Present Value of Perpetuity Calculator

The present value of a perpetuity is the value today of an infinite series of equal payments received at regular intervals. Because money received in the future is worth less than money received today, each future payment is discounted to its value today.

The sum of all these infinite payments results in a finite amount. This tool allows you to calculate the present value of a fixed perpetuity or growing perpetuity and takes into account the frequency and timing of payments. It can also be used to discount cash flows and discount rates.

What is a lifetime pension?

A perpetuity is a series of regular payments that go on forever. It's an annuity payment that pays out the same amount at fixed intervals, with the first payment made after a certain period of time and then continuing indefinitely. In reality, there are few financial products that actually last forever, but some products are considered to be perpetuities. British consols were government bonds without a maturity date, which were first issued by the UK government in the 18th century and paid interest until they were finally redeemed in 2015. Preferred shares are often considered to be perpetuities because they pay out a fixed dividend before common stockholders. Real estate investments that generate steady income, such as rental properties, are also valued on this concept, as are university endowments and charitable foundations. Yale University holds one of five surviving Dutch water authority perpetuities issued in 1648, which still pays interest today.

Why does an infinite stream of payments have a finite value?

It seems impossible to add up an infinite number of payments and get a finite number, but this is what the time value of money represents. A dollar today is worth more than a dollar received in the future because you can invest that dollar today and earn interest on it over that time. Therefore, future payments are discounted, with the present value being lower the further into the future they occur. A payment due many centuries from now would be heavily discounted and have an almost negligible present value. These payments form a convergent geometric series, where each term is a constant factor times the previous term, so that it gets progressively smaller and approaches zero, with the sum approaching a finite limit. This limit is the present value of the lifetime annuity.

So, here's how you use this tool.

Enter the cash flow for each period and the annual discount rate. Select how often payments are made and whether they occur at the end or beginning of each period. The present value will be shown instantly. If the "growing annuity" option is selected, then the payment increases by a certain percentage every year. By selecting the sensitivity analysis section, you can display low and high values with respect to the current interest rate. This tool also allows for solving in reverse. If one of the values (present value, cash flow or discount rate) is left blank and the other two are entered, then the corresponding field will be automatically filled in. You can start with any of the values.

Formula for present value of an annuity:

For an annuity where a fixed amount is paid at the end of each period, the present value is equal to the cash flow divided by the discount rate.

C is the cash flow per period and r is the discount rate per period.

Lets take an example:

The perpetuity in today's value is calculated as :

The present value and the Perpetuity are equivalent views:

Where does the value sit?

Consider another example, suppose a bond pays out $10 every year forever, and the discount rate is 5 percent.

So the present value of this annuity is 200 today. If the discount rate were increased to 8 percent, while keeping the annual payment at 10, then the value would be only 125. This is because higher interest rates cause each future payment to be discounted more heavily. When payments are made at the beginning of each period, this is an annuity due, and each payment will be discounted by one less time period, so it is multiplied by one plus the discount rate for each period. The following table shows the individual symbols and their example values.

If you look at the same fixed annuity with different discount rates, you can see how dramatically interest rates affect the outcome. In each row, the cash flow stays at $10 while only r is changed.

Growth dependent perpetual annuity and the Gordon growth model

Not all infinite cash flows are the same. A growing perpetuity is a cash flow that grows at a constant rate g forever. This compensates for some of the discounting effect and is suitable for rents or dividends that increase over time. The Gordon growth model is used to calculate the present value.

where C is the first payment, r is the discount rate and g is the growth rate. The only strict requirement is that the discount rate must be greater than the growth rate. If the speed of increase in payments is greater than the speed of discounting, then each payment's value in today's money will equal the previous one, and the series does not converge, so the value goes to infinity. As long as g remains less than r, the value is finite. Suppose a dividend is 10, the discount rate is 8 percent, and it currently increases by 2 percent each year.

A 2% growth increases the value from $125 to about $167. This is a better investment with the same initial payment. The perpetuity growing at a constant rate is a core component of the dividend discount model used for valuing stocks, and it is also used in discounted cash flow analysis for terminal value.

Consider cash flow as income from an investment.

This formula can also be interpreted differently. If we establish the relationship that the present value (PV) is equal to C divided by r, then C is equal to the product of PV and r. Imagine you deposit an amount P into an account with an annual interest rate of r. Every year, you can withdraw the interest earned on your investment, which is equal to P multiplied by r, while leaving the principal amount intact and earning more interest over time. This constant withdrawal pattern is known as a perpetuity, and the present value represents the initial investment required to fund it. The size of the fund is determined in a similar way. To generate a fixed annual income stream, you would need to invest an amount equal to the desired income divided by the expected rate of return on your investment. The rate of return, or the ratio of cash flow to price, is known as the "cap rate" used by real estate and income investors. For a constant perpetuity, this rate of return is simply r, while for a growing perpetuity it is r minus g.

Perpetuities in reality

A typical example is a perpetuity bond. The UK government has issued bonds that have no maturity date so investors receive a steady return but the capital is never repaid. The only way to liquidate them is to sell the bond on to another person. Preference shares are similar when dividends are fixed and expected to continue. Property is often valued as an annuity forever. If an investor expects $50,000 in net rental income each year and uses a 4 percent interest rate then the value of the property is the result of dividing $50,000 by .04 which equals $1,250,000. Funds or trusts that are set up to support an organization indefinitely also use this same mathematical method. It's important to note that in most cases the actual cash flow doesn't really last forever. As economic conditions, tenants and interest rates change the value of a perpetuity is not a guarantee but rather a simplified estimate.

Factors that affect present value

There are three factors that affect the result. The discount rate has the strongest effect. The higher the interest rate, the more future payments will be discounted, resulting in a lower value. A lower interest rate results in a higher value, and if the interest rate goes to zero, the value rises infinitely. The size of the cash flows has the opposite effect: the larger the payments are, the higher the present value is. Growth counteracts the discounting effect, reducing the difference between r and g and thus increasing the value. If g equals r, the value diverges. Timing also matters. A perpetuity due pays at the beginning of each period, so its value will always be slightly higher than that of an ordinary perpetuity if the payment amounts are equal.

Comparison of perpetuities and annuities

An annuity is a series of equal payments over a limited period of time. A perpetuity is simply the infinite continuation of that concept. A perpetuity is the limit of an annuity as the number of payments becomes infinitely large. Since future payments far in the future have little impact on value, the present value of a long-term annuity already approximates the value of a perpetuity. This also explains why the formula for a perpetuity is much simpler than that for an annuity, since the term "number of payment periods" completely disappears. In the convergence chart of this tool, the cumulative present value continues to rise and then approaches the upper limit of the perpetuity. On the other hand, the original total payment grows linearly and indefinitely.