In investment, an **annuity** is a series of payments made at equal intervals.^{ [1] } Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by mathematical functions known as "annuity functions".

- Types
- Timing of payments
- Contingency of payments
- Variability of payments
- Deferral of payments
- Valuation
- Annuity-certain
- Life annuities
- Amortization calculations
- Example calculations
- Legal regimes
- See also
- References
- Other sources

An annuity which provides for payments for the remainder of a person's lifetime is a life annuity. An annuity which continues indefinitely is a perpetuity.

Annuities may be classified in several ways.

Payments of an *annuity-immediate* are made at the end of payment periods, so that interest accrues between the issue of the annuity and the first payment. Payments of an *annuity-due* are made at the beginning of payment periods, so a payment is made immediately on issue.

Annuities that provide payments that will be paid over a period known in advance are *annuities certain* or *guaranteed annuities.* Annuities paid only under certain circumstances are *contingent annuities*. A common example is a life annuity, which is paid over the remaining lifetime of the annuitant. *Certain and life annuities* are guaranteed to be paid for a number of years and then become contingent on the annuitant being alive.

**Fixed annuities**– These are annuities with fixed payments. If provided by an insurance company, the company guarantees a fixed return on the initial investment. Fixed annuities are not regulated by the Securities and Exchange Commission.**Variable annuities**– Registered products that are regulated by the SEC in the United States of America. They allow direct investment into various funds that are specially created for Variable annuities. Typically, the insurance company guarantees a certain death benefit or lifetime withdrawal benefits.**Equity-indexed annuities**– Annuities with payments linked to an index. Typically, the minimum payment will be 0% and the maximum will be predetermined. The performance of an index determines whether the minimum, the maximum or something in between is credited to the customer.

An annuity that begins payments only after a period is a *deferred annuity* (usually after retirement). An annuity that begins payments as soon as the customer has paid, without a deferral period is an *immediate annuity*.^{[ citation needed ]}

Valuation of an annuity entails calculation of the present value of the future annuity payments. The valuation of an annuity entails concepts such as time value of money, interest rate, and future value.^{ [2] }

If the number of payments is known in advance, the annuity is an *annuity certain* or *guaranteed annuity*. Valuation of annuities certain may be calculated using formulas depending on the timing of payments.

If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an *annuity-immediate*, or *ordinary annuity*. Mortgage payments are annuity-immediate, interest is earned before being paid. What is Annuity Due? Annuity due refers to a series of equal payments made at the same interval at the beginning of each period. Periods can be monthly, quarterly, semi-annually, annually, or any other defined period. Examples of annuity due payments include rentals, leases, and insurance payments, which are made to cover services provided in the period following the payment.

↓ | ↓ | ... | ↓ | payments | |

——— | ——— | ——— | ——— | — | |

0 | 1 | 2 | ... | n | periods |

The *present value* of an annuity is the value of a stream of payments, discounted by the interest rate to account for the fact that payments are being made at various moments in the future. The present value is given in actuarial notation by:

where is the number of terms and is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or *rent* is:

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest is stated as a nominal interest rate, and .

The *future value* of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by:

where is the number of terms and is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or *rent* is:

**Example:** The present value of a 5-year annuity with a nominal annual interest rate of 12% and monthly payments of $100 is:

The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the *principal* of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal.

Future and present values are related since:

and

To calculate present value, the *k*-th payment must be discounted to the present by dividing by the interest, compounded by *k* terms. Hence the contribution of the *k*-th payment *R* would be . Just considering *R* to be 1, then:

which gives us the result as required.

Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (*n* − 1) years. Therefore,

An *annuity-due* is an annuity whose payments are made at the beginning of each period.^{ [3] } Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.

↓ | ↓ | ... | ↓ | payments | |

——— | ——— | ——— | ——— | — | |

0 | 1 | ... | n − 1 | n | periods |

Each annuity payment is allowed to compound for one extra period. Thus, the present and future values of an annuity-due can be calculated.

where is the number of terms, is the per-term interest rate, and is the effective rate of discount given by .

The future and present values for annuities due are related since:

**Example:** The final value of a 7-year annuity-due with a nominal annual interest rate of 9% and monthly payments of $100 can be calculated by:

In Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due.

An annuity-due with *n* payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity. Thus we have:

- . The value at the time of the first of
*n*payments of 1. - . The value one period after the time of the last of
*n*payments of 1.

A *perpetuity* is an annuity for which the payments continue forever. Observe that

Therefore a perpetuity has a finite present value when there is a non-zero discount rate. The formulae for a perpetuity are

where is the interest rate and is the effective discount rate.

Valuation of life annuities may be performed by calculating the actuarial present value of the future life contingent payments. Life tables are used to calculate the probability that the annuitant lives to each future payment period. Valuation of life annuities also depends on the timing of payments just as with annuities certain, however life annuities may not be calculated with similar formulas because actuarial present value accounts for the probability of death at each age.

If an annuity is for repaying a debt *P* with interest, the amount owed after *n* payments is

Because the scheme is equivalent with borrowing the amount to create a perpetuity with coupon , and putting of that borrowed amount in the bank to grow with interest .

Also, this can be thought of as the present value of the remaining payments

See also fixed rate mortgage.

Formula for finding the periodic payment *R*, given *A*:

Examples:

- Find the periodic payment of an annuity due of $70,000, payable annually for 3 years at 15% compounded annually.
*R*= 70,000/(1+〖(1-(1+((.15)/1) )〗^(-(3-1))/((.15)/1))- R = 70,000/2.625708885
- R = $26659.46724

Find PVOA factor as. 1) find *r* as, (1 ÷ 1.15)= 0.8695652174 2) find *r* × (*r*^{n} − 1) ÷ (*r* − 1) 08695652174 × (−0.3424837676)÷ (−1304347826) = 2.2832251175 70000÷ 2.2832251175= $30658.3873 is the correct value

- Find the periodic payment of an annuity due of $250,700, payable quarterly for 8 years at 5% compounded quarterly.
- R= 250,700/(1+〖(1-(1+((.05)/4) )〗^(-(32-1))/((.05)/4))
- R = 250,700/26.5692901
- R = $9,435.71

Finding the Periodic Payment(R), Given S:

R = S\,/((〖((1+(j/m) )〗^(n+1)-1)/(j/m)-1)

Examples:

- Find the periodic payment of an accumulated value of $55,000, payable monthly for 3 years at 15% compounded monthly.
- R=55,000/((〖((1+((.15)/12) )〗^(36+1)-1)/((.15)/12)-1)
- R = 55,000/45.67944932
- R = $1,204.04

- Find the periodic payment of an accumulated value of $1,600,000, payable annually for 3 years at 9% compounded annually.
- R=1,600,000/((〖((1+((.09)/1) )〗^(3+1)-1)/((.09)/1)-1)
- R = 1,600,000/3.573129
- R = $447,786.80

In finance, **discounting** is a mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. This transaction is based on the fact that most people prefer current interest to delayed interest because of mortality effects, impatience effects, and salience effects. The **discount**, or **charge**, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

The **net present value** (**NPV**) or **net present worth** (**NPW**) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the annual effective discount rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.

In economics and finance, **present value** (**PV**), also known as **present discounted value**, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be equal or more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater than tomorrow. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of borrowed funds is less than the total amount of money paid to the lender.

In finance and economics, **interest** is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum, at a particular rate. It is distinct from a fee which the borrower may pay to the lender or some third party. It is also distinct from dividend which is paid by a company to its shareholders (owners) from its profit or reserve, but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by risk taking entrepreneurs when the revenue earned exceeds the total costs.

The **time value of money** is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.

In finance, an **interest rate swap** (**IRS**) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations with forward rate agreements (FRAs), and with zero coupon swaps (ZCSs).

In finance, a **perpetuity** is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as consols and were all finally redeemed in 2015.

**Compound interest** is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

**Future value** is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return; it is the present value multiplied by the accumulation function. The value does not include corrections for inflation or other factors that affect the true value of money in the future. This is used in time value of money calculations.

In finance, the **duration** of a financial asset that consists of fixed cash flows, such as a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield, or the percentage change in price for a parallel shift in yields.

**Actuarial notation** is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.

The term **annual percentage rate of charge** (**APR**), corresponding sometimes to a **nominal APR** and sometimes to an **effective APR** (**EAPR**), is the interest rate for a whole year (annualized), rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate. Those terms have formal, legal definitions in some countries or legal jurisdictions, but in the United States:

**Retirement annuity plan** is a financial product that ensures regular income to retirees in later years most often issued and distributed by an insurance organization. The main idea behind this product is to provide retirees the opportunity to attain income after retirement. A 'Retirement annuity plan (RAP) is a type of retirement plan similar to IRA that provides a stream of regular (single) distributions to an insured retiree. Time intervals between distributions as well as their amount are defined by conditions and type of the annuity between issuer organization and client. Nowadays many types of retirement annuities are offered on the market.

An **amortization calculator** is used to determine the periodic payment amount due on a loan, based on the amortization process.

The **actuarial present value** (**APV**) is the expected value of the present value of a contingent cash flow stream. Actuarial present values are typically calculated for the benefit-payment or series of payments associated with life insurance and life annuities. The probability of a future payment is based on assumptions about the person's future mortality which is typically estimated using a life table.

In insurance, an **actuarial reserve** is a reserve set aside for future insurance liabilities. It is generally equal to the actuarial present value of the future cash flows of a contingent event. In the insurance context an actuarial reserve is the present value of the future cash flows of an insurance policy and the total liability of the insurer is the sum of the actuarial reserves for every individual policy. Regulated insurers are required to keep offsetting assets to pay off this future liability.

In banking and finance, an **amortizing loan** is a loan where the principal of the loan is paid down over the life of the loan according to an amortization schedule, typically through equal payments.

Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) **continuous repayment mortgage** is a mortgage loan paid by means of a continuous annuity.

The **sum of perpetuities method** (SPM) is a way of valuing a business assuming that investors discount the future earnings of a firm regardless of whether earnings are paid as dividends or retained. SPM is an alternative to the Gordon growth model (GGM) and can be applied to business or stock valuation if the business is assumed to have constant earnings and/or dividend growth. The variables are:

In finance, a **zero coupon swap** (**ZCS**) is an interest rate derivative (IRD). In particular it is a linear IRD, that in its specification is very similar to the much more widely traded interest rate swap (IRS).

- ↑ Kellison, Stephen G. (1970).
*The Theory of Interest*. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 - ↑ Lasher, William (2008).
*Practical financial management*. Mason, Ohio: Thomson South-Western. p. 230. ISBN 0-324-42262-8.. - ↑ Jordan, Bradford D.; Ross, Stephen David; Westerfield, Randolph (2000).
*Fundamentals of corporate finance*. Boston: Irwin/McGraw-Hill. p. 175. ISBN 0-07-231289-0.

- Samuel A. Broverman (2010).
*Mathematics of Investment and Credit, 5th Edition*. ACTEX Academic Series. ACTEX Publications. ISBN 978-1-56698-767-7. - Stephen Kellison (2008).
*Theory of Interest, 3rd Edition*. McGraw-Hill/Irwin. ISBN 978-0-07-338244-9. - Sprague, Thomas Bond (1878). .
*Encyclopædia Britannica*. Vol. II (9th ed.). pp. 72–89.

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