Problem 39 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the compound interest formulas, \(A=P\left(1+\frac{r}{n}\right)^{n-1}\) and \(A=P e^{n t},\) to solve Exercises \(39-42 .\) Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 10,000\) for 5 years at an interest rate of \(5.5 \%\) if the money is a. compounded semiannually; b. compounded monthly; c. compounded continuously.

Step-by-Step Solution

Answer

The accumulated value of the investment after 5 years would be approximately \$13,659.22 if compounded semiannually, \$13,693.66 if compounded monthly, and \$13,730.90 if compounded continuously.

Step 1: Calculate Accumulated Value for a Semiannual Interest Compounding

To determine the accumulated value when the interest is compounded semiannually, we need to use the formula \(A=P(1+\frac{r}{n})^{nt}\). Here, P is the principal amount, r is the annual interest rate (expressed as a decimal), n is the number of times the interest is compounded in a year, and t is the time period in years. Substituting the given values, we get: \(A = 10000(1+\frac{0.055}{2})^{2 * 5}\)

Step 2: Calculate Accumulated Value for a Monthly Interest Compounding

Next, we determine the accumulated value when the interest is compounded monthly. The calculation involves substituting the given values into the same formula used in Step 1, adjusting the value of n to account for monthly compounding. We find: \(A = 10000(1+\frac{0.055}{12})^{12*5}\)

Step 3: Calculate Accumulated Value for Continuous Interest Compounding

Finally, we calculate the accumulated value when the interest is compounded continuously. In this case, we need to use the formula \(A=Pe^{rt}\), where P is the principal amount, r is the annual interest rate (expressed as a decimal), and t is the time period in years. Substituting the given values, we have: \(A= 10000* e^{0.055 * 5}\)

Key Concepts

Interest Rate

The interest rate is a fundamental concept in any investment's growth. It represents the cost of borrowing money or the profit from lending or investing money, typically expressed as a percentage. In a financial sense, the interest rate can significantly affect how quickly your investment grows over time.
When dealing with compound interest, the interest rate is usually described annually. For example, in the original exercise, there is an interest rate of 5.5%. This means each year, 5.5% of your investment or loan balance is added to the principal, significantly influenced by how frequently it is compounded.
Remember, a higher interest rate usually results in more money earned over a period. But it also means you might be paying more if you're on the borrowing side. Understanding how to convert annual rates into different compounding periods is crucial for effective financial planning.

Semiannual Compounding

Semiannual compounding refers to the process of interest being calculated and added to the principal twice a year. This means that every six months, any interest earned is reinvested, and future interest calculations include these earnings.
In our exercise, semiannual compounding uses the formula: \[A = P \left( 1 + \frac{r}{n} \right)^{nt}\]

• P is the principal (initial investment),
• r is the annual interest rate as a decimal,
• n is the number of compounding periods per year (2 for semiannual), and
• t is the number of years.

Every time interest is compounded, your principal grows, and so does the interest. This compounding effect accelerates the growth of the investment, particularly when the interest rate is higher or the compounding frequency is greater.

Monthly Compounding

Monthly compounding means interest is calculated and added to the balance twelve times a year - once each month. Like semiannual compounding, monthly calculates interest on the already earned interest, speeding up growth.
The formula to calculate the future investment value when compounded monthly is:\[A = P \left( 1 + \frac{r}{n} \right)^{nt}\]

• P is the initial investment amount,
• r is the annual interest rate (as a decimal),
• n is 12 (the number of months), and
• t is the time in years.

Each month, more frequent compounding periods lead to slightly higher earnings compared to less frequent compounding. This can make a noticeable difference in the accumulated value over time, especially in long-term investments.

Continuous Compounding

Continuous compounding presents a mathematical idealization where interest is added to the principal at every possible instant. In reality, banks and financial institutions cannot compound continuously, but it’s an important concept to understand because it maximizes the effect of compounding.
The formula used for continuous compounding is different:\[A = Pe^{rt}\]

• P is the principal amount (initial investment),
• r is the annual interest rate (as a decimal), and
• t is the time in years.

Each infinitesimal addition of interest builds on the previous amount, causing the most rapid growth possible. This method reveals the theoretical maximum amount of interest that can be earned within a given period and is a useful comparison point for finite compounding methods.