We need to determine the annual interest rate required for an investment to grow from $210,000 to $1,000,000 over 19 years, assuming no additional contributions.

This problem can be solved using the future value formula for compound interest:

$FV = PV \times (1 + r)^t$

Where:

• $FV$ is the future value ($1,000,000),
• $PV$ is the present value ($210,000),
• $r$ is the annual interest rate (which we need to find),
• $t$ is the time period in years (19 years).

Rearranging the formula to solve for $r$:

$r = \left( \frac{FV}{PV} \right)^{\frac{1}{t}} - 1$

Now, let's substitute the given values:

$r = \left( \frac{1,000,000}{210,000} \right)^{\frac{1}{19}} - 1$

I'll calculate this now.The annual interest rate required for the investment to grow from $210,000 to $1,000,000 over 19 years is approximately 8.56%.

Do you need further details or have any questions?

Here are 5 related questions you might consider:

1. What if the retirement period is reduced to 15 years? How would the required interest rate change?
2. How would additional yearly contributions affect the required interest rate?
3. What is the effect of inflation on the future value in today's dollars?
4. How does compounding frequency (e.g., monthly or quarterly) affect the required interest rate?
5. What if they have more than $210,000 currently—how would that change the interest rate?

Tip: The more frequently interest is compounded, the less the required nominal interest rate for the same future value.