Simple and Compound Interest Calculations

The Time Value of Money

At its heart, finance is about valuing and comparing cash flows that occur at different points in time. The core idea is that money available today is worth more than the same amount in the future. This isn't just a matter of preference; it's due to the opportunity to invest that money and earn a return.

To formalize this idea, we'll start with a foundational reading. This chapter will introduce you to why money has a time value and define the key terms we'll be using throughout this course.

For a more intuitive feel of how interest rates connect present and future amounts of money, watch the beginning of the following video. It uses a simple scenario to illustrate the concept of indifference between receiving money now versus later.

Calculating Future Value with Simple Interest

The most straightforward way to calculate interest is the simple interest method. Here, interest is earned only on the original amount of money invested, known as the principal. It does not account for the re-investment of interest earnings.

The formula for simple interest is:

Where:

• $I$ is the total interest earned
• $P$ is the principal
• $R$ is the annual interest rate
• $T$ is the time in years

The total future value (FV), or simple amount (S), is the principal plus the interest:

Let's see this in action. The following video provides a clear, practical example of calculating simple interest.

Calculating Future Value with Compound Interest

While simple interest is easy to understand, most financial instruments use compound interest. This is where things get more powerful. With compound interest, you earn "interest on interest." The interest earned in each period is added to the principal, and subsequent interest calculations are based on this new, larger principal.

This leads to exponential growth, a concept you'll be familiar with from your statistics background. The formula for the future value with compounding is:

Where:

• $FV$ is the future value
• $PV$ is the present value (the initial principal)
• $i$ is the interest rate per period
• $n$ is the number of compounding periods

The image below shows a typical compound interest problem. Notice how the formula $A=P(1+r{)}^t$ is applied directly to find the total amount owed on a car loan after 5 years.

To see a direct comparison of simple versus compound interest on the same investment, let's return to the Khan Academy video.

Compounding Frequency

Interest can be compounded more than once a year (e.g., semi-annually, quarterly, or monthly). The more frequent the compounding, the faster your investment grows.

The general formula is adjusted to account for this:

Where:

• $m$ is the number of compounding periods per year
• $n$ is the number of years
• $i$ is the nominal annual interest rate

The following reading explains this concept in detail and provides worked examples for different compounding frequencies. As you review it, focus on how the variables $i$ and $n$ in the basic formula are adjusted to become the rate per period ($i/m$) and the total number of periods ($m\times n$).

Calculating Present Value

Just as we can "compound" a present value into the future, we can "discount" a future value back to the present. This is essential for understanding what a future cash flow is worth today. Calculating the present value (PV) is simply a rearrangement of the compound interest formula.

The formula for present value is:

Here, the interest rate $i$ is often referred to as the discount rate.

Let's look at one final reading to solidify this concept.

For extra practice, you can explore the resource "CHAPTER 4 : SIMPLE & COMPOUND INTEREST" (LINK). It contains a large number of solved problems and practice questions covering all the concepts from this lesson, which is excellent preparation for the exam's numerical focus.

Conclusion

In this introductory lesson, we have established the crucial concept of the time value of money and learned the essential skills to quantify it.

Key Takeaways:

• Money has time value because of the opportunity to earn a return on it.
• Simple interest is calculated only on the initial principal.
• Compound interest is calculated on the principal and accumulated interest, leading to exponential growth.
• The frequency of compounding significantly impacts the future value of an investment.
• Present Value (PV) is the current worth of a future sum of money, found by discounting the future value back to the present.

These calculations are the fundamental building blocks for financial analysis. In our next lesson, we will extend these principles to handle more complex situations involving multiple cash flows over time, specifically by learning to calculate the present value of annuities and perpetuities.