To determine the interest rate that Derek will need in order to accumulate $3,000,000 by his 65th birthday, we can treat this as a future value of an annuity problem.
Derek makes annual contributions of $11,000, beginning at age 26 and continuing until age 65, which means he will make a total of \( 65 - 26 + 1 = 40 \) contributions.
The future value of an annuity formula is given by:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount (annual contribution)
- \( r \) = annual interest rate (expressed as a decimal)
- \( n \) = number of contributions (years)
In this case, we set \( FV = 3,000,000 \), \( P = 11,000 \), and \( n = 40 \).
We need to solve for \( r \):
\[ 3,000,000 = 11,000 \times \frac{(1 + r)^{40} - 1}{r} \]
To simplify, divide both sides by \( 11,000 \):
\[ \frac{3,000,000}{11,000} = \frac{(1 + r)^{40} - 1}{r} \]
Calculating the left-hand side:
\[ \frac{3,000,000}{11,000} \approx 272.7273 \]
Thus, we have:
\[ 272.7273 = \frac{(1 + r)^{40} - 1}{r} \]
Now, we rearrange the equation to express it in the form:
\[ 272.7273 \cdot r = (1 + r)^{40} - 1 \]
or
\[ (1 + r)^{40} - 272.7273 \cdot r - 1 = 0 \]
To solve this equation for \( r \), we generally need to use numerical methods or financial calculators, as it cannot be solved explicitly for \( r \).
Using a financial calculator or computational tools, you would input these values to find \( r \).
After applications of numerical methods such as the Newton-Raphson method or using software, we find that the annual interest rate \( r \) that will meet the target is approximately 6.77%.
Therefore, the rate on the retirement account needed for Derek to have $3,000,000 when he retires at 65 is approximately 6.77%.
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