Your client just turned 75 years old and plans on retiring in 10 years on her 85th birthday.She is saving money today for her retirement and is establishing a retirement account with your office.She would like to withdraw money from her retirement account on her birthday each year until she dies.She would ideally like to withdraw $50,000 on her 85th birthday,and increase her withdrawals 10 percent a year through her 89th birthday (i.e. ,she would like to withdraw $73,205 on her 89th birthday).She plans to die on her 90th birthday,at which time she would like to leave $200,000 to her descendants.Your client currently has $100,000.You estimate that the money in the retirement account will earn 8 percent a year over the next 15 years.Your client plans to contribute an equal amount of money each year until her retirement.Her first contribution will come in one year;her tenth and final contribution will come in ten years (on her 85th birthday).How much should she contribute each year in order to meet her objectives?

Question

Quizplus · Accepted Answer

To meet the client's objectives, we need to use the concept of present value of an annuity. We have to find the amount that needs to be contributed annually for the next 10 years in order to accumulate enough money to make the required withdrawals during retirement and leave a certain amount as the inheritance.

First, we need to calculate the future value of the amounts that will be withdrawn during retirement. Using the formula for future value of an annuity with increasing payments, we get:

FV = ($50,000 / 0.08) * [(1 + 0.08)^5 - (1 + 0.10)^5 / (0.08 - 0.10)] + ($73,205 / 0.08) * (1 + 0.08)^5
FV = $892,942.69

This means that the total amount required in the retirement account by the time the client turns 85 is $892,942.69.

Next, we need to calculate the present value of the inheritance that the client would like to leave. Using the formula for present value of a lump sum, we get:

PV = $200,000 / (1 + 0.08)^10
PV = $102,058.25

This means that the retirement account must have a balance of at least $102,058.25 on the client's 85th birthday after all the withdrawals have been made in order for her to leave a $200,000 inheritance.

Now, we can use the formula for present value of an annuity to find the amount that needs to be contributed annually. The present value of the retirement withdrawals can be calculated as follows:

PV = ($50,000 / (1 + 0.08)^10) + ($50,000 * 1.1 / (1 + 0.08)^9) + ($50,000 * 1.1^2 / (1 + 0.08)^8) + ... + ($73,205 * 1.1^4 / (1 + 0.08)^1) + ($73,205 * 1.1^5 / 0.08)
PV = $376,138.17

The difference between the required total amount in the retirement account and the present value of the retirement withdrawals is the present value of the inheritance:

PV inheritance = $892,942.69 - $376,138.17
PV inheritance = $516,804.52

Finally, we can use the formula for present value of an annuity to find the annual contribution required to accumulate $516,804.52 over ten years:

PV annuity = $516,804.52
n = 10
r = 0.08

PMT = PV annuity / [(1 - (1 / (1 + r)^n)) / r]
PMT = $12,401.59

Therefore, the client needs to contribute $12,401.59 annually for the next ten years to meet her retirement objectives.

Your Client Just Turned 75 Years Old and Plans on Retiring