## Understand

Perpetuity is a type of an annuity in which equal cash flows are received with an infinite period of time at regular intervals. It is also called perpetual annuity because it has equal installments but with no end. Although it seems illogical that infinite cash flows have finite present value, but according to the time value of money principle where each payment becomes a fraction of last, the present value of a perpetuity is finite and is a sum of discounted values of infinite payments. British consol is a perfect example of it where bond holder receives infinite payments at regular intervals.

## Perpetuity Formula

The value of infinite payments may vary because of change in interest rate which is being used as a discount factor in the formula further it is supposed in the formula that the cash flows per period never change.

Present value of perpetuity = A/r

Where,

**A** is fixed amount of installment being paid after regular intervals, and

**r** is a discount factor or interest rate.

## How is the formula derived?

The present value of the formula can be tested theoretically by using ordinary annuity formula which is

Present value annuity = A[1-(1+r)^{-n}/r]

Where,

A is fixed amount of installment being paid after regular intervals,

n is the total number of annuity payment or number of the equal interval in which equal payments are made,

as n gets larger, (1+r)^{-n} get smaller and when n approaches towards infinity, (1+r)^{-n }becomes zero. After adding zero in (1+r)^{-n }in formula, present value of annuity formula changes as under

Present value annuity or perpetual annuity= A[1-0 /r]

After simplification,

Present value of perpetual annuity = A/r

## Example 1

Suppose, you want to secure your retirement years by receiving regular payments of $20 per year till infinity which will be discounted using the interest rate of 5%. How many dollars would you need to invest today so that you can get $20 per year till infinity?

Solution

Perpetuity = A/r

Perpetuity = $20/0.05

Perpetuity = $400

This means that value of the infinite stream of payments is $400 and the price of that annuity should be $400.

## Example 2

Calculate the present value of an annuity, paying $100 per year till infinity and currently banks offers an interest rate of 7% on loans.

Solution

P = A/r

Where,

P is the value of perpetuity

A = $100

R = 7%

Perpetuity = $100/0.07

Perpetuity = $1428.57