Question

Derek can deposit $11,000 on each birthday beginning with his 26th and ending with his 65th. What will the rate on the retirement account need to be for him to have $3,000,000 in it when he retires on his 65th birthday?

Answer 1

what vaue do i enter for N, i/y, PV, PMT, and FV on a finance calculator?

Answer 2

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%**.

Answer 3

To solve for the interest rate (i/y) on a financial calculator using the future value of an annuity formula, you would input the following values:

- **N (Number of Periods)**: This is the total number of contributions Derek will make, which is 40 (from age 26 to 65 inclusive).

- **I/Y (Interest Rate per Year)**: This is what we are solving for, so leave this blank or set it to a placeholder value.

- **PV (Present Value)**: This represents the present value of the annuity. Since Derek is only making deposits and we can consider the beginning value to be $0 (because the account starts accumulating after the first deposit), you would enter **0** here.

- **PMT (Payment Amount)**: This is the annual deposit amount that Derek will make. Enter **11,000**.

- **FV (Future Value)**: This is the desired amount in the account at retirement. Enter **3,000,000**.

### Summary of inputs for the financial calculator:
- **N (Number of deposits)**: 40
- **I/Y (Interest rate)**: (leave this blank or set a placeholder, e.g., 5.00)
- **PV (Present Value)**: 0
- **PMT (Payment)**: 11,000
- **FV (Future Value)**: 3,000,000

Once you enter these values, you can calculate "I/Y" to find the interest rate required for the account to reach $3,000,000 by age 65.