Problem 53 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find how much money should be deposited in a bank paying interest at the rate of \(8.5 \% /\) year compounded quarterly so that, at the end of \(5 \mathrm{yr}\), the accumulated amount will be $$\$ 40,000$$.

Step-by-Step Solution

Answer

To find the initial deposit that will result in a future value of $40,000 after 5 years with an annual interest rate of 8.5% compounded quarterly, we use the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\). Plugging in the given values and solving for P, we get \(P = \frac{40,000}{(1 + \frac{0.085}{4})^{20}} \approx 25,489.90\). Therefore, the initial deposit should be approximately $25,489.90.

Step 1: Understand the compound interest formula

The formula for compound interest is given by: \[A = P(1 + \frac{r}{n})^{nt}\] Where: A = the future value of the investment/loan, including interest P = the principal investment amount (the initial deposit) r = the annual interest rate (decimal) n = the number of times that interest is compounded per unit t t = the time the money is invested or borrowed for, in years In this problem, we need to find P, given A, r, n, and t.

Step 2: Convert the given data into the appropriate form

We are given the following data: A = $40,000 r = 8.5% per year (convert it to decimal by dividing by 100) n = compounded quarterly (4 times per year) t = 5 years Now, convert the annual interest rate to a decimal: r = 8.5% / 100 = 0.085

Step 3: Set up the formula with our given data

Now we can plug in the given values into the compound interest formula: \(40,000 = P(1 + \frac{0.085}{4})^{4 \times 5}\)

Step 4: Solve for the initial deposit (P)

To find the value of P, we need to first simplify the equation: \(40,000 = P(1 + \frac{0.085}{4})^{20}\) Now, we isolate P by dividing both sides of the equation by \((1 + \frac{0.085}{4})^{20}\): \(P = \frac{40,000}{(1 + \frac{0.085}{4})^{20}}\) Now, calculate the required initial deposit (P): \(P = \frac{40,000}{(1 + 0.02125)^{20}} = \frac{40,000}{1.569051938}\) \(P = 25,489.90\)

Step 5: Final answer

The initial amount that should be deposited in the bank is approximately $25,489.90.

Key Concepts

Future Value of Investment

The concept of the future value of an investment is a key aspect of financial planning and investment analysis. It refers to the value of a current asset at a specified date in the future when interest is applied over a period of time. The future value can be calculated using the compound interest formula:
\[A = P(1 + \frac{r}{n})^{nt}\]
Understanding the future value is crucial for investors as it allows them to predict how much their current investments will grow over time, helping them to make informed decisions regarding their financial goals. Compound interest, which is interest calculated on the initial principal and also on the accumulated interest of previous periods, can significantly increase the future value of an investment, especially over a long term.
To optimize comprehension, remember that compounding frequency (the value of 'n' in the formula) can enhance the accumulated amount, as the interest is being calculated more frequently. For example, if the interest is compounded quarterly rather than annually, the investment will grow faster due to more frequent application of interest.

Initial Deposit Calculation

Calculating the initial deposit, denoted by 'P', is essential when you're planning to reach a certain future value with a given interest rate and compounding frequency. This is often the case when you're saving up for a future goal, like education or retirement. The formula to find 'P' when 'A', 'r', 'n', and 't' are given is derived from rearranging the compound interest formula:
\[P = \frac{A}{(1 + \frac{r}{n})^{nt}}\]
In our exercise, we applied this to find the necessary initial deposit to reach $40,000 at the end of 5 years with an 8.5% annual interest rate and quarterly compounding. The calculation involves reverse-engineering the formula to determine how much one needs to initially invest. This type of calculation allows individuals to plan their savings accordingly, ensuring they begin with the correct deposit to achieve their desired future value.

Interest Rate Conversion

Interest rate conversion from a percentage to a decimal is a fundamental step in the process of calculating both the future value of an investment and the initial deposit required. This conversion is simple yet vital. The annual interest rate given as a percentage must be converted into a decimal form before being plugged into the formula. This is done by dividing the percentage by 100:
\[r = \frac{\text{Interest Rate in Percentage}}{100}\]
For instance, an interest rate of 8.5% becomes 0.085 in decimal. This conversion allows for more effortless calculations and avoids common errors associated with not converting percentages correctly. Accurately converting and understanding the interest rate as a decimal is essential for financial calculations related to investments and loans.