Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the accumulated amount \(A\) if the principal \(P\) is invested at the interest rate of \(r /\) year for \(t\) yr. $$ P=\$ 150,000, r=14 \%, t=4, \text { compounded monthly } $$

Step-by-Step Solution

Answer

The accumulated amount $A$ after 4 years is approximately $218,493.05 when the principal $P$ is $150,000, the interest rate $r$ is 14%, compounded monthly. We used the compound interest formula \(A = P(1 + \frac{r}{n})^{nt}\) with given values and calculated the amount.

Step 1: Convert given values to the correct format

We are given the values: P = $150,000 r = 14% (decimal format: 0.14) t = 4 years n = compounded monthly (12 times a year) Now, let's use these values in the compound interest formula.

Step 2: Plug the values in the compound interest formula

Now we need to use the given values in the formula: $$ A = P(1 + \frac{r}{n})^{nt} $$ Plugging in the values, we get: $$ A = 150000(1 + \frac{0.14}{12})^{(12)(4)} $$

Step 3: Calculate the accumulated amount

Let's now simplify the expression by performing the calculations: $$ A = 150000(1 + \frac{0.14}{12})^{(12)(4)} $$ $$ A = 150000(1 + 0.01166667)^{48} $$ $$ A = 150000(1.01166667)^{48} $$ $$ A \approx 218493.05 $$ Therefore, the accumulated amount A after 4 years is approximately $218,493.05.

Key Concepts

Accumulated Amount Calculation

Understanding how to calculate the accumulated amount of an investment is fundamental in planning for financial growth. The accumulated amount is the sum of the principal and the interest earned over a certain period, considering that the interest is compounded at specific intervals. For our example, where a principal amount of \(150,000 is invested at an annual interest rate of 14% for 4 years, compounded monthly, the formula used for computing the accumulated amount is expressed as \( A = P(1 + \frac{r}{n})^{nt} \).

By decomposing the elements, we identify \( P \) as the principal amount, \( r \) as the annual interest rate in decimal form, \( n \) as the number of times the interest is compounded per year, and \( t \) as the time in years. It's important to convert the percentage interest rate to a decimal by dividing it by 100 and to note the frequency of compounding to accurately calculate the total amount accumulated after the specified time. In our example, the calculated accumulated amount is approximately \)218,493.05 after 4 years.

Financial Mathematics

Financial mathematics is a field of applied mathematics that analyzes financial markets and tools to help individuals and businesses make informed financial decisions. It encompasses concepts like the calculation of accumulated amounts, present and future values, annuities, and mortgage payments. Compound interest, a key aspect of financial mathematics, represents the growth of an investment or loan amount over time, with interest calculated on the initial principal as well as the accumulated interest.

In our textbook example, using financial mathematics enables the calculation of the future value of an investment given specifics like the principal, interest rate, time, and compounding frequency. Grasping these concepts also aids in understanding the broader financial implications, such as the impact of different interest rates and compounding frequencies on investments or debts.

Interest Rates

Interest rates are central to financial mathematics and influence a wide range of economic decisions. They represent the cost of borrowing money or the benefit of saving it; hence, they can affect consumer spending, business investment, and the overall economic health. The interest rate, commonly expressed as a percentage, can be calculated as simple or compound interest, with compound interest being the more accurate reflection of how most financial systems operate.

In the compounding context, the rate is applied repeatedly over the investment period, leading to exponential growth. When dealing with compound interest, like in our example with a 14% annual rate compounded monthly, the way the rate interacts with the compounding frequency significantly alters the final accumulated amount. The precise calculation of this impact requires attention to detail to avoid underestimating or overestimating the influence of the interest rate on the accumulated amount.

Time Value of Money

The time value of money is a concept suggesting that a sum of money is worth more now than the same amount in the future, due to its earning potential. It's the foundation for understanding the compound interest formula, as the formula essentially calculates the future value of money invested over a period at a given rate of return. This concept implies that when money presents the opportunity to earn interest, there's an intrinsic benefit in having that money immediately rather than later.

Using our compound interest example, if you were to invest \(150,000 today at an interest rate of 14% compounded monthly for four years, the future sum's increase reflects the earning potential realized over time. Had the \)150,000 simply been kept without earning interest, its purchasing power could also potentially decrease due to inflation. Understanding the time value of money guides investors on the importance of putting their money to work at the earliest opportunity and highlights the benefits of compound interest in financial growth.