Problem 10 - Calculus Concepts

Question

Investment You have \(\$ 5000\) to invest, and you have two options: Option A: \(4.9 \%\) compounded monthly Option B: \(4.8 \%\) compounded continuously. a. Calculate the annual percentage yield for each option. Which is the better option? b. By how much would the two investments differ after 3.5 years?

Step-by-Step Solution

Answer

Option A is better with a higher APY. After 3.5 years, Option A will be \$10.56 ahead.

Step 1: Calculate APY for Option A

The annual percentage yield (APY) for compounded interest is calculated using the formula:\[ APY = \left(1 + \frac{r}{n}\right)^n - 1 \]where \( r \) is the nominal interest rate and \( n \) is the number of compounding periods per year. For Option A, \( r = 0.049 \) and \( n = 12 \):\[ APY = \left(1 + \frac{0.049}{12}\right)^{12} - 1 \approx 0.050028 \approx 5.0028\% \]

Step 2: Calculate APY for Option B

For continuous compounding, the APY is calculated using the formula:\[ APY = e^r - 1 \]where \( r \) is the nominal interest rate. For Option B, \( r = 0.048 \):\[ APY = e^{0.048} - 1 \approx 0.049015 \approx 4.9015\% \]

Step 3: Compare APY Values

Compare the two APYs calculated:- Option A APY: \(5.0028\%\)- Option B APY: \(4.9015\%\)Since Option A has a higher APY, it is the better investment option based on yield.

Step 4: Calculate Future Value for Option A

Use the formula for compound interest to calculate the future value:\[ FV = P\left(1 + \frac{r}{n}\right)^{nt} \]where \( P = 5000 \), \( r = 0.049 \), \( n = 12 \), and \( t = 3.5 \):\[ FV = 5000\left(1 + \frac{0.049}{12}\right)^{12 \times 3.5} \approx 5944.48 \]

Step 5: Calculate Future Value for Option B

Use the formula for continuous compounding to calculate the future value:\[ FV = Pe^{rt} \]where \( P = 5000 \), \( r = 0.048 \), and \( t = 3.5 \):\[ FV = 5000e^{0.048 \times 3.5} \approx 5933.92 \]

Step 6: Calculate Difference in Future Values

Subtract the future value of Option B from the future value of Option A:\[ \text{Difference} = 5944.48 - 5933.92 = 10.56 \]The investments differ by approximately \$10.56 after 3.5 years.

Key Concepts

Compounded Interest

Compounded interest is a powerful concept in investment calculations. It means that the interest earned on an investment is reinvested to earn more interest. This cycle continues as long as the investment is held, which can lead to exponential growth of the investment over time.

There are different compounding periods, such as daily, monthly, or annually. This period determines how often the interest is added to the principal amount. The more frequently compounding occurs, the greater the total interest and future value of the investment can become. For example, in Option A from the exercise above, the interest is compounded monthly. This means that, every month, the interest earned is added to the principal, allowing the next month's interest to be calculated on a larger amount.

Understanding compounded interest is key to making informed investment decisions as it directly impacts the growth of your investment.

Annual Percentage Yield (APY)

The Annual Percentage Yield or APY is an essential metric in finance. It represents the annual interest earned on an investment, including the effect of compounding.

* **Formula for Compounded Interest:** The APY for investments with periodic compounding can be calculated using \[ APY = \left(1 + \frac{r}{n}\right)^n - 1 \] Here, \( r \) is the nominal interest rate, and \( n \) is the number of times the interest compounds per year.* **Continuous Compounding Formula:** For continuously compounded investments, the formula changes to \[ APY = e^r - 1 \] where \( e \) is the mathematical constant approximately equal to 2.71828.

In our exercise, Option A has an APY of roughly 5.0028%, higher than Option B’s at 4.9015%. This makes Option A the preferable choice if one is solely considering yield.

Continuous Compounding

Continuous compounding is a fascinating concept that can maximize the interest earned on an investment.

In contrast to periodic compounding where interest is credited at regular intervals, continuous compounding assumes that interest is constantly being added to the principal, mathematically expressed using the exponential function. The formula to calculate the future value using continuous compounding is \[ FV = Pe^{rt} \].Here, \( P \) represents the initial investment, \( r \) the annual interest rate, and \( t \) the time in years.

Although continuous compounding is theoretical and not typically possible in practical terms, it serves as an idealization for comparing investment efficiency. In our example, comparing the continuously compounded option revealed that it is slightly less beneficial than the investment compounded monthly over 3.5 years.

Future Value Calculation

Calculating the future value (FV) of an investment is crucial because it allows investors to determine how much their investment will be worth in the future.

* **Formula for Periodic Compounding:** The formula used for calculating the future value with periodic compounding is \[ FV = P\left(1 + \frac{r}{n}\right)^{nt} \]. * \( P \) is the initial principal balance. * \( r \) is the annual interest rate. * \( n \) is the number of compounding periods per year. * \( t \) is the time in years.* **Formula for Continuous Compounding:** When the interest compounds continuously, use \[ FV = Pe^{rt} \].

As seen in the exercise, after 3.5 years, the future value of the investment with monthly compounding was \(5944.48, slightly more than the \)5933.92 calculated for continuous compounding. Understanding future value calculations helps investors to choose the option that best suits their financial goals.