## Future value of Annuity

Either payment or receipts in form of sequence then it is said to be 'Annuity'. The annuity can be sub-divided into three categories, they are

1. Deferred Annuity / End of the period
2. Annuity Due / Beginning of the period
3. Sinking Fund Method

1. Annuity at the end of the period

Future Value of the Annuity at End of the Period (FVAEP) is product of Principal amount (A) and Future Value Interest factor Annuity (FVIFA). In view of formula

FVAEP = A*FVIFA

Where

A = Principal Amount
FVIFA = [(1+k)^n]-1] / k, where k = rate of Interest and n = No. of years

For example

Assuming that Mr. X deposits Rs.1,000 annually in a bank for 5 years at the rate of 10%.

Then the value of the deposits after 5 years based on deferred annuity is

FVAEP = Rs.1,000(FVIFA) for 10% and 5 year = 1,000*(((1 .10)^5 -1)/0.10) = 1,000 × 6.105 = Rs.6,105

2. Annuity at the beginning of the period

Future Value of the Annuity at the Beginning of the Period (FVABP) is product of Future Value of Annuity at End of the Period (FVAEP) and principal amount added up with yearly interest. In view of formula

FVABP = FVAEP * (A+I)

Where

A = Principal Amount
I = Rate of Interest
FVAEP = A*FVIFA
FVIFA = [(1+k)^n]-1] / k
k = rate of Interest and
n = No. of years.

For Example

Assuming Mr. X invest Rs 1,000 at the beginning of every year, for five years at the rate of 10%.

Then the value of the investment after five years is

FVABF = Rs.1,000 × 6.7155= Rs.6,715.5

3. Sinking Fund Factor Method

Sinking Fund factor method one of the most popular method used by investors. Investors knows the proposed value of investment they require in future, accordingly they see to that, the fund made available by investing/depositing now viz the amount deposited with regular time intervals for fetching returns and those returns are accumulated for future needs till certain point of time.

The amount deposited with regular time interval is calculated based on the time availability, interest of deposit, required amount. This can be expressed in formula

A = FVA [(K/(1+K)^n) -1]

Where,
A = Amount to be deposited
FVA = Future Value of deposited Amount
K = Interest
n= Number of years

For example

Mr. X is running a business and planning to expand his business after 10 years and he arrives the value of proposed project is Rs. 20,000. The interest rate prevailing in market is 12%

Thus the Amount to be deposited annually is = Rs.20,000 [(0.12/(1+.12)^10) -1]
= 20,000*(0.05698) = Rs. 1,139.60. Therefore by depositing Rs 1,1,39.60 annually for 10 years yields the required amount of Rs. 20,000.