CompoundVision

Compound Interest

The powerful financial principle of compound interest is something everyone can benefit from once they understand it. With compound interest, you can make your money work for you instead of the other way around. Allegedly, Albert Einstein called compound interest the eighth wonder of the world.

The way compound interest works is that you earn interest not only on your initial amount (principal) but also interest on the previously accumulated amount of interest over time. One common way to explain compound interest is like making a snowball. In the beginning, when rolling a snowball in the snow it grows slowly but the more snow it picks up, the larger the snowball becomes and the faster it grows in size. That is why compound interest is often spoken of as the snowball effect.

You can compare compound interest to simple interest where you only earn interest on the initial amount. Given you don't save any more money, you will earn the same amount of interest every interval. The money will grow linearly instead of exponentially which it does with compound interest.

When it comes to compound interest, the most important factor is the time you save and allow your saved money to compound. “The best time to plant a tree was 20 years ago, the second best time is now.” To really get to experience the effects of compound interest, it is important to start saving early. To achieve the same returns when saving smaller amounts over a longer period, you would need to significantly increase the initial amount if saving over a shorter period. So, it is worth to save a small amount every month over a period of decades.

Mathematically, the formula for compound interest is an exponential function that grows at an increasing rate for each interval (see more in “The Formula”).

There are different compound interest intervals, for instance annually, quarterly, and monthly. The more frequently you compound your savings, the higher the returns over time. Annual is used in CompoundVision, which was chosen because it is the most common frequency.

Monthly contributions, which you can include in the CompoundVision calculator, will also greatly impact the long-term return. This is because you will earn compound interest on these contributions as well in addition to the initial amount. As for the frequency, the monthly contributions are compounded annually.

Example:

Let's say your initial amount is $1000, and the interest rate each year is 10%.

Year 1 (one year after you started):
Interest: $1000 * 10% = $100
Total: $1000 + $100 = $1100

Year 2:
Interest: $1100 * 10% = $110
Total: $1100 + $110 = $1210

Year 3:
Interest: $1210 * 10% = $121
Total: $1210 + $121 = $1331

As you can see the amount of interest grows each year.

Debt:

Compound interest also applies to debt. If you have a loan and don't pay the interest, this will be added to the debt and simply increase the amount of interest you need to pay in the future. You could say it is the reverse as if you saved money with compound interest. The debt will increasingly grow, and it gets harder to pay off. That's why it's crucial to at least pay off the interest as quickly as possible.

The Formula

The following formulas are used in CompoundVision to calculate the future value.

$FV=P\cdot(1+r)^{t}$

where:

$FV$ - is the future value
$P$ - is the principal
$r$ - is the annual interest rate (decimal)
$t$ - is the number of years of applied annual interest

The formula is an exponential function that increasingly grows the greater $t$ becomes (increasingly grows over time). This formula is derived from a more complex one, where the frequency has been set to one year.

Monthly contributions included:

$FV=\sum\nolimits_{i = 1}^t {M\cdot12\cdot}$$(1+g)^{i-1}\cdot(1+r)^{t-i}$

where:

$M$ - is the monthly contribution
$g$ - is the yearly increase rate of the monthly contribution (decimal)

This formula calculates the compound interest of continuous monthly deposits over $t$ years. When $g$ is greater than 0, a yearly increase in the monthly contributions is applied to the calculation. This formula is derived from a more complex one, where the compound frequency has been set to one year. Additionally, the formula is based on monthly contributions instead of quarterly, semi-annual, or other possible contribution intervals. Note also that this formula calculates the compound at the end of the year.

Combined formula:

By adding the above formulas, you can calculate the compound interest on an initial amount along with monthly contributions, resulting in the total future value.

Inflation

With the CompoundVision calculator, you can enter an annual inflation rate. This really shows how much of an impact inflation has on each person's future savings. There is a common overlooked misconception when considering the future value of money, which is also intended to be emphasized with this calculator.

Inflation is a general increase in prices of goods and services over time. This leads to a loss of value of the savings you have. Let’s say you have $100, and there's a product you want to buy that costs exactly $100. However, when you return to buy the product later, it now costs $102 (2% increase). Even though you still have the same $100, you can no longer afford the product because its price has increased. This means the purchasing power of your money has decreased, and that you need more money to buy the same product.

There are two key terms used to describe future monetary value: nominal and real. Nominal refers to the future value without accounting for inflation, while real value is adjusted for inflation, reflecting the true purchasing power.

Without any interest on your capital, it will decrease in value over time. Try the calculator with an interest rate of 0% and some inflation rate (e.g., 2%), and you will see the real future value of your savings. To counteract the impact of inflation, it is good if you can obtain a minimum of the same annual interest rate on your savings as the national annual inflation rate. With compound interest, it will be easier to beat inflation.