Investment Showdown: Benjamin Vs. Zachary

by Alex Johnson 42 views

Hey everyone! Today, we're diving into a fascinating financial puzzle. We're comparing two investment scenarios to see who comes out on top. Specifically, we're looking at Benjamin's investment versus Zachary's investment. It's a classic example of how different interest rates and compounding periods can impact your returns. So, grab your calculators (or your thinking caps!), and let's break it down. We'll find out who's investment grows faster and by how much, and we'll compare the results of the two different strategies. This is a great way to understand the power of compound interest and how it can help you reach your financial goals faster.

The Investment Details: Benjamin's Investment

Let's start with Benjamin's investment. Benjamin was the first investor, and he made a significant move by investing a cool $99,000. He chose an account that offered an interest rate of 7.5% per year. Now, the special thing about Benjamin's account is that the interest is compounded continuously. What does that mean? It means the interest is constantly being added to the principal, and then it earns more interest. Think of it like a snowball effect – the bigger the snowball gets, the faster it rolls down the hill. This continuous compounding is powerful because it maximizes the growth of Benjamin's investment. In other words, because of the continuously compounded interest the amount will grow larger than if the interest was compounded periodically. The result of this can be seen over longer periods of time. The amount of the investment is also very significant, which is why it will be worth investigating. The interest rate is also a significant amount, making this a very profitable investment. The continuous compounding means that the return will be better, the more often the interest is added to the principal.

The formula for continuous compound interest is: A = Pe^(rt), where:

  • A = the amount of money accumulated after n years, including interest.
  • P = principal amount (the initial amount of money).
  • r = annual interest rate (as a decimal).
  • t = the time the money is invested or borrowed for, in years.
  • e = Euler's number (approximately 2.71828).

Let's plug in Benjamin's numbers: P = $99,000, r = 0.075 (7.5% as a decimal).

The Investment Details: Zachary's Investment

Now, let's turn our attention to Zachary's investment. Zachary also invested $99,000, the same amount as Benjamin. However, Zachary's investment account offered a slightly higher interest rate of 7.875% per year. But the key difference is how the interest is compounded. Zachary's interest is compounded quarterly. This means that the interest is calculated and added to the principal four times a year. While the difference in the interest rate might seem small, the compounding frequency can significantly affect the investment's growth over time. Since this is only quarterly, it will be added at a lesser rate than Benjamin's continuous compounding. Nevertheless, Zachary's investment is still a great choice. Zachary's investment is also significant, and over time it will increase. The difference is the time it takes to see the investment's return. Zachary's investment, since the interest is compounded quarterly will take longer to realize the same profit as Benjamin's investment. We are talking about two very smart and strategic investors, each taking on a different approach to their investment.

The formula for compound interest is: A = P (1 + r/n)^(nt), where:

  • A = the future value of the investment/loan, including interest
  • P = the principal investment amount (the initial deposit or loan amount)
  • r = the annual interest rate (as a decimal)
  • n = the number of times that interest is compounded per year
  • t = the number of years the money is invested or borrowed for.

Let's plug in Zachary's numbers: P = $99,000, r = 0.07875 (7.875% as a decimal), n = 4 (compounded quarterly).

Calculating the Time Difference

To figure out how much longer it takes for one investment to reach a certain amount compared to the other, we need to solve for t (time). Since we are not given a specific target amount, we need to compare how long it takes for each investment to grow to the same amount. Therefore, we need to find the time it takes for a certain amount to reach a certain level. To simplify things and have a common point of reference, let's calculate the time it takes for each investment to double. Let's calculate the doubling time for both Benjamin and Zachary's investment. The result will give us the time it takes for their investment to double in value. We can then compare the results to see which account will double first. Keep in mind, this is just a way to illustrate the difference between the two investment options. Keep in mind that continuous compounding is generally more beneficial than quarterly compounding because it accounts for the interest growth more frequently.

For Benjamin: We want to find the time it takes for the investment to double. So, A = $198,000 (double the initial investment), P = $99,000, and r = 0.075. Using the formula A = Pe^(rt), we can solve for t: $198,000 = $99,000 * e^(0.075t).

For Zachary: We also want to find the time it takes for the investment to double. So, A = $198,000 (double the initial investment), P = $99,000, r = 0.07875, and n = 4. Using the formula A = P (1 + r/n)^(nt), we can solve for t: $198,000 = $99,000 * (1 + 0.07875/4)^(4t).

Solving for Time: Benjamin's Investment

Let's calculate the time it takes for Benjamin's investment to double. First, divide both sides of the equation by $99,000: 2 = e^(0.075t). Next, take the natural logarithm (ln) of both sides: ln(2) = 0.075t. Finally, divide by 0.075 to solve for t: t = ln(2) / 0.075.

Using a calculator, t ≈ 9.24 years. This means Benjamin's investment doubles in about 9.24 years. We can now compare the result to Zachary's investment. This gives us a reference point to compare the two investments.

Solving for Time: Zachary's Investment

Let's calculate the time it takes for Zachary's investment to double. First, divide both sides of the equation by $99,000: 2 = (1 + 0.07875/4)^(4t). Simplify: 2 = (1.0196875)^(4t). Take the natural logarithm (ln) of both sides: ln(2) = 4t * ln(1.0196875). Now, divide by 4 * ln(1.0196875) to solve for t: t = ln(2) / (4 * ln(1.0196875)).

Using a calculator, t ≈ 8.80 years. This means Zachary's investment doubles in about 8.80 years. Now we know, how long it takes for both investments to double. Let's compare the results.

Comparing the Results

Now, let's compare the results. Benjamin's investment doubles in approximately 9.24 years, while Zachary's investment doubles in approximately 8.80 years. This means Zachary's investment grows faster, even though Benjamin's investment has a slightly lower interest rate. Now, let's calculate how much longer it would take for Benjamin's investment to double. To do this, we need to subtract Zachary's time from Benjamin's time: 9.24 years - 8.80 years = 0.44 years. So, to the nearest hundredth of a year, it takes Benjamin's investment 0.44 years longer to double.

Conclusion: The Investment Showdown Winner

In this investment showdown, Zachary's investment comes out slightly ahead in terms of doubling time, despite the lower interest rate. The difference may not seem significant at first, but over longer periods, the slight advantage in compounding frequency can lead to a larger overall return. While continuous compounding is powerful, the slightly higher interest rate on Zachary's investment, along with the quarterly compounding, allowed Zachary's investment to grow faster. This demonstrates how crucial it is to consider both the interest rate and the compounding frequency when choosing an investment. It’s also a great reminder of the power of compound interest and how it can help you reach your financial goals faster. Remember to carefully consider the interest rate, the compounding frequency, and your own investment goals when making financial decisions. Thanks for joining me on this investment journey! Now, go out there and make those investments work for you!

For more information on compound interest and investment strategies, check out these resources: