Annuity Calculator( Future and Present Value of Annuity Calculator)

An annuity is a series of payments made at equal intervals. it is a sequence of payments or receipt made or received at regular intervals. Examples of annuity are savings accounts, pension payments, hire purchase payment

The difference between annuity and lump sum is that annuity provides regular payments over time while a lump sum provides a one-time payment of the entire amount. That is, an annuity offers a steady income stream, while a lump sum provide immediate access to the full sum.

An annuity can be classified into ordinary annuity and annuity due. It can also be categorized into immediate and deferred annuity. Furthermore, annuity can be categorized into fixed and variable annuity. We also have a special type of annuity called perpetuity.

An ordinary annuity is an annuity where the series of equal payments or cash flows are made at the end of consecutive periods over a fixed length of time. That is, in an ordinary annuity, the payments are made at the end of each period. For example, if the annuity has a monthly interval, the payments would be made at the end of each month.

An ordinary annuity is an annuity where the series of equal payments or cash flows are made at the beginning of consecutive periods over a fixed length of time. That is, in an annuity due, the payments are made at the beginning of each period. For example, if the annuity has a quarterly interval, the payments would be made at the beginning of each month.

A deferred annuity is a type of annuity in which the income payments to the annuitant are postponed until a future date. During the deferral period, the money invested in the annuity grows either through interest, investment returns, or a specified rate of return. Deferred annuities usually have two phases: the accumulation phase and the payout phase. The accumulation phase is when you make one-time or recurring deposits into the annuity and the money grows on a tax-deferred basis. The payout phase is when you start receiving income payments from the annuity, either for a set period of time or for your entire life.

An immediate annuity is a type of annuity where the annuitant (the person who owns the annuity) begins to receive payments shortly after a lump-sum premium is paid. Unlike deferred annuities, there is no accumulation phase; the distribution or payout phase begins almost immediately.

A perpetuity is a continuous and never-ending series of equal payments. In other words, it is an annuity with no end date, where the payments go on indefinitely. In annuity, the stream of payment continue indefinitely.

The present value of perpetuities can be calculated by dividing the periodic payments by the discount rate.

\(PV = \frac{C}{r}\)

Where PV is the present value of the perpetuity, C is the constant payment or cash flow, r is the discount rate.

Let illustrate this with an example. if a perpetuity pays $100 per year and the discount rate is 5%, the present value of the perpetuity is:

\(\text {PV} = \frac {100} {0.05}=2000\)

This means that the buyer must pay $2000 for the perpetuity that pays $100 per year forever.

The present value of an annuity is the amount of money that would be needed today to fund a series of future payments from the annuity, given a certain interest rate or discount rate. It is the amount that, if invested at the beginning of the first period of an annuity, would be enough to pay out the annuity is the present value of the annuity. To illustrate, suppose that for the next 7 years,  €700 is deposited at the end of each year in an account that will earn 10% interest compounded annually.
The present value of the annuity would be the amount that must be deposited in one lump payment today to produce the same amount after 7 years, assuming the same interest rate.

If the annuity is an ordinary annuity, then the present value can be calculated thus:

\(PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]\)

If it is an annuity due, then it can be calculated as follows:

\(PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]\left(1+\frac{i}{m}\right)\)

Where C is the cash flow or payment amount, i is the interest rate, m is the compounding frequency, n is the number of years

Compounding frequency is the number of times per year that the interest rate is applied to the accumulated balance of the annuity. The compounding frequency can be annually(m=1), bi-annually(m=2), quarterly(m=4), monthly(m=12) or weekly(m=52). Compounding frequency affects the future value and present of the annuity because the more frequently the interest is compounded, the higher the future value and the lower the present value of an annuity, and vice versa.

The difference between compounding frequency and payment frequency in annuity is that compounding frequency is the number of times per year that the interest rate is applied to the accumulated balance of the annuity, while payment frequency is the number of times per year that the annuity payments are made or received.

An annuity where the compounding frequency is the same as its payment frequency is called simple annuity.

On the other hand, an annuity where the compounding frequency is different from its payment frequency is called general annuity.

The future value of an annuity is the total value of a group of recurring payments at a certain date in the future, assuming a particular rate of return, or discount rate. It is the future value of a stream of equal payments or received.

If the annuity is an ordinary annuity, then It can be calculated as follows:

\(FV=C\left[\frac{\left(1+\frac{i}{m}\right)^{n m}-1}{\frac{i}{m}}\right]\)

However, if the annuity is an annuity due, then it can be calculated as follows:

\(FV=C\left[\frac{\left(1+\frac{i}{m}\right)^{n m}-1}{\frac{i}{m}}\right]\left(1+\frac{i}{m}\right)\)