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 need to calculate the future value for both options.

1. **Compounded Monthly:**

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.
- $P$ = principal amount (the initial amount of money).
- $r$ = annual interest rate (decimal).
- $n$ = number of times that interest is compounded per year.
- $t$ = the number of years the money is invested for.

For this case:
- $P = 14,000$
- $r = 0.0405$ (4.05%)
- $n = 12$ (monthly)
- $t = 1$

Plugging in the values:

\[
A = 14000 \left(1 + \frac{0.0405}{12}\right)^{12 \times 1}
\]
\[
A = 14000 \left(1 + 0.003375\right)^{12}
\]
\[
A = 14000 \left(1.003375\right)^{12}
\]
Calculating $A$:

\[
A \approx 14000 \times 1.041318 \approx 14678.45
\]

2. **Compounded Continuously:**

The formula for continuous compounding is:

\[
A = Pe^{rt}
\]

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

For this case:
- $P = 14,000$
- $r = 0.04$ (4.0%)
- $t = 1$

Plugging in the values:

\[
A = 14000 \times e^{0.04 \times 1}
\]
\[
A = 14000 \times e^{0.04}
\]
Calculating $e^{0.04} \approx 1.04081$:

\[
A \approx 14000 \times 1.04081 \approx 14673.34
\]

### Conclusion:
- Amount from 4.05% compounded monthly: **$14,678.45**
- Amount from 4.0% compounded continuously: **$14,673.34**

**Result:** The investment at **4.05% compounded monthly** yields a larger amount in one year.

Do you want further details on the calculations or any other questions? Here are some related questions to consider:

1. How does compounding frequency affect overall returns?
2. What would be the outcome if the investment period were longer than one year?
3. How do different interest rates compare when compounded continuously versus monthly?
4. What are some other common investment options and their compounding methods?
5. How can you calculate the effective annual rate (EAR) for different compounding methods?

**Tip:** Always consider the effect of compounding frequency when comparing different investment rates.

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. The calculations reveal $14,678.45 for monthly compounding and $14,673.34 for continuous compounding.

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