2.1 Introduction to Compound Interest

A. Simple and Compound Interest Comparison 

In the previous chapter, we focused on simple interest, a method where interest is calculated only on the principal amount. This unit shifts to compound interest. Let’s review what was discussed before. When money is borrowed or lent, a fee, known as interest, is charged. This fee is akin to rent on the borrowed amount.

There are two main ways to calculate interest:

• Simple Interest: the interest is calculated only on the initial amount borrowed or invested. With this method, the growth of money is linear, meaning it increases at a constant rate. Simple interest is typically applied to short-term financial products, such as short-term personal loans and treasury bills.
• Compound Interest: the interest earned in earlier periods, such as previous months, is added to the principal amount. This accumulated total then earns interest in subsequent periods. As a result, money grows exponentially, or at an increasingly rapid rate, under compound interest. This method is commonly applied to long-term financial products like mortgages and car loans.

To understand the difference between simple and compound interest, let’s use an example. Imagine $1000 is invested at a 10% annual interest rate for three years. We’ll compare two scenarios: one where the investment earns simple interest and another where it earns compound interest. To calculate the interest amount, we will use the formula [asciimath]I=P*r*t[/asciimath] provided in Chapter 1.

Simple Interest: $1000 invested at 10% p.a.	Interest earned during year 1: [asciimath]I_1=($1000)(10%)(1 \ "year")[/asciimath] [asciimath]=$100[/asciimath] Interest earned during year 2: [asciimath]I_2=(1000)(10%)(1)[/asciimath] [asciimath]=$100[/asciimath] Interest earned during year 3: [asciimath]I_3=(1000)(10%)(1)[/asciimath] [asciimath]=$100[/asciimath] Total interest earned after 3 years: [asciimath]I_t=$100+$100+$100[/asciimath] [asciimath]=$300[/asciimath]
Compound Interest: $1000 invested at 10% p.a. compounded annually.	Interest earned during year 1: [asciimath]I_1=($1000)(10%)(1 \ "year")[/asciimath] [asciimath]=$100[/asciimath] Interest earned during year 2: [asciimath]I_2=ubrace((1100))_(New\ P=1000+100)(10%)(1)[/asciimath] [asciimath]=$110[/asciimath] Interest earned during year 2: [asciimath]I_3=ubrace((1210))_(New\ P=1100+110)(10%)(1)[/asciimath] [asciimath]=$121[/asciimath] Total interest earned after 3 years: [asciimath]I_t=$100+$110+$121[/asciimath][asciimath]=$331[/asciimath]

Note that in the compound interest scenario, the interest earned during previous periods (for example, the previous year) is added to the original principal. This results in a higher amount on which interest is calculated in subsequent periods (such as the following years). As a result, the same investment accrues more interest and grows quicker over the same time frame with compound interest than it would with simple interest.

B. Determining Compounding Periods

The compounding period, also known as the conversion period, refers to the time interval between successive calculations of compound interest. Essentially, it’s the duration between one interest calculation and the next. Compounding frequency (C/Y), on the other hand, indicates how many times interest is compounded within a year. Table 2.1.1 lists various common frequencies for compounding interest.

Table 2.1.1 Compounding Frequencies for Common Compounding Periods

The more frequently interest is compounded within a year, the greater the maturity value of the investment will be, given the same principal, rate, and time period. Figure 2.1.1 illustrates this by comparing the graph of the future value of $1000 as a function of time over 20 years for simple interest and compound interest with the compound interest calculated at varying frequencies. 

Figure 2.1.1 Comparison of the future value of $1000 as a function of time for simple interest and compounded interest at different compounding frequencies

When working on problems involving compound interest, compounding frequency is typically presented in words, like “monthly”. Your task is to identify this term and convert it into a numerical value, as outlined in Table 2.1.1, before proceeding with calculations. The relevant information in a problem is typically presented in the format “[asciimath]I//Y[/asciimath] compounded [asciimath]C//Y[/asciimath] “.

Where [asciimath]I//Y[/asciimath] represents the nominal interest rate, and [asciimath]C//Y[/asciimath] is the compounding frequency expressed in words. For example, in the statement “an investment earning 3.5% compounded semi-annually”, the nominal interest rate is 3.5%, and the compounding frequency, “semi-annually”, corresponds to [asciimath]C//Y = 2[/asciimath].

The periodic interest rate, which is the interest rate per compounding period is given by

[asciimath]i=(I//Y)/(C//Y)[/asciimath]Formula 2.1a

The total number of compounding periods over the entire term of an investment or loan is calculated by multiplying the compounding frequency per year by the total number of years in the term. This is represented by

 [asciimath]N=C//Y*t[/asciimath]Formula 2.2a

Where [asciimath]N[/asciimath] is the total number of compounding periods, [asciimath]C//Y[/asciimath] is the compounding frequency per year, and [asciimath]t[/asciimath] is the total number of years in the term.

Like simple interest problem solving, this chapter will discuss how to use formulas to compute the future value, present value, and the other related variables. Additionally, this chapter will show you how to use a financial calculator (e.g., Texas Instrument BA II Plus) for the calculations. A financial calculator has all the formulas built into it. It is suggested that you learn both approaches.

Section 2.1 Exercises