Question

Write a function that computes the balance of a bank account with a given initial balance and interest rate, after a given number of years. Assume interest is compounded yearly.

Short answer

The function calculates compound interest with the formula \( A = P(1 + r)^n \), returning the balance after the specified years.

Step-by-step solution

Understand the Problem

The problem asks us to write a function that calculates the balance of a bank account with compound interest. We are given an initial balance, an interest rate, and the number of years over which the interest is compounded. The output should be the account balance after the given number of years.

Define the Function

Let's define a function named `compute_balance` that takes three parameters: `initial_balance`, `interest_rate`, and `years`. These parameters represent the starting amount of money, the annual interest rate as a decimal, and the number of years the interest is compounded, respectively.

Understand the Compound Interest Formula

Compound interest is calculated using the formula: \\[ A = P(1 + r)^n \] \where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount (initial balance), \( r \) is the annual interest rate, and \( n \) is the number of years.

Implement the Formula in the Function

Inside the function, use the compound interest formula to calculate the balance after the given number of years. Use the formula \( A = P(1 + r)^n \) where `P` is `initial_balance`, `r` is `interest_rate`, and `n` is `years`. Return the calculated balance.

Write the Function Code

Here's the complete code for the function: ```python def compute_balance(initial_balance, interest_rate, years): # Calculate the balance after the specified number of years final_balance = initial_balance * (1 + interest_rate) ** years return final_balance ```

Test the Function

Test the function with sample inputs to ensure it calculates the balance correctly. For example, you can test the function with inputs: `compute_balance(1000, 0.05, 5)` and check if the output matches the expected balance.

Key concepts

Balance Calculation

In the context of financial mathematics, calculating the balance of a bank account involves determining how much money you will have after a certain number of years, given an initial deposit and a fixed interest rate. When dealing with compound interest, the amount grows exponentially rather than linearly. This means your money doesn't just earn interest on the initial deposit—it also earns interest on the accumulated interest from previous periods. Here's how this works:

• Your initial deposit is called the "principal".
• Each year, the amount of money in the account grows by the interest rate, applied to the current balance.
• The formula to calculate your final balance is \( A = P(1 + r)^n \).

In this formula, \( P \) is the initial balance, \( r \) is the annual interest rate, and \( n \) is the number of years the interest is applied. This calculation allows you to predict the future balance based on these variables, providing a clear picture of long-term financial growth.

Python Functions

Python functions are a fundamental aspect of coding in Python. A function allows you to group a set of statements and executions into a single, reusable block of code. For our compound balance calculation, we create a function called `compute_balance`. This enables us to easily calculate the future balance with different initial amounts, interest rates, and time spans, without rewriting the logic. Here's how it works:

• The function is defined using the `def` keyword, followed by its name and parentheses that enclose parameters.
• In this example, the parameters are `initial_balance`, `interest_rate`, and `years`.
• The body of the function contains the logic to compute the balance.
• Finally, the result is returned using the `return` statement.

Such functions are beneficial because they make the code cleaner, reusable, and easier to understand, especially for recurring calculations such as balance computation with varying inputs.

Interest Rate

The interest rate is a crucial part of financial growth through compounding. It's the percentage at which your balance grows per year. Understanding its role in financial mathematics is important for making informed decisions about investments and savings. When dealing with compound interest:

• The rate is usually expressed as a decimal in calculations. For instance, 5% interest is written as 0.05.
• It determines how much more of the balance will be added each year.
• Higher rates mean faster growth of your savings.

The interest rate affects how much interest you gain on not only your initial deposit but also on the interest accumulated from previous periods. Selecting the right rate is critical in long-term financial planning, as it vastly influences the final outcome.

Financial Mathematics

Financial mathematics encompasses a wide range of concepts that help us analyze and understand financial situations. One such key concept is the calculation of compound interest, which is a common scenario when dealing with savings and investments.Compound interest can be understood by:

• Recognizing it as a method where interest is added to the principal so that interest is earned on interest.
• Using the formula \( A = P(1 + r)^n \) to predict outcomes of investments over time.
• Applying this knowledge to make better decisions concerning loans, savings accounts, and investment portfolios.

By employing financial mathematics, individuals can forecast their financial future, plan retirement funds, or set savings goals more accurately. Understanding these processes ensures individuals can manage their finances wisely and profitably.