Relevance of the Theme: Financial Mathematics and Interest

The study of Financial Mathematics, specifically Interest and Value Change Over Time, is a fundamental pillar for understanding the financial world around us. From the simplest act of saving to the most complex investments and loans, all actions in the financial realm involve an addition of values over time. This phenomenon is known as compound interest and its understanding is essential for making financial decisions, both personal and professional.

• Interest of the Theme: This theme is crucial in everyday mathematics and, therefore, is highly consumable and applicable. The ability to understand and calculate interest on investments or loans, for example, provides students with a practical skill that can be used throughout their lives. Moreover, the study of compound interest and how values change over time is a challenging and interesting concept.

• Connection with Other Themes: Financial Mathematics, in general, and Interest and Value Change Over Time, in particular, are inextricably linked to many other components of the mathematical curriculum. Mastery of these topics can facilitate the understanding of subjects such as Functions and Progressions, which also involve the idea of how quantities change over time. Furthermore, understanding interest and constantly changing values is essential for building solid knowledge in Economics and Finance, which are highly interconnected fields.

Theoretical Development

• Components:

  • Capital: It is the initial amount of money being considered in a financial application. In the context of compound interest, capital is the starting point from which interest is calculated over time.

  • Interest: Represents the remuneration of capital. They are calculated from a rate (indicated as a percentage) and act as a "feedback" of time. There are two types of interest: simple interest and compound interest.

  • Interest Rate: It is the percentage of the capital that will be added to it during a certain period of time. This rate is vital for the calculation of interest and, therefore, for the change of values over time.

  • Period: Refers to the time interval during which interest is calculated. The longer the period, the greater the change in financial values.

• Key Terms:

  • Simple Interest: It is the type of interest where the remuneration only affects the initial capital. In other words, the interest value is the same throughout the period.

  • Compound Interest: In this type of interest, the interest rate affects both the initial capital and the previous interest. It is this characteristic that generates an accelerated change in values over time.

  • Amount: It is the final value adding the initial capital and the interest. In the case of compound interest, the amount increases more rapidly than in the case of simple interest.

• Examples and Cases:

  1. Simple Interest: If a loan of R$1000.00 is made at an interest rate of 10% per year and the interest charge is simple, the interest at the end of one year will be R$100.00. The next year, the 10% rate will be applied again on the initial capital, generating another R$100.00 of interest. Therefore, the change in value over time is constant: R$100.00 per year.

  2. Compound Interest: Using the same example of a loan of R$1000.00 at an interest rate of 10% per year, but with compound interest, the calculation changes. In the first year, the interest will be R$100.00, as in the previous example. But, in the second year, the 10% rate will be applied on the final capital of the first year, which already includes the interest. This will result in R$110.00 of interest in the second year. Thus, the change in value over time will be R$100.00 in the first year and R$110.00 in the second, illustrating the acceleration of change over time in relation to simple interest.

Detailed Summary

• Relevant Points:

  • Interest: Key element in Financial Mathematics, representing the remuneration of capital over time. They are calculated from a rate, which acts as a "feedback" of time.

  • Capital: Starting point for the calculation of interest. It is the initial value being considered in a financial application.

  • Interest Rate: Determines the percentage of the capital that will be added to it during a given period of time. This rate directly influences the change of financial values over time.

  • Period: Time interval during which interest is calculated. It has a direct influence on the change of financial values.

  • Simple Interest: Type of interest in which the remuneration only affects the initial capital. That is, the interest value is the same throughout the period.

  • Compound Interest: Interest that affects both the initial capital and the accumulated interest. The amount increases more rapidly in relation to simple interest.

  • Amount: Final value that includes the initial capital and the interest. In the case of compound interest, the amount increases more rapidly in relation to simple interest.

• Conclusions:

  • The understanding and application of the concepts of interest and change of values over time are fundamental for making assertive financial decisions.

  • Compound interest is preferred in investment situations, due to the acceleration of the amount over time.

  • The principles of interest and change of values over time are applicable in various financial scenarios, from simple applications such as savings, to the management of large financial assets.

• Exercises:

  1. Exercise 1: Calculate the amount of an investment of R$500.00 at an annual rate of 12.5% after 5 years of application, considering the regime of compound interest.

  2. Exercise 2: Compare the amount of a loan of R$1000.00 at an interest rate of 10% per year after 3 years of charging, considering the regime of simple interest and compound interest.

These exercises will test the understanding of the concepts of compound and simple interest, in addition to practicing their application in real situations.