## What is an annuity?

Strictly
speaking, an **annuity** is simply a series of cash inflows or
outflows whose timing and amounts are governed by a contractual
agreement. By this definition, a mortgage loan contract with specified
monthly payments may be called an annuity, and a bond purchase
that provides semi-annual interest payments to the bond holder are
also one. This article defines and illustrates the term **annuity**, kinds of annuities, and associated calculations, in the context of related terms.

As the term is usually used and understood, however, it refers
to a class of financial service products designed to deliver
an income stream. Annuities of this kind are created and issued by
insurance companies. They may be purchased directly from
the issuing insurance company (the issuer), but they can also be purchased through distributors—typically banks or brokerage houses. In any case, the person buying one is the **contract holder**, and a person receiving payments (a beneficiary) is an **annuitant**.

Contract holder and receiver may be the same person as when, for instance, people buy them early in life to provide themselves with retirement income later. Or, the contract holder may purchase one to provide income for a child, widow, widower, or someone else.

Annuity issuers normally earn income for the annuitant (and themselves) by using funds paid in by the contract holder as investment principal, to be re-invested in stock shares, bonds, or other securities.

Contracts describe the life of an annuity, or its duration, the
time period over which the contract holder pays in,
and the time period over which the annuitant receives income payments. The
specified time between these payments is the ** period**. Contracts normally specify monthly, quarterly, semiannual, or annual periods.

Note that earnings may be retained in the investment account so as to add to the investment principal and produce additional earnings of their own, while actual fund withdrawal by the annuitant may occur later.

Annuity contracts also specify conditions for surrender of the investment during its life, that is, conditions under which the contract holder may withdraw all or part of the original investment and cancel the annuity. Surrender typically brings additional fees to the contract holder, but these are usually set by a fee schedule that decreases surrender costs as the end of annuity life approaches.

## Contents

- What is an annuity?
- How do annuities serve as investments?
- What are the different kinds of annuities on the market?
- What are the most frequently used annuity calculations?
- Fixed annuity due future value FV
_{AD}formulas and calculations. - Fixed ordinary annuity future value FV
_{OA}formulas and calculations. - Fixed ordinary annuity future value FV
_{OA}multiple periods per year. - Annuity notation: Fixed ordinary annuity future value FV
_{OA}. - Annuity notation: Fixed annuity due future value FV
_{AD}. - Variable annuities future value calculations, formulas and examples.

- Fixed annuity due future value FV

## How do annuities serve as investments?

Annuities of the kind described above are properly viewed as *financial investments*.
The potential buyer can anticipate the amounts and timings of
cash outflows and cash inflows and, for this reason, these
investments can be evaluated and compared using many of
the same cash flow metrics that apply to a broad range of financial investments—metrics such as future value (FV), internal rate of return (IRR), simple return on investment (ROI), effective annual yield, and annual percentage rate (APR).

Most annuities have another characteristic in common with other kinds of investments, namely an element of risk. The value of expected income may be less than absolutely certain for several reasons.

- Annuity income depends on the issuer's ability to maintain a healthy business and meet its own financial obligations. These investments, however, typically lack the kind of government guarantees that often stand behind other investments such as bank savings account deposits, so that issuer failure probably means investment failure as well.
- Another risk comes with so-called variable annuities, where income payments to the beneficiary depend on the issuer's ability to bring a good return from its own investments. That is, in fact, the reason that income from these investments varies from period to period.
- In contrast to variable instruments, most fixed investments allow the contract holder to "lock in" a rate of return for the recipient. However, if the investment is purchased when inflation and interest rates are relatively low, income will also be lower than if it had been bought when inflation and interest rates were higher. That is, when rates rise after the purchase, the annuitant may be locked into lower returns than would now be available from other investments.
- The beneficiary may die before receiving the all of the income that was contracted.

The purchase contract may include various clauses meant to mitigate such risks, such as an **indexing clause** that adjust return rates as securities market prices (and inflation and interest rates) change, or a death-benefit clause
that transfers income payments to the annuitant's estate or
beneficiaries in case that person dies before annuity
end. However, as with risk-reward situations generally, the lower
risk provided by such clauses comes at a cost, in the form of lower
returns than comparable investments without them.

## What are the different kinds of annuities on the market?

The potential buyer must select from a very large set of annuity kinds and classes, where each product is shaped by a long list of features and characteristics. Just a few of the many possibilities are illustrated below.

### What are the differences between fixed and variable (pay out) annuity payments?

Annuities are classified first as either fixed or variable, referring to the periodic payments beneficiaries receive:

- For the fixed kinds, each income payment to the annuitant is the same, period to period.
- For the variable kind, income earned can vary from period to period.

Fixed annuities normally pay at a fixed, guaranteed rate. For these, the issuer usually invests the contract holder's principal in low risk government bonds, high grade corporate stock or bonds, or other relatively "safe" securities. While such investments are low risk in their own right, the buyer should remember that annuity income will depend on the issuer's ability to make payments (i.e., the issuing insurance company's ability to service its claims).

With the variable form, the recipient's income can vary from period to period because income depends on the performance of the issuer's investments in stocks, bonds, money market funds, mutual funds, and other financial instruments with a market price that changes. Investing in the variable form is, in fact, not very different from investing directly in these instruments: issuers typically offer a choice between income based on conservative, relatively safe investment portfolios, and annuity income based aggressive portfolios with more potential gain, but also come with higher risk.

### What are the differences betweeen single and multiple (pay in) annuity payments?

The timing of the buyer's payments into the investment plays a significant role in determining financial metrics for an annuity, such as future value, effective annual yield, or internal rate of return (for more on metrics, see annuity calculations, below). Annuities may be set up to require payment from the contract holder in different ways:

- For the single-payment form, the buyer (contract holder) pays in just one lump sum payment
- For the multiple-payment form (or regular payment annuity), the buyer contributes to the investment principal through a series of payments over time.

### How long is annuity duration?

Contract duration also plays a significant role in determining financial metrics for an annuity, such as future value, effective annual yield, or internal rate of return (for more on these metrics, see the section Calculations, below). Durations fall into essentially three classes:

- Fixed duration: These annuities provide income payments for a fixed number of periods, or specific duration (e.g., 10 years or 20 years).
- Life duration: Life annuities provide income payments for the life of the beneficiary. Depending on the specific contract, the income stream may simply terminate with the recipient s death, or the income stream or a lump-sum payout may be transferred to that person's beneficiaries.
- Perpetual (or perpetuity) annuities: These provide an income stream that continues forever. In reality, insurance companies and governments no longer create and sell perpetuities as financial service products. Most recently, the British government sold war bonds called consoles in the 18th and 19th centuries which are essentially perpetuities. Consoles that were issued then are still traded and still providing income to their current owners, albeit at a low rate. Although perpetual bonds are no longer issued, some other forms of investment can provide income essentially in perpetuity, such a real estate investments, or shares of preferred stock.

### What is the timing of annuity payments within period?

Annuities are also divided into two classes depending on payment timing, i.e., whether the payments are made at the beginning or the end of each period.

- End of period payments: When payments are scheduled for the end of each period, the instrument is called an "ordinary" contract or annuity-immediate.
- Start of period payments: When payments are made at the start of period, the instrument is known as an annuity-due.

The difference is of small consequence for long duration contracts, such as life contracts or perpetuities. The difference, however, has a more noticeable impact on future value and other financial metrics for shorter duration contracts, e.g., annuities with a 10-year life.

### What are deferred tax annuities, indexed annuities, and guaranteed return annuities?

Annuities can differ with respect to way they create tax liabilities for the investor, the way payment rates are calculated, and the kinds of guarantees the issuer provides for the buyer. For example:

- Tax deferred: "Tax deferred" means that the income tax liability on these earnings is deferred until the beneficiary actually takes possession of the funds. If that person wishes to leave periodic income "on deposit" in the annuity account for some time, income accumulates and earns income on itself, but the beneficiary owes no taxes on it until funds are withdrawn.
- Index (or indexed equity): An index contract pays annuitant income based on a return rate that is linked to a securities market index, such as the S&P 500 in the United States or the FTSE 100 in London. Regarding potential gains (and the accompanying risk), index forms score higher than comparable non-indexed forms, but lower than direct investments in securities.
- Guaranteed return annuities (GRA): These are sold with an issuer guarantee that the contract holder will never receive back less than 100% of the principal invested, no matter what happens in the securities markets and no matter how interest rates change—even if the instrument is surrendered (the principal is withdrawn and the annuity closed) during its life.

## What are the most frequently used annuity calculations?

Calculations are designed to compare different annuities, or to compare these investments with other forms of investment. These calculations—metrics—serve to answer questions about value to beneficiaries (annuitants) and costs to the buyer, while recognizing time-value of money concepts.

An annuity can be described mathematically as series of cash inflows and/or cash outflows continuing across a specified series of time periods. As such, the calculations below apply not only to income-producing financial service products, but also to the broader range of contracts as defined in the first paragraph of this article, including loan repayments and bond investments.

Annuity metrics apply the same time-value-of money concepts that underlie discounted cash flow analysis and compound interest calculations. For that reason, the examples below are
intentionally shown first with the same time-value-of money symbols
and notation used elsewhere in this encyclopedia. However, because
these instruments are issued by insurance companies, and because they view
contracts and payments as actuarial exercises, the (insurance) industry
describes annuity calculations with a special ** notation** (or **actuarial notation**). Therefore, example calculations below are also shown using actuarial notation.

#### Annuity calculations defined, explained, and calculated below are the following

- Fixed annuity due future value FV
_{AD}formulas and calculations. - Fixed ordinary annuity future value FV
_{OA}formulas and calculations. - Fixed ordinary annuity future value FV
_{OA}multiple periods per year. - Annuity notation: Fixed ordinary annuity future value FV
_{OA}. - Annuity notation: Fixed annuity due future value FV
_{AD}. - Variable annuities future value calculations, formulas and examples.

### Fixed annuity due future value FV_{AD} formulas and calculations

Future value calculations address questions like this: What value will the annuity have at the end of its life? Formulas [3] and [4] in Exhibit 1 below produce the future value of fixed forms with a specified number of periods.

Exhibit 1. Formulas and symbols for calculating fixed annuity future values."

#### Example: Future value, fixed annuity due (FV_{AD})

What will be the value of a fixed contract due at the end of its life? Consider an annuity paying the beneficiary $100 annually, for five years, with payments coming at the (annual) period start (payment at period start makes this an annuity due). Use a nominal (annual) interest rate of 8.0%, and re-invest all incoming payments for the life of the contract at the same rate. Using the symbols above,

PMT = $100

* i* = 8.0% = 0.08

* q* = 1

* Y* = 5

*n* = *Yq* = (5)(1) = 5

The resulting cash flow stream is shown graphically in Exhibit 2:

Exhibit 2. Cash flow stream for a fixed annuity due (payments appear at start of each period). When payments arrive at period start, they earn interest for the period they arrive as well as subsequent periods.

To see interest compounding at work, consider first just the initial $100 payment. That payment arrives at the start of Period 1 (Year 1), and earns compound interest for 5 periods. The future value of just that payment at annuity end is given by formula [1] above:

FV = PMT ( 1+ ( i /q ) )^{n}

= $100 ( 1+ ( 0.8 / 1 ) )^{5}

= $100 (1.08 )^{5 } = $100 (1.4693) = $149,93

For
the full payment stream for this annuity due, one payment is
compounded five times (as shown), another payment four
times before the end of the annuity, another payment is compounded 3
times, and so on. Formula [2} produces the future value (FV or FV_{AD}) of the entire annuity due:

FV_{AD} = $100 ( 1+ ( 0.8 / 1 ) )^{1 }+ $100 ( 1+ ( 0.8 / 1 ) )^{2 }+ $100 ( 1+ ( 0.8 / 1 ) )^{3 }+ $100 ( 1+ ( 0.8 / 1 ) )^{4 }+ $100 ( 1+ ( 0.8 / 1 ) )^{5 }

= $100 (1.08)^{ }+ $100 (1.1664)^{ }+ $100 (1.2597) + $100 (1.3605) + $100 (1.4693)^{ }

= $633.59

#### Fixed ordinary annuity future value FV_{OA} formulas and calculations

What will be the value of a fixed ordinary annuity at the end of its life? For the annuity due above, each incoming payment earns interest in the period it arrives and in subsequent periods. However, in the more common ordinary annuity, (or annuity-immediate) payments arrive at period end and therefore do not begin earning interest until the following period. Formula [3] produces future value for an ordinary annuity. Formula [3} simply subtracts one compounding cycle from each term in formula [2], by reducing each exponent by 1. Exhibit 3, below, shows the timing of cash flow payments in the ordinary annuity version of this example:

Exhibit 3. Cash flow stream for a fixed annuity—ordinary annuity (payments appear at end of each period). When payments arrive at period end, they do not earn interest the next period.

Formula [2} produces the future value (FV or** FV _{OA}**) of the ordinary annuity:

FV_{OA} = $100 ( 1+ ( 0.8 / 1 ) )^{0 }+ $100 ( 1+ ( 0.8 / 1 ) )^{1 }+ $100 ( 1+ ( 0.8 / 1 ) )^{2 }+ $100 ( 1+ ( 0.8 / 1 ) )^{3 }+ $100 ( 1+ ( 0.8 / 1 ) )^{4 }

= $100 (1.0)^{ }+ $100 (1.08)^{ }+ $100 (1.1664) + $100 (1.2597) + $100 (1.3605)^{ }

= $586.66

In this case, with 5-period annuities, the difference between the annuity due (FV_{AD} = $633.59) and the ordinary annuity (FV_{OA}
= $586.66) is relatively large. However, as the number of periods
increases (e.g., as with a life annuity or a 30-year mortgage loan), the
overall impact of payment timing within the period becomes less
important.

### Future value, fixed ordinary annuity FV_{OA} multiple periods per year

What will be the value of a fixed ordinary annuity at the end of its life, if payments are paid and compounded quarterly? The examples above used
annual periods, but consider now another 5-year ordinary annuity with quarterly (3-month) periods (*q* = 4), with four $25 payments per year, for a total of 20 periods (*n* = 20) The same nominal interest rate, 8.0% per year, represents
a 2.0% interest rate for each quarterly period (i.e., *i* / q = 0.08/4 = 0.02). That is, the input variables for formula [3] are now:

PMT = $25

* i* = 8.0% = 0.08

* q* = 4

* Y* = 5

*n* = *Yq* = (5)(4) = 20

By formula [3], the future value is now:

FV_{OA} = PMT ( 1+ (* i* / *q* ) )^{0 }+ PMT ( 1+ ( 0.8 / 4 ) )^{1 }+ **...**^{ }+ PMT ( 1+ ( .08 / 4 ) )^{(20 – 1) }^{ }

= $25(1.0) + 25 (1.02)^{1} + $25 (1.02)^{2} + $25 (1.02)^{3} + $25 (1.02)^{4} + **... **

**...**+ $25 (1.02)

^{17}+ $25 (1.02)

^{18}+$25 (1.02)

^{19 }= $25 (1.0) + $25 (1.02)

^{1}+ $25 (1.02)

^{2}+ $25 (1.02)

^{3}+ $25 (1.02)

^{4}+

**...**

**...**+ $25 (1.02)

^{17}+ $25 (1.02)

^{18}+ $25 (1.02)

^{19 }= $25 (1.0000) + $25 (1.0200) + $25 (1.0404) + $25 (1.0612) + $25 (1.0824) +

**...**

**...**+ $25 (1.4002) + $25 (1.4282) + $25 (1.4568)

= $25 (24.2974)

= $607.43

(Not all terms are shown on intermediate steps above.) The shorter periods in this example compared to the previous example (3 months vs. 1 year) lead to a higher future value($607.43 vs. $586.66). Both examples use the same nominal interest rate (8.0%), but the more frequent payments gives the second example a higher effective interest rate (see the encyclopedia entry interest).

** Annuity notation: Fixed ordinary annuity future value FV _{OA} fomulas and calculations**

Annuities
that are financial service products are issued by insurance companies.
In the insurance industry, annuity metrics are viewed as belonging
to the larger body of actuarial calculations. From actuaries, annuity
calculations are presented in a special **annuity notation** (a subset of **actuarial notation**). Annuity notation examples are shown below in Exhibits 4, 5, and 6.

Exhibit 4. Symbols and formulas for calculating Fixed annuity future values, in annuity notation."

#### Annuity Notation Example: Fixed ordinary annuity future value FV_{OA}

How
do you find the value of a fixed ordinary annuity at the end of
its life, using annuity notation? Future value formula [ 4 ] is applied
to the same FV_{OA} example just shown above, for a fixed
ordinary annuity, extending five years, with a nominal annual interest
rate of 8.0%, with quarterly $25 payments reinvested for the life
of the annuity. That is,

PMT = $25

* i* = 8.0% = 0.08

* q* = 4

* Y* = 5

*n* = *Yq* = (5)(4) = 20

i / q = 2.0% = 0.02

Using formula [ 4 ]

FV_{OA} = PMT [ ( (1 + (* i* / *q* )* ^{n}* – 1 ) / (

*i*/

*q*) ]

= $25 [ ( ( 1 + 0.02)* ^{20}* – 1 ) / 0.02 ]

= $25 [ (0.485947) / 0.02 ]

= $607.43

This is the same result obtained earlier by computing 20 individual terms in the conventional future value formula [ 3 ].

Annuity notation for future value of an annuity due (FV_{AD}) is just a simple modification of the future FV_{OA}
formula for an ordinary annuity. With an annuity due, payments arrive
at the start of each period, thereby earning one more period's interest
than a comparable ordinary annuity (where payments arrive at period
end). The FV_{AD} annuity notation formula is thus just the FV_{OA} formula multiplied by (1+periodic interest rate), to add the extra compounding period.

Exhibit 5. Formulas and symbols for calculation future values for a fixed annuity due, in annuity notation.

### Annuity notation: Fixed annuity due future value FV_{AD}

How do you find the value of a fixed annuity due (FV_{AD})
at the end of its life, using annuity notation? For the annuity
due, payments arrive at period start rather than period end, and formula
[ 7 ] above captures the extra compounding cycle with the term (1 + *i* / *q*
) to the right of the expression in brackets. Consider now the
annuity from the previous example, but now cast as an annuity due:
the annuity extends five years, $25 payments are made quarterly, and
reinvested at a nominal (annual) interest rate of 8.0% for the remaining
life of the annuity. Again, the interest rate per period, *i* / *q*, is 8.0% / 4 , or 0.02. That is, ...

PMT = $25

* i* = 8.0% = 0.08

* q* = 4

* Y* = 5

*n* = *Yq* = (5)(4) = 20

i / q = 2.0% = 0.02

Using formula [ 7 ]

FV_{AD} = PMT [ ( (1 + (* i* / *q* )* ^{n}* – 1 ) / (

*i*/

*q*) ] (1 +

*i*/

*q*)

= $25 [ ( ( 1 + 0.02)* ^{20}* – 1 ) / 0.02 ] (1 + 0.08/ 4 )

= $25 [ (0.485947) / 0.02 ] (1 + 0.02)

= $619.58

A
comparison of the last two examples shows different FV results due to
payment timing (payments either at period start or end). The annuity due
has a slight larger future value (FV_{AD }) = $619.58 compared to the ordinary annuity future value (FV_{OA}= $607.43).

### Variable annuities future value formulas and calculations

What is value of a variable annuity at the end of its life? This is the same future value question addressed above for fixed annuities and—in principle—the question is answered with the same kind of cash flow analysis shown above for fixed annuities.

Remember, however, that variable annuities have variable payments from period to period because payment for many of these annuities is tied to investment performance, or market prices for money market funds, bonds, or other securities. For these annuities, future payments cannot be known with certainty. For such variable annuities, It is better not to think of variable annuity payments and future values as forecasts, but rather as *target values* that will result if investment performance goals are met.

Exhibit 6. Formulas and symbols for calculating variable annuity future values.