Suppose an investor wants to have $10 million in 40 years in the retirement account. Suppose the stock market will grow 6% compounded continuously and the investor contributes a fixed amount P to the stock market at the end of each month for 40 years. Find P .

To find the fixed monthly contribution $ P $ needed for the investor to have $10 million in 40 years with a continuous compound interest rate of 6%, we can use the formula for continuous compounding combined with regular contributions:

\[
A = P \cdot \left( \frac{e^{rt} - 1}{e^{r/n} - 1} \right)
\]

Where:
- $ A $ is the final amount desired ($10,000,000),
- $ P $ is the monthly contribution (what we're solving for),
- $ r $ is the continuous compound interest rate (0.06 for 6%),
- $ t $ is the time in years (40 years),
- $ n $ is the number of contributions per year (12 since contributions are monthly),
- $ e $ is Euler's number (approximately 2.71828).

### Steps:

1. First, calculate the exponential growth factor:
\[
e^{rt} = e^{0.06 \times 40}
\]

2. Next, we plug these values into the formula to solve for $ P $.

Let me calculate this.The fixed monthly contribution $ P $ required for the investor to reach $10 million in 40 years, with a 6% continuously compounded interest, is approximately **$5000.93**.

Would you like further details or have any questions about this? Here are some related questions to deepen your understanding:

1. How would the monthly contribution change if the interest rate increased to 7%?
2. What is the effect of contributing bi-weekly instead of monthly?
3. How much would the investor have in 20 years with the same monthly contribution?
4. What if the investor wanted to reach $5 million instead of $10 million? How would $ P $ change?
5. How does continuous compounding differ from annual compounding in terms of growth?

**Tip:** When contributing regularly, even small increases in interest rates significantly impact long-term growth due to the compounding effect.

Suppose an investor wants to have $10 million in 40 years in the retirement account. Suppose the stock market will grow 6% compounded continuously and the investor contributes a fixed amount P to the stock market at the end of each month for 40 years. Find P.

Calculate the fixed monthly contribution required for an investor to reach $10 million in 40 years with a 6% continuously compounded interest rate. This problem uses the continuous compounding formula and includes exponential growth with regular monthly contributions. Ideal for financial mathematics students and long-term investment planning.

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