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Compound interest is the reinvestment of an asset's earnings, including capital appreciation or interest, to generate additional income over time.
Compound interest is the reinvestment of an asset's earnings, including capital appreciation or interest, to generate additional income over time. This growth is calculated through an exponential function, since an investment generates returns not only from the initial capital but also from previous accumulated earnings.
Therefore, compound interest is different from linear growth, which is simply the principal earning interest each period.
Compound interest generally refers to the increase in the value of an asset due to the interest earned on the principal and accumulated interest. This phenomenon is a direct manifestation of the concept of "time value", also known as compound interest.
Compound interest is crucial in finance, as the returns it brings are the motivation behind many investment strategies. For example, many companies offer Dividend Reinvestment Plans (DRIPs), which allow investors to reinvest their cash dividends into purchasing more shares. Reinvesting in these dividend-paying stocks accelerates investors' returns because the increased share count will consistently boost future dividend income, assuming dividends remain stable.
Investing in dividend growth stocks adds another layer of compounding to this strategy on top of reinvesting the dividends, which some investors call "double compounding." In this case, not only are the dividends reinvested to purchase more shares, but the per-share dividends from these dividend growth stocks also increase.
The future value (FV) formula for a current asset relies on the concept of compound interest. It considers the present value of the asset, the annual interest rate, the frequency of compounding each year (or the number of compounding periods), and the total number of years. The general formula for compound interest is:
<!-- @@@ -->This formula assumes no changes to the original principal balance other than interest.
536,870,912: Want to know what 100% daily compound interest looks like? Demi's folk tale "A Grain of Rice" revolves around a reward, one grain of rice on the first day and then double the number of grains of rice each day for 30 days. By the last day of the month, the cumulative rewards will exceed 536 million grains of rice. [2]
The effect of compound interest increases with the frequency of compound interest. Assume that one year is a time period. The more compounding cycles there are, the higher the future value of the investment; obviously two compounding cycles per year are better than one, and four compounding cycles is better than two.
To illustrate this effect, consider the following example. Assume an investment of $1 million has an annual rate of return of 20%. The future value based on different number of compounding periods is as follows:
1,200,000
1,210,000
1,215,506
1,219,391
1,220,934
1,221,336
Apparently, despite the significant increase in the number of compound interest cycles, future value growth has tended to remain stable. The frequency of compounding over a period of time has a limited impact on the growth of an investment. This limit is called "continuous compounding" and can be calculated using the following formula:
<!-- @@@ -->In the above example, the future value with continuous compounding is: FV =
<!-- @@@ -->1,221,404.
Note: Compound interest is an example of the "snowball effect," in which small effects gradually accumulate into a larger, more severe condition.
Compound interest works on both assets and liabilities. While compound interest quickly increases the value of an asset, it can also increase the loan amount as unpaid principal and previous interest accumulate to accrue more interest. Even if loan repayments are made, compounding interest may cause the total amount owed to increase in the future.
The concept of compound interest is especially tricky on credit card balances. Not only is the interest rate on credit card debt high, but the interest charges may be added to the principal balance, resulting in interest in the future. For this reason, the effects of compound interest are not necessarily "good" or "bad." The effects of compounding may vary depending on an investor's specific financial situation.
To illustrate how compound interest works, assume you have 10,500, reflecting the 10,000 principal. In the second year, the account achieves 5% growth on the original principal and interest on the first year's 525 in the second year, bringing the total balance to $11,025.
| Compound interest example | |||
|---|---|---|---|
| Compounding cycle | Starting balance | Interest | Ending balance |
| 1 |
500.00 |
<!-- @@@ -->10,500.00 |
<!-- @@@ -->11,025.00 | | 3 |
<!-- @@@ -->551.25 |
<!-- @@@ -->11,576.25 |
<!-- @@@ -->12,155.06 | | 5 |
<!-- @@@ -->607.75 |
<!-- @@@ -->12,762.82 |
<!-- @@@ -->13,400.96 | | 7 |
<!-- @@@ -->670.05 |
<!-- @@@ -->14,071.00 |
<!-- @@@ -->14,774.55 | | 9 |
<!-- @@@ -->738.73 |
<!-- @@@ -->15,513.28 |
<!-- @@@ -->16,288.95 |
After 10 years, with no withdrawals and a steady annual interest rate of 5%, the account would have grown to 500 in the first period to $775.66 in the tenth period.
Furthermore, after 10 years, our investment has grown by 5,000 per year (a total of $500 over 10 years).
The Rule of 72 is a heuristic for estimating the time it would take for an investment or savings to double given compound interest (or continued returns). The rule says that the number of years required is 72 divided by the interest rate. At a 5% interest rate with compounding, it would take approximately 14 years and 5 months to double. [4]
Simple interest pays interest only on the principal amount invested or deposited. For example, if you deposit 50 in interest each year. However, compound interest pays "interest on interest," so in the first year you will receive 52.50 ($1,050 × 0.05), and so on.
In addition to compound interest, investors can also earn compound returns by reinvesting dividends. This means that the dividend cash received is used to purchase additional shares of the company, which will also pay dividends in the future.
There are several average calculations used in finance. When calculating the average return on an investment or savings account with compound interest, it's best to use the geometric mean. In finance, this is sometimes called the time-weighted average return or compound annual growth rate (CAGR).
High-yield savings accounts are a great example of compound interest. Let's say you deposit $1,000 in a savings account. In the first year, you will earn a certain amount of interest. If you don't spend any of the money in the account, and the interest rate remains at least the same as it was the previous year, your interest earnings will be higher in the second year. This is because a savings account adds the interest earned to the cash balance, resulting in higher interest.
Compound interest and interest play an important role in an investor's financial success. If you take advantage of compound interest, you will make more money faster; if you take on compound interest debt, you will be stuck in larger and larger debts for a long time. Through compound interest, financial balances can grow exponentially faster than linear interest.
[1] Donna Kirk et al. “Contemporary Mathematics: 6.4 Compound Interest.” OpenStax, 2023.
[2] Barnes & Noble. “One Grain of Rice: a Mathematical Folktale.”
[3] Gilbert Strang et al. “Calculus Volume 1: 1.5 Exponential and Logarithmic Functions.” OpenStax, 2016.
[4] Investor.gov. “What Is Compound Interest?”
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该公式假定除了利息之外，对原始本金余额没有其他的变动。
536,870,912: 想知道100%每天复利是什么样子吗？德米的民间故事《一粒米》围绕着一个奖励展开，第一天奖励一粒米，之后每天奖励的米粒数加倍，持续30天。到月底最后一天，累计奖励将超过536百万粒米。[2]
复利的效果随着复利频率的增加而增强。假设一年为一个时间周期。复利周期越多，投资的未来价值就越高；显然，每年两个复利周期要优于一个，而四个复利周期又优于两个。
为了说明这一效果，考虑以下例子。假设投资100万美元，年收益率为20%。基于不同复利周期数的未来价值如下：
因此，复利不同于线性增长，后者仅是本金每个周期获得利息。
复利通常指因为本金和累积利息所产生的利息而增加的资产价值。这一现象是“时间价值”概念的直接体现，也被称为复利。
复利在金融中至关重要，因其带来的收益是许多投资策略背后的动机。例如，许多公司提供红利再投资计划（DRIPs），允许投资者将其现金红利再投资，购买更多股票。再投资这些支付红利的股票会使投资者的回报加速，因为增加的股票数量将一致地提升未来的红利收入，假设红利保持稳定。
在再投资红利的基础上，投资红利增长股票会为这一策略增添另一层复利，有些投资者称之为“双重复利”。在这种情况下，不仅红利被再投资以购买更多股票，而且这些红利增长股票的每股分红也在增加。
当前资产的未来价值（FV）公式依赖于复利的概念。它考虑资产的现值、年利率、每年复利的频率（或复利周期数）和总的年数。复利的一般公式是：
1,210,000
该公式假定除了利息之外，对原始本金余额没有其他的变动。
536,870,912: 想知道100%每天复利是什么样子吗？德米的民间故事《一粒米》围绕着一个奖励展开，第一天奖励一粒米，之后每天奖励的米粒数加倍，持续30天。到月底最后一天，累计奖励将超过536百万粒米。[2]
复利的效果随着复利频率的增加而增强。假设一年为一个时间周期。复利周期越多，投资的未来价值就越高；显然，每年两个复利周期要优于一个，而四个复利周期又优于两个。
为了说明这一效果，考虑以下例子。假设投资100万美元，年收益率为20%。基于不同复利周期数的未来价值如下：
显然，尽管复利周期的数量显著增加，未来价值增长幅度反而趋于平稳。复利在一定时间内的频率对投资的增长影响有限。这个极限被称为“连续复利”，可以用以下公式计算：
1,210,000
注意： 复利是一种“雪球效应”的例子，其中小的影响逐渐累积，形成更大的、更加严重的状态。
复利在资产和负债上都发挥作用。虽然复利迅速提升资产的价值，但它也可能增加贷款金额，因为未支付的本金和之前的利息会累积产生更多利息。即使进行贷款还款，复利利息也可能导致未来的欠款总额增加。
复利的概念对信用卡余额尤其棘手。信用卡债务的利率不仅高，而且利息费用可能会被添加到本金余额中，从而将来产生利息。由于这个原因，复利的影响不一定是“好”或“坏”。复利的效果可能会根据投资者的具体财务状况而有所不同。
为了说明复利的运作，假设账户中有10,000美元，年利率为5%。在第一个复利期结束时，账户中的总额上升到10,500美元，反映了500美元的利息增加至10,000美元本金。第二年，账户分别针对原始本金和第一年500美元的利息实现5%的增长，第二年的收益为525美元，总余额达到11,025美元。
| 复利示例 | |||
|---|---|---|---|
| 复利周期 | 起始余额 | 利息 | 结束余额 |
| 1 | $10,000.00 | $1,215,506 |
显然，尽管复利周期的数量显著增加，未来价值增长幅度反而趋于平稳。复利在一定时间内的频率对投资的增长影响有限。这个极限被称为“连续复利”，可以用以下公式计算：
12,762.82 |
在上述例子中，连续复利的未来价值为：FV = 670.05 | 1,221,404。
注意： 复利是一种“雪球效应”的例子，其中小的影响逐渐累积，形成更大的、更加严重的状态。
复利在资产和负债上都发挥作用。虽然复利迅速提升资产的价值，但它也可能增加贷款金额，因为未支付的本金和之前的利息会累积产生更多利息。即使进行贷款还款，复利利息也可能导致未来的欠款总额增加。
复利的概念对信用卡余额尤其棘手。信用卡债务的利率不仅高，而且利息费用可能会被添加到本金余额中，从而将来产生利息。由于这个原因，复利的影响不一定是“好”或“坏”。复利的效果可能会根据投资者的具体财务状况而有所不同。
为了说明复利的运作，假设账户中有10,000美元，年利率为5%。在第一个复利期结束时，账户中的总额上升到10,500美元，反映了500美元的利息增加至10,000美元本金。第二年，账户分别针对原始本金和第一年500美元的利息实现5%的增长，第二年的收益为525美元，总余额达到11,025美元。
| 复利示例 | |||
|---|---|---|---|
| 复利周期 | 起始余额 | 利息 | 结束余额 |
| 1 | $703.55 | $14,774.55 | |
| 9 | $14,774.55 | $500.00 | $15,513.28 |
| 10 | $15,513.28 | $775.66 | $10,500.00 |
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