# Perpetuity Calculator

Determine the net present value of a never-ending stream of payments, given its amount, growth rate, and discount rate.

The Present Value of never-ending stream of 100\$ payments, with a discount rate of 2%, is 5.000\$.

The Perpetuity Calculator helps you compute the present value (PV) of a financial instrument that provides its holder with a never-ending flow of payments.

Inputs Required:

1. Payment amount: the cash payment that the investor is expected to receive (each year) for an infinite amount of time.

2. Discount rate: it represents the 'time value of money', today's money is worth more than the same amount of money in the future. This is because money can earn interest or be invested to generate returns over time, so a dollar today is worth more than a dollar in the future. Conversely, and by the same token, if you delay receiving a dollar in the future, its present value is lower due to the opportunity cost of not having access to that money today.

3. Growth rate: the rate at which the payment amount is expected to grow in the future, which offsets the negative effect of the discount rate, virtually compensating the investor for 'waiting' for a cash flow in the future.

As long as the discount rate is larger than the growth rate, the series is converging and he Present Value then is computed as follows:

`Present Value = Payment amount ``/ (Discount rate - Growth rate)`

A practical example:

 Payment amount 100 Discount rate 6% Growth rate 2%

Calculating PV:

`Present Value = 100\$ ``/ (6% - 2%) = 2.500\$`

In this example, the PV of an annuity that rewards the investor with 100\$ per year at a growing rate of 2%, with a discount rate of 6%, is 2.500\$.

The discount rate used in the calculation is typically the prevailing market rate for similar investments, as it represents the opportunity cost of investing in perpetuity. The concept of converging series is also important in perpetuity calculations, as it ensures that the sum of the discounted cash flows approaches a finite limit as the number of cash flows approaches infinity. This is necessary to ensure that the perpetuity has a well-defined present value.

In the context of growing interest rates, the opportunity cost of money increases (as better options are available outside), and the value of perpetuity decreases.

A noteworthy example of a financial perpetuity is a consol bond. Consols were first issued by the British government in the 18th century and were widely used in the 19th century. These bonds had no maturity or redemption date specified. The only way for an investor to receive the principal back was to sell the bond on the secondary market. Consols were popular with investors seeking a steady stream of income without the risk of their principal being returned prematurely.

The concept of perpetuity is also widely used in real estate investments to estimate the value of a property based on the expected future cash flows generated. If the rental income is expected to remain stable and predictable for an indefinite period of time, then the investor may treat this as perpetuity and calculate the present value of the expected future cash flows using a discount rate that reflects the time value of money.

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