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Today is your 24th birthday, and you decide to save $12,000 each year (with the first deposit one year from now), in an account paying 7% p.a., compounded annually. You will make your last deposit 41 years from now when you retire at age 65. You have a life expectancy of 95 years. How much money would you be able to withdraw per year after your retirement, if the first withdrawal occurs on your 66th birthday? (Round your answer in dollars to 2 decimal places, e.g. put 1204.42 if your answer is 1204.4243.)

Question

Today is your 24th birthday, and you decide to save $12,000 each year (with the first deposit one year from now), in an account paying 7% p.a., compounded annually. You will make your last deposit 41 years from now when you retire at age 65. You have a life expectancy of 95 years. How much money would you be able to withdraw per year after your retirement, if the first withdrawal occurs on your 66th birthday? (Round your answer in dollars to 2 decimal places, e.g. put 1204.42 if your answer is 1204.4243.)

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Solution

First, we need to calculate the total amount of money you will have saved by the time you retire at age 65. This is the future value of an ordinary annuity, which can be calculated using the formula:

FV = P * ((1 + r)^n - 1) / r

Where: FV = Future value (total savings at retirement) P = Annual deposit ($12,000) r = Interest rate per period (7% p.a. = 0.07) n = Number of periods (41 years)

FV = 12,000((1+0.07)411)/0.07FV12,000 * ((1 + 0.07)^41 - 1) / 0.07 FV ≈ 12,000 * 131.501 / 0.07 FV ≈ $2,257,214.43

Next, we need to calculate the amount of money you can withdraw per year after retirement. This is the annual payment of an ordinary annuity, which can be calculated using the formula:

P = FV * (r / ((1 + r)^n - 1))

Where: P = Annual payment (annual withdrawal) FV = Future value (total savings at retirement) r = Interest rate per period (7% p.a. = 0.07) n = Number of periods (95 years - 65 years = 30 years)

P = 2,257,214.43(0.07/((1+0.07)301))P2,257,214.43 * (0.07 / ((1 + 0.07)^30 - 1)) P ≈ 2,257,214.43 * 0.07 / 7.612 P ≈ $20,770.13

Therefore, you would be able to withdraw approximately $20,770.13 per year after your retirement.

This problem has been solved

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