervals. The most common payment intervals are yearly (once a year), semi-annually (twice a year), quarterly (four times a year), and monthly (once a month). Some eamples of annuities! "ortgages, #ar payments, $ent, %ension fund payments, &nsurance premiums.
TYPES OF ANNUITIES Ordinary Annuity: An 'rdinary Annuity has the following characteristics! cha racteristics! • •
The payments are always made at the end of each interval The interest rate compounds at the same interval as the payment interval
or calculating the sum of a series of regular payments the following formula should be used! S n
=
R
(- + i ) − n
i
Example: Alan decides to set aside %* at the end of each month for his child+s college educat education ion.. &f the child child were were to be born born today today,, how much will will be availa available ble for its college college education when she turns years old/ Assume an interest rate of 0 compounded monthly. Solution: irst, we assign all the terms!
$1 %* i1 *.*2 or *.**344 n1 5 2, or 24 6ow substituting into our formula, we have! R
Formula for calculating preent !alue of a imple annuity: An
[ − (- + i ) − ] = i n
R -
Example: Alan as8s you to help him determine the appropriate price to pay for an annuity offering a retirement income of %,*** a month for * years. Assume the interest rate is 40 compounded monthly. Solution: Substituting into our formula, we have!
$ 1 %,*** i 1 *.*4 2 or *.** n 1 2 *, or 2* An
= -*** - − (- + *.** ) *.**
−-2*
A n 1 %*,*79.3 Annuity Due: &n an annuity due, the payments occur at the beginning of the payment period.
or calculating the sum of a series of regular payments the following formula should be used! S n ( due)
=
R (-
+ i ) (- + i ) − n
i
Example: Alan wants to deposit %9** into a fund at the beginning of each month. &f he can earn *0 compounded interest monthly, how much amount will be there in the fund at the end of 4 years/ Solution:
$ 1 %9** i 1 *.*2 or *.**5999 n 1 2 4 or 72 Substituting into our formula yields!
%9**(:*.*2);(:*.*2)<72-= S n (due) 1 >>>>>>>>>>>>>>>> *.*2 S n (due) 1 %9**(5.9) S n (due) 1 %2,47 Formula for calculating preent !alue of an annuity due:
$(:i);-(:i)<-n= A n(due) 1 >>>>>>>>-1 i Example: The monthly rent on an apartment is %* per month payable at the beginning of each month. &f the current interest is 20 compounded monthly, what single payment 2 months in advance would be equal to a year+s rent/ Solution: $1 %* i1 *.22 or *.* n1 2
Substituting into the formula gives! %*(:*.*9);-(:*.*)<-2= A n(due) 1 >>>>>>>>>>>>>>> *.* A n(due) 1 %*(.97) A n(due) 1 %*,5*.*
