Annuity is an investment tool where fixed amount of payment or receipt is obtained as per the pre-determined contract between the parties. In simple language, it is an agreement to make series of payment of certain amount of money to specific party for a predetermined period of time. This kind of the payment can be made on the end the period or could be done at the beginning of the period. The initial one is called ordinary annuity and the latter one is called annuity due.
The present value of the annuity can be determined by the formula
PV of annuity = PMT*[1/k – 1/k(1+k) n]
Where PMT = fixed predetermined cash payment or receipt
n is number of periods and k is the rate per periods
Whereas in terms of the annuity due there is a level of stream of payment starting immediately. And annuity due is worth (1+ k) times the value of ordinary annuity. However, if the annuity is expected to grow with rate g than, PV of growing annuity is equivalent to:
= PMT * [1-(1+g) n/(1+k)n]
Let’s take a general example to understand how to calculate the annuity.
Other parameters are: -
Discount rate(k)= 10%
Maturity period(n)= 5
Number of compounding in a year(m)=1
PMT
|
Year
|
15,000
|
0
|
15,000
|
1
|
15,000
|
2
|
15,000
|
3
|
15,000
|
4
|
Future Value = PMT*((1+k/m) ^((m*n) +1) - (1+k/m))/(k/m))
= 15000*((1+0.1/1) ^ ((1*5) +1) -(1+0.1/1))/(0.1/1)
=100,734.2
Therefore, the value of the annuity is $100,734.2 and the interest earning is $25,734.00.
Alternatively, the future value of this annuity due can be calculated as:-
= PMT[((1+k) ^n-1)/k] * (1+k)
= 15000[((1+.1) ^5-1)/0.1 * 1.1
=100,734.2
Hence the total future value of the annuity having equal annual payment of 15000 per year at a discounting rate of 10% for 5 years will be $100,734.2.
Similarly, the present value of the annuity with following parameters can be calculated as:
Discount rate(k)= 10%
Maturity period(n)= 5
Number of compounding in a year(m)=1, PMT=15000 can be calculated as:-
= PMT * ((1 - (1 / (1 + r) ^ n)) / r)
= 15000* ((1-(1/(1+0.1)^5))/0.1)
= 56,861.8
The above mentions are the ways to calculate the future and the present value of annuity. These helps the investor to make rational decision by predicting the present and future value of the annuity and make the financial decisions accordingly.
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