# What is Compound Interest: Formula, Stocks, & Pension # What is compound interest?

When you save money in a savings account or invest it in funds or stocks, the money can grow, and they do so by giving you interest or returns. In a savings account, you can get interest, and when you invest money in funds or shares, you can get a return by raising the price or by dividends.

Normally, your total return consists of both price growth and dividends over time. The compound interest effect makes the money grow faster because you simply earn interest on already earned interest.

# How does compound interest work?

Compound interest effect means that you get a return on your saved amount plus earned interest – and in that way the money grows exponentially. It is often called the snowball effect and means that even small amounts in the long run can grow to large sums.

Example
Imagine that you deposit \$1000 in a savings account on January 1 and that there is 1% interest on the savings account. The interest is calculated annually, so when the year is over you have \$1010 in your savings account.

You still have the money in the savings account throughout year two as well, and the interest rate is still 1%. But in year two, you earn 1% interest on the entire amount in your savings account, ie \$1010.

So at the end of year two, you will have \$1021 in your savings account. If you do the same in year three, you will have \$1033 in the account at the end of that year.

Calculation:

Year 1: 1000 x 1.01 = 1010

Year 2: 1010 x 1.01 = 1021

Year 3: 1021 x 1.01 = 1033

### Another way of explaining how compound interest works:

A classic example of the compound interest effect is the one dollar, which doubles every day for a month. Suppose you have 1 dollar on day one, which is doubled to 2 dollars and then to 4 dollars and so on, for one month.

On day 30, the figure will amount to more than 1 billion.

It is conceivable that half of the sum was generated between day 29 and day 30. This is a powerful example of the exponential growth, ie the interest-on-interest effect.

# Does compound interest work on stocks and mutual funds as well?

Yes, the compound interest effect works exactly the same, regardless of whether you have the money in a savings account or invest it in funds or shares.

If the funds or shares rise in value, the increase in value will take place exponentially through the magic of compound interest.

Remember to always calculate the effective interest rate before entering it into an investment calculation.

This differs from the nominal interest rate, which is often what is stated before taking into account any fees and whether payment is made annually or monthly.

# Formula for calculating compound interest

To make your own calculations, it is easiest to use a formula. However, it is important to choose the right formula depending on what the savings look like. In the case of a one-time deposit which is then remunerated over time, the upper formula is used. If it is instead an annual saving where new money is invested, the lower formula is used:

One-time deposit formula:
Money after x number of years = initial capital ⋅ (1 + interest) number of years.

In this case, a sum of 100,000, which is saved for 10 years at 5% interest, would be as follows:

Saved money = 100,000 ⋅ 1.0510 = 162889.46

• Sn are saved money
• a1 are the annual deposits
• kn, the interest rate is raised in number of years
• k interest rate
• In this case, a provision of 10,000 per year for 10 years, at an interest rate of 5%, would be as follows:

Saved money = 10000 ⋅ (1.0510-1) / 0.05 = 125 778.92

It is also possible to use the method for other more advanced investments, then it is usually called the final value method.

# The factors that affect when you want to calculate compound interest

If you simplify it a bit, you can say that a money machine with a compound interest rate is affected by the following factors:

• Time the money is allowed to work in peace. A reasonable time horizon is 10 years or more.
• The return (or interest rate) they work for. A reasonable annual average return is 8 percent per year, as that is what you can get with an index fund over long periods of time.
• The initial starting amount that you put into the money machine.
• The tax you need to pay to the state

It is these “knobs” on the money machine that you can change.

# Calculate compound interest with simple memory rules

It can often be difficult to calculate the value of compound interest in my head and then I usually use the following memory rules:

• 72 through the interest rate gives the number of years for when the money is doubled. For example. 10% / year doubles every 7.2 years
• About 5 times the money in 20 years at an annual return of 8 percent per year

Or:

• A saved weekly expense gives 750 times the money in 10 years
• A saved monthly expense gives 187 times the money in 10 years

The above is calculated at 8 percent interest as an annual average return. The reason I am counting on 8 percent is that this is what an index fund on the stock exchange with a low or non-existent fee can provide.

That is, if the alternative is to buy something today, then I can think that if I instead save that money, then I can in 10 years with compound interest instead have several times the money instead.

# Conclusion

## Use compound interest for your pension

If you are looking for a smart way to save for your pension, the interest on the interest method is something to invest in.

The money you put into an investment works by itself and with the help of the time and the return, you will eventually hopefully have large amounts invested.

How securities such as funds and equities have performed historically does not guarantee what it will be like in the future.

## Sources

https://www.jstor.org/stable/43236859

https://www.cambridge.org/core/journals/journal-of-the-institute-of-actuaries/article/abs/an-early-book-on-compound-interest-richard-witts-arithmeticall-questions/EC890F3D408B8B035B63095CE01A5D0E

https://www.emerald.com/insight/content/doi/10.1108/01443580010341853/full/html

https://heinonline.org/HOL/LandingPage?handle=hein.journals/taxlr38&div=21&id=&page=

https://www.cambridge.org/core/journals/journal-of-the-institute-of-actuaries/article/abs/compound-interest-in-the-seventeenth-century/8477E0D1B7219B7D29197BB78724FA08

https://www.cambridge.org/core/journals/journal-of-the-institute-of-actuaries/article/abs/of-compound-interest/2E6137CC68196DC8C0779DB6DB2CACC1

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1307878