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Compound interest is a way to calculate the additional compensation due on both the initial principal and all the accumulated interest from the previous time periods. For the purpose of calculation, we will derive it using the simple interest formula.

Recall: Simple Interest ($) = Principal ($) x Interest Rate (%) x Time (Years)

Before we proceed further, we shall adopt the following acronyms to simplify the equation:

Simple Interest = P x i x N

Since compound interest considers both additional compensation from the initial principal and all the accumulated interest, we will modify the formula like this:

Derivation of Formula
Derivation of Formula

Let’s analyse the formula from the first year to the third year. For each time period, we are able to find a pattern in the total value at the end of the period. To point out, we noticed that

Total Value at end of Year N = P (1 + i)^N

Furthermore, this formula works across all periods, where N is a positive integer. For this purpose, the formula to calculate compound interest becomes

Compound Interest = Total Value at end of Period – Initial Principal

= P (1 + i)^N – P = P [ (1 + i)^N – 1].

Example 1 (Using Compound Interest)

I invested $100,000 at ta rate of 10% per annum for 6 years.

Compound Interest = $100,000 [ (1 + 0.1)^6 – 1] = $77,156.10

Altogether, I will receive $100,000 + $77,156.10 = $177,156.10 at the end of 6 years for my investment.

Example 1 (Using Simple Interest)

Simple Interest = $100,000 x 10% x 6 = $60,000

Accordingly, we will have missed out $77,156.10 – $60,000 = $17,156.10. This is the interest earned on top of the interest accumulated from the previous time period.

Real Life Application

As an illustration, the local banks give us an interest rate of 0.05% per annum for the deposit that we placed with them.

Let’s keep $10,000 in the bank for 10 years. Accordingly,

Compound Interest = $10,000 [ (1 + 0.0005)^10 – 1] = $50.11 (rounded to 2 decimal places)

In sum, we will have $10,000 + $50.11 = $10,050.11 for saving our money with the bank.

P.S. How happy are you with receiving a mere $50.11 for keeping your money in the bank for a decade? 🙃

Conclusion

When we compute the returns using compound interest, its yield will always be larger as compared to the returns generated by simple interest. This is because interest is earned on top of all the accumulated interest in the time period.

In the real world, there are many applications, e.g. calculation for bank interest, investment return.

All in all, our money is working harder for us when our returns is computed using compound interest. Of course, this is if and only if we keep the capital and the interest earned within the account!

Thoughts of the Day 💭

1️⃣ When will compound interest work against us?

2️⃣ How many years will it take to double our money by keeping it in the bank?

3️⃣ How do you grow your hard-earned money?


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