Question

Today is Derek’s 25th birthday. Derek has been advised that he needs to have $3,816,699.00 in his retirement account the day he turns 65. He estimates his retirement account will pay 10.00% interest. Assume he chooses not to deposit anything today. Rather he chooses to make annual deposits into the retirement account starting on his 29.00th birthday and ending on his 65th birthday. How much must those deposits be?

Answer 1

To determine how much Derek needs to deposit annually into his retirement account starting on his 29th birthday and ending on his 65th birthday, we can use the future value of an annuity formula.

1. Identify the variables:

   • Future Value (FV) = $3,816,699.00
   • Interest rate (r) = 10% or 0.10
   • Number of deposits (n) = 65 - 29 = 36 years

2. Future Value of an Annuity Formula: The future value of an annuity formula is: \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \] where:

   • \( PMT \) = annual payment (which we need to find)
   • \( r \) = interest rate per period
   • \( n \) = number of periods

3. Rearranging for PMT: We can rearrange the formula to find \( PMT \): \[ PMT = \frac{FV \times r}{(1 + r)^n - 1} \]

4. Substituting known values:

   • \( FV = 3,816,699.00 \)
   • \( r = 0.10 \)
   • \( n = 36 \)

   Therefore, substituting into the formula: \[ PMT = \frac{3,816,699 \times 0.10}{(1 + 0.10)^{36} - 1} \]

5. Calculating \((1 + 0.10)^{36}\): \[ (1.10)^{36} \approx 36.194 \]

   • Thus, \( (1.10)^{36} - 1 \approx 36.194 - 1 \approx 35.194 \)

6. Substituting back into the PMT formula: \[ PMT = \frac{3,816,699 \times 0.10}{35.194} \approx \frac{381,669.90}{35.194} \approx 10,828.099 \]

   Therefore, the annual deposit \( PMT \approx 10,828.10 \).

So, Derek must make annual deposits of approximately $10,828.10 starting on his 29th birthday and ending on his 65th birthday in order to accumulate $3,816,699 by the time he turns 65.

Answer 2

what value to enter for N, i/y, PV, PMT, and FV on the finance calculator

Answer 3

To use a financial calculator to solve for the annual payment Derek would need to make, you'll want to set the following values based on the information we have:

• N (Number of Periods): The number of deposits Derek will make from age 29 to age 65. Since he will be making deposits for 36 years, enter:

  • N = 36

• I/Y (Interest Rate per Year): The annual interest rate for the retirement account. Given that the interest rate is 10%, enter:

  • I/Y = 10

• PV (Present Value): The present value of the retirement account at the beginning of Derek's contributions. Since he is not making any initial deposits (or the present value is zero), enter:

  • PV = 0

• PMT (Payment): This is what we are solving for. Leave this value blank or enter 0, depending on the calculator.

• FV (Future Value): The amount Derek wants to have in the retirement account by the time he turns 65, which is given as $3,816,699. Enter:

  • FV = 3,816,699

Summary:

• N = 36
• I/Y = 10
• PV = 0
• PMT = ? (leave this blank to calculate)
• FV = 3,816,699

Once you have entered these values into the financial calculator, you can compute \( PMT \) to find the annual deposit amount. The calculator should provide the annual deposit amount as approximately $10,828.10.