Problem 9 - Algebra and Trigonometry

Question

Funding an Annuity How much money must be invested now at 9\(\%\) per year, compounded semiannully, to fund an annuity of 20 payments of \(\$ 200\) each, paid every 6 months, the first payment being 6 months from now?

Step-by-Step Solution

Answer

Approximately $2416 must be invested now.

Step 1: Understand the Problem

We want to find the present value of an annuity, which consists of 20 semiannual payments of $200 each, paid every 6 months. The annuity is funded with an investment that earns 9% annually, compounded semiannually.

Step 2: Determine the Interest Rate per Period

Since interest is compounded semiannually, we must convert the annual interest rate to a semiannual rate. If the annual rate is 9%, then the semiannual rate is 9% / 2 = 4.5%, or 0.045 in decimal form.

Step 3: Determine the Number of Periods

The annuity consists of 20 payments, and each period corresponds to one of these semiannual payments, so there are 20 periods in total.

Step 4: Use the Present Value of Annuity Formula

The present value (PV) of an annuity can be calculated using the formula: \[ PV = P \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \] where \( P \) is the payment amount, \( r \) is the interest rate per period, and \( n \) is the number of periods.

Step 5: Plug in the Values

Here, \( P = 200 \), \( r = 0.045 \), and \( n = 20 \). Substituting these into the formula: \[ PV = 200 \times \left(\frac{1 - (1 + 0.045)^{-20}}{0.045}\right) \]

Step 6: Calculate the Present Value

First, calculate \((1 + 0.045)^{-20} \approx 0.4564\). Then substitute this value back into the expression to find the present value: \[ PV = 200 \times \left(\frac{1 - 0.4564}{0.045}\right) = 200 \times \left( \frac{0.5436}{0.045} \right) \approx 200 \times 12.080 \approx 2416 \]

Step 7: Conclusion

The present amount that needs to be invested to fund the annuity is approximately $2416.

Key Concepts

Semiannual Compounding

Semiannual compounding refers to the process where interest is calculated and added to the investment balance twice a year. This affects the growth of your investment as interest is earned more frequently compared to annual compounding.
When an annual interest rate is given and compounding occurs semiannually, the interest rate must be divided by two to find the semiannual rate.
For example:

• If the annual interest rate is 9%, you divide it by 2, giving a semiannual rate of 4.5%.
• This rate is typically converted into decimal form for calculations: 0.045.

Understanding how the compounding period affects your computations is crucial for calculating the present value of an investment or annuity.

Annuity Payments

An annuity is a series of equal payments made at regular intervals. In our exercise, these payments occur every six months, with each payment being $200.
Such regular intervals make annuities a popular choice for planning future investments or expenses. Paying close attention to the schedule of payments helps in applying the correct formulas to determine the present value or future value of an annuity.
Key characteristics include:

• Frequency: Semiannual payments mean twice per year.
• Number of payments: Our example involves 20 payments, resulting in an investment period of 10 years.
• First payment timing: Payments start after an initial delay, commonly called the deferred period. Here, the first payment is 6 months from the start of the investment.

Understanding these details helps inform calculations regarding how much to invest now to meet future payment obligations.

Interest Rate Per Period

The interest rate per period indicates how much your investment will grow in each compounding interval. In financial mathematics, the correct rate per period is crucial for accurate calculations.
To calculate it:

• Identify the annual interest rate, e.g., 9%.
• Divide it by the number of compounding periods in a year (semiannual means two periods), leading to 4.5% per period.
• Convert this percentage into decimal format for use in formulas: 0.045.

This adjusted rate then applies to each compounding cycle and it impacts the future and present value calculations of investments.

Financial Mathematics

Financial mathematics is the field concerned with applying mathematical methods to solve problems related to finance, such as calculating the present value of annuities. This field incorporates compound interest, annuity payments, and interest rate per period.
Present value calculations in financial mathematics help determine how much should be invested now to achieve a desired annuity in the future.

• The present value (PV) formula helps establish how an investment with constant payments grows over time, accounting for interest and time.
• PV calculations incorporate factors like period interest rate, the number of periods, and payment size.
• Using formulas and methodologies is essential for informed decision-making in finance.

Understanding these concepts ensures accuracy in calculations for personal or business investments, providing a more secure financial planning foundation.