# Delayed Perpetuity Examples and Types

When individuals or organizations invest, part of the things they consider is the return they will get on their investment (ROI). Annuities are fast becoming a common investment option for a lot of individuals and organizations who are seeking an investment that provides a reasonable ROI over time.

An annuity is a contract between an individual or an organization and an insurance company. The individual or organization invests either through a series of payments or a one-time lump payment and in exchange for the investment, the insurance company pays the investor a steady income either for a specified period or indefinitely.

A delayed perpetuity is a type of investment that works in a similar way as annuities, however, in the case of a delayed perpetuity, the payment that the investor will receive does not start immediately. It will usually be deferred for a specified period before the investor starts receiving payments.

Delayed perpetuity examples can be found in various practical aspects of life such as pension payment plans, endowment funds, and corporate finance. Before we discuss the delayed perpetuity examples, let us define what a delayed perpetuity means.

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## What is delayed perpetuity?

In financial terms, perpetuity refers to a reoccurring series of payments that are received by an individual or an organization over time with no ending date. This, joined with delayed conveys the intrinsic feature of a delayed perpetuity; a series of payments that will begin after a specified period with no ending date.

For example, if Mr. Titus makes an investment with Adriel Insurance in 2023 and the terms of the investment indicate that he will start receiving $1,000 annually starting from 2028 without a termination date, we can view the investment as a delayed perpetuity.

This is because Mr. Titus does not immediately receive a return on his investment, instead, a predetermined date has been stated which is 2028. Additionally, the $1,000 annual payment he will receive continues indefinitely without a termination date.

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## Types of delayed perpetuity

- Joint life
- Fixed-term
- Index-linked

Delayed perpetuities can be broadly classified into any of the above based on the terms of the financial arrangement. Payments from delayed perpetuities may occur quarterly, semi-annually, or annually depending on the terms of the agreement.

Let us see how each type of delayed perpetuity works.

### Joint life delayed perpetuity

This type of delayed perpetuity is an arrangement between two individuals who are usually a married couple and the insurance company. It provides payment to the beneficiaries after a predetermined period for the rest of their lives.

A major benefit of the joint life delayed perpetuity is that irrespective of the death of either of the couples, the other living partner will continue receiving payment until they die.

Thus, this type of delayed perpetuity serves as a sure income source for couples for the rest of their lives. Additionally, it protects either of them from income challenges that may arise after the death of their partner.

### Fixed-term delayed perpetuity

Fixed-term delayed perpetuity is the most common type of delayed perpetuity. With this type, the investor is guaranteed a steady income stream indefinitely once the specific years of nonpayment have been exceeded.

Depending on the perpetuity arrangement, some fixed-term delayed perpetuity may be inherited by the beneficiary’s next of kin or a relative at their death.

The fixed term delayed perpetuity therefore provides dual benefit to an investor:

- A guaranteed income stream that lasts indefinitely once the initial period of nonpayment has passed.
- A means of income for the next of kin or relative. This is especially beneficial if the relative is dependent on the investor.

### Index-linked delayed perpetuity

This delayed perpetuity type works in a similar way as the fixed-term delayed perpetuity in that payments begin after a previously determined date and continue indefinitely. It however differs in one significant way by being linked to an inflation index such as the Consumer Price Index (CPI).

This means that the payments a beneficiary receives will be dependent on the fluctuations or stability of the index to which the delayed perpetuity has been linked.

For example, an increase in the index leads to an increase in the payment that will be received by the beneficiary while a decrease in the index will result in a decrease in the amount received.

The key benefit of the index-linked delayed perpetuity is that being linked to an inflation index protects the beneficiary’s investment from inflation which may erode the value of the investor’s income over time.

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## What is the delayed perpetuity formula?

Where:

FV = Future value

C = Amount of periodic payment

r = Discount rate

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## Delayed perpetuity examples

### Example 1

For example, if Miss. Joy will receive an annual payment of $10,000 and the interest rate is 10%, the future value of the delayed perpetuity can be calculated using FV = C/r

FV = $10,000/0.1

FV = $100,000

This means that the future value of Miss. Joy’s investment is $100,000

### Example 2

If a deferred perpetuity provides $10,000 payments annually for life at a discount rate of 20% with the first payment delayed until the end of the sixth year. Calculate the future value of this perpetuity.

The future value of the deferred perpetuity can be calculated using FV = C/r

C = $10,000

r = 20%

FV = $10,000/20%

FV = $50,000

Thus, the future value of the deferred perpetuity is $50,000

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## Present value of delayed perpetuity

The Present Value (PV) of a delayed perpetuity refers to the current value of the future payments that an individual or organization will receive based on a specified discount rate or rate of return.

The discount rate or rate of return is usually a percentage and usually determines the current value of the future cash flow before the time a beneficiary starts getting payout.

### Present value of a delayed perpetuity formula

The formula for calculating the present value of a delayed perpetuity is given by PV = FV x (1/1 + r)^t-1 where

PV = Present value

FV = Future value of cash flow which is given by the amount of periodic payment divided by the discount rate expressed as C/r

r = Discount rate

t = Time when the payout will begin

Therefore, the present value of delayed perpetuity can also be expressed as PV = C/r x (1/1 + r)^t-1

The present value formula is used when a person wants to discount the future payments they will receive to get its present value. The lower the discount rate, the higher the present value of future payments while the higher the discount rate, the lower the present value of future payments.

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## Present value of delayed perpetuity examples

### Example 1

Assuming Mr. Victor invests in the preferred shares of Financial Falconet in 2016. Based on the purchase agreement, the company will pay him $1,000 from the end of year 6 (2022). Suppose the discount rate is 5% and the payment will be infinite; calculate the present value of Mr. Victor’s delayed perpetuity.

To calculate the present value of Mr. Victor’s delayed perpetuity, we will use PV = C/r x (1/1 + r)^t-1

C = $1,000

r = 5% 0r 0.05

t = 6 years

PV = $1,000/0.05 x (1/1 + 0.05)^6-1

PV = $2,000 X (1/1.05)^5

PV = $2,000 X (0.952380952)^5

PV = $2,000 X 0.783526166

PV = $1,567.05

Thus, the present value of Mr. Victor’s preferred stock at Financial Falconet is $1,567.05

### Example 2

Mrs. Maureen invested a lump sum of $1,000,000 in a pension fund that will pay her $10,000 annually for the rest of her life. If the interest rate is 5% and she will start receiving payments when she retires in 10 years. Calculate the present value of her investment.

The present value of her investment = FV x (1/1 + r)^t-1

FV = C/r = $10,000/0.05 = $200,000

r = 5%

t = 10 years

PV = $200,000 x (1/1 + 0.05)^10-1

PV = $200,000 x (1/1.05)^9

PV = $200,000 x (0.952380952)^9

PV = $200,000 x 0.644608916

PV = $128,921.78

This indicates that the present value of Mrs. Maureen’s investment is approximately $128,921

### Example 3

Assuming a stock pays investors an annual dividend of $15 in perpetuity beginning 8 years from the time they purchased the stock. If the discount rate is 7%, we can calculate the present value of the stock using PV = C/r x (1/1 + r)^t-1

C = $15

r = 7%

t = 8 years

PV = $15/7% x (1/1+0.07)^8-1

PV = $214.285714 x (1/1.07)^7

PV = $214.285714 x (0.934579439)^7

PV = $214.285714 x 0.622749742

PV = $133.446373

Therefore, the future value of the stock is $133.446373

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## Conclusion

We have looked at the different delayed perpetuity examples, using the delayed perpetuity formulas to calculate the future value and the present value.

A delayed perpetuity is an investment option that pays the investor a certain amount of money indefinitely after a predetermined date. Thus, investing in a delayed perpetuity can be beneficial in several ways.

One of the benefits of investing in a delayed perpetuity is that an investor can make small contributions to the investment over a period of time before the payout begins.

It is also a source of steady income over time since most delayed perpetuity investments make payments to investors infinitely.