Problem 9 - Calculus Concepts

Question

Investment You have \(\$ 1000\) to invest, and you have two options: Option A: \(4.725 \%\) compounded semiannully Option B: \(4.675 \%\) compounded continuously. a. Calculate the annual percentage yield for each option. Which is the better option? b. Calculate the future value of each investment after 2 years and after 5 years. Does your choice of option depend on the number of years you leave the money invested?

Step-by-Step Solution

Answer

Option A has a higher APY and better future value for 2 years and 5 years, making it the better option overall.

Step 1: Understanding the Problem

We need to calculate the annual percentage yield (APY) for both options and then find out the future value of the investments after 2 and 5 years. This will help us determine which investment is better.

Step 1: Calculate APY for Option A

Option A compounds semiannually at a rate of 4.725%. First, determine the effective rate by using the formula for compound interest: \[(1 + \frac{0.04725}{2})^2 - 1\]Calculate the result to get the APY.

Step 2: Calculate APY for Option B

Option B uses continuous compounding with a rate of 4.675%. The formula for continuous compounding is: \[e^r - 1\]where \( r = 0.04675 \). Substitute the value to find the APY.

Step 3: Compare APYs to Determine the Better Option

Compare the APYs calculated from Step 1 and Step 2. The option with the higher APY is considered better for maximizing returns.

Step 4: Calculate Future Value for Option A after 2 Years

Use the compound interest formula for semiannual compounding: \[FV = 1000 \times (1 + \frac{0.04725}{2})^{2 \times 2}\]Calculate the future value after 2 years.

Step 5: Calculate Future Value for Option B after 2 Years

Use the formula for continuous compounding: \[FV = 1000 \times e^{0.04675 \times 2}\]Compute the future value after 2 years.

Step 6: Calculate Future Value for Option A after 5 Years

Use the same formula as in Step 4 but adjust for 5 years: \[FV = 1000 \times (1 + \frac{0.04725}{2})^{2 \times 5}\]Calculate this value.

Step 7: Calculate Future Value for Option B after 5 Years

Use the continuous compounding formula as in Step 5 but for 5 years: \[FV = 1000 \times e^{0.04675 \times 5}\]Compute the future value.

Step 9: Final Decision

After calculating all future values, compare the results for 2 years and 5 years. Determine if the initially better option (based on APY) still holds or if durations influence which option is optimal.

Key Concepts

Continuous Compounding

Continuous compounding is a method of calculating interest where the frequency of compounding is infinite. This means that interest is being added to the principal constantly and thereby earns interest on itself. It's different from periodic compounding methods, such as annually, semiannually, or quarterly where interest is added at specific intervals.
The formula for continuous compounding is:

• Future Value (FV) = Principal (P) × \(e^{rt}\)

where:

• \(P\) is the initial principal (amount of money).
• \(r\) is the annual interest rate (expressed as a decimal).
• \(t\) is the time in years.
• \(e\) is the base of the natural logarithm, approximately 2.71828.

This method maximizes the interest accrued over time. In the context of our exercise, Option B with its continuous compounding at 4.675% is an attractive choice because it ensures uninterrupted growth of your investment.

Annual Percentage Yield (APY)

The Annual Percentage Yield (APY) represents the real rate of return on an investment over a one-year period, accounting for the effects of compounding interest. Unlike the nominal rate, which simply states the annual interest rate without compounding, APY provides investors with a clear picture of how much they'll actually earn.

• For periodic compounding (like semiannual), the APY is calculated as:
  \((1 + \frac{r}{n})^n - 1\)
  where \(n\) is the number of compounding periods per year.
• For continuous compounding, the APY formula becomes:
  \(e^r - 1\).

In our exercise, Option A (semiannual compounding) and Option B (continuous compounding) have to be compared using APY to understand which gives a better yield. APY is critical because it reflects how an investment grows due to compounding over time.

Future Value

The future value of an investment represents the amount of money an investment will grow to over a period of time at a specific interest rate. This concept helps investors project how much they might earn in the future.

• For semiannual compounding, use:
  \(FV = P \times (1 + \frac{r}{n})^{nt}\)
  where \(n\) is the number of times interest is compounded per year.
• For continuous compounding, apply:
  \(FV = P \times e^{rt}\).

In this scenario, we calculate the future value of both investment options over periods of 2 years and 5 years. The choice between semiannual and continuous compounding could affect which option generates a higher return based on the investment period. By evaluating both options, we can identify which provides the optimal future value depending on investment duration.