Problem 3 - Applied Calculus

Question

Suppose \$ 1000\( is invested in an account paying interest at a rate of \)5.5 \%$ per year. How much is in the account after 8 years if the interest is compounded (a) Annually? (b) Continuously?

Step-by-Step Solution

Answer

(a) $1484.81; (b) $1552.71.

Step 1: Understanding Compound Interest Formula for Annual Compounding

For annual compounding, the formula to calculate the future value of an investment is given by \( A = P(1 + r)^n \), where: \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (\$1000), \( r \) is the annual interest rate (5.5\% or 0.055), and \( n \) is the number of years (8).

Step 2: Calculating the Future Value (Annual Compounding)

Substitute the given values into the compound interest formula: \( A = 1000(1 + 0.055)^8 \). Calculate the expression: \( A = 1000(1.055)^8 \). Using a calculator, \( (1.055)^8 \approx 1.4848127 \). Thus, \( A = 1000 \times 1.4848127 \approx 1484.81 \).

Step 3: Understanding Continuous Compounding Formula

For continuous compounding, the future value is calculated using the formula \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated, \( P \) is the initial principal balance, \( r \) is the rate of interest per period, \( t \) is the number of periods the money is invested for, and \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 4: Calculating the Future Value (Continuous Compounding)

Substitute the values into the continuous compounding formula: \( A = 1000e^{0.055 \times 8} \). Calculate the exponent: \( 0.055 \times 8 = 0.44 \). Now find \( A = 1000e^{0.44} \). Using a calculator, \( e^{0.44} \approx 1.552707 \). Thus, \( A = 1000 \times 1.552707 \approx 1552.71 \).

Key Concepts

Annual Compounding

When you invest money with annual compounding, the interest earns on the initial amount at specific intervals—every year, in this case. Imagine putting your money in a box. At the end of every year, the amount increases as interest is added, like putting more cash in the box with the amount from the previous year.

• The formula to calculate annual compounding is:
  \( A = P(1 + r)^n \) where:
• \(A\) = Future value after interest is added
• \(P\) = Initial investment or principal
• \(r\) = Annual interest rate (in decimal form)
• \(n\) = Number of years

Applying the numbers from our example (\( P = 1000, r = 0.055, = 8 \)), you end up with about \$1,484.81 after 8 years. This way, your money grows steadily as each year's interest builds on the last.

Continuous Compounding

Continuous compounding is a bit like magic; at least, it can feel that way! Instead of calculating interest at specific intervals, continuous compounding grows your money continuously. Imagine your money growing every moment, every second; it is as if the interest is being calculated and added an infinite number of times within a year.

• Its formula looks like this:
  \( A = Pe^{rt} \), where:
• \(A\) = Future value with continuous interest
• \(P\) = Initial principal balance
• \(r\) = Interest rate
• \(t\) = Time, or investment length
• \(e\) = A constant approximated to 2.71828

For our scenario, this means plugging in \( P = 1000 \), \( r = 0.055 \), and \( t = 8 \). The future value grows to around \$1,552.71. This process, albeit more complex, results in slightly higher returns than annual compounding.

Interest Rate

The interest rate is the magic percentage that dictates how much your investment will grow over a period of time. Whether it’s compounding annually or continuously, the rate serves as the engine for your money’s growth.

• In our example, the annual interest rate is 5.5%, which is expressed as 0.055 in decimal form.
• It dictates how quickly and vastly your investment flourishes.

A higher interest rate means your money can grow faster, but it's essential to consider factors such as risk and market stability when seeking higher rates. In this scenario, 5.5% is a moderate rate, reflecting reasonable growth over time, demonstrated by how both annual and continuous compounding increase your initial investment.

Future Value

Future Value (FV) defines how much an investment is worth after a certain period, considering the effects of interest rates and compounding. Think of it as the end-value of your invested money, factoring in the interest accumulated over time.

• This concept is applicable for any investment or savings that accumulate interest over time.
• Using either annual or continuous compounding, future value showcases the monetary growth and is instrumental in planning financial goals.

In our given example, the future value if compounded annually comes out to \\(1,484.81, while for continuous compounding, it's around \\)1,552.71. This illustrates how the compounding method used can significantly impact the final outcome. Understanding and calculating future value help you make informed saving and investment decisions.