The Financial Formula for Annual Annuity

An annuity is an agreement with an insurance company in which you make payments that are either one-time or over a longer term. In return, you will receive regular payments that can either begin immediately or later, such as when you retire. You can choose to receive your payments on different payment schedules that range from annual to monthly. But how do you calculate an annual annuity?

With an annuity, there is a point in the future when the payments will end. If you are using an annuity for retirement, hopefully, it will outlast you. The value of money depreciates over time, and it can be said that ​$10​ today is worth more than ​$10​ tomorrow.

Things like inflation and interest rates affect the real value of money over time. When you are purchasing an annuity, you want to know what the value of the payments will be in the future. This is called the present value of money.

Here is the annual annuity formula. In its simplest form, the present value of the money can be expressed as pv=c/r. For an annuity, the pv=c/r formula is usually expressed as:

PV = PMT * 1-(1/(1+r)^n))/r

PV = present value

PMT = amount of each annuity payment

r = interest or discount rate

n = number of payment periods

Let’s say you have an annuity that pays ​$50,000​ per year for ​25​ years at a discount rate of ​6​ percent.

PV = $50,000 * 1- (1/(1+0.06)^25))/0.06

PV = $639,168

A perpetuity is like an annuity, only there is no end to the payments. They continue forever with no end date. A perpetuity considers the present value of future payments. It is similar to owning a rental apartment.

You could potentially rent the apartment forever and receive income from it. In reality, you probably will not own it for life and the rent will not stay fixed. Other things, like repairs, maintenance or property values in the area might affect what the apartment is worth in the future.

One main factor that makes perpetuities different from annuities is that perpetuity payments do not stay the same, but they grow over time. If the growth rate of the perpetuity is ​10​ percent, each payment will be ​10​ percent larger than the last payment.

What is a good example of a perpetuity? This all sounds good, but unfortunately, there are few real-life examples of companies that offer perpetuities.

There is one real-life example. At the end of World War I, the United Kingdom sold perpetuities to fund the war, but it eventually bought them all back. The concept of perpetuity is used in theory to calculate the valuation of an annuity, stock or another investment instrument.

Calculating the present value of a perpetuity is similar to calculating the present value for an annuity. The formula for a perpetuity is the same as that for an annuity, with the time constraints removed. It uses the simple formula pv=c/r. Let’s see an example. How much is ​$100​ at ​10​ percent interest at the end of each year forever worth today?

If we use PV=C/R, it becomes:

PV = $100/10%

PV = $1,000

These are only simple examples for understanding the difference between an annuity and perpetuity. Many different types of annuities exist, and the calculations are different. For instance, the calculations for a fixed or variable annuity must consider different factors.