Problem 10 - Precalculus: Mathematical for Calculus

Question

Funding an Annuity A 55 -year-old man deposits \(\$ 50,000\) to fund an annuity with an insurance company. The money will be invested at \(8 \%\) per year, compounded semiannually. He is to draw semiannual payments until he reaches age \(65 .\) What is the amount of each payment?

Step-by-Step Solution

Answer

The semiannual payment amount is approximately $1,679.42.

Step 1: Understand the Problem

The problem involves an annuity funded with an initial deposit. The goal is to find out the amount of each semiannual payment such that the account is fully depleted when the annuitant reaches the age of 65.

Step 2: Identify Key Information

The initial deposit is $50,000. The interest rate is 8% per year, compounded semiannually, thus each compounding period (half a year) has an interest rate of \( \frac{8\%}{2} = 4\% \). The time frame is 10 years, from age 55 to 65, providing \( 10 \times 2 = 20 \) semiannual periods.

Step 3: Use the Annuity Formula

The future value of an annuity formula \( A = P \times \frac{(1 + r)^n - 1}{r} \) is rearranged to solve for the payment \( P \). Here, \( A = 50,000 \), \( r = 0.04 \), and \( n = 20 \).

Step 4: Calculate the Factor

Calculate the annuity factor using \( \frac{(1 + r)^n - 1}{r} \). Thus, \( \frac{(1 + 0.04)^{20} - 1}{0.04} = \frac{(1.04)^{20} - 1}{0.04} \). Calculate the value: \((1.04)^{20} = 2.191123\), hence the factor is \( \frac{2.191123 - 1}{0.04} \approx 29.7781 \).

Step 5: Solve for the Payment Amount

Plug the annuity factor back into the rearranged formula \( P = \frac{A}{\text{factor}} \). Therefore, \( P = \frac{50,000}{29.7781} \approx 1679.42 \).

Step 6: Conclusion

The semiannual payment amount required to deplete the annuity over 10 years, with 4% interest per period, is approximately $1,679.42.

Key Concepts

Compound Interest

Understanding compound interest is crucial in calculating annuities. Unlike simple interest, which is computed on the original principal only, compound interest considers both the principal and the accumulated interest from previous periods. This results in interest earning on interest, which can significantly increase the investment over time.

In the context of our exercise, when a 55-year-old deposits $50,000 at an 8% annual rate, compounded semiannually, the interest is applied twice a year. Here's how it works:

• The annual rate of 8% is divided into two periods of 4% each (since it is compounded semiannually).
• With each period, the interest is calculated on the increased total from previous compounding.

This means that by the end of the 10-year period, the man's deposited funds will grow exponentially more than if only simple interest was employed.

Future Value of Annuity

The future value of an annuity is a financial concept used to calculate the value of a stream of payments at a specific point in the future, taking into account compound interest. For our problem, understanding this concept helps determine how the initial \(50,000 deposit can be structured into semiannual payments over 10 years.

The formula for the future value of an annuity is:\[A = P \times \frac{(1 + r)^n - 1}{r}\]In this setting:

• A is the future value of the annuity, originally given as \)50,000.
• P is the payment amount.
• r is the interest rate per period (4% or 0.04).
• n is the total number of payments (20 periods).

This formula helps us back-calculate the semiannual payment (P) that depletes the annuity fully by the end of the period, allowing the individual to optimize his withdrawals based on the compound growth of his initial investment.

Semiannual Payments

Semiannual payments involve splitting interest calculation and payments into two equal parts over a year. Understanding this concept is essential when calculating annuities with a compounding structure like the one in our problem.

Key points to remember with semiannual payments include:

• The interest rate for each period is half of the annual rate.
• Payments are made at the end of each period, adapting to the compounding frequency of the investment.

In our example, the individual must understand that his $50,000 cannot simply be divided by the number of periods for equal payments. Instead, calculations must account for the compounding influence of the interest every 6 months. By doing so, each payment guarantees the full depletion of the annuity over the desired timeframe, following the calculated semiannual amount derived from our annuity formula.