Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the amount (future value) of each ordinary annuity. $$\$ 500 /$$ semiannual period for \(12 \mathrm{yr}\) at \(11 \% / \mathrm{year}\) compounded semiannually

Step-by-Step Solution

Answer

The future value of the ordinary annuity is approximately $15,297.61, with a semiannual payment of $500 for 12 years and an 11% annual interest rate compounded semiannually.

Step 1: Calculate the interest rate per period and the number of periods

As the annual interest rate is 11% compounded semiannually, we can find the interest rate per period (r) by dividing the annual interest rate by 2: \[r = \frac{11}{100*2} = 0.055\] The annuity is being paid for 12 years, and there are 2 payments each year (semiannual). So, the number of periods (n) can be found by multiplying the number of years by the number of payments per year: \[n = 12 * 2 = 24\]

Step 2: Apply the future value of an ordinary annuity formula

Now that we have the values for r and n, we can use the formula to find the FV of the annuity: \[FV = P * \frac{(1+r)^n - 1}{r}\] Let P = $500. We can plug in the values and calculate the FV: \[FV = 500 * \frac{(1+0.055)^{24} - 1}{0.055}\]

Step 3: Calculate the future value of the annuity

Finally, let's perform the calculations: \[FV = 500 * \frac{(1.055)^{24} - 1}{0.055}\] \[FV = 500 * \frac{2.678319 - 1}{0.055}\] \[FV = 500 * \frac{1.678319}{0.055}\] \[FV \approx 15297.61\] Thus, the future value of the ordinary annuity is approximately $15,297.61.

Key Concepts

Compounded Interest

Compounded interest is a crucial financial concept that allows your money to grow more rapidly than simple interest. Unlike simple interest, which is calculated only on the principal amount, compounded interest is calculated on both the initial principal and the accumulated interest from previous periods. This means that you earn interest on your interest, enabling exponential growth.
Understanding how interest is compounded is essential for accurately calculating future values of investments or savings. Compounded semiannually means the interest is calculated twice a year. For instance, if you have an annual interest rate of 11%, it would be divided equally over two periods, meaning each period has an interest rate of 5.5%. This ensures that the interest compounds effectively every six months.
To benefit from compound interest:

• Start investing early to allow more time for your money to grow.
• Choose investments that compound frequently—ideally more than once per year.

Ordinary Annuity Formula

An ordinary annuity is a sequence of equal payments made at the end of each period. The future value (FV) of an ordinary annuity can be determined using a specific formula. This formula helps you calculate how much your series of regular payments will grow over time when invested at a certain compounded interest rate.
The formula for calculating the future value of an ordinary annuity is given by:\[ FV = P \times \frac{(1+r)^n - 1}{r} \]where:

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

This formula accounts for each payment's contribution to the final sum by considering both the principal and the interest accumulated.

Interest Rate Per Period

The interest rate per period is a key component in calculating the growth of an investment. It represents the interest applied for each compounding interval.
To find the periodic interest rate, you divide the annual interest rate by the number of compounding periods per year. For example, with an annual rate of 11% and semiannual compounding, the interest rate per period is calculated as:\[ r = \frac{11}{100 \times 2} = 0.055 \]This means 5.5% interest is applied every six months. Clearly distinguishing each compounding period's rate helps ensure precise calculations for the growth of an investment or loan.

Number of Periods Calculation

The number of periods is another pivotal factor in future value calculations. It determines how many times the interest will be applied over an investment's life.
To calculate the number of periods for an annuity or investment, multiply the number of years by the frequency of compounding per year. For instance, with an investment over 12 years compounded semiannually, the total number of periods would be:\[ n = 12 \times 2 = 24 \]This means the interest is compounded, and payments are made a total of 24 times over the life of the annuity. This step is vital as it feeds into other calculations involving compounded interest and future value.