Present value of a perpetuity

This present value of perpetuity calculator computes the present value of a perpetuity. By default, it calculates the present value of a $500 cash flow at a periodic interest rate (I/Y) of 3.5%. The growth rate of the cash flows is assumed to be 0%. To compute a different present value, modify at least one of the following fields: Cash flow, Discount rate (%), or Growth rate (%). As you make changes to these values, the calculator will automatically update the present value field. It’s important to bear in mind that the growth rate must be lower than the discount rate for accurate results.

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What is a perpetuity?

A perpetuity is a type of financial instrument that promises to pay a fixed amount of money at regular intervals forever. It is essentially a form of infinite annuity, where the payments continue indefinitely rather than for a fixed number of years.

A common example of a perpetuity is a government bond that pays a fixed interest rate forever, with no maturity date. Another example is a real estate investment that generates rental income, with no end date.

Another interesting aspect of perpetuities is that they are rare in practice, as most financial instruments have a finite lifespan. However, there have been instances of financial instruments that exhibit perpetuity-like characteristics or have been referred to as perpetuities due to their indefinite nature of cash flows. Here are a couple of examples:

1. Consols (Consolidated annuities): These were a type of government bond issued by the British government in the 18th and 19th centuries. Consols paid a fixed interest (coupon) to bondholders indefinitely, without a maturity date. While not true perpetuities, they were often referred to as such due to their long-lasting nature.
2. Certain preferred stocks: Some preferred stocks issued by companies have no fixed maturity date and pay a fixed dividend to shareholders indefinitely. While these stocks do have the characteristics of perpetuities, they are not true perpetuities since the issuing company could choose to redeem them at a future date or under certain conditions.

It’s important to note that even in these cases, circumstances can change, and issuers might decide to buy back or redeem these securities, ending the cash flows.

The concept of perpetuities is more theoretical and serves as a model to understand the valuation of assets with long-duration cash flows, rather than a common occurrence in the financial markets. Perpetuities can still be useful in certain contexts, such as for valuing long-term income streams or for creating trusts that generate income for future generations.

An example

One interesting aspect of perpetuities is that their value can be calculated using a simple formula:

\text{Present Value} = \frac{\text{Payment}}{\text{Discount Rate} - \text{Growth Rate}}

where:

• Payment is the amount of money received at each interval.
• Discount Rate is the interest rate used to discount future payments to their present value.
• Growth Rate is the rate at which the cash flows of the perpetuity grow.

This means that the value of a perpetuity increases as the payment amount or the growth rate increases, but decreases as the discount rate (the rate of return required by investors) increases.

\text{Present Value} = \frac{\$100}{0.05 - 0.00}=\$2,000

It is important to note that the perpetuity formula assumes that payments are made at a constant interval forever, and that the discount and growth rates remain constant over time. In reality, both of these assumptions may not hold true, which could affect the accuracy of the calculated present value.