Introduction to Financial Mathematics: Compound Interest

Relevance of the Topic

Financial Mathematics is a discipline always present in our lives and permeates various aspects of daily life, from personal situations to business decision-making. The study of compound interest, in particular, is fundamental for understanding how money can grow (or decrease) over time. Through compound interest, we can make future predictions about investments, understand the dynamics of debts, and, more importantly, make more informed and conscious decisions regarding our money.

Contextualization

In the vast field of Financial Mathematics, compound interest is one of the key concepts that form the basis of understanding the time value of money. Without mastering compound interest, many other topics in Financial Mathematics may seem obscure or complicated. Therefore, we will begin our study with compound interest, exploring how it differs from simple interest and deepening the analysis of how it works over time. This topic, besides being a crucial step in Financial Mathematics, always finds intersections with other topics, such as discount and inflation, and can be applied to a wide range of real-life scenarios, making it an essential study in the Mathematics curriculum.

Theoretical Development

Components

• Initial Capital (C): It is the amount of money initially invested or loaned. In compound interest, this value increases over time.

• Interest Rate (i): Represents the percentage of the initial capital that will be added (or charged, if applicable) periodically. It is a crucial factor for the growth of compound interest.

• Time (t): It is the period of time during which the initial capital will generate interest. It can be expressed in months, years, bimesters, etc.

• Amount (M): It is the final value that the initial capital will reach after the addition of compound interest. It is the sum of the initial capital and the interest.

• Compound Interest Formula: M = C(1 + i)^t, where M is the amount, C is the initial capital, i is the interest rate, and t is the time.

Key Terms

• Compound: Comes from the Latin "componere", which means "to put together". In the context of interest, it denotes the action of periodically adding the interest to the initial capital, forming a larger whole.

• Interest on Interest: Represents the main characteristic of compound interest - the interest generated each period is incorporated into the initial capital, causing the amount to grow increasingly faster.

Examples and Cases

1. Long-Term Investment: Suppose we invest R$1,000.00 in an application with an interest rate of 5% per month. After 10 months, using the compound interest formula, the final amount will be M = 1000 * (1+0.05)^10 = R$1,628.89. Notice that, unlike simple interest, the rate is applied to the accumulated amount month by month, resulting in a significantly larger final amount.

2. Interest Charges on Credit Cards: This is an example of how compound interest can lead to rapidly growing debts. Suppose the credit card interest rate is 10% per month and we owe R$1,000.00. If we do not pay the debt for 6 months, the amount to be paid will be M = 1000 * (1+0.10)^6 = R$1,771.56. Notice how the amount owed increases each month, due to the charging of interest on interest.

These examples illustrate the importance of understanding and using compound interest in our daily lives, both for making investments and for taking loans or financing. Notice that the same amount of capital with the same interest rate can generate quite different results if we use simple or compound interest, reinforcing the need to master this concept in the study of Financial Mathematics. We will offer more examples, exercises, and cases to enhance the understanding of these topics.

Detailed Summary

Relevant Points

• Difference between Simple Interest and Compound Interest: While simple interest is calculated only on the initial capital, compound interest is calculated on the initial capital plus previously accumulated interest. This causes the amount to grow more sharply over time in the compound interest regime.

• Capital Composition Over Time: In compound interest, the initial capital "changes" each calculated period, as it incorporates the previously accumulated interest. This means that, with each new period, the initial capital is larger than the previous one, which explains the process of accelerating the amount over time.

• Importance of the Interest Rate: The interest rate in a compound interest regime defines the pace of growth of the initial capital. Higher rates result in faster growth of the amount, while lower rates are synonymous with slower growth.

• Applicability of Compound Interest: Compound interest is present in various real-life situations, being used in financial investments, in the charging of loans and financing, and even in monetary correction.

Conclusions

• Exponential Effect of Compound Interest: When delving into the calculations of compound interest, the exponential effect they have on the growth of capital is noticeable. This implies that, the longer the investment time, the greater the difference between the amounts of an application with simple and compound interest.

• Influence of Time and Interest Rate: Time and the interest rate are two factors that have a significant impact on the final amount in a compound interest regime. The longer the investment time and/or the interest rate, the greater the amount obtained.

• Need for Financial Planning: Understanding compound interest is a key tool for efficient financial planning. Knowing how the variables (time, interest rate, initial capital) interact can make the difference between good and bad financial decisions.

Suggested Exercises

1. Exercise 1: If you deposit R$ 500.00 in a savings account that pays 1% interest per month, what will be the value of your investment after 2 years?

2. Exercise 2: A loan of R$ 10,000.00 was granted at an interest rate of 5% per month. If the borrower opts to pay the loan after 1 year, how much will he have to pay?

3. Exercise 3: Pedro decided to invest R$ 2,000.00 in an application with an interest rate of 2.5% per semester. After 4 semesters, what will be the amount that Pedro will have accumulated in his account?