什么是年金表？

What is an annuity table?

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An annuity table is a tool used to determine the present value of an annuity or other structured series of payments. This tool, commonly used by accountants, actuaries, and other insurance professionals, takes into account the amount invested in the annuity and its age to determine the amount payable to the annuity purchaser or annuitant.

The present value of any future annuity amount can also be calculated using a financial calculator or specially designed software.

Key Points

• An annuity table is a tool used to determine the present value of an annuity.
• Annuity tables calculate the present value of an annuity by applying a discount rate to future payments.
• An annuity table uses the discount rate and the number of payment periods to give the appropriate factors.
• Using an annuity table, you will multiply the amount of the periodic payment by a given factor.

How an annuity watch works

Annuity tables provide a factor, based on time and a discount rate (interest rate), by which the present value of an annuity payment is determined. For example, if the expected interest rate is 3%, you can use an annuity table to calculate the present value of an annuity that pays $10,000 per year for 15 years.

Important Note: According to the concept of time value of money, a lump sum paid now is worth more than the same amount received in the future.

Having $10,000 today is more valuable than receiving$1,000 every year for the next 10 years because the money can be invested to earn interest. At the end of the 10-year period, the present value of $10,000 will exceed the sum of all annual payments, even if invested at the same interest rate.

Annuity table and present value of annuity

The present value formula for an ordinary annuity (different from an annuity due) is as follows:

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Suppose someone has the opportunity to receive an annuity that pays $50,000 annually for the next 25 years at a discount rate of 6$650,000. He needed to identify more reasonable options. Using the above formula, the present value of the annuity is:

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639,168\&\textbf{Where:}\&\text{PVA}=\text{Present value of annuity}\end{aligned} $$

Based on this information, the annuity is worth less than $10,832 from a time-adjusted perspective, so the person should choose a lump sum payment.

Note that this formula works for an ordinary annuity, where payments are made at the end of each period. In the above example, each payment of $50,000 would be made at the end of each year for 25 years. For an annuity due, payments are made at the beginning of the period. To calculate the present value of an annuity due, multiply the above formula by a factor of (1 + r):

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If the above example were an annuity due, its value would be:

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677,518\end{aligned} $$

In this case, the person should choose the maturity annuity because it is worth $27,518 more than the lump sum payment.

Instead of calculating via the above formula, you can also use an annuity table. Annuity tables simplify calculations by automatically providing a factor for the second half of the formula. For example, an ordinary annuity table will give you a number (called a factor) that is preset for the (1 - (1 + r) ^ - n) / r) part of the formula.

The determination of the factor depends on the interest rate (r in the formula) and the number of payment periods (n in the formula). In an annuity table, the number of periods is usually listed on the left side, and the interest rate is usually listed on the top row. Simply select the correct interest rate and number of periods to find the factors in the intersection cells. This factor is then multiplied by the dollar amount of the annuity payment to arrive at the present value of an ordinary annuity.

The following is an example of an ordinary annuity present value table:

If we take the 6% interest rate and 25-year example mentioned above, you will find that factor = 12.7834. If you multiply the 12.7834 factor from this annuity table by the $50,000 payment amount, you get$639,170, which is almost the same as the $639,168 calculated by the previous formula. The small difference between the numbers reflects rounding of the number 12.7834 in the annuity table.

There is a separate table for annuities due which gives the correct factors based on the second formula.

What is an annuity?

An annuity is an insurance contract that provides an income stream, usually during retirement. Annuities may be fixed, variable, or indexed. It has two phases: first the accumulation (savings) phase, and then the disbursement (income) phase. Payment may be immediate or delayed.

What is the difference between an ordinary annuity and an annuity due?

An ordinary annuity is one that generates payments at the end of the annuity period, while a maturity annuity is an annuity that is expected or paid at the beginning of the payment period.

Can lottery winners use an annuity table?

Lottery winners can use an annuity table to determine whether it makes more financial sense to receive their lottery winnings in a lump sum today or to pay them in installments over many years. However, lottery winnings are a relatively rare form of annuity. More commonly, an annuity is a type of investment used to provide an individual with a steady stream of pension income.

Summary

Annuity tables are used primarily by accounting, insurance, or other financial professionals to determine the present value of an annuity. It takes into account the amount invested in the annuity and its length of existence, thus facilitating the determination of the amount payable to the annuity purchaser or annuitant.

About TheQuantHackers

TheQuantHackers is a cutting-edge community composed of a group of people from the world's top universities and quantitative finance practitioners, dedicated to exploring the infinite possibilities in the fields of artificial intelligence (AI) and quantification (Quant). Our team members come from world-renowned universities such as the University of Cambridge, Oxford University, Harvard University, ETH Zurich, Peking University, and the University of Science and Technology of China. Our external consultants come from first-class companies such as Microsoft, HSBC, Citadel, Man Group, Citi, Jump Trading, and top domestic private equity companies.$\begin{aligned}&\text{P} =\text{PMT}\times\frac{ 1 - (1 + r) ^ -n}{r}\\&\textbf{其中：}\\&\text{P} = \text{年金流的现值}\\&\text{PMT} =\text{每笔年金支付的金额}\\&r = \text{利率（即折现率）}\\&n = \text{支付期数}\end{aligned}$

假设某人有机会获得一项在未来25年每年支付50,000美元的年金，折现率为6%，或者获得650,000美元的一次性付款。他需要确定更合理的选择。使用上述公式，年金的现值为：

$$ \begin{aligned}&\text{PVA} = $50,000 \times \frac{1 - (1 + 0.06) ^ -25}{0.06} = $639,168\&\textbf{其中：}\&\text{PVA}=\text{年金的现值}\end{aligned} $$$50,000 \times \frac{1 - (1 + 0.06) ^ -25}{0.06} = $ 年金表是一种用于确定年金或其他结构化支付系列现值的工具。这种工具通常被会计师、精算师和其他保险人员使用，它考虑了投入年金的金额及其存在的时间，以确定应支付给年金购买者或年金领取人的金额。

任何未来年金金额的现值也可以通过金融计算器或专门设计的软件进行计算。

关键要点

• 年金表是用于确定年金现值的工具。
• 年金表通过将折现率应用于未来支付款项来计算年金的现值。
• 年金表使用折现率和支付期数来给出适当的因子。
• 使用年金表，您将把定期支付的金额与给定因子相乘。

年金表的工作原理

年金表提供一个因子，该因子基于时间和折现率（利率），通过将年金支付乘以该因子即可确定其现值。例如，如果预计利率为3%，则可使用年金表计算一项在15年内每年支付10,000美元的年金的现值。

重要提示： 根据货币时间价值的概念，当前一次性支付的金额比未来收到相同金额的价值更高。

今天拥有10,000美元比未来10年每年收到1,000美元更有价值，因为这笔钱可以投资以获取利息。在10年期结束时，10,000美元的现值将超过所有年度支付的总和，即使按照相同的利率进行投资。

年金表与年金的现值

普通年金的现值公式（与到期年金不同）如下：

$\begin{aligned}&\text{P} = \text{PMT} \times\left(\frac{1 - (1 + r) ^ -n}{r}\right) \times (1 + r)\end{aligned}$

如果上述例子是一个到期年金，则其价值为：

$$ \begin{aligned}&\text{P}= $50,000\&\quad \times\left( \frac{1 - (1 + 0.06) ^ -25}{0.06}\right)\times (1 + 0.06) = $677,518\end{aligned} $$$50,000\&\quad \times\left( \frac{1 - (1 + 0.06) ^ -25}{0.06}\right)\times (1 + 0.06) = $

假设某人有机会获得一项在未来25年每年支付50,000美元的年金，折现率为6%，或者获得650,000美元的一次性付款。他需要确定更合理的选择。使用上述公式，年金的现值为：