# What is compound interest? Calculation of compound interest Compound interest is one of the fundamental and most important laws in the world of finance. Also called interest capitalization. With the help of this law, one can both become fabulously rich and go bankrupt. Any financially literate person should be able to make the correct calculation of compound interest. To do this, it is necessary, first, to understand its essence, and second, to understand the computational formulas and be able to use them correctly.

## A simple example of interest capitalization

Two investors simultaneously opened accounts and invested $10,000 each at an interest rate of 30% per annum. A year later, they earned$3,000 profit. The first investor withdraws this profit from the account and puts it under the mattress, and the second leaves it on the account. As a result, the first investor will have the same $10,000, and the second will have$13,000. A year later, the first investor will have a profit of $3,000 again, and the second will have$3,900. This is because the first has profit from $10,000 (10,000 * 30% = 3,000), while the second has from$13,000 (13,000 * 30% = 3,900). Due to compound interest, the second investor earned more.

With each subsequent year, the capital of the first investor will grow linearly, and of the second – exponentially, if the second investor continues to keep the profit, letting it work. After 20 years, the first investor will have $70,000, and the second will have$1,900,496. This is 27.15 times more!

Let’s look at the chart and the table. At first, the effect of compound interest is not so noticeable, but over time, the growth becomes astronomical:

Period Simple interest Compound interest
Total amount Profit Total amount Profit
0 10,000.00   10,000.00
1 13,000.00 3,000.00 13,000.00 3,000.00
2 16,000.00 3,000.00 16,900.00 3,900.00
3 19,000.00 3,000.00 21,970.00 5,070.00
4 22,000.00 3,000.00 28,561.00 6,591.00
5 25,000.00 3,000.00 37,129.30 8,568.30
6 28,000.00 3,000.00 48,268.09 11,138.79
7 31,000.00 3,000.00 62,748.52 14,480.43
8 34,000.00 3,000.00 81,573.07 18,824.56
9 37,000.00 3,000.00 106,044.99 24,471.92
10 40,000.00 3,000.00 137,858.49 31,813.50
11 43,000.00 3,000.00 179,216.04 41,357.55
12 46,000.00 3,000.00 232,980.85 53,764.81
13 49,000.00 3,000.00 302,875.11 69,894.26
14 52,000.00 3,000.00 393,737.64 90,862.53
15 55,000.00 3,000.00 511,858.93 118,121.29
16 58,000.00 3,000.00 665,416.61 153,557.68
17 61,000.00 3,000.00 865,041.59 199,624.98
18 64,000.00 3,000.00 1,124,554.07 259,512.48
19 67,000.00 3,000.00 1,461,920.29 337,366.22
20 70,000.00 3,000.00 1,900,496.38 438,576.09

With a simple interest, the graph of capital increase is linear, because the earned profit is withdrawn and does not participate in the generation of new profit. However, when the earned profit is added to the main amount of capital and later participates in the creation of a new profit, the graph turns out to be exponential, growing like an avalanche. Against the background of exponential graphics, linear looks almost horizontal. The difference, as they say there. It is much more profitable to use interest capitalization!

## Formulas for calculating compound interest

The formula for calculating compound interest is as follows:

$\mathrm{FV}=\mathrm{PV}{\left(1+\frac{r}{n}\right)}^{nt}$

, where: FV – the final sum; PV – the initial amount; r – the interest rate; n – the number of charges for the period; t – the number of periods.

The formula for calculating the initial amount is expressed as:

$\mathrm{PV}=\frac{\mathrm{FV}}{{\left(1+\frac{r}{n}\right)}^{nt}}$

The formula for calculating interest rate is as follows:

$r=n\left[\sqrt[nt]{\frac{\mathrm{FV}}{\mathrm{PV}}}-1\right]$

The formula for calculating the number of periods is:

$t=\frac{\mathrm{ln}\left(\frac{\mathrm{FV}}{\mathrm{PV}}\right)}{n\left[\mathrm{ln}\left(1+\frac{r}{n}\right)\right]}$

## Examples of calculating compound interest

The investor invested $10,000 at the rate of 30% per annum. What amount will be after 20 years? $\mathrm{FV}=10,000{\left(1+30%\right)}^{20}=1,900,496.38$ ### Calculation example 2 The investor invested$10,000 at the rate of 30% per annum, which is charged monthly. What amount will be after 20 years?

$\mathrm{FV}=10,000{\left(1+\frac{30%}{12}\right)}^{12×20}=3,747,379.65$

What should be the initial amount in order to have $100,000 over 10 years at a 30% annual capitalization rate? $\mathrm{PV}=\frac{100,000}{{\left(1+30%\right)}^{10}}=7,253.82$ ### Calculation example 4 What should be the initial amount in order to have$100,000 over 10 years at a 30% annual capitalization rate, charged monthly?

$\mathrm{PV}=\frac{100,000}{{\left(1+\frac{30%}{12}\right)}^{12×10}}=5,165.78$

### Calculation example 5

What should be the annual interest rate of capitalization for $10,000 to grow to$500,000 over 10 years?

$r=\sqrt{\frac{500,000}{10,000}}-1=47.88%$

### Calculation example 6

What should be the annual interest rate of capitalization, charged monthly for $10,000 to grow to$500,000 over 10 years?

$r=12\left[\sqrt[12×10]{\frac{500,000}{10,000}}-1\right]=39.76%$

### Calculation example 7

How many years will it take for $10,000 to grow to$500,000 at an annual 30% rate?

$t=\frac{\mathrm{ln}\left(\frac{500,000}{10,000}\right)}{\mathrm{ln}\left(1+30%\right)}=14.91$

### Calculation example 8

How many years will it take for $10,000 to grow to$500,000 with an annual 30% rate charged monthly?

$t=\frac{\mathrm{ln}\left(\frac{500,000}{10,000}\right)}{12\left[\mathrm{ln}\left(1+\frac{30%}{12}\right)\right]}=13.2$

## Calculation of the load on compound interest

It so happens that it is necessary to withdraw part of the profit, and not completely reinvest it. For example, to spend part of the income on living. Will compound interest sustain a similar load without destroying capitalization?

Let’s imagine two initial amounts of $10,000. One grows according to the rule of capitalization of interest, and the other grows according to the rule of simple interest. The yield is 30% per annum. At year 5, part of the amount of$10,000 is annually withdrawn. Let’s see what happens:

Period Simple interest Compound interest
Total amount Profit Withdrawn amount Total amount Profit Withdrawn amount
0 10,000.00     10,000.00
1 13,000.00 3,000.00   13,000.00 3,000.00
2 16,000.00 3,000.00   16,900.00 3,900.00
3 19,000.00 3,000.00   21,970.00 5,070.00
4 22,000.00 3,000.00   28,561.00 6,591.00
5 15,000.00 3,000.00 10,000.00 27,129.30 8,568.30 10,000.00
6 8,000.00 3,000.00 10,000.00 25,268.09 8,138.79 10,000.00
7 400.00 2,400.00 10,000.00 22,848.52 7,580.43 10,000.00
8 520.00 120.00   19,703.07 6,854.56 10,000.00
9 676.00 156.00   15,613.99 5,910.92 10,000.00
10 878.80 202.80   10,298.19 4,684.20 10,000.00
11 1,142.44 263.64   13,387.65 3,089.46
12 1,485.17 342.73   17,403.94 4,016.29
13 1,930.72 445.55   22,625.13 5,221.18
14 2,509.94 579.22   29,412.67 6,787.54
15 3,262.92 752.98   38,236.47 8,823.80
16 4,241.80 978.88   49,707.41 11,470.94
17 5,514.34 1,272.54   64,619.63 14,912.22
18 7,168.64 1,654.30   84,005.51 19,385.89
19 9,319.23 2,150.59   109,207.17 25,201.65
20 12,115.00 2,795.77   141,969.32 32,762.15
• Capital, growing according to the compound interest rule, withstood 6 withdrawals of $10,000 each and was able to recover, breaking through a maximum of$28,561 and increasing to $141,969.32; • Capital, growing according to the simple interest rule, withstood only 3 withdrawals and of the remaining$400 could not recover to a maximum of $22,000, remaining at$12,115.

Compound interest with their power can withstand the load in the form of withdrawals of a part of the sums, allowing capitalization to further increase capital. Simple interest did not cope with this task, which speaks of their inefficiency.

## The calculation of the negative rate of compound interest

Interest capitalization can be not only positive when welfare is multiplied, but also negative. At a negative rate, capitalization works against the investor, destroying capital. A vivid example of such destruction is inflation.

Let’s consider an example of calculation where the initial amount of \$10,000 is destroyed at a negative rate of 30% per annum:

The first 5 years are the most devastating – less than 20% of the amount remains. Next is the slow destruction of the remaining capital. At a negative rate, the effect is strongest in the initial periods. Similarly, inflation affects the purchasing power of money, destroying it. Therefore, the yield should be higher than inflation.

## Conclusion

With a positive rate, the effect of compound interest is not immediately noticeable. Then the difference is noticeable and increases like a snowball. The higher the rate and the longer the time horizon, the stronger the interest capitalization manifests itself. With a negative rate, the destructive effect appears immediately and gradually fades. Losing and destroying is much easier and faster than earning and building.

In contrast to simple interest, compound interest withstands the load when the investor starts to take and spend part of the profits. Capital continues to grow and multiply.

To achieve maximum results, reinvest all profits at the maximum possible positive interest rate of return, taking into account the risks. Even an insignificant additional 1–2% over a long period gives a big difference, so fight for every percentage of profitability. Avoid disruptive negative inflation rates. The better the result, the faster you will achieve financial freedom.