Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 73.00. 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 73.0 when he fully retires, he will wants to have $2,537,531.00 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 5.00% interest rate.

1 answer

To find out how much Derek needs to contribute to his retirement account, we can break down the problem into several steps:

  1. Determine the time frames:

    • Derek will work and contribute from age 26 to 65, which is 39 years.
    • He will then not make any contributions from age 65 to 73, a period of 8 years.
    • After age 73, he will want to have $2,537,531 in his account one year later (age 74).
  2. Calculate the time until retirement at age 74:

    • From age 65 to age 74 (the year he wants the total): 9 years.
  3. Future value calculation:

    • To find out how much he needs to have at age 65 to grow to $2,537,531 in 9 years at 5% interest, we can use the Future Value formula:

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

    Here:

    • \( FV \) = future value ($2,537,531)
    • \( PV \) = present value (the amount at age 65)
    • \( r \) = interest rate (0.05)
    • \( n \) = number of years (9)

    Rearranging to solve for \( PV \):

    \[ PV = \frac{FV}{(1 + r)^n} \] \[ PV = \frac{2,537,531}{(1 + 0.05)^9} \]

    Calculate \( (1 + 0.05)^9 \):

    \[ (1 + 0.05)^9 \approx 1.5513 \]

    Now substitute back to find \( PV \):

    \[ PV = \frac{2,537,531}{1.5513} \approx 1,632,126.47 \]

  4. Calculate the contributions made from age 26 to 65:

    • Now we need to determine how much he needs to contribute annually from age 26 to 65 (39 years) to accumulate approximately $1,632,126.47 by age 65.

    We use the Future Value of an Annuity formula:

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

    Where:

    • \( FV \) = future value we found ($1,632,126.47)
    • \( P \) = annual contribution (what we want to find)
    • \( r \) = interest rate (0.05)
    • \( n \) = number of contributions (39 years)

    Rearranging to solve for \( P \):

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

    Substitute into the equation:

    \[ P = \frac{1,632,126.47 \times 0.05}{(1 + 0.05)^{39} - 1} \]

    Compute \( (1 + 0.05)^{39} \):

    \[ (1 + 0.05)^{39} \approx 5.4013 \]

    Now plug this back into the formula:

    \[ P = \frac{1,632,126.47 \times 0.05}{5.4013 - 1} \] \[ P = \frac{81,606.32}{4.4013} \approx 18,527.54 \]

Thus, Derek must contribute approximately $18,527.54 annually from his 26th birthday to his 65th birthday in order to reach his retirement goal by the time he turns 74.