Question

You have just turned 22 years​ old, received your​ bachelor's degree, and accepted your first job. Now you must decide how much money to put into your retirement plan. The plan works as​ follows: Every dollar in the plan earns 6.6 % per year. You cannot make withdrawals until you retire on your 65th birthday. After​ that, you can make withdrawals as you see fit. You decide that you will plan to live to 100 and work until you turn 65. You estimate that to live comfortably in​ retirement, you will need $ 115 comma 000 per​ year, starting at the end of the first year of retirement and ending on your 100th birthday. You will contribute the same amount to the plan at the end of every year that you work. How much do you need to contribute each year to fund your​ retirement? Your annual contribution should be ​$ nothing.

Answer 1

To fund your retirement, you should determine the total amount you need by calculating the present value of the $115,000 annual withdrawal over 35 years with a 6.6% interest rate, and then calculate the annual contribution needed using the future value of an annuity formula to accumulate that lump sum over your 43-year working period.

Step-by-step explanation:

To determine how much you need to contribute each year to fund your retirement, we can employ a strategy that combines both the future value of an annuity (to calculate the total amount you'll accumulate by the time you retire) and the present value of an annuity (to determine how much you'll need to withdraw annually during retirement). This approach considers the impact of compound interest on your contributions and the retirement income needs you've specified.

First, we calculate the present value of the annual withdrawal you desire in retirement. This is the amount you need upon retirement to be able to withdraw $115,000 per year, considering a 6.6% annual interest rate, over a period of 35 years (from age 65 to 100). We use the present value of an annuity formula for this calculation:

Present Value of Annuity = Payment x ((1 - (1 + r)⁽⁻ⁿ⁾) / r)

Where Payment is the annual withdrawal ($115,000), r is the annual interest rate (6.6% or 0.066), and n is the number of years of retirement (35).

Second, once we have the lump sum needed at retirement, we calculate the annual contribution using the future value of an annuity formula, to find out how much you should save annually to reach that lump sum. This formula considers the same 6.6% interest rate and compounds over your 43-year working period:

Annual Contribution = Future Value / (((1 + r)ⁿ⁻¹) / r)

By calculating these two values, you can then determine your annual contribution.

Answer 2

Your annual contribution=$87,809.24

Step-by-step explanation:

Step 1: Determine total amount needed in retirement

Total amount=Expenditure per year×number of years after retirement

where;

Expenditure per year=$ 115,000

Number of years after retirement=(100-65)=35 years

replacing;

Total amount=(115,000×35)=$4,025,000

You need a total amount of $4,025,00 for your plan

Step 2: Determine contribution per year

The contribution per year is the same as principal amount per year

Total amount needed per year=4,025,000/(65-22)=93,604.65 per year

Using the formula

A=P(1+r)^n

where;

A=Future value per year=$93,604.65

P=unknown=p

r=6.6%=6.6/100=0.066

n=1

replacing;

93,604.65=p(1+0.066)^1

93,604.65=1.066 p

p=93,604.65/1.066

p=87,809.24

Your annual contribution=$87,809.24