Problem 4 - Precalculus: Mathematics for Calculus

Question

Annuity Find the amount of an annuity that consists of 24 monthly payments of \(\$ 500\) each into an account that pays \(8 \%\) interest per year, compounded monthly.

Step-by-Step Solution

Answer

The future value of the annuity is approximately \$12,679.85.

Step 1: Identify Key Components

For the annuity calculation, we need to identify the number of payment periods (\(n\)), the periodic payment amount (\(P\)), the interest rate per period (\(r\)), and the total amount of the annuity at the end. Given:- Monthly payments: \(P = \$500\)- Total number of payments: \(n = 24\)- Annual interest rate: \(8\%\)- Since interest is compounded monthly, the monthly interest rate is \(\frac{8\%}{12} = 0.0067\).

Step 2: Define the Future Value of An Annuity Formula

The future value of an ordinary annuity can be calculated using the formula:\[FV = P \left( \frac{(1 + r)^n - 1}{r} \right)\]where:- \(FV\) is the future value of the annuity,- \(P\) is the periodic payment,- \(r\) is the interest rate per period,- \(n\) is the total number of payments.

Step 3: Substitute Known Values Into the Formula

Substitute the known values into the formula:- \(P = 500\)- \(r = 0.0067\)- \(n = 24\).\[FV = 500 \left( \frac{(1 + 0.0067)^{24} - 1}{0.0067} \right)\]

Step 4: Calculate the Future Value

First, calculate \((1 + 0.0067)^{24}\):\[(1 + 0.0067)^{24} \approx 1.1697\]Then compute the fraction:\[\frac{1.1697 - 1}{0.0067} \approx 25.3597\]Finally, calculate the future value:\[FV = 500 \times 25.3597 \approx 12679.85\]

Step 5: Interpret the Result

The future value of the annuity after 24 monthly payments of \\(500 each, with an interest rate of 8% per year compounded monthly, is approximately \\)12,679.85.

Key Concepts

Future Value of Annuity

Understanding the future value of an annuity is essential when you want to find out how much money you will have accumulated at the end of a series of regular payments. This concept is used extensively for retirement planning, savings plans, and loan analysis.
The formula is generally written as:\[FV = P \left( \frac{(1 + r)^n - 1}{r} \right)\]This formula helps you calculate the total amount you will have after making consistent payments over a specific period. Here:

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

In our exercise, after making 24 monthly payments of \(\\(500\), you will end up with around \(\\)12,679.85\). This number is the future value of your annuity, including the compounded interest over the 24 months.

Compound Interest

Compound interest is a powerful concept that significantly affects the growth of an annuity. It refers to the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. This means your interest earns interest.
The formula used to calculate compound interest in the context of an annuity is reflected in the power \((1 + r)^n\), where:

• \(r\) is the interest rate per period.
• \(n\) is the total number of compounding periods.

For example, if you invest money in an account that pays an annual interest rate of 8%, compounded monthly, your money doesn't just earn a one-time 8% return. Instead, it's earning interest on top of interest every month, thereby accelerating your total growth by the end of the term.

Periodic Payment

A periodic payment is the amount of money paid or received at each interval in an annuity plan. It is a consistent and predictable amount that allows for managing finances effectively.
In the exercise above, the periodic payment was \(\$500\) per month, which was paid into an interest-bearing account for 24 months.
This steady flow of identical payments works well with the compound interest to maximize your future value. For budget planning, remembering your periodic payments is crucial for predicting savings growth and managing your cash flow accurately.

Interest Rate per Period

The interest rate per period is an essential factor in calculating the future value of an annuity. It determines how much interest your payments will generate per period.
When the nominal annual interest rate is given, like 8%, and it is compounded more frequently than once a year, you must divide it by the number of compounding periods per year.
For monthly compounding, as with our exercise, you divide the annual rate of 8% by 12 months, which gives you an interest rate per period of 0.0067.
Understanding and correctly applying this rate is crucial for accurate annuity calculations, ensuring your predictions align with how banks and financial systems calculate interest.