Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the periodic payment \(R\) required to accumulate a sum of \(S\) dollars over \(t\) yr with interest earned at the rate of \(r \% /\) year compounded \(m\) times a year. $$ S=20,000, r=4, t=6, m=2 $$

Step-by-Step Solution

Answer

So, the periodic payment R required to accumulate a sum of \(20,000\) over \(6\) years with an annual interest rate of \(4\%\) compounded semi-annually is approximately \(1,245.67\).

Step 1: Understanding the Future Value of an Annuity Formula

The future value of an annuity (FV) is given by the formula: $$ FV = R * \frac{(1 + i)^n - 1}{i} $$ Where: - R is the periodic payment - i is the interest rate per period (i = \(\frac{r}{100m}\)) - n is the number of periods (n = m * t) In this case, we are given the future value (S) and we are looking to solve for the periodic payment R.

Step 2: Substitute the given values into the formula

We are given the values: S = 20,000, r = 4, t = 6, and m = 2. Our first step will be to calculate i and n: i = \(\frac{r}{100m}\) = \(\frac{4}{100*2}\) = 0.02 n = m * t = 2 * 6 = 12 Now, we substitute these values into the future value of an annuity formula: $$ 20,000 = R * \frac{(1 + 0.02)^{12} - 1}{0.02} $$

Step 3: Solving for the periodic payment R

Now we have an equation with one variable, R, which we can solve: $$ 20,000 = R * \frac{(1.02)^{12} - 1}{0.02} $$ First, we calculate the value of the expression: $$ \frac{(1.02)^{12} - 1}{0.02} \approx 16.0535 $$ Then, we divide both sides of the equation by this value to solve for R: $$ R = \frac{20,000}{16.0535} \approx 1,245.67 $$

Step 4: Final Answer

So, the periodic payment R required to accumulate a sum of \(20,000 over 6 years with an annual interest rate of 4% compounded semi-annually is approximately \)1,245.67.

Key Concepts

Periodic Payment Calculation

Calculating periodic payments is a core aspect of understanding annuities. When you desire a specific sum of money in the future, you need to know how much to pay periodically. This is termed the 'periodic payment'. For our case:

• We aim to accumulate $20,000 over 6 years.
• The interest rate is 4% annually, compounded semi-annually.

The future value of an annuity formula helps us find this periodic payment. First, calculate the interest per period and the total number of periods.
For this example:

• Interest per period, \(i = \frac{4}{100 \times 2} = 0.02\)
• Number of periods, \(n = 2 \times 6 = 12\)

Next, we substitute these values in the future value formula and solve for \(R\), the periodic payment.

Compound Interest Formula

Compound interest involves earning or paying interest on previously accumulated interest. This process accelerates the growth of your savings or the expense of your loans. In financial calculations, like our annuity example, the compound interest formula is a fundamental piece.
The formula for interest per period is given by:

• \(i = \frac{r}{100m}\)

With this formula:

• \(r = 4\), annual rate expressed as a percentage
• \(m = 2\), number of compounding times per year

We calculate \(i\), which we use to determine how much interest adds to each payment period, compounding as payments continue over 12 periods for the 6 years the annuity lasts.

Annuity Payment Formula

The annuity payment formula provides the structure to solve for the desired payment. An annuity is a series of payments made over a certain number of periods. The future value of an annuity formula is:

• \(FV = R \cdot \frac{(1 + i)^n - 1}{i}\)

It's essential for determining the amount you need to pay periodically to reach a specific future value, \(FV\). For fixed interest and payment periods:

• \(R\) is calculated by rearranging the formula once \(FV\), \(i\), and \(n\) are known.

In this scenario, we enter \(20,000\) as \(FV\) and solve for \(R\), finding the exact periodic payment of approximately \(1,245.67\). The result is your financial roadmap, defining each step toward reaching your financial goal.

Financial Mathematics

Financial mathematics encompasses the study of these mathematical techniques used in financial and investment decisions. Concepts such as calculating periodic payments, understanding compound interest, and using annuity formulas are crucial skills.
Mastery of these calculations helps in:

• Determining how much to save for future needs.
• Understanding the impact of interest rates over time.
• Making informed decisions about loans and investments.

In our exercise, we connect these elements to solve a specific real-world problem, aiding financial literacy and practical application. By using formulas effectively, anyone can make better financial choices and plan for the future with more certainty.