Annuity Calculator

An annuity is a series of equal payments made at regular intervals. This calculator looks at this cash flow from two perspectives: the present value - that is, how much these payments are worth today - and the future value - that is, what amount will be achieved if invested over a period of time. It supports both ordinary and annuity due annuities, allows you to split the annual interest rate into monthly or quarterly payments, and can handle cases where the payment increases annually. In addition, by leaving one field blank, you can work out either the payment amount, the interest rate, or the number of years.

What is a pension?

In finance, an annuity refers to a series of equal payments made over a specified period of time. Examples include rent, car loan or mortgage payments, retirement income, small claims, or regular savings account deposits. To determine the value of this cash flow, two values must be considered: the present value and the future value. The present value discounts each future payment to its current worth, as money received later is worth less than money received today. The future value calculates how much each investment would grow by compounding interest at the end of the period, showing the resulting balance. Both are based on the time value of money.

The term "annuity" can also refer to retirement products sold by insurance companies. These are products where a lump sum is paid in return for future income. Such contracts may include additional features such as lifetime payments, death benefits or fixed or variable returns and are divided into immediate and deferred annuities. The calculations on this page are based on the financial annuity, i.e. a series of equal payments, an interest rate and a period of time that underlie all these products.

Ordinary pension versus prepayment pension

The result depends on the timing of payment. An ordinary annuity pays at the end of each period and is common for mortgages, car loans, and most bonds. An annuity due pays at the beginning of each period and is typically used for rent, lease payments, and many insurance premiums. Because each payment in an annuity due occurs one period sooner, it earns additional interest and has less discounting. Thus, the value of an annuity due will always be higher than that of an ordinary annuity by a factor of 1 + rate. You can compare both types of annuities by toggling the "timing" field.

Instructions for using this calculator:

Enter the payment amount, annual interest rate and number of years. Select the frequency of payments and whether they are at the end or beginning of each period. The present value and future value will be displayed immediately. By activating "Increasing Payments" you can model cash flows that increase by a certain percentage annually. By activating the "Sensitivity" section, you can check the low and high future values based on the entered interest rate. This calculator can also work in reverse. If you leave one of the fields (payment amount, interest rate, number of years or present value) blank and fill in the other three, you can directly determine the missing value.

Formula for pension calculations:

In a regular annuity the future value is calculated by compounding each payment to the end of the time period and then adding up all of those results.

Conversely, in present value each payment is discounted to its current value.

PMT is the payment per period, i is the interest rate per period and n is the number of periods. If payments are made at the beginning of each period, multiply one of the results by (1 + interest rate per period) to get the value of an annuity due. Even if the interest rate is given as an annual rate, you will need to divide it by the number of payments per year and multiply n by the number of years if the payment frequency is higher than once a year. If there are m payments per year, then i is the annual rate divided by m and n is the number of years multiplied by m.

Every year at the end of the year an amount of $1,000 is deposited and this continues for 5 years. Assume that the return on capital is 8 percent.

A similar annuity with a term of 5 years and annual payments of $1,000 has a present value of about $3,992.71 and a future value of about $5,866.60. The total payment is $5,000 but the compounding interest increases it by $866.60. For an annuity due these two values are increased to about $4,312.13 and $6,335.93 respectively. The individual symbols and example values are shown in the following table.

Increasing pensions and eternal pensions

Not all cash flows are the same. In a growing annuity each payment is increased by some percentage rate g. This would be true for an annuity where payments were adjusted for cost of living increases, or for rent which goes up every year. The present value of this is given by:

If the cash flows continue forever, then the annuity becomes a perpetuity. As the number of payments increases, the present value approaches a finite upper limit. This is because payments far into the future are heavily discounted and contribute little to the value.

A growing perpetuity converges only if the interest rate exceeds the growth rate. Otherwise its value is infinite. This calculator shows the value of a growing perpetuity next to the results for a fixed term, so you can judge how close a long-term annuity is to an infinite cash flow.

Factors that affect the result:

There are three factors that change both values at the same time. The most important factor is the interest rate. The higher the interest rate, the greater the future value growth through compounding and the lower the present value due to discounting. So an increasing interest rate increases the difference between the two values. A decreasing interest rate causes the two values to converge. Payment amounts and number of payments also have a proportional effect since the higher the payments and the larger the amount, the more valuable the result is. Timing is also important. An annuity due is better than an ordinary annuity under similar conditions. Companies that buy structured settlements usually use a discount rate in the range of 9 to 18 percent which explains why the lump sum payment is often much lower than the total amount.

Present value and future value:

These two values answer different questions about the same cash flow. The present value is useful when you need to decide whether to accept a payment in installments or as a lump sum, or if you want to determine how much you would pay today for a future cash flow that will generate income. The future value is useful when you want to save for a specific goal and know what ending balance you'll have. This calculator shows both values at the same time so you can evaluate your payment amount and savings plan simultaneously without switching between tools.

This calculator is for educational and planning purposes only and does not constitute financial advice. Interest rates, yields, costs and tax implications vary and actual results will depend on the conditions stated. Please consult a qualified professional before making any decisions about retirement, comparisons or annuity products.