What Are Annuities Used For

Series of payments made at equal intervals

An annuity is a series of payments made at equal intervals.^[one] Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and alimony payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at whatsoever other regular interval of time. Annuities may exist calculated by mathematical functions known equally "annuity functions".

An annuity which provides for payments for the remainder of a person'due south lifetime is a life annuity.

Types [edit]

Annuities may be classified in several ways.

Timing of payments [edit]

Payments of an annuity-immediate are fabricated at the end of payment periods, so that interest accrues between the result of the annuity and the first payment. Payments of an annuity-due are made at the offset of payment periods, so a payment is fabricated immediately on issueter.

Contingency of payments [edit]

Annuities that provide payments that will be paid over a period known in advance are annuities sure or guaranteed annuities. Annuities paid simply nether certain circumstances are contingent annuities. A mutual example is a life annuity, which is paid over the remaining lifetime of the annuitant. Certain and life annuities are guaranteed to exist paid for a number of years and and then go contingent on the annuitant being alive.

Variability of payments [edit]

• Fixed annuities – These are annuities with fixed payments. If provided by an insurance visitor, the company guarantees a fixed render on the initial investment. Fixed annuities are not regulated by the Securities and Exchange Commission.
• Variable annuities – Registered products that are regulated past the SEC in the The states. They allow direct investment into various funds that are specially created for Variable annuities. Typically, the insurance company guarantees a certain expiry 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 functioning of an index determines whether the minimum, the maximum or something in between is credited to the client.

Deferral of payments [edit]

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

Valuation [edit]

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

Annuity-certain [edit]

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.

Annuity-immediate [edit]

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

	↓	↓	...	↓	payments
———	———	———	———	—
0	1	2	...	northward	periods

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

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

where ${\displaystyle n}$ is the number of terms and ${\displaystyle i}$ is the per catamenia involvement rate. Present value is linear in the amount of payments, therefore the nowadays value for payments, or hire ${\displaystyle R}$ is:

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

In do, oftentimes loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest ${\displaystyle I}$ is stated as a nominal involvement charge per unit, and ${\textstyle i=I/12}$ .

The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments fabricated to an interest-bearing account. For an annuity-firsthand, it is the value immediately after the n-thursday payment. The hereafter value is given by:

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

where ${\displaystyle n}$ is the number of terms and ${\displaystyle i}$ is the per menstruation involvement charge per unit. Future value is linear in the corporeality of payments, therefore the future value for payments, or rent ${\displaystyle R}$ is:

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

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

${\displaystyle {\text{PV}}\left({\frac {0.12}{12}},5\times 12,\$100\right)=\$100\times a_{{\overline {lx}}|0.01}=\$4,495.l}$

The rent is understood as either the amount paid at the end of each flow in return for an corporeality PV borrowed at time cypher, the principal of the loan, or the amount paid out by an involvement-bearing business relationship at the end of each menstruum when the amount PV is invested at time zero, and the account becomes zero with the north-th withdrawal.

Future and present values are related since:

${\displaystyle s_{{\overline {n}}|i}=(1+i)^{northward}\times a_{{\overline {due north}}|i}}$

and

${\displaystyle {\frac {ane}{a_{{\overline {n}}|i}}}-{\frac {1}{s_{{\overline {due north}}|i}}}=i}$

Proof of annuity-immediate formula [edit]

To calculate present value, the k-th payment must be discounted to the present past dividing by the interest, compounded past k terms. Hence the contribution of the thou-th payment R would be ${\displaystyle {\frac {R}{(1+i)^{k}}}}$ . Just because R to be i, and so:

${\displaystyle {\brainstorm{aligned}a_{{\overline {northward}}|i}&=\sum _{one thousand=ane}^{n}{\frac {1}{(1+i)^{k}}}={\frac {i}{1+i}}\sum _{chiliad=0}^{due north-ane}\left({\frac {i}{1+i}}\right)^{grand}\\[5pt]&={\frac {1}{1+i}}\left({\frac {i-(one+i)^{-north}}{ane-(ane+i)^{-1}}}\correct)\quad \quad {\text{by using the equation for the sum of a geometric series}}\\[5pt]&={\frac {1-(1+i)^{-northward}}{1+i-1}}\\[5pt]&={\frac {i-\left({\frac {1}{1+i}}\right)^{due north}}{i}},\finish{aligned}}}$

which gives us the consequence every bit required.

Similarly, we tin testify the formula for the future value. The payment made at the stop of the last yr would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (due north − i) years. Therefore,

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

Annuity-due [edit]

An annuity-due is an annuity whose payments are made at the kickoff of each period.^[three] Deposits in savings, rent or charter payments, and insurance premiums are examples of annuities due.

↓	↓	...	↓		payments
———	———	———	———	—
0	one	...	n − i	due north	periods

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

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

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

where ${\displaystyle n}$ is the number of terms, ${\displaystyle i}$ is the per-term involvement rate, and ${\displaystyle d}$ is the effective charge per unit of discount given by ${\displaystyle d={\frac {i}{i+i}}}$ .

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

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

${\displaystyle {\frac {one}{{\ddot {a}}_{{\overline {northward}}|i}}}-{\frac {1}{{\ddot {southward}}_{{\overline {n}}|i}}}=d.}$

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

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

In Excel, the PV and FV functions take on optional 5th argument which selects from annuity-immediate or annuity-due.

An annuity-due with north payments is the sum of ane annuity payment now and an ordinary annuity with one payment less, and as well equal, with a time shift, to an ordinary annuity. Thus nosotros have:

${\displaystyle {\ddot {a}}_{{\overline {n|}}i}=a_{{\overline {north}}|i}(one+i)=a_{{\overline {n-i|}}i}+i}$ . The value at the fourth dimension of the first of n payments of ane.

${\displaystyle {\ddot {s}}_{{\overline {n|}}i}=s_{{\overline {n}}|i}(one+i)=s_{{\overline {due north+ane|}}i}-1}$ . The value one period after the time of the terminal of n payments of 1.

Perpetuity [edit]

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

${\displaystyle \lim _{north\,\rightarrow \,\infty }{\text{PV}}(i,north,R)=\lim _{n\,\rightarrow \,\infty }R\times a_{{\overline {northward}}|i}=\lim _{n\,\rightarrow \,\infty }R\times {\frac {1-\left(ane+i\correct)^{-n}}{i}}=\,{\frac {R}{i}}.}$

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

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

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

Life annuities [edit]

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

Amortization calculations [edit]

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

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

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

Also, this can exist idea of as the present value of the remaining payments

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

See also fixed rate mortgage.

Example calculations [edit]

Formula for finding the periodic payment R, given A:

${\displaystyle R={\frac {A}{ane+\left(i-\left(1+{\frac {j}{1000}}\right)\correct)^{-{\frac {(n-ane)}{j/one thousand}}}}}}$

Examples:

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

Find PVOA factor every bit. 1) discover r as, (1 ÷ one.15)= 0.8695652174 two) find r × (r ^north − ane) ÷ (r − 1) 08695652174 × (−0.3424837676)÷ (−1304347826) = 2.2832251175 70000÷ 2.2832251175= $30658.3873 is the correct value

1. Discover the periodic payment of an annuity due of $250,700, payable quarterly for eight years at five% compounded quarterly.
   • R= 250,700/(1+〖(i-(ane+((.05)/4) )〗^(-(32-1))/((.05)/4))
   • R = 250,700/26.5692901
   • R = $ix,435.71

Finding the Periodic Payment(R), Given South:

R = Due south\,/((〖((1+(j/m) )〗^(n+one)-i)/(j/m)-1)

Examples:

1. Find the periodic payment of an accumulated value of $55,000, payable monthly for three years at 15% compounded monthly.
   • R=55,000/((〖((1+((.fifteen)/12) )〗^(36+1)-i)/((.15)/12)-1)
   • R = 55,000/45.67944932
   • R = $1,204.04
2. Find the periodic payment of an accumulated value of $ane,600,000, payable annually for 3 years at ix% compounded annually.
   • R=1,600,000/((〖((1+((.09)/1) )〗^(3+i)-ane)/((.09)/ane)-1)
   • R = 1,600,000/3.573129
   • R = $447,786.eighty

Legal regimes [edit]

• Annuities under American constabulary
• Annuities under European constabulary
• Annuities under Swiss police force

Meet besides [edit]

• Amortization computer
• Stock-still rate mortgage
• Life annuity
• Perpetuity
• Time value of money

References [edit]

1. ^ Kellison, Stephen G. (1970). The Theory of Involvement. Homewood, Illinois: Richard D. Irwin, Inc. p. 45
2. ^ Lasher, William (2008). Applied financial management. Bricklayer, Ohio: Thomson S-Western. p. 230. ISBN0-324-42262-viii. .
3. ^ Jordan, Bradford D.; Ross, Stephen David; Westerfield, Randolph (2000). Fundamentals of corporate finance . Boston: Irwin/McGraw-Hill. p. 175. ISBN0-07-231289-0.

• Samuel A. Broverman (2010). Mathematics of Investment and Credit, 5th Edition. ACTEX Academic Serial. ACTEX Publications. ISBN978-ane-56698-767-seven.
• Stephen Kellison (2008). Theory of Involvement, 3rd Edition. McGraw-Hill/Irwin. ISBN978-0-07-338244-nine.

What Are Annuities Used For,

Source: https://en.wikipedia.org/wiki/Annuity

Posted by: lightliess1983.blogspot.com

Share this post