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Annuity

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Series of payments made at equal intervals

For other uses, seeAnnuity (disambiguation).

Ininvestment, anannuity is a series of payments made at equal intervals based on a contract with a lump sum of money.^[1] Examples of annuities are regular deposits to asavings account, monthlyhome mortgage payments, monthlyinsurance payments andpension 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 bymathematical functions known as "annuity functions".

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

Types

Annuities may be classified in several ways.

Timing of payments

Payments of anannuity-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 anannuity-due are made at the beginning of payment periods, so a payment is made immediately on issue.

Contingency of payments

Annuities that provide payments that will be paid over a period known in advance areannuities certain orguaranteed annuities. Annuities paid only under certain circumstances arecontingent annuities. A common example is alife 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.

Variability of payments

Fixed annuities – These are annuities with fixed payments. If provided by an insurance company, the company guarantees a fixed return on the initial investment. In the United States, fixed annuities are not regulated by theSecurities and Exchange Commission.^{[citation needed]}
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.

Deferral of payments

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

Valuation

Valuation of an annuity entails calculation of thepresent value of the future annuity payments. The valuation of an annuity entails concepts such astime value of money,interest rate, andfuture value.^[2]

Annuity-certain

If the number of payments is known in advance, the annuity is anannuity certain orguaranteed annuity. Valuation of annuities certain may be calculated using formulas depending on the timing of payments.

Annuity-immediate

If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called anannuity-immediate, orordinary annuity. Mortgage payments are annuity-immediate, interest is earned before being paid.

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

Thepresent 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 inactuarial notation by:

a_{{\overline {n}}|i}={\frac {1-(1+i)^{-n}}{i}},

where $n {\displaystyle n}$ is the number of terms and $i {\displaystyle i}$ is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, orrent $R {\displaystyle R}$ is:

{\text{PV}}(i,n,R)=R\times a_{{\overline {n}}|i}.

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest $I {\displaystyle I}$ is stated as anominal interest rate, and ${\textstyle i=I/12}$ .

Thefuture 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:

s_{{\overline {n}}|i}={\frac {(1+i)^{n}-1}{i}},

where $n {\displaystyle n}$ is the number of terms and $i {\displaystyle i}$ is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, orrent $R {\displaystyle R}$ is:

{\text{FV}}(i,n,R)=R\times s_{{\overline {n}}|i}

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

{\text{PV}}\left({\frac {0.12}{12}},5\times 12,\$100\right)=\$100\times a_{{\overline {60}}|0.01}=\$4,495.50

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, theprincipal 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:

s_{{\overline {n}}|i}=(1+i)^{n}\times a_{{\overline {n}}|i}

and

{\frac {1}{a_{{\overline {n}}|i}}}-{\frac {1}{s_{{\overline {n}}|i}}}=i

Proof of annuity-immediate formula

To calculate present value, thek-th payment must be discounted to the present by dividing by the interest, compounded byk terms. Hence the contribution of thek-th paymentR would be ${\frac {R}{(1+i)^{k}}}$ . Just consideringR to be 1, then:

{\begin{aligned}a_{{\overline {n}}|i}&=\sum _{k=1}^{n}{\frac {1}{(1+i)^{k}}}={\frac {1}{1+i}}\sum _{k=0}^{n-1}\left({\frac {1}{1+i}}\right)^{k}\\[5pt]&={\frac {1}{1+i}}\left({\frac {1-(1+i)^{-n}}{1-(1+i)^{-1}}}\right)\quad \quad {\text{by using the equation for the sum of a geometric series}}\\[5pt]&={\frac {1-(1+i)^{-n}}{1+i-1}}\\[5pt]&={\frac {1-\left({\frac {1}{1+i}}\right)^{n}}{i}},\end{aligned}}

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,

s_{{\overline {n}}|i}=1+(1+i)+(1+i)^{2}+\cdots +(1+i)^{n-1}=(1+i)^{n}a_{{\overline {n}}|i}={\frac {(1+i)^{n}-1}{i}}.

Annuity-due

Anannuity-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.

{\ddot {a}}_{{\overline {n|}}i}=(1+i)\times a_{{\overline {n|}}i}={\frac {1-(1+i)^{-n}}{d}},

{\ddot {s}}_{{\overline {n|}}i}=(1+i)\times s_{{\overline {n|}}i}={\frac {(1+i)^{n}-1}{d}},

where $n {\displaystyle n}$ is the number of terms, $i {\displaystyle i}$ is the per-term interest rate, and $d {\displaystyle d}$ is theeffective rate of discount given by $d={\frac {i}{i+1}}$ .

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

{\ddot {s}}_{{\overline {n}}|i}=(1+i)^{n}\times {\ddot {a}}_{{\overline {n}}|i},

{\frac {1}{{\ddot {a}}_{{\overline {n}}|i}}}-{\frac {1}{{\ddot {s}}_{{\overline {n}}|i}}}=d.

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:

{\text{FV}}_{\text{due}}\left({\frac {0.09}{12}},7\times 12,\$100\right)=\$100\times {\ddot {s}}_{{\overline {84}}|0.0075}=\$11,730.01.

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

An annuity-due withn 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:

{\ddot {a}}_{{\overline {n|}}i}=a_{{\overline {n}}|i}(1+i)=a_{{\overline {n-1|}}i}+1

. The value at the time of the first ofn payments of 1.

{\ddot {s}}_{{\overline {n|}}i}=s_{{\overline {n}}|i}(1+i)=s_{{\overline {n+1|}}i}-1

. The value one period after the time of the last ofn payments of 1.

Perpetuity

Aperpetuity is an annuity for which the payments continue forever. Observe that

\lim _{n\,\rightarrow \,\infty }{\text{PV}}(i,n,R)=\lim _{n\,\rightarrow \,\infty }R\times a_{{\overline {n}}|i}=\lim _{n\,\rightarrow \,\infty }R\times {\frac {1-\left(1+i\right)^{-n}}{i}}=\,{\frac {R}{i}}.

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

a_{{\overline {\infty }}|i}={\frac {1}{i}}{\text{  and  }}{\ddot {a}}_{{\overline {\infty }}|i}={\frac {1}{d}},

where $i {\displaystyle i}$ is the interest rate and $d={\frac {i}{1+i}}$ is the effective discount rate.

Life annuities

Valuation oflife annuities may be performed by calculating theactuarial present value of the future life contingent payments.Life tables are used to calculate theprobability 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.

Amortization calculations

If an annuity is for repaying a debtP with interest, the amount owed aftern payments is

{\frac {R}{i}}-(1+i)^{n}\left({\frac {R}{i}}-P\right).

Because the scheme is equivalent with borrowing the amount ${\frac {R}{i}}$ to create a perpetuity with coupon $R {\displaystyle R}$ , and putting ${\frac {R}{i}}-P$ of that borrowed amount in the bank to grow with interest $i {\displaystyle i}$ .

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

R\left[{\frac {1}{i}}-{\frac {(i+1)^{n-N}}{i}}\right]=R\times a_{{\overline {N-n}}|i}.

See alsofixed rate mortgage.

Example calculations

Formula for finding the periodic paymentR, givenA:

R={\frac {A}{1+\left(1-\left(1+{\frac {j}{m}}\right)\right)^{-{\frac {(n-1)}{j/m}}}}}

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) findr as, (1 ÷ 1.15)= 0.86956521742) findr × (rⁿ − 1) ÷ (r − 1)08695652174 × (−0.3424837676)÷ (−1304347826) = 2.283225117570000÷ 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

Legal regimes

See also

References

^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.

Other sources

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)."Annuities" .Encyclopædia Britannica. Vol. II (9th ed.). pp. 72–89.

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