# Perpetuity

A perpetuity refers to a security that is paid for an infinite amount of time. In the sphere of finance perpetuity means an ongoing flow of similar cash streams which do not end.The idea of an infitite term is used in some financial theories. for example — the dividend discount model contains references about perpetuity.

## Definition of Perpetuity

To begin with, such a term as an annuity represents a stream of cash flows. A perpetuity itself is an everlasting kind of annuity that flocks into infinity. The flow of cash streams lasts for a perpetual amount of time. People, who are involved in finance, turn to the calculation of perpetuity in value methodology in order to find the current value of a company's cash flows.

British obligations (or consols) can be a clear example of financial operations with perpetual cash flows, which were dismissed by the Bank in Engand in 2015. As the bondholders acquired consoles from the government of the UK, they have a right to keep on getting annyual percent payments endlessly.

Although it may seem a bit counterintuitive, ongoing series of cash can have a limited present value. It is determined because of the time value of currency, and thus each payment is only part of the previous one.

In particular, this formula of perpetuity defines the sum of cash flows in the final year of the company's operation. When valuing a company, it is stated to be an operating entity, which means that its operations continue forever. That’s why the final year is perpetual, and analysts utilize the formula of perpetuity to determine its value.

## Present Value of Perpetuity

The formula to determine the present value of a perpetuity, or the state valuable paper with perpetual cash flows, is as follows:

The main method used to determine the indefinite term is to split the money flows by a particular interest rate. The formula that is used to define the finite value in the cash flow for valuation is a bit more complex.

It is the calculation of cash flows in year 10 of the business, redoubled by one + a logterm growth rate of the company, and then the given sum is split with the difference between the growth rate and the capital cost.

In other words, the final value means a certain amount of money flows divided by a particular interest rate — the main formula for defining a perpetuity.

## Practical calculation of Perpetuity

For example, if it is predicted that the enterprise earns $ 100,000 in the 10th year, and the cost of the company's capital is 8% with a long-term growth rate of 3%, the perpetuity value is as follows:

This signifies that $100,000 paid in perpetuity, subject to the growth rate of 3% at the capital cost of 8%, is worth $2.06 million over a decade. Now, people have to define the value of 2.06 million dollars today. To do that, economists take another formula known as the present value of an indefinite contract.

## Growing Perpetuities

The net present value of a perpetuity is not as great as it seems because the temporal money value undermines the value of dollars far into the distant future (for example, due to the process of inflation). Consequently, the cash flows taken from a fixed perpetuity a long time from now possibly will become insignificant from the perspecrive of future buying power.

The increasing perpetual payment regulates the amount of payments each period according to the level of inflation, providing a stable level of purchasing power over the years. Thus, the current value of a growing perpetual contract will be higher than a “pegged” or non-growing perpetual contract. The greater the increase rate of payments for the period in the forseeable future, the bigger the present value.

The method of calculating the perpetuity growth is almost similar to the basic formula, but deducts the inflation rate(e.g. the growth rate — g) from the rate of discount, r, in the denominator:

Note that the growth rates in a growing perpetuum remain fixed throughout its infinite life, which makes it able to include only an approximate estimate of what inflation on average may be in the long term.

## Perpetuity in real life

This financial instrument allows a flow of cash stream in perpetuity to be without end. Previously, England suggested to the government the idea of a bond called a “consol” , and it was arranged in the form of perpetuity, despite the fact that these instruments have since been disbanded. In comparison with other bonds, perpetuity doesn’t have any fixated repayment period, however it implies paying percentages forever.

The second example relates to the property sector, when the owner acquires a real estate and then leases it. The owner has access to an endless cash flow from the lease as long as the real estate continues to exist (provided that the tenant continues to lease).

Another example is preferred shares, where an indefinite settlement assumes that the company will keep on working on the market indefinitely and continue to pay dividends.

## Perpetuity value

At the first sight, it seems as though an instrument that offers an endless stream of cash flows would be almost always valuable, but this is not like that. In the view of mathematics, the value of the instrument is finite, and its validity can be defined by making a discount for its future cash flows to the current with a use of specified discount rate.

This proceeding, also called discounted cash flow (DCF) analysis, is also exctensively used to evaluate other security types, such as stocks, bonds and property investments.

## Perpetuity vs. annuity

Perpetual contract and annuity are indentical instruments in the sense that both offer a fixed set of cash flows over time. However, the main difference between the two is that annuities have a pre-determined end date, known as the “maturity date,” while perpetual contracts are designed to be used forever. It is important to note that both annuities and perpetual obligations can be evaluated using DCF analysis.

## Longevity of Perpetuity

Perpetuity lasts forever. Perpetuities are literally investments which provide payments endlessly, without any maturity or expiration date. In fact, they are annuitues which never end. Such financial products as perpetuity are quite uncommon today, but in general the idea of a perpetuity and the defining its current value (by dividing the cash flow amount by the rate of discount) remains a crucial method in finance.