Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 75.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 75.0 when he fully retires, he will wants to have $2,506,064.00 in his retirement account. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 4.00% interest rate.

Answer 1

To determine the amount Derek must contribute to his retirement account from his 26th to his 65th birthday so that he can reach $2,506,064.00 by the end of year following his 75th birthday, we need to perform several calculations involving future values of annuities and investment growth.

Step 1: Determine the Future Value Requirement

Derek wants to have $2,506,064.00 in his retirement account at the end of the year following his 75th birthday. This means he needs this amount when he is 76 years old.

Step 2: Working Back to His 65th Birthday

Since Derek's investments will grow at an interest rate of 4.00%, we can calculate how much he needs when he turns 65. He will not make any more contributions after age 65, but the money in the account will continue to grow until he is 76.

Calculate the Growth of the Account from Age 65 to Age 76:

The time between age 65 and age 76 is 11 years. We can use the future value formula to determine how much Derek needs to have in his account when he turns 65 to reach his goal at age 76:

\[ FV = PV \times (1 + r)^n \]

Where:

• \( FV = 2,506,064.00 \)
• \( r = 0.04 \) (the interest rate)
• \( n = 11 \) years

Rearranging the formula to find \( PV \) (the amount he needs at age 65):

\[ PV = \frac{FV}{(1 + r)^n} \]

Plug in the numbers:

\[ PV = \frac{2,506,064.00}{(1 + 0.04)^{11}} = \frac{2,506,064.00}{(1.04)^{11}} = \frac{2,506,064.00}{1.48024} \approx 1,692,456.92 \]

So, Derek must have approximately $1,692,456.92 in his retirement account when he turns 65.

Step 3: Determine Contributions from Age 26 to Age 65

Next, we need to determine how much Derek should contribute annually to reach this amount by his 65th birthday. He will contribute from age 26 to age 65, which is a total of \( 65 - 26 + 1 = 40 \) years.

We can use the future value of an annuity formula for contributions:

\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]

Where:

• \( FV = 1,692,456.92 \) (the amount needed at age 65)
• \( PMT \) = annual contribution
• \( r = 0.04 \)
• \( n = 40 \)

Rearranging the formula to find \( PMT \):

\[ PMT = \frac{FV \times r}{(1 + r)^n - 1} \]

Now plug in the numbers:

\[ PMT = \frac{1,692,456.92 \times 0.04}{(1 + 0.04)^{40} - 1} \]

First, calculate \( (1 + 0.04)^{40} \):

\[ (1.04)^{40} \approx 5.21068 \]

Now calculate \( (1.04)^{40} - 1 \):

\[ 5.21068 - 1 = 4.21068 \]

Now plug everything back into the equation for \( PMT \):

\[ PMT = \frac{1,692,456.92 \times 0.04}{4.21068} \approx \frac{67,698.2768}{4.21068} \approx 16,069.31 \]

Conclusion

Derek must contribute approximately $16,069.31 annually from his 26th birthday to his 65th birthday in order to reach his retirement goal of $2,506,064.00 by the time he is 76 years old.

Answer 2

What values do I enter for N, i/y, PV, PMT, and FV to solve on a finance calculator?

Answer 3

To solve for the annual contributions (\(PMT\)) using a finance calculator, you will need to understand what each variable (N, i/y, PV, PMT, FV) represents.

Here's how to set up the calculations:

1. N (Number of Periods): This represents the total number of years over which you will be making contributions. Since Derek will contribute from age 26 to age 65, N should be 40.

2. i/y (Interest Rate per Year): This represents the annual interest rate expressed as a percentage. Since the interest rate is 4%, you should enter it as 4.

3. PV (Present Value): This is the future amount that needs to be achieved at the end of the contribution period, expressed as a negative value since it's an amount that you want to accumulate. So, for Derek's case, you'll enter -1,692,456.92.

4. PMT (Payment per Period): This is what you're solving for (the annual contribution) so you can leave this blank or enter 0.

5. FV (Future Value): Since you’re calculating how much to contribute to accumulate to $1,692,456.92, you will leave this as 0 because this is at age 65 and you will have no value owed at that time.

Summary of Inputs:

• N = 40
• i/y = 4
• PV = -1,692,456.92
• PMT = 0 (or leave blank)
• FV = 0

Solver Steps:

1. Enter these values into your financial calculator.
2. Compute the \(PMT\).

After completing these steps, your calculator should output the required annual payment (\(PMT\)), which is the amount Derek needs to contribute each year from age 26 to 65.