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Perpetuity Calculator

Table of contents

What is a perpetuity? — Perpetuity definitionThe time value of moneyHow to calculate perpetuity? — Present value of perpetuity calculatorPerpetuity formula — an exampleGrowing perpetuity formulaHow to calculate perpetuity? — Growing perpetuity examplePerpetuity in real life — examples

Perpetuity calculator is a helpful tool when determining the present value of a perpetuity. To say that something lasts in perpetuity means that it continues forever. Visit our present value calculator to explore present value further.

An annuity is a series of fixed payments made at equal intervals for a specified period of time. In finance, a perpetuity is a type of an annuity, but with one difference - regular payments will be made indefinitely.

Read on if you want to find out how to define perpetuity, how to calculate perpetuity, and explore some real-life examples.

What is a perpetuity? — Perpetuity definition

We can define perpetuity as a stream of regular, never-ending fixed payments. Let's break these terms down a bit:

  • Regular — payments will occur after a specified time. They can be monthly or yearly.
  • Fixed — means that the amount paid will always be the same.
  • Indefinitely — the payments will continue without end.

By now, you are probably thinking about one or two things: Infinite value? Well, not exactly. Sounds too good to be true? Yes, there is a catch. Allow us to explain:

Indeed, the payments will not end. You need to keep in mind, however, that even though the amount paid will be the same, each subsequent payment will be worth less than the previous one. How so? In finance, there is the concept of the time value of money. Simply put, one dollar you have now is worth more than one dollar you will have tomorrow.

The time value of money

The future value of money, covered in our future value calculator, is lower than its present value. This happens because money has earning potential — the sooner you have it, the sooner you can use it to get more — for example, by investing or generating interest. Over time, inflation also leads to a general price increase and the lowering of the buying power of money. Visit our buying power calculator and inflation calculator to learn more.

The further away payment is, the lower its present value will be. What about these $100 payments done a couple of hundred years ago? They're not worth as much as the $100 you will get next month. In fact, their value will be close to zero. Infinite payments have a finite value and thus can be calculated. That is why it is useful to figure out the present value when looking at perpetuity payments — it will tell you how much your payments are actually worth.

To learn more about this concept, consult our time value of money calculator!

How to calculate perpetuity? — Present value of perpetuity calculator

The fact that the fixed payments you will receive have a constantly decreasing present value allows us to calculate perpetuity. The present value of a perpetuity is equal to the regular payment divided by the discount rate and can be expressed with the following perpetuity formula:

PV = D / R

where:

  • PV is the present value of perpetuity — how much the perpetuity is worth,
  • D is the dividend or regular payment — the amount of cash flow received every period,
  • R is the discount rate — a percentage amount representing the time value of money concept. It lowers the value of future cash flows.

Do you know the present value and dividend and want to find the discount rate? Remember that our calculator can do reverse calculations as well. It does not matter which two variables you start with. The perpetuity calculator will automatically compute the third.

Perpetuity formula — an example

In this example, you will see how to calculate perpetuity step by step. You are offered a bond that pays a $10 dividend yearly and carries on indefinitely. Assuming a 5% discount rate, how much would such a perpetuity be worth? Let's calculate:

PV = $10 / 5% = $200

In this case, the present value of perpetuity would be $200, which should be the bond's price.

Let's try another example with the same dividend of $10. What would happen to the present value if the discount rate changed to 8%? Let's find out:

PV = $10 / 8% = $125

The present value would equal $125, less than the first case. Higher discount rates lead to a faster decrease in present value. In both cases, the $10 you receive yearly loses value over time. In the second case, this loss happens faster.

Growing perpetuity formula

From its definition, we know that a perpetuity is a stream of indefinite fixed payments paid out at regular intervals. You also found out that the value of those payments constantly decreases. To offset this decrease, there is a so-called growing perpetuity.

A growing perpetuity involves payments that do not remain fixed. Instead, they grow at a constant rate. If the growth rate is 4%, each payment will be 4% higher than the previous one. This is called compound interest. Despite the growth, the loss of value will also happen here, as in normal perpetuity, but it will be smaller.

To calculate the present value of growing perpetuity, you can use the growing perpetuity formula:

PV = D / (R − G)

where, as previously:

  • PVPresent value of perpetuity;
  • DDividend; and
  • RDiscount rate.

and the new variable is:

  • G which represents the growth rate of payments. Just like the discount rate, it is also a percentage value.

One important condition is that the growth rate has to be smaller than the discount rate (R > G). This is necessary for the definition of perpetuity to be true. A growth rate higher than the discount rate would lead to each subsequent payment growing in present value. In such a case, the growing perpetuity would have an infinite value, making it impossible to calculate.

How to calculate perpetuity? — Growing perpetuity example

Let's use our $10 dividend payments from the last example. Suppose the discount rate is still 8%, but this time, the dividends are supposed to grow at a rate of 2%.

PV = 10 / (8% − 2%) = $166.67

Without the growth rate, the present value of $10 dividends at an 8% discount rate was $125. The 2% growth rate of dividends helped to increase the present value to about $167, making it a better investment.

Perpetuity in real life — examples

Actual perpetuities are hard to find in the real world. However, there are many investments based on selected features of perpetuities. In the case of real estate, the owner is entitled to recurring rental payments — so the real estate is valued as a perpetuity, but it is not guaranteed.

Preferred stocks offer a fixed dividend and can be calculated as perpetuity. Those dividends can, however, sometimes be withheld, or the company may go bankrupt.

The United Kingdom first issued Consols in the eighteenth century — a special kind of bond used to finance government debt. They were fully redeemed in 2015.

Yale University is the owner of one of five known Dutch perpetuities from 1648, and in 2015 collected 12 years' worth interest on the bond.

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