## Calculating Perpetuities

This section discusses calculating perpetuities, a special type of annuity where the stream of payments never ends.

The present value of a perpetuity is simply the payment size divided by the interest rate and there is no future value.

#### LEARNING OBJECTIVE

• Calculate the present value of a perpetuity

#### KEY TAKEAWAYS

##### Key Points
• Perpetuities are a special type of annuity; a perpetuity is an annuity that has no end, or a stream of cash payments that continues forever.
• To find the future value of a perpetuity requires having a future date, which effectively converts the perpetuity to an ordinary annuity until that point.
• Perpetuities with growing payments are called Growing Perpetuities; the growth rate is subtracted from the interest rate in the present value equation.

##### Key Terms
• growth rate: The percentage by which the payments grow each period.

Perpetuities are a special type of annuity; a perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. Essentially, they are ordinary annuities, but have no end date. There aren't many actual perpetuities, but the United Kingdom has issued them in the past.

Since there is no end date, the annuity formulas we have explored don't apply here. There is no end date, so there is no future value formula. To find the $FV$ of a perpetuity would require setting a number of periods which would mean that the perpetuity up to that point can be treated as an ordinary annuity.

There is, however, a $PV$ formula for perpetuities . The $PV$ is simply the payment size ($A$) divided by the interest rate ($r$). Notice that there is no $n$, or number of periods. More accurately, is what results when you take the limit of the ordinary annuity $PV$ formula as $n → ∞$.

It is also possible that an annuity has payments that grow at a certain rate per period. The rate at which the payments change is fittingly called the growth rate ($g$). The $PV$ of a growing perpetuity is represented as $\mathrm{PVGP}=\frac{\mathrm{A}}{(\mathrm{i}-\mathrm{g})}$. It is essentially the same as in except that the growth rate is subtracted from the interest rate. Another way to think about it is that for a normal perpetuity, the growth rate is just 0, so the formula boils down to the payment size divided by $r$.

Source: Boundless