# Delayed Perpetuity

Perpetuity is the series of fixed payments that will last forever. The delayed perpetuity is the series of payments that will start at a specific date in the future. It also knowns as deferred perpetuity. For example, the fixed dividend of preferred shares will begin in the next three years from now, rather than next year. This series of payments is known as delayed perpetuity.

The delay perpetuity has a lower value than the normal perpetuity as the payment start later than normal.

## Present Value of Delayed Perpetuity

\[PV \space of\space Normal\space Perpetuity = {C \over r}\] \[PV \space of\space Delayed\space Perpetuity = {1 \over (1+r)^{T-1}}\ *{C \over r}\]

- C: annual cash flow
- r: annual interest rate
- T: year delay from today till the first payment

We have to discount the present value of normal perpetuity as it is the present value when the first payment is made which is in the future. So we have to discount again by using the present value formula. And the formula above is the combination of both normal perpetuity and present value.

## Example of Delayed Perpetuity

In 2020, Mr. A invests in a preferred share in Company XYZ. Based on the term and conditions, he will receive a fixed payment of $ 5,000 from the end of year 3 (2023). The payment will be infinite. Suppose the bank pays an annual interest rate of 5%. Calculate the present value of delayed perpetuity.

PV of delayed perpetuity = $ 90,703

If we use the normal perpetuity formula the present value is $100,000 (5,000/5%) but it is the present value as at 3^{rd} year (2023). In order to get PV as of today (First year), we have to discount it again by using the normal present value. So the present value would be $ 90,703.