Question

An investment of the same amount is made in three different accounts with the same interest rate. One account uses simple interest annually, another account uses compounded interest semi-annually, and the last account uses continuously compounded interest annually. How do the investments compare over time?

Answer 1

To compare the three different accounts with the same principal investment amount and interest rate, we need to understand how each type of interest works: simple interest, compound interest (semi-annually), and continuous compound interest.

Let’s denote:

• \( P \): the principal amount (initial investment)
• \( r \): the annual nominal interest rate (as a decimal)
• \( t \): the time in years
• \( A_s \): the amount in the simple interest account
• \( A_c \): the amount in the compounded interest account (semi-annually)
• \( A_e \): the amount in the continuously compounded account

1. Simple Interest

For simple interest, the formula for the total amount after time \( t \) is given by: \[ A_s = P(1 + rt) \]

2. Compounded Interest (Semi-Annually)

For compound interest that is compounded semi-annually, the formula is: \[ A_c = P\left(1 + \frac{r}{2}\right)^{2t} \] This formula accounts for the fact that the interest is added twice a year.

3. Continuous Compounding

For continuously compounded interest, the formula is: \[ A_e = Pe^{rt} \] where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Comparison Over Time

To see how the investments compare over time, we can write each formula clearly. For a given investment amount \( P \) and interest rate \( r \):

1. Simple Interest: \[ A_s = P(1 + rt) \]

2. Compounded Interest (Semi-Annually): \[ A_c = P\left(1 + \frac{r}{2}\right)^{2t} \]

3. Continuous Compounding: \[ A_e = Pe^{rt} \]

Key Observations

• At \( t = 0 \): All three accounts will just have the principal amount \( P \).
• As \( t \) increases:
  • The simple interest grows linearly, which means it will always lag behind the other two accounts.
  • The compounded interest will grow more rapidly than the simple interest due to the compounding effect, but it will grow slower than continuously compounded interest.
  • The continuously compounded interest will always yield the highest amount because it considers interest compounding at every possible instant.

Conclusion

Over time, you can expect the following order of investment growth:

1. Simple Interest: Lowest growth.
2. Compounded Interest (Semi-Annually): Moderate growth, outperforming simple interest.
3. Continuous Compounding: Highest growth, outperforming both simple and semi-annual compounding.

As time goes on, the differences in growth between these account types will become more pronounced, particularly favoring the accounts that utilize some form of compounding.