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

Question

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 begin to make annual withdrawals of $180,752.00 from his retirement account until he turns 91.00. 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 6.00% interest rate.

Answer 1

To solve for the annual contributions Derek needs to make to his retirement account from his 26th birthday to his 65th birthday, we will break down the question into two parts: the amount needed at retirement to fund his withdrawals and the contributions required to reach that amount.

Part 1: Determine the Future Value Needed at Retirement

1. Withdrawal Amount: Derek will withdraw $180,752 annually for 16 years (from age 76 to 91).
2. Interest Rate (i/y): 6% (0.06 as a decimal).
3. Number of Withdrawals (N): 16 years (from age 76 to 91).
4. Future Value (FV): $0 (since he will fully deplete the account).
5. Payment (PMT): -180752 (as it is a cash outflow).

Using these values, we will first calculate the Present Value (PV) of the annuity at age 75:

\[ PV = PMT \times \frac{(1 - (1 + r)^{-n})}{r} \] Substituting in the values:

• PMT = -180752
• r = 0.06
• n = 16

Using the finance calculator to find PV:

• N = 16
• PMT = -180752
• i/y = 6
• FV = 0

This calculation will give you the Present Value (PV) at age 75 of the withdrawals that need to be funded.

Part 2: Determine the Annual Contributions from Age 26 to 65

Now, we know the amount of money Derek will need at the end of the 65th year (age 65) to fund the withdrawals starting at age 76.

1. Let's call the future value we calculated from the first part as \( FV \) (the amount needed at age 65).
2. Interest Rate (i/y): 6% (0.06 as a decimal).
3. Number of Contributions (N): 40 years (from age 26 to 65).
4. Payment (PMT): Unknown (what we are solving for).
5. Present Value (PV): 0 (since he starts saving from nothing).

Now we calculate the amount he needs to contribute annually in order to accumulate the future value \( FV \) by age 65.

Using the future value of an annuity formula: \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]

Rearranging to solve for PMT gives us: \[ PMT = \frac{FV \cdot r}{(1 + r)^n - 1} \]

Using the finance calculator to find PMT (the amount he needs to contribute each year):

1. N = 40
2. i/y = 6
3. PV = 0
4. FV = \( FV \) (the value computed in the first part)

So, once you've calculated \( FV \) from the first part, you can enter those values into the calculator to find the required PMT.

Summary of Inputs for Finance Calculator

1. For Withdrawals (to find PV) at age 75:

   • \( N = 16 \)
   • \( PMT = -180752 \)
   • \( i/y = 6 \)
   • \( FV = 0 \)

2. For Contributions (to find PMT from age 26 to 65):

   • \( N = 40 \)
   • \( i/y = 6 \)
   • \( PV = 0 \)
   • \( FV = [Value calculated from the first part] \)

This systematic approach will allow you to find the annual contributions that Derek must make to his retirement account.