What are annuities and how to calculate annuities?

Annuity is a fixed sum of money paid to the investor or insurer to a series of annual sums for specific length of time or the rest of their life. In the case of reverse mortgage loan provided to senior citizen against security of their home is another example where loan amount is used to purchase of a life annuity.

The annuities are paid according to the contract, usually at a frequency of monthly, quarterly, half yearly, or annually. There are two types of annuities viz. ordinary annuities and annuities due. In ordinary annuity, payments of annuities are made at the end of each period. In annuity due, payments of annuity are made at the beginning of the period. Rent is an example of annuity due.

Calculation of Annuities

Time value of money invested in annuity scheme:

Present Value (PV) of money: Present Value (PV) of money is the value at the current point of time.

Future value (FV) of money: The value of money may go up at a specified time in the future, assuming a certain interest rate. This is known as Future Value (FV) of money. This information is useful for finding out what would have been the accumulated amount at given interest rate. Similarly, if we are making payment of loan, the future value is useful in determining the total cost of loan.

As the present and future value calculations for ordinary annuities – and annuities due are slightly different – we will first find out the present and future value calculation for ordinary annuities.

In order to calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let us assume that you are receiving Rs.10000 every year for the next five years and you invest each payment at 7% compounded annually.

The future value that these payments should have at the end of the five-year period is calculated as under:

Formula: P=PMT [{(1+R)^N-1}/R]

Where:

P = The future value of the annuity stream to be paid in the future,

PMT = The amount of each annuity payment,

R = The interest rate,

N = The number of periods over which payments are made

Thus, the future value of the annuity  is

P = 10000 [{(1 + 0.07)⁵ – 1} / 0.07]

P = Rs.575, 07.40

As another example, what if the interest on the investment compounded monthly instead of annually, and the amount invested were Rs.10, 000 at the end of month? The calculation is:

P = 10,000 [((1 +0 .005833)60 – 1) / .005833]

The .005833 interest rate used in the last example is 1/12th of the full 7% annual interest rate.

P = 715921.60

Types of Annuity – There are 2 types in terms of valuation of annuity:

(NOTE – Basic classification of Annuity is – Ordinary Annuity & Annuity Due)

Calculation in the case of annuity due:

 If the payments are made at the beginning of each period it is called annuity due. In this case, each annuity payment is allowed to compound for one extra period .

Note that, since each payment in this series is made 1 period earlier, we have to discount the formula for 1 period. We get the following formula for Annuity Due,

FV of Annuity Due = FV of Ordinary Annuity x (1+R) = [P(X-1)/R] x (1+R)

Similarly, we can find out the  PV of Annuity Due = FV of Annuity Due / X = FV of Annuity Due x DF

[Note: compounding factor (1+R)^N  is symbolized as  X], P= Deposit amount]

Discounting Factor (DF): To get the present value from future value of a certain sum of money, we have to discount (e.g., filter out the interests gained) the future value of that sum of money. This process is known as Discounting. We use the formula to calculate present value PV = FV x DF

Formula for DF is DF = 1 / (1+R)^N

where R=rate of interest, and N=no. of period

[ Note, that we have taken Compounding Factor X=(1+R)^N while calculating EMI. Discounting Factor (DF) is reciprocal of the compounding factor].

Similarly, we can calculate with the following formulations;

PV = FV x DF;  FV = PV / DF, where DF = 1 / (1+R)^N

or, PV = FV / X; FV = PV x X, where X = (1+R)^N