Problem 11 - 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. $$ S=100,000, r=4.5, t=20, m=6 $$

Step-by-Step Solution

Answer

The periodic payment R required to accumulate a sum of 100,000 dollars over 20 years with an interest rate of 4.5% per year compounded 6 times a year is approximately \(281.75\).

Step 1: 1. Write down the formula for the future value of an annuity

The future value of an annuity formula is as follows: $$ S = R\frac{(1+i)^{mt}-1}{i} $$ Where S is the future value (the accumulated sum), R is the periodic payment, i is the periodic interest rate, m is the number of times interest compounded per year, and t is the total number of years.

Step 2: 2. Convert the annual interest rate to a periodic interest rate

We need to convert the annual interest rate (r) to a periodic interest rate (i). The formula for the periodic interest rate is: $$ i = \frac{r}{m \times 100} $$ Plugging in the values from the exercise, we get: $$ i = \frac{4.5}{6 \times 100} = \frac{9}{1200} = 0.0075 $$

Step 3: 3. Substituting the known values into the annuity formula

Now, we will substitute the known values into the annuity formula and solve for R. $$ 100,000 = R\frac{(1 + 0.0075)^{6 \times 20}-1}{0.0075} $$

Step 4: 4. Calculate the numerator and denominator of the fraction in the formula

Calculate the values of the numerator and denominator of the fraction by performing the calculations: $$ (1 + 0.0075)^{6 \times 20} - 1 = (1.0075)^{120} - 1 \approx 2.661045 $$ Thus, the fraction becomes: $$ \frac{2.661045}{0.0075} \approx 354.806 $$

Step 5: 5. Solve for the periodic payment (R)

Divide both sides of the equation by the fraction to solve for R: $$ R = \frac{100,000}{354.806} \approx 281.75 $$ The periodic payment R required to accumulate a sum of 100,000 dollars over 20 years with an interest rate of 4.5% per year compounded 6 times a year is approximately $281.75.

Key Concepts

Periodic Payment Calculation

When planning to accumulate a certain sum in the future, determining the periodic payment is crucial. Periodic payment refers to the consistent amount of money one deposits at regular intervals over a specified period to achieve a financial goal. This amount depends on several factors like the interest rate, the compounding frequency, the time period, and the desired future sum. To calculate the periodic payment, we should know:

• The future sum (S) we want to accumulate.
• The interest rate per period (i).
• The total number of compounding periods (mt).

In the example exercise, the goal is to determine the periodic payment required to achieve $100,000 in 20 years with the specified conditions. The calculated payment in this exercise notably emphasizes how lower regular payments over a long period can still lead to a significant accumulation.

Compound Interest

Compound interest is the process where the interest earned on a sum of money in a savings or investment account is reinvested. It enables you to earn interest on interest. This powerful concept is one of the key components that drive the accumulation of funds in an annuity. The formula used in our exercise leverages compound interest by spreading out interest accrual over multiple periods.To manage compounding effectively, consider:

• Compounding frequency (m): More frequent compounding increases the final amount due to more frequent interest application.
• The interest rate conversion: Convert the annual interest rate to a periodic rate (i) using the formula \( i = \frac{r}{m \times 100} \) to accurately factor in this frequency.

In the exercise example, the annual interest rate and a compounding frequency of 6 times a year are utilized, which impacts the effectiveness of the periodic payments towards reaching $100,000.

Annuity Formula

The annuity formula is crucial in calculating both periodic payments and the future value of a series of cash flows. It combines periodic payments with compound interest to determine how much money you need to invest per interval to reach a future sum (the future value). The formula used is:\[ S = R\frac{(1+i)^{mt}-1}{i} \]Where

• \( S \) is the intended future value.
• \( R \) is the periodic payment.
• \( i \) is the periodic interest rate.
• \( mt \) is the total number of interest compounding periods.

By rearranging this formula, you can solve for \( R \), or the periodic payment. Understanding the annuity formula helps in making informed financial decisions. In our exercise, we applied this formula to find the required monthly payment to reach a cash inflow target by utilizing specific interest rate data. The process demonstrates the importance of understanding each component for accurate financial planning.