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Godfrey has just recovered from a serious illness and is planning to start his early retirement at the age of 55 (i.e., start the retirement right after his 55th birthday).
According to Godfrey’s calculations, he would need an average of $40,000 per month for the living expenses during the retirement years. The monthly withdrawal will be made at the beginning of each (retirement) month. Godfrey would also like to leave his teenage daughter $20m which she would be able to inherit in 10 years after his retirement.
The retirement plan Godfrey purchases will provide a fixed interest return of 12%, compounded monthly.
a) How much money (the “target amount”) will Godfrey need to have in his retirement savings account at his 55th birthday so that he meet all expected expenses if he expects a 15-year retirement life?
To meet the “target amount”, the retirement plan provides Godfrey with 2 choices/plans:
Plan (1): make a single deposit of $4,086,356.21 into the retirement plan account (that earns a fixed interest return of 12%, compounded monthly, as stated above); or
Plan (2): make 72 month-end deposits of $50,000 each into the retirement plan account (the last of which will be made exactly at his 55th birthday) PLUS one single payment at his 55th birthday.
b) If Godfrey chooses Plan (1), when at the latest (in terms of age) will be required to make the lump-sum deposit in order to have enough money to meet the retirement expenses?
c) If Godfrey chooses Plan (2), what is the size of the single payment that he will be required to make at his 55th birthday in order to have enough money to meet the retirement expenses?

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JohnPositivity
a) To calculate Godfrey's retirement savings needs, we use the future value formula for an annuity with monthly compounding:

Future Value (FV) = Payment (P) × ((1 + r)^(nt) - 1) / r + Payment (P) × (1 + r)^(nt) × t

Where:
- FV is the future value (target amount) needed for retirement.
- P is the monthly withdrawal amount ($40,000).
- r is the monthly interest rate (12% / 12 = 1% or 0.01).
- n is the number of times the interest is compounded per period (monthly).
- t is the total number of periods (15 years * 12 months = 180 months).

By plugging in the values:

FV = 40,000 × ((1 + 0.01)^180 - 1) / 0.01 + 40,000 × (1 + 0.01)^180 × 15

FV ≈ 40,000 × (385.3212731) + 40,000 × (8.938248032)

FV ≈ 15,412,851.25 + 357,529.9213

FV ≈ 15,770,381.17

So, the target amount for Godfrey's retirement savings is approximately $15,770,381.17.

b) For Plan (1), Godfrey would need to make a single deposit of $4,086,356.21.

c) For Plan (2), we need to find the single payment required at his 55th birthday.
First, we find the present value (PV) of the annuity payments:

PV_annuity ≈ 72 × ((1 - (1 + 0.01)^-72) / 0.01)

PV_annuity ≈ 72 × (47.525154)

PV_annuity ≈ 3,421,114.35

Then, we find the remaining balance to reach the target amount:

Remaining Balance = 15,770,381.17 - 3,421,114.35

Remaining Balance ≈ 12,349,266.82

Finally, we calculate the single payment:

Single Payment ≈ 12,349,266.82 / (1 + 0.01)^72

Single Payment ≈ 12,349,266.82 / 5.653297705

Single Payment ≈ 2,183,026.89

So, for Plan (2), the single payment required at his 55th birthday is approximately $2,183,026.89.

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