Question

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 1500, \quad r=2 \%, \quad T=10 \text { years } $$

Short answer

The future value of the annuity is the amount \( A \), which is computed using the formula.

Step-by-step solution

Identify the Parameters

The constants are given in the problem as follows: the periodic income \( c(t) = $1500 \), the interest rate \( r = 2% = 0.02 \) per period (assumed to be annual), and the term of the annuity \( T = 10 \) years.

Apply the Formula for the Future Value of an Annuity

The future value \( A \) of an annuity is given by the formula: \( A = c(t) * \frac{(1 + r)^T - 1}{r} \)

Apply the Given Values

By substituting the given values into the equation, it becomes: \( A = 1500 * \frac{(1 + 0.02)^{10} - 1}{0.02} \)

Compute the Arithmetic Operations

Perform the operations in the equation with the prescribed order of operations (brackets, powers, multiplication and division, addition and subtraction).

Calculate the future value

After performing all arithmetic operations, the value of \( A \) will be the future value of the annuity.

Key concepts

Income Function

In the context of annuities, the income function, often denoted as \( c(t) \), represents the periodic payment amount, which is the amount of money contributed to the annuity at regular intervals. For this example, it is given that \( c(t) = \\(1500 \), meaning each period, \\)1500 is added to the annuity. Understanding the income function is crucial because it determines how much will be accumulated over time. The regularity and size of these payments play a significant role in the total amount accumulated in the annuity. It's like setting aside a portion of your income regularly to build up significant savings over time.

• The periodic payment is fixed and regular.
• Consistency in payments helps grow the annuity's future value.

By knowing the value of \( c(t) \), you can plan effectively for future financial goals.

Interest Rate

The interest rate is a percentage that represents the cost of borrowing money or the gain from investing money. In the context of annuities, the interest rate, denoted as \( r \), signifies how much the periodic contributions grow each year. For this exercise, an interest rate of 2%, or \( r = 0.02 \), is assumed to be annual. Interest rates are vital because even a small rate compounded over many periods can significantly increase the annuity's future value. Think of interest as a reward for allowing your money to remain invested over time.

• 2% annual rate implies each year's contributions grow by 2%.
• The compounding effect takes place, increasing the total value.

A good understanding of the interest rate helps better anticipate how much you'll actually have at the end of the annuity's term.

Annuity Term

The annuity term, often indicated by \( T \), is the duration over which the annuity payments are made. For this specific situation, the term is \( T = 10 \) years. This term is critical because it determines the number of periods the contributions will be compounding interest. The longer the term, the more opportunities for the interest to work its magic through the power of compounding.

• Longer terms typically lead to higher future values due to more compounding periods.
• Each period builds on the previous one's growth.

Understanding the term gives you a clear picture of how long you need to invest regularly to meet desired financial outcomes.

Arithmetic Operations

To find the future value of an annuity, several arithmetic operations must be carefully performed. Calculating this value involves basic mathematical concepts applied systematically. The formula for the future value of an annuity is: \[ A = c(t) \times \frac{(1 + r)^T - 1}{r} \]Breaking down the operations:

• Exponential Calculation: First, calculate \((1 + r)^T\), which allows for the compounding effect of interest over the period.
• Subtraction: Subtract one from the result of the exponential calculation to find how much the annuity has grown beyond the original contributions.
• Division: Divide the result by \( r \), the interest rate, to normalize the exponential growth.
• Multiplication: Multiply by \( c(t) \), the periodic contribution, to compute the future value.

Performing these operations in the correct sequence is essential to accurate and meaningful results.