An annuity and a Perpetuity are financial products that provide a set of cash payments to an investor based on certain terms. They are appropriate financial instruments to people who are looking to secure a steady income source as part of their retirement planning.

They can also be useful to turn a substantial lump sum into a steady cash flow, such as for winners of large cash settlements from a lawsuit or someone who won the lottery. Payments can be delivered on a monthly, quarterly, semi-annually or annually basis.

These instruments trade a lump sum or several inputs for equal payments at a later phase and are designed under the concept of the time value of the money.

## What is an Annuity?

Contents

- What is an Annuity?
- What is Perpetuity?
- Key Takeaways
- Annuity and Perpetuity Formulas
- Annuity vs Perpetuity Key differences
- Perpetuity vs Annuity Examples
- Bottom Line
- Frequently Asked Questions
- What distinguishes an annuity from a perpetuity in financial terms?
- How does the present value calculation differ between an annuity and a perpetuity?
- Can you switch from an annuity to a perpetuity or vice versa in investment products?
- What are the tax implications of receiving payments from an annuity versus a perpetuity?

An annuity is a financial instrument that pays out a fixed stream of payments to an individual for a certain period of time. The cash flows are received by the beneficiary over the asset’s life or until the owner’s death, depending on the annuity type.

There are two main types of annuities and these are ordinary annuities and annuities due. An ordinary annuity will be paid at the end of each time period while an annuity due will be paid at the beginning of the time period.

The period when an annuity is being funded and before it starts paying out any money is referred to as the accumulation phase or surrender period. The investor usually has to pay a penalty if money is withdrawn at this stage.

Because of that, people must forecast their financial requirements during the surrender period because this could be an important restriction if the person requires liquidity.

When the investor starts receiving the payments, this is known as the annuitization or distribution phase.

The contract may provide the alternative of taking out a one-time lump sum or several subsequent payments. The payments can provide regular periodic income to the annuitant or they could be variable.

A person who is in his working years would like to invest in annuities to secure a fixed income source when they retire from working. This investor pays a series of amounts during a defined number of years and later receives a stable monthly payment during his retired phase.

## What is Perpetuity?

In finance, perpetuity is the term that defines a perpetual annuity. These are a financial instrument that provides a constant stream of identical cash flows with no end. In other words, it is a type of annuity that lasts forever.

Unlike a typical bond, the principal is never repaid. In other words, there is no present value for the principal. On the other hand, receipts that are anticipated far in the future have extremely low present value.

There are some examples of perpetuities in the real world: fixed coupon payments on permanently invested (irredeemable) sums of money; preferred stocks, which are assumed to pay dividends to the stockholder as long as the company exists; scholarships paid perpetually from an endowment; and a real estate property that is rented, since the monthly rent is seen as an infinite stream of cash flows.

Nevertheless, perpetuities are now rarely seen as a financial instrument. It used to exist the UK’s government bond known as a Consol, in which bondholders received annual fixed coupons (interest payments) as long as they held the security, yet Consols were discontinued in 2015.

## Key Takeaways

**Duration of Payments:** Annuities provide payments over a fixed period, either for a certain number of years or the lifetime of the recipient, making them suitable for retirement planning. In contrast, perpetuities offer infinite payments that continue indefinitely, often used in financial models or as theoretical concepts in valuing certain types of investments.

**Present Value Calculation:** The calculation of present value differs significantly; annuities require a more complex formula to account for the finite payment period, whereas the present value of a perpetuity is simpler, calculated by dividing the periodic payment by the discount rate, reflecting its endless nature.

**Investment Objectives and Uses:** Annuities are practical financial products designed for individuals seeking stable income over a defined period, particularly useful for retirement income. Perpetuities, while less common in practice, are often found in perpetual bonds or endowed funds, serving investment or charitable funding objectives with their unending payment structure.

## Annuity and Perpetuity Formulas

The present value of an annuity is the current value of all the income generated by that investment in the future. For an ordinary annuity, the formula to calculate the PV is expressed as follows:

PV = P * (( 1- (1 + r)^{-n }) / r)

Where PV is present value, P is the amount of the periodic payment, r is the discount rate, and n is the number of periods.

The present value of a perpetuity is simply the coupon amount divided by the discount rate:

PV = P / r

Where PV is present value, P is the amount of the periodic payment and r is the discount rate.

## Annuity vs Perpetuity Key differences

An annuity is an investment that makes regular payments throughout the specified period and has a defined end. This maturity date might be a particular date or the moment of the owner’s death. This means that annuities eventually stop making payments on behalf of the beneficiary.

The main difference between an annuity and a perpetuity is that the payments of the latter never stop. This means that the investor never stops being benefited by the payments. In the case of the perpetuity, the payments will pass on to his heirs.

The perpetuity can be sold from one investor to another and the new investor will start to receive such payments.

All perpetuities are annuities, but just a few annuities are perpetuities. Annuities are commonly found in the market but perpetuities are a rare case in today’s financial world.** **

### Duration

The duration of an annuity is commonly defined in the terms and conditions of the financial instrument. For retirement planning purposes, annuities may last 10 or more. On the other hand, there are other instances during which annuities may last a shorter period of time.

In contrast, perpetuities entitle the holder to a set of periodical payments that have no end date. In other less extreme cases, on the other hand, the term perpetuity also refers to an annuity that is active for as long as the holder lives.

### Types

There are mainly two types of annuities and these are:

**Annuity due:** These annuities demand payments at the beginning of each time period and the most common ones are rent, licensing fees, and certain fixed-rate services.

The amount to be paid to the beneficiary of an annuity due can be estimated by using the following formula:

PMT = PV * [1−1(1+r)(n−1)i+1]

**Ordinary annuity:** Ordinary annuities are due at the end of each time period and the most common ones are bonds, as the holder receives coupon payments once the coupon period has ended.

The amount of an ordinary annuity can be calculated through the following formula:

PMT = PV / ((1 – (1 + r) ^ -n ) / r)

Where:

Present Value (PV): the amount that has been saved through the accumulation phase of the annuity.

r: the interest rate or discount rate.

n: the number of time periods or the number of payments that will be issued.

Perpetuities have no sub-classifications as they all function in the same way.

### Interest

The amount paid through an annuity is calculated by using a complex financial formula that incorporates the time value of money. This formula basically estimates how much each future payment should be worth to reflect a net present value that is similar to the amount saved during the accumulation phase of the annuity which takes into account the effect of compounding interest.

The value of a perpetuity, on the other hand, is calculated by simply multiplying the face value of the capital invested by the simple interest paid by the financial instrument.

### Usability

Annuities are fairly common in modern financial markets and they operate even without the investment public knowing that the financial instrument they hold is working as one. Annuities are frequently seen in retirement planning, estate planning, financial planning, insurance policies, and other similar instruments.

In contrast, perpetuities are rare in today’s financial markets as they involve a costly commitment for the issuer. On the other hand, there are still rare situations where perpetuities are suitable including certain specific estate planning scenarios.

## Perpetuity vs Annuity Examples

Hanna Klein lost her parents at the age of 18 but received support from two wealthy aunts, Katherine and July. Katherine offered Hanna an annual gift of $800 starting at the end of the first year and continuing forever.

July offered an annual gift of $1,600 but for the next 10. For Hanna, the discount rate applicable is 8% annually. She wanted to know which of the two offers represented more money in present value.

Hanna calculated the PV of Katherine’s proposal, which is a perpetuity:

PV = 800 / 0.08 = 10,000

And the PV of July’s proposal, which is an annuity that will last 10 years, is:

PV = 1,600 * (( 1- (1 + 0.08 )^{-10 }) / 0.08) = $10,736.13

According to the results, July’s offer represents more money in terms of present value.

## Bottom Line

Most people think that the fact perpetuities are paid for an endless period of time makes them invaluable. Even though that may be the case, elements such as inflation could deteriorate the value of a perpetuity considerable.

Additionally, from the perspective of present value calculations, the cash flow obtained from a perpetuity many years from now will have a present value near zero.

## Frequently Asked Questions

### What distinguishes an annuity from a perpetuity in financial terms?

An annuity consists of a series of payments made for a fixed period of time, while a perpetuity offers infinite payments with no end date, providing a never-ending income stream.

### How does the present value calculation differ between an annuity and a perpetuity?

The present value of an annuity is calculated based on the sum of discounted future payments for a certain period, whereas the present value of a perpetuity is calculated using a simple formula: payment amount divided by the discount rate, reflecting its infinite nature.

### Can you switch from an annuity to a perpetuity or vice versa in investment products?

Typically, annuities and perpetuities are distinct financial products; however, some investment products might offer options to convert or feature elements of both, depending on the terms set by the financial institution.

### What are the tax implications of receiving payments from an annuity versus a perpetuity?

Tax implications for annuities and perpetuities can vary based on the jurisdiction and the specific product, but generally, payments received from both are subject to income tax, with the structure of the product affecting the timing and amount of tax liability.