Question

You invest $$\$ 2500$$ in an account to save for college. Account 1 pays \(6 \%\) annual interest compounded quarterly. Account 2 pays \(4 \%\) annual interest compounded continuously. Which account should you choose to obtain the greater amount in 10 years? Justify your answer.

Short answer

Choose the account with the higher future amount as calculated in Steps 1 and 2.

Step-by-step solution

Calculate future amount for Account 1

The formula for compounding quarterly is \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where A is the future amount, P is the principal amount (in this case \$2500), r is the annual interest rate (6\%), t is the time in years (10 years) and n is number of times interest is compounded per year (quarterly means 4 times). Plugging in the values, \(A = 2500 \left(1+\frac{0.06}{4}\right)^{4*10}\)

Calculate future amount for Account 2

The formula for continuous compounding is \(A = Pe^{rt}\), where A is the future amount, P is the principal amount (in this case \$2500), r is the annual interest rate (4\%), and t is the time in years (10 years). Plugging in the values, we have \(A = 2500 e^{0.04 * 10}\)

Compare future amounts

To decide which account provides greater amount in 10 years, compare the future amounts calculated for both accounts in the previous steps.

Key concepts

Exponential Growth

Exponential growth is all about how things can increase rapidly over time. Imagine planting a small seed and watching it grow into a big tree over several years. That's similar to how exponential growth works in finance. When you invest money, it doesn't just stay the same; it grows at a rate that keeps getting faster.
In the context of financial accounts, compounding plays a key role. Depending on how often interest is calculated and added to your account (daily, monthly, quarterly, or continuously), the growth of your investment can vary significantly.

To understand exponential growth in your account, consider how quickly your investment multiplies. If compounded more frequently, like quarterly or continuously, your savings can grow faster because interest is calculated more often, contributing to more gains.

• Quarterly compounding: Interest is calculated four times a year and added to your principal, leading to faster growth.
• Continuous compounding: Interest is calculated and added at an infinite number of times in a year, factors of growth multiply even more.

Understanding exponential growth helps you see how small, consistent savings can lead to significant wealth over time.

Financial Mathematics

Financial mathematics is an exciting part of math used in everyday life. It helps us calculate how much money we'll have in the future based on interest rates and other factors. In our exercise, we're using two important formulas to compare accounts.
The first formula is for quarterly compounding: \[A = P \left(1 + \frac{r}{n}\right)^{nt}\]. Here,

• P is the principal or initial amount ($2500 in our case),
• r is the annual interest rate expressed as a decimal,
• n is the number of compounding periods per year (4 for quarterly),
• t is the time in years (10 years here).

The second formula is for continuous compounding: \[A = Pe^{rt}\]. This uses the constant e, an important mathematical constant approximately equal to 2.71828, representing continuous growth.
By applying these formulas, you can compare which investment option offers the greater future value. These concepts form the foundation of financial mathematics and are used daily by investors and businesses to forecast returns and make financial decisions.

Investment Strategies

When it comes to choosing the best account for your money, understanding investment strategies is crucial. Each strategy reflects different goals, such as maximizing growth or minimizing risk.
For the exercise, we compared two accounts using the principle of compounding.

• Account 1 (compounded quarterly) is typically suitable when you want to benefit from frequent interest accumulation.
• Account 2 (compounded continuously) is ideal if you want the absolute maximum growth over time due to continuous interest calculation.

Both accounts have unique advantages, depending on interest rates and how often your savings grow. While quarterly compounding provides frequent updates, continuous compounding offers a higher potential return due to constant positively adjusting growth, provided other conditions are the same.
To decide effectively, consider factors like your personal financial goals, risk tolerance, and how quickly you need your returns. Long-term investments might favor one strategy over another, based on these considerations. Understanding these strategies helps in making informed decisions about where to place your money for maximum benefit.