## Compound Interest: An Overview

As a society, we all dream of achieving financial freedom or independence. However, most people do not take the essential steps towards achieving this goal.

One of the steps that can be taken is investing in assets that accrue interest over time. The concept of interest can be overwhelming, but it’s essential to understand the principle to invest wisely.

Compounding interest is the interest earned on the reinvestment of previously earned interest. Essentially, compound interest means that interest is earned on interest.

It’s like a snowball effect, where interest earned grows exponentially over time. In this article, we will delve into the practical aspect of compound interest.

We’ll talk about the compound interest formula and how it’s used to calculate an investment’s ending value. We will use a practical example to illustrate the concept of annual compound interest and display how to calculate and show the ending value of an investment at the end of each year.

## Compound Interest Formula and Calculation

### The formula for calculating the ending value of an investment after a specific period is as follows:

`A=P(1+r/n)^(nt)`

### Where:

- A= Ending value of investment
- P= Initial investment
- r= Annual interest rate
- n= Number of times the interest compounds per year
- t= Number of years

The formula can look intimidating, but it’s simple to understand. The variables represent the following: P is the principal sum, i.e., the amount of money initially invested.

R is the interest rate, and t is the amount of time the investment is compounded. N represents the frequency of compounding, i.e., the number of times the interest is applied to the interest earned.

For a better understanding, let’s use an example. Suppose you invest $10,000 with an interest rate of 5% compounded quarterly for a period of ten years.

To calculate the ending value of the investment, we’ll plug in the variables in the formula as follows:

`A= $10,000(1+0.05/4)^(4x10)`

### The calculation will be as follows:

```
A= $10,000 (1+0.0125)^40
A= $10,000 (1.0125)^40
A= $16,386.86
```

Therefore, the investment will be worth $16, 386.86 after ten years.

## Function for displaying ending investment after each period

Calculating the ending investment value is essential, but it’s equally important to display how much the investment is worth at the end of each period. This can be done using a display function.

A display function is a function that shows how much an investment is worth at the end of a specified period. This function provides a summary of the investment’s progress, making it easier to track the investment.

### The formula for the display function is as follows:

`FV = PV x (1 + r )^n`

### Where:

- FV= Future value of investment
- PV= Present value of investment
- r= Annual interest rate
- n= Number of compounding periods

For example, using the same investment details from the previous example, we can represent the final value of the investment after each year of the investment’s ten-year period with a display function. The function will be:

- Year 1: FV = $10,000 * (1 + (0.05 / 4))^(4 * 1) = $10,512.50
- Year 2: FV = $10,512.50 * (1 + (0.05 / 4))^(4 * 1) = $11,054.08
- Year 3: FV = $11,054.08 * (1 + (0.05 / 4))^(4 * 1) = $11,625.05
- Year 4: FV = $11,625.05 * (1 + (0.05 / 4))^(4 * 1) = $12,226.05
- Year 5: FV = $12,226.05 * (1 + (0.05 / 4))^(4 * 1) = $12,858.85
- Year 6: FV = $12,858.85 * (1 + (0.05 / 4))^(4 * 1) = $13,524.34
- Year 7: FV = $13,524.34 * (1 + (0.05 / 4))^(4 * 1) = $14,223.52
- Year 8: FV = $14,223.52 * (1 + (0.05 / 4))^(4 * 1) = $14,957.48
- Year 9: FV = $14,957.48 * (1 + (0.05 / 4))^(4 * 1) = $15,727.43
- Year 10: FV = $15,727.43 * (1 + (0.05 / 4))^(4 * 1) = $16,534.70

Hence, the investment will earn the investor a return of around $16,534.70 at the end of ten years.

## Example 1: Annual Compound Interest

Calculating compound interest is complex, but aspects such as the frequency of compounding and interest rates make the calculation complex. For our illustration purposes, let’s use an annual interest rate to make the calculation less daunting.

Suppose you invest $5,000 at an interest rate of 5% per annum for ten years. What will be the ending value of the investment at the end of the ten-year period?

### The calculation will be calculated as follows:

```
A = $5,000(1+0.05)^10
A = $5,000(1.05)^10
A = $8,132.93
```

Therefore, the investment will be worth $8,132.93 at the end of the ten-year period. Display of ending investment after each year during 10-year period.

- Year 1: FV = $5,000 * (1 + (0.05 * 1))^(1) = $5,250.00
- Year 2: FV = $5,250.00 * (1 + (0.05 * 1))^(1) = $5,512.50
- Year 3: FV = $5,512.50 * (1 + (0.05 * 1))^(1) = $5,787.13
- Year 4: FV = $5,787.13 * (1 + (0.05 * 1))^(1) = $6,074.49
- Year 5: FV = $6,074.49 * (1 + (0.05 * 1))^(1) = $6,375.23
- Year 6: FV = $6,375.23 * (1 + (0.05 * 1))^(1) = $6,689.99
- Year 7: FV = $6,689.99 * (1 + (0.05 * 1))^(1) = $7,019.53
- Year 8: FV = $7,019.53 * (1 + (0.05 * 1))^(1) = $7,364.65
- Year 9: FV = $7,364.65 * (1 + (0.05 * 1))^(1) = $7,726.24
- Year 10: FV = $7,726.24 * (1 + (0.05 * 1))^(1) = $8,105.26

Hence, the investment will earn the investor a return of around $8,105.26 at the end of ten years.

## Conclusion

Compound interest is a powerful tool that everyone should explore as they invest in their future. The snowball effect, where interest earned grows exponentially over time, is a concept that can have a significant impact on your investment’s value.

Understanding the compound interest formula and using a display function to illustrate how the investment will grow will guide an investor to make informed investment decisions. Compound interest may seem complex, but it’s a tool that might help one achieve their financial goals if given the time and proper investment choice.

## Example 2: Monthly Compound Interest

In addition to understanding annual compound interest, it’s also essential to dive into monthly compounding. Compounding interest monthly varies from compounding interest annually in that the interest is calculated and added to the principal balance every month.

This means that investors have the opportunity to earn more interest monthly than they would earn annually with the same investment.

### Calculation for Investment Compounded Monthly

To calculate the value of a monthly compounding investment, we will use the same formula, but instead of calculating the investment based on an annual rate, we will calculate it based on the monthly rate:

`A=P(1+r/n)^(nt)`

### Where:

- A= Ending value of investment
- P= Initial investment
- r= Annual interest rate
- n= Number of times the interest compounds per year
- t= Number of years

The formula for calculating the monthly compound interest is the same as that of an annual compound interest, but the frequency of interest is what varies. The frequency of compounding (n) in a monthly compounding scenario is 12, and this means that the annual interest rate (r) will be divided by twelve (12) to get the monthly interest rate.

For example, suppose we invest $10,000 at a monthly interest rate of 0.5% for five years. To calculate the investment’s ending value, we’ll plug in the variables in the formula as follows:

`A= $10,000(1+0.005/12)^(12x5)`

### The calculation will be as follows:

```
A= $10,000 (1+0.00416)^60
A= $10,000 (1.2802477)
A= $12,802.48
```

Therefore, the investment will be worth $12,802.48 at the end of the five-year term.

## Calculation for Investment Compounded Daily

Compounding interest daily is another way to maximize an investor’s returns. In this scenario, the interest is compounded daily by adding a small amount of interest to the principal balance every day.

### The formula for calculating daily compounding interest is as follows:

`A=P(1+r/n)^(nt)`

### Where:

- A= Ending value of investment
- P= Initial investment
- r= Annual interest rate
- n= Number of times the interest compounds per year
- t= Number of years

The frequency of compounding is what varies, and in a daily compounding investment, the frequency (n) will be 365. Let’s use an example to illustrate this better.

Suppose we invest $10,000 at a daily interest rate of 0.1% for ten years. To calculate the investment’s ending value, we’ll plug in the variables in the formula as follows:

`A= $10,000(1+0.001/365)^(365x10)`

### The calculation will be as follows:

```
A= $10,000 (1+0.0000274)^3650
A= $10,000 (1.1159370)
A= $11,159.37
```

Therefore, the investment will be worth $11,159.37 at the end of the ten-year term.

## Calculation for Investment Compounded over 15 Years

Investing in a long-term investment is a crucial financial move as it allows investors to earn more compound interest. For example, suppose we invest $10,000 at an interest rate of 4% compounding annually for 15 years.

### Placing the values in the formula gives:

```
A= $10,000(1+0.04)^15
A= $10,000(1.748858)
A= $17,488.58
```

Hence, the investment is worth $17,488.58 at the end of the 15-year period.

## Additional Resources

There are plenty of materials available to help investors learn different investment strategies and tools. The internet has a vast library of resources that investors can access to improve their investment knowledge.

### Here are some resources worth checking out:

- Investopedia – It’s a financial education website that offers investors access to essential investment-related advice, provided by financial experts.
- The Wall Street Journal – It’s a newspaper that provides detailed information on stocks, bonds, commodities, and markets.
- National Association of Investors Corporation (NAIC) – It’s a non-profit organization dedicated to educating members about investing and aware of good investment practices.
- Nerdwallet – It’s a website that provides banking and investment services reviews.
- The Simple Dollar – It’s a website that provides everyday people with personal finance advice.

In conclusion, investing in compound interest may increase an investor’s wealth significantly, in the long run. It’s essential to understand how to calculate the investment’s ending value, the frequency of compounding, and the impact of the diversification strategy on the investment’s risk and returns.

By arming oneself with this knowledge, investors can make informed investment decisions that can provide a reliable financial foundation for future years. In conclusion, understanding compound interest is crucial for anyone interested in long-term investments.

Whether it’s calculating the investment’s ending value, using monthly or daily compounding, or investing over a specific period, compound interest can play a critical role in generating significant wealth. By using the compound interest formula and display function, investors can make informed decisions when planning their investment strategies.

The importance of diversification and knowledge of investment options can enable investors to mitigate risks and maximize returns. In short, compound interest is an essential tool in achieving financial freedom, and understanding the concept can lead to informed investment decisions.