Understanding the magic of compounding interest

Not being sure of the catchphrase of my title, I will start directly with a quote from the famous Abert Einstein.

“Compound interest is the eighth wonder of the world. Whoever can understand compound interest benefits from it, whoever doesn’t… pays for it.”

Albert Einstein, Nobel Prize laureate and physician.

While there is no formal proof that this sentence actually came from Mr. Einstein, what is certain is the power of compound interest and our inability (usually) to fully understand it as human beings. But why do we have trouble understanding how it works?

Linear and exponential function

Generally most people have no trouble understanding and visualizing linear behaviors. This is probably due to the fact that most people around us behave in a linear way, as do most natural phenomena. Let’s take some examples of linear phenomenom’s:

  • Monthly transfer to your savings account: you transfer CHF 300.00 each month. In one year, you have transferred CHF 3,600 (12 x CHF 300).
  • Fuel consumption: your car consumes 7 liters per 100 km on average. After 800 km, you will have consumed 56 liters (8 x 7 liters).

I think the concept is simple and obvious to understand. In mathematical language, it is perhaps even simpler:

f(x) -> ax , where “a” is a real number called coefficient of the function (or proportionality)
– If “a” > 0, “f” will increase
– If “a” < 0, “f” will decrease

In short, it is quite basic so far. If we had to represent this behaviour graphically, we could have something like this:

Fonctions linéaires

However, compound interest follows an exponential function. Here are some examples:

  • Bacteria have an exponential growth (they multiply themselves).
  • The number of people on the planet is growing exponentially (about 1.12% per year)1.
  • The capitalization of money in a bank account (this is what we are looking).

In mathematical language, it is a bit more complicated and I will not explain the different possible representations and properties of this type of function. A visualization will be more meaningful:

Fonctions exponentielles

We can imagine an exponential function as a constant % growth with the same multiplicative factor. Let’s take the case we are interested in, the interest rate of a bank account. Here are the values :

Example with a bank account

Let’s get to the heart of the subject with this basic example.

  • Capital on the account on January 1, Year 0: CHF 1000.00
  • Account interest rate: 5.0 %.
  • Interest calculation: annual
  • Annual deposits: 0.- CHF
  • Period : = 1 year

This is a very basic and certainly slightly theoretical case, because there are no savings accounts in Switzerland with 5% interest and it is uncommon to not deposit any money in this account for 45 years. In addition, you should also subtract the account management fees. But the objective here is to understand the power of compound interest.

Question: What will be the amount in this account in 45 years?

To calculate this dynamically, the formula would be : f(x) = 1000 * 1.050x , x is number of years. Entering this function into Excel will give us the following values:

What is important to understand here is the exponential evolution of interest. As the column “interest for the period” shows, each year the placement yields more interest. After 5 years, the initial investment of CHF 1,000.00 generates CHF 60.78 in interest, whereas after 45 years, without having made a single additional deposit, our investment of CHF 1,000.00 generates CHF 427.86 in interest! This is due to the fact that the interest generated each year is accumulated on the initial CHF 1,000 and that the interest rate (or profit) remains the same. At the end, our investment of 1000.00 CHF allows us to have, after 45 years and under the theoretical hypothetical conditions presented, 8’895.01 CHF!

These values may seem unbelievable, but this is the magic of compound interest. It is true that in Switzerland, you will not find a savings account with a rate of 5.0%, but this can be true in the case of stock market investments. Note that in real life, the calculation of interest can be done monthly, semi-annually, quarterly or annually. This also influences the calculation slightly.

You can see clearly on this graph representing the total equity on the account over the years, that the curve is not linear, but well and truly exponential. The longer you leave the money on the account, the more interest you will earn each year! The orange curve (line) shows that there were no other payments during all these years and the two other curves have obviously a more or less similar growth since the total equity (blue curve) is composed of the initial balance added to the interest accumulated each year.

Same calculation with an annual deposit

The previous simulation represents a case where no capital is added for 45 years. Imagine the calculation if you deposit 1000 CHF per year into this account for 45 years. We assume that the account already had CHF 1000 at the beginning. Without interest, you will have 46’000 CHF on this account (initial 1000 CHF + 45 x 1000 CHF). To simplify the calculation, we will say that you pay 1’000.00 CHF on January 1st and that the interest is calculated once a year on December 31st.

Amazing, isn’t it? Here is what it looks like visually:

Still not convinced? One last calculation for the journey.

Let’s take Tom’s example. Tom is 20 years old and has just started working. Since his salary is not very high, he cannot save more than CHF 50 per month. However, Tom has an older friend who has repeatedly explained to him the importance of starting to invest his money as soon as possible. Tom asks his friend to help him invest CHF 50/month in a 3a retirement savings account with 60% invested in stocks. According to the averages and various studies of the institution where Tom invests his money, the average annualized return is 3.5% after deducting all fees.

How much will Tom have in that account when he retires, say at age 65?

He has therefore invested CHF 50/month for 45 years, i.e. CHF 600/year at an average annualized rate of 3.5%. In total, he will have deposited CHF 27,000, but his account will have a total balance of about CHF 64,480. This means that the interest generated represents 58% of the total capital! So Tom has saved more money through compound interest than through his own savings.

If this still doesn’t convince you, don’t hesitate to contact me and I’ll be happy to discuss the subject with you!

Conclusions

What should we remember from this article? Whether it is classic savings with low interest rates or at best a personal retirement saving account in a 3a invested in the stock market for example, apart from the fact that the average expected return will potentially be very different, the compound interest will do its job and will be more and more efficient with time. Time is the only variable you cannot control. Over your life, your salary will change, your investment and therefore savings potential will increase and interest rates will fluctuate.

If I were to give one advice to someone who doesn’t yet have a clear vision of personal savings, it would be this:

Start saving (investing) now! Not tomorrow, but today and do it regularly! It doesn’t matter how much you put in. Whether you put in CHF 50.00 / month or CHF 3000.00, your money should work for you!

It is obvious that in order to obtain a performance rate of 5% and more, you will not be able to do it with classic bank accounts such as savings or salary. If you want to reach this kind of objective, you will have to turn to stock market investments and the rate will obviously not be guaranteed. However, stock market investments are no longer reserved for the elite and already rich people. An article on this subject will be available soon. In any case, before you start investing in the stock market, I advise you to put the maximum on your 3rd pillar 3a (6’883.- CHF in 2021). I invite you to read my article on the subject which allows you to make your money grow, while saving taxes.

Other articles that might interest you

Sources

1 : https://www.worldometers.info/fr/population-mondiale/

Leave a Reply

Your email address will not be published. Required fields are marked *