Suppose that you have $ 14 comma 000 to invest. Which of the two rates would yield the larger amount in 1 year​: 4.05​% compounded monthly or 4.0 % compounded​ continuously?


To determine which investment yields a larger amount in one year, we can calculate the future value for both scenarios.

### 1. Compounded Monthly (4.05%):
The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ = the amount of money accumulated after n years, including interest.
- $ P $ = the principal amount ($14,000).
- $ r $ = annual interest rate (decimal).
- $ n $ = number of times that interest is compounded per year.
- $ t $ = number of years the money is invested or borrowed.

For this case:
- $ P = 14000 $
- $ r = 0.0405 $
- $ n = 12 $
- $ t = 1 $

Substituting the values in:

\[
A = 14000 \left(1 + \frac{0.0405}{12}\right)^{12 \times 1}
\]

Calculating:

\[
A = 14000 \left(1 + 0.003375\right)^{12} \approx 14000 \left(1.042155\right) \approx 14688.17
\]

### 2. Compounded Continuously (4.0%):
The formula for continuous compounding is:

\[
A = Pe^{rt}
\]

Where:
- $ e $ is the base of the natural logarithm (approximately 2.71828).

For this case:
- $ P = 14000 $
- $ r = 0.04 $
- $ t = 1 $

Substituting the values in:

\[
A = 14000 \times e^{0.04 \times 1} \approx 14000 \times e^{0.04} \approx 14000 \times 1.04081 \approx 14571.34
\]

### Conclusion:
- **Monthly Compounding at 4.05% yields approximately:** $14,688.17
- **Continuous Compounding at 4.0% yields approximately:** $14,571.34

Thus, **4.05% compounded monthly yields the larger amount after one year.**

Would you like further details or have any questions? Here are some related questions you might consider:

1. How does the frequency of compounding affect the overall return?
2. What would be the future values for different principal amounts at these rates?
3. How would a change in the interest rate affect the outcome?
4. What is the impact of inflation on the real return of these investments?
5. How do these investments compare to other investment options like stocks or bonds?

**Tip:** Always compare the effective annual rate (EAR) when evaluating different compounding frequencies.

Suppose that you have $14,000 to invest. Which of the two rates would yield the larger amount in 1 year: 4.05% compounded monthly or 4.0% compounded continuously?

Discover which investment yields more in one year: $14,000 at 4.05% compounded monthly or 4.0% compounded continuously. Learn the calculations for both scenarios and see the results.

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