The concepts of compound interest and time value of money are the foundations of finance. Not only your personal finance, but all the Finance, with a capital F. World economy, the wealth of nations, banks, pensions, everything is based on those concepts. You will know everything about compound interest, and how to use it to build your wealth. I will start by showing the power of compound interest. Then I will show you the simple math used for computing compound interest.
And as a bonus, at the end of this post, I will tell you how prepaying your mortgage or any other debt relates to compound interest, and how it makes you save a lot more the prepaid amount!
If you have a question, want to share a thought, or tell me how great this post is, please comment below!
The eighth wonder of the world!
Einstein said: “Compound interest is the eighth wonder of the world. He who understands it earns it… he who doesn’t… pays it.” Einstein, a brilliant person, I think you agree with me on that, did not say that his theory of general relativity was the eighth wonder of the world. No, it is compound interest. A financial concept! Let’s see why with an example first!
The power of compound interest
We have Bob and Alice. They both have $10,000, and they find a great saving account that pays them 10% per year.
Bob earns each year $1,000, and he spends it: restaurants, newest phones… He does not spend his initial investment though, so he keeps getting $1,000 each year.
Alice, on the other hand, does not spend it. On the contrary, she keeps reinvesting the interest earned! So at the end of the first year, she had her initial $10,000 plus $1,000. This $11,000 earned her $1,100 at the end of the second year. $100 more than what Bob got. And at the end of the third year, she earned $1,210 — $110 more than what she got the previous year. So no only Alice earns more and more each year, but what she earns is growing faster and faster. Can you guess how much money she has when she retires? Let’s say in 30 years. She has nearly $174,500! Her $10,000 grew 17 times.
Bob got only $30,000 from his initial investment during the same period, $1,000 a year for 30 years.
The difference is that Bob only had a simple interest, the interest you get after a single period of interest payments. Alice took advantage of compound interest.
This is why you should always start to invest as early as possible!
In the long run, when you take into account dividends and growth, stock markets produce nearly a 10% return per year on average!
The importance of saving for your retirement!
So how does it affect your nest egg for your retirement?
Imagine you can save $5,000 a year when you start your career, and you can increase the amount you save by 2% each year, so you save $5,100 in year 2, $5,202 in year 3, and so on. You invest in stocks, and you get an average return of 10%. Over 40 years, you contributed toward your nest egg about $300,000. Can you guess how much this money grew? Do you have $500,000, maybe a million dollars?
No, you have now around $2,700,000!
So did you start to build your nest egg for retirement? How do you invest to benefit from compound interest?
The math of compound interest
Now let’s talk about the math to compute compound interest. If you are not interested in the math, you can jump to the end of the post to learn how prepaying a debt is making you money.
The math involved in compound interest formulas is straightforward. I am sure you can follow it. And I promise it will be quick.
Compounding over “t” years
You have a principal P (the amount of money you have), a yearly interest rate r in decimal form, and you want to compound it over t years, to get the new amount A you have, the compound interest formula is:
Compound interest formula
Usually, r is stated as the interest rate per year, but frequently, the interest may be paid or received by multiple installments over the year. For example, most bonds pay interest semi-annually, which means you get your interest payments twice a year. In the U.S., it is usual for dividend-paying stocks to pay a dividend quarterly, which means you receive four payments each year. And think about your mortgage, you pay interest each month, so twelve times a year.
We can generalize the compound interest formula to take into account that interest can be compounded multiple times during a year, we call n the number of times the interest is compounded per year:
Effective Annual Interest Rate Formula
Now if you have an interest rate r, compounded n times over a year, you can quickly get the Equivalent, or Effective, Annual Yield, which is the interest rate that produces the same earnings but with no compounding during the year.
Let’s take an example. If you have 10% compounded two times a year, the Equivalent Annual Yield is equal to 10.25%.
And if you compound every month, the Equivalent Annual Yield is equal to 10.47%.
Don’t be confused, but the Equivalent Annual Yield is also called Effective Annual Yield, Annual Equivalent Rate (AER), or Effective Annual Interest Rate. They are all the same thing.
Continuous Compounding Formula
You saw with our example that the more you compound an annual interest rate, higher is the Effective Annual Interest Rate. One question you can ask is: is it growing to infinity the more you compound or has it a limit?
The famous mathematician Jacob Bernoulli had the same question, and in 1683 he discovered the famous constant ⅇ by studying this question!
If you compound continuously, which means an infinite number of compounding periods, the continuously compounded interest formula is:
Continuous Compounding Effective Annual Yield Formula
And last formula, the Effective Annual Yield is:
Bonus: prepaying debt and compound interest
Now time for the bonus: how prepaying a debt is related to compound interest and how compound interest helps you!
Let’s assume you just bought a house, and you have now a 30-year mortgage at a 4% interest rate with a principal of $400,000. Your monthly payment is $1,910. You have $4,000 extra cash. So it is roughly two monthly payments. If you decide to use it to prepay your mortgage, what is the impact on your mortgage? Will it shorten your mortgage by two months? No, it will reduce your mortgage by nearly seven months!
Why? Because it is precisely the same as $4,000 invested in a saving account with an interest rate of 4% over 30 years! The $4,000 prepayment today is like about $13,000 in 30 years! So each time you prepay a debt, it is like saving money, at the interest rate of your debt! So if you can, prepay your most toxic debts, like high-interest credit card debts as soon as possible!
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Questions of the day:
Did you start to build your nest egg for retirement? How do you invest to benefit from compound interest? If you have any question or thought about compound interest, write it in the comments below!