Annuity

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An an annuity is a series of payments made at the same interval. Examples of annuities are regular deposits to savings accounts, monthly home mortgage payments, monthly insurance payments, and pension payments. Annuities can be classified by frequency of payment date. Payments (deposits) can be made weekly, monthly, quarterly, yearly, or at other regular time intervals.

An annuity that provides payments for the rest of a person's lifetime is a lifetime annuity.

Video Annuity

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Annuities can be classified in several ways.

Payment time

Payments from annuity-soon are made at the end of the payment period, so interest arises between the annuity issue and the first payment. Payment of annuity maturity is made at the beginning of the payment period, so an immediate payment is made on the issue.

Payment contents

Annuities that provide payments to be paid over the period previously known are specific annuities or guaranteed annuities. Annuities paid only under certain circumstances are contingent annuities . A common example is a lifetime annuity, paid for the remainder of the annuitant's lifetime. Certain annuities and lives are guaranteed to be paid for several years and then become dependent on a live annuitant.

Diverse payout

• Fixed annuity - This is an annuity with fixed payments. If provided by an insurance company, the company guarantees a fixed profit on the initial investment. Annuities remain unregulated by the Securities and Exchange Commission.
• Variable annuities - Registered products regulated by the SEC in the United States. They allow direct investment into a variety of funds specifically created for Variable annuities. Typically, insurance companies guarantee certain death benefits or lifetime withdrawal benefits.
• Annuity indexed equity - Annuities with payouts associated with the index. Typically, the minimum payment will be 0% and the maximum will be predetermined. Index performance determines whether the minimum, maximum or something is credited to the customer.

Payment delay

Annuities that start payments only after the period are suspension of annuities . Annuities that begin payments without a period of delay are immediate annuities .

Maps Annuity

Assessment

An annuity assessment requires the calculation of the present value of future annuity payments. Annuity assessment requires concepts such as time value of money, interest rates, and future value.

Annuity-certain

If the amount of payment is known in advance, the annuity is specific annuity or annuity guarantee . Annuity assessment can be calculated using the formula depending on the timing of the payment.

Annuity-immediate

If the payment is made at the end of the time period, so the interest accumulates before the payment, the annuity is called annuity-soon , or the usual annuity . The mortgage payment is an immediate annuity, interest earned before being paid.

The current value of the annuity is the value of the payout flow, discounted by the interest rate to explain the fact that payments are being made at various moments in the future. The present value is provided in actuarial notes by:

${\ displaystyle a _ {{overline {n}} | i} = {\ frac {1- \ left (1 i \ right) ^ {- n}} { i}},}  ÂÂ$

di mana ${\ displaystyle n}  ÂÂ$ adalah jumlah istilah dan ${\ displaystyle i}  ÂÂ$ adalah suku bunga per periode. Nilai sekarang adalah linier dalam jumlah pembayaran, oleh karena itu nilai sekarang untuk pembayaran, atau sewa ${\ displaystyle R}  ÂÂ$ adalah:

${\ displaystyle PV (i, n, R) = R \ kali sebuah _ {{overline {n}} | i}}  ÂÂ$

Dalam prakteknya, sering pinjaman dinyatakan per tahun sementara bunga diperparah dan pembayaran dilakukan setiap bulan. Dalam hal ini, minat ${\ displaystyle I}  ÂÂ$ dinyatakan sebagai suku bunga nominal, dan ${\ displaystyle i = I/12}  ÂÂ$ .

The future value of the annuity is the sum of the accumulated, including payments and interest, from the payment flow made to the flower account. For immediate annuity, it is immediate value after n-th payment. Future values ​​are given by:

${\ displaystyle s _ {{overline {n}} | i} = {\ frac {(1 i) ^ {n} -1} {i}}} Â Â$

di mana ${\ displaystyle n}  ÂÂ$ adalah jumlah istilah dan ${\ displaystyle i}  ÂÂ$ adalah suku bunga per periode. Nilai masa depan adalah linier dalam jumlah pembayaran, oleh karena itu nilai masa depan untuk pembayaran, atau sewa ${\ displaystyle R}  ÂÂ$ adalah:

${\ displaystyle FV (i, n, R) = R \ kali s _ {{overline {n}} | i}}  ÂÂ$

Contoh: Nilai sekarang dari anuitas 5 tahun dengan suku bunga tahunan nominal 12% dan pembayaran bulanan $ 100 adalah:

${\ displaystyle PV (0,12/12,5 \ kali 12, \ $ 100) = \ $ 100 \ kali a _ {{\ overline {60}} | 0,01} = \ $ 4 \ , 495,50}  ÂÂ$

This lease is understood as the amount paid at the end of each period in return for the amount of PV borrowed at the time of zero, the principal of the loan, or the amount paid by the account of interest. at the end of each period when the PV amount is invested at zero, and the account becomes zero with nth withdrawal.

Nilai-nilai masa depan dan masa sekarang terkait sebagai:

${\ displaystyle s _ {{overline {n}} | i} = (1 i) ^ {n} \ kali _ {{\ overline {n}} | i }}  ÂÂ$

dan

${\ displaystyle {\ frac {1} {a _ {{overline {n}} | i}}} - {\ frac {1} {s _ {{\ overline { n}} | i}}} = i}  ÂÂ$

Bukti rumus anuitas-segera

which is the desired result.

Similarly, we can prove the formula for future value. Payments made at the end of last year will not collect interest and payments made at the end of the first year will collect interest for the total ( n -1) years. Therefore,

${\ displaystyle s _ {{overline {n}} | i} = 1 (1 i) (1 i) ^ {2} \ cdots (1 i) ^ {n-1} = (1 i) ^ {n} a _ {{overline {n}} | i} = {\ frac {(1 i) ^ {n} -1} {i}}} Â Â$

Annuity Annuity

An annuity because is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.

Setiap pembayaran anuitas diizinkan untuk digabungkan untuk satu periode tambahan. Dengan demikian, nilai saat ini dan masa mendatang dari anuitas dapat dihitung melalui rumus:

${\ displaystyle {\ ddot {a}} _ {{\ overline {n |}} i} = (1 i) \ kali a _ {{\ overline {n | }} i} = {\ frac {1- \ left (1 i \ right) ^ {- n}} {d}}}  ÂÂ$

dan

${\ displaystyle {\ ddot {s}} _ {{\ overline {n |}} i} = (1 i) \ kali s _ {{overline {n | {| }} i} = {\ frac {(1 i) ^ {n} -1} {d}}}  ÂÂ$

di mana ${\ displaystyle n}  ÂÂ$ adalah jumlah istilah, ${\ displaystyle i}  ÂÂ$ adalah suku bunga jangka per, dan ${\ displaystyle d}  ÂÂ$ adalah tingkat diskon efektif yang diberikan oleh ${\ displaystyle d = i/(i 1)}  ÂÂ$ .

Nilai masa depan dan sekarang untuk tunjangan hari tua terkait sebagai:

${\ displaystyle {\ ddot {s}} _ {{overline {n}} | i} = (1 i) ^ {n} \ kali {\ ddot { a}} _ {{\ overline {n}} | i}}  ÂÂ$

dan

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

Contoh: Nilai akhir dari anuitas 7 tahun yang jatuh tempo dengan suku bunga tahunan nominal 9% dan pembayaran bulanan $ 100:

$            {\displaystyle         F        V_{            d            u            e          }        (        \mathrm{0,09}                  /                12        ,        7        Ã\AE \text{'}\; -        12        ,        \mathrm{\$}        100        )        =        \mathrm{\$}        100        Ã\AE \text{'}\; -        {                          \stackrel{¨}{s}                      }_{                          \stackrel{¯\;               }{84}                                      |                        \mathrm{0,0075}          }        =        \mathrm{\$}        11                \mathrm{730,01.}      }        \{\backslash \; displaystyle\; FV\_\; \{due\}\; (0,09/12,7\; \backslash \; 12\; kali,\; \backslash \; \$\; 100)\; =\; \backslash \; \$\; 100\; \backslash \; kali\; \{\backslash \; ddot\; \{s\}\}\; \_\; \{\{\backslash \; overline\; \{\; 84\}\}\; |\; 0,0075\}\; =\; \backslash \; \$\; 11\; \backslash ,\; 730,01.\}\;  ÂÂ$

Note that in Excel, the PV and FV functions take the optional fifth argument that selects from an immediate annuity or a maturity benefit.

A perpetuity adalah anuitas yang pembayarannya terus berlanjut selamanya. Sejak:

${\ displaystyle \ lim _ {n \, \ rightarrow \, \ infty} \, PV (i, n, R) \, = \, {\ frac {R} {i}}}  ÂÂ$

bahkan lamanya memiliki nilai sekarang yang terbatas ketika ada tingkat diskonto non-nol. Rumus untuk lamanya adalah: