Problem 91 - Algebra and Trigonometry

Question

Don contributes \(\$ 500\) at the end of each quarter to a tax-sheltered annuity (TSA). What will the value of the TSA be after the 80 th deposit ( 20 years) if the per annum rate of return is assumed to be \(5 \%\) compounded quarterly?

Step-by-Step Solution

Answer

The TSA will be valued at approximately \$68,624.\

Step 1: Understand the Given Information

Identify the key pieces of information. Don contributes \$500 at the end of each quarter, the annual interest rate is assumed to be \(5\%\) compounded quarterly, and the total number of deposits is 80.

Step 2: Convert Annual Interest Rate to Quarterly Rate

The annual interest rate is \(5\%\). Since the interest is compounded quarterly, divide the annual rate by 4 to get the quarterly rate: \( \frac{5\%}{4} = 1.25\%\). In decimal form, the quarterly rate is \( 0.0125 \).

Step 3: Identify the Number of Periods

Since Don makes deposits at the end of each quarter, and there are 20 years with 4 quarters in a year, the total number of deposits (periods) is \( 20 \times 4 = 80 \).

Step 4: Use the Future Value of an Annuity Formula

The Future Value (FV) of an annuity formula for regular payments made at the end of each period is given by \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]where: \( P = 500 \) (quarterly payment), \( r = 0.0125 \) (quarterly interest rate), and \( n = 80 \) (total number of contributions).

Step 5: Substitute Values into the Formula

Substitute the values into the future value formula: \[ FV = 500 \times \frac{(1 + 0.0125)^{80} - 1}{0.0125} \].

Step 6: Calculate the Future Value

Calculate the expression inside the parentheses first: \[ 1 + 0.0125 = 1.0125 \].Next, calculate \[ 1.0125^{80} \approx 2.7156 \].Then, find \[ 2.7156 - 1 = 1.7156 \].Finally, divide by the interest rate and multiply by the payment amount: \[ FV = 500 \times \frac{1.7156}{0.0125} = 500 \times 137.248 = 68624 \].

Key Concepts

Compounding Interest

When you see the term 'compounding interest,' it's talking about how interest is calculated on both the initial principal and on the accumulated interest from previous periods. In this exercise, the interest is compounded quarterly, meaning every three months. This makes calculations a bit more detailed but also means the annuity grows faster compared to just annually compounded interest. For instance, with an annual interest rate of 5%, you divide it by 4 to get the quarterly interest rate of 1.25%. This smaller, more frequent application of interest allows your investment to grow at a progressively accelerating rate.

Quarterly Contributions

Quarterly contributions mean you add money to your annuity every three months. For Don, he contributes $500 each quarter. Over 20 years, that makes a total of 80 contributions (since 20 years × 4 quarters per year = 80). Regular contributions help your investment grow more consistently because you're not waiting a whole year before adding funds. By contributing quarterly, you take advantage of the compound interest effect multiple times within a year. This steady stream of investments, combined with quarterly compounding, leads to greater growth over time.

Annuity Formula

The annuity formula used here helps calculate the future value of a series of regular payments made at the end of each period. The formula is: \( FV = P \times \frac{(1 + r)^n - 1}{r} \) Here, \( P \) is the regular quarterly payment (\$ 500), \( r \) is the quarterly interest rate ( 0.0125), and \( n \) is the total number of quarterly payments (80). Applying these values, you calculate the future value. First, calculate the term \( (1 + r)^n \). For our exercise, this becomes \(1.0125^{80} \). Then, subtract 1 from the result and divide by the quarterly interest rate. Finally, multiply by \( P \) to get the future value. Using this formula ensures that all aspects of compound growth and regular contributions are included, giving you an accurate measure of how much your investment will be worth.