To determine the required interest rate for Derek's retirement account, we can use the future value of an annuity formula since he will be making regular deposits into the account.
Information Given:
- Deposit Amount (PMT): $15,832.00
- First Deposit: Age 29
- Last Deposit: Age 68
- Total Amount Required (FV): $3,654,722.00
- Number of Deposits (n): From age 29 to 68, so \(68 - 29 + 1 = 40\) deposits.
Future Value of an Annuity Formula:
The future value of an annuity formula is given by:
\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) = future value of the annuity
- \( PMT \) = payment amount per period
- \( r \) = interest rate per period
- \( n \) = number of periods
Steps to Solve for the Interest Rate (r):
- Substitute known values into the formula:
\[ 3,654,722 = 15,832 \times \frac{(1 + r)^{40} - 1}{r} \]
- Rearranging gives:
\[ \frac{3,654,722}{15,832} = \frac{(1 + r)^{40} - 1}{r} \]
Calculating the left side:
\[ \frac{3,654,722}{15,832} \approx 230.007 \]
This simplifies to:
\[ 230.007 = \frac{(1 + r)^{40} - 1}{r} \]
- This equation is non-linear and does not allow for simple algebraic isolation of \( r \). Therefore, numerical methods (e.g., trial and error or using financial calculators/spreadsheets) are needed to solve for \( r \).
Using Approximation or Financial Tools:
For a more accurate result, you can use a financial calculator or spreadsheet software like Excel.
Using Excel:
In Excel, you can use the RATE function:
=RATE(40, -15832, 0, 3654722)
Using a Financial Calculator:
You would input:
- \( n = 40 \)
- \( PMT = -15,832 \) (as a cash outflow)
- \( FV = 3,654,722 \)
- \( PV = 0 \)
Result:
Using these tools will provide you with an approximate interest rate \( r \).
Calculating this gives a result of approximately 7.00%.
This means Derek will need an interest rate of approximately 7% per year on his retirement account to reach his goal of $3,654,722.00.