Problem 70 - A Graphical Approach to College Algebra

Question

Find the future value of each annuity. Payments of \(\$ 800\) at the end of each year for 12 years at \(3 \%\) interest compounded annually

Step-by-Step Solution

Answer

The future value is $11,353.60.

Step 1: Understand the Problem

We need to find the future value of an annuity, which is a series of equal payments made at regular intervals. In this problem, \(P=800\) dollars are paid at the end of each year for 12 years with an interest rate of \(3\%\) compounded annually.

Step 2: Use the Future Value of Annuity Formula

For an annuity due at the end of each period, the future value is given by \[FV = P \times \frac{(1 + r)^n - 1}{r}\]where \(P=800\) is the payment amount, \(r=0.03\) is the interest rate, and \(n=12\) is the number of periods.

Step 3: Plug in the Values

Substitute \(P=800\), \(r=0.03\), and \(n=12\) into the formula: \\[FV = 800 \times \frac{(1 + 0.03)^{12} - 1}{0.03}\]

Step 4: Calculate the Future Value

First, calculate \(1 + 0.03 = 1.03\). Then, raise 1.03 to the 12th power: \(1.03^{12} = 1.42576\). Next, subtract 1: \(1.42576 - 1 = 0.42576\). Finally, divide by the interest rate: \[\frac{0.42576}{0.03} = 14.192\]. Multiply this by the payment amount:\[800 \times 14.192 = 11,353.6\].

Key Concepts

Compound Interest

Compound interest is a crucial concept in financial mathematics. It essentially refers to the process of earning interest on both the initial principal and the accumulated interest from previous periods. This means that each time interest is calculated, it's added to the original principal amount. Over time, the principal grows along with the interest, allowing you to earn more in each period.
For example, in our scenario, a an interest rate of 3% is compounded annually. This means that each year, the interest is calculated and added to the existing amount of money, not just the principal but also on the accumulated interest to date. This leads to an exponential growth of the investment.
When calculating compound interest, the formula used is \[ A = P(1 + r/n)^{nt} \] where:

• \( A \) is the amount of money accumulated after \( n \) years, including interest.
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (decimal format).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested for, in years.

Understanding how compound interest works helps in planning better financial strategies, whether you're saving for retirement or managing an investment portfolio.

Annuity Formula

The annuity formula is used to calculate the future value or the total value of a series of equal payments made at regular intervals, such as monthly or annually. An annuity is commonly used in retirement planning, loan repayments, and investment portfolios to ensure regular cash inflows or outflows.
To determine the future value of an annuity, you can apply the formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] In this formula:

• \( FV \) represents the future value of the annuity.
• \( P \) is the payment amount per period.
• \( r \) is the interest rate per period.
• \( n \) is the total number of payments.

The formula calculates the sum of all payments, including the interest earned on them by the end of the payment period. In the exercise provided, payments of \(800 are made annually for 12 years at an interest rate of 3%. By plugging these values into the formula, we find the future value to be approximately \)11,353.60. This value indicates the amount accumulated after completing all payments and compounding interest.
Understanding how the annuity formula works aids in financial planning by giving a clear picture of potential returns on investments that involve regular payments.

Financial Mathematics

Financial mathematics is a field of applied mathematics focused on financial markets, investments, and risk assessment. It encompasses a wide range of mathematical methods and models used in managing financial operations.
Some essential areas of financial mathematics include:

• **Interest Calculations:** Understanding simple and compound interest is fundamental in loans and investments.
• **Annuities:** These provide a mathematical model for regular payments or receipts and are important for calculating loans, savings plans, and retirement funds.
• **Risk Management:** Models for assessing financial risk help in creating strategies to minimize and manage it effectively.
• **Valuations of Assets:** Determining the present or future value of financial assets or investments is essential for making informed decisions.

In the original exercise, we utilized financial mathematics to determine the future value of an annuity. This involves combining our understanding of compound interest and the annuity formula to ensure accurate financial predictions and efficient planning.
By leveraging these concepts, financial mathematics equips individuals with the tools needed for smart financial decision making at both personal and professional levels.