Question

You plan to make contributions to your retirement account for the next 20 years. After the last contribution, you will retire and begin withdrawing $3000 each month, and you want the money to last an additional 20 years. Assume your account earns 8% interest compounded monthly.

a.How much do you need to have saved in 20 years in order to withdraw according to plan?

b) How much do you need to deposit into the savings account for the next 20 years in order to reach the goal you set in part (a)?

c) Both the savings phase and the withdrawal phase last for 20 years. Why is the amount you need to contribute each month not equal to the amount you plan to withdraw each month? In other words, why is the answer to part (b) less than the $2000 monthly withdrawal amount?

the answer to a should be solved through a payout annuity equation, atleast i think so. which would be

p0= d(1-(1+r/k)^-nk)/(r/k)

(P0 is the balance in the account at the beginning (starting amount, or principal).

d is the regular withdrawal (the amount you take out each year, each month, etc.)

r is the annual interest rate (in decimal form. Example: 5% = 0.05)

k is the number of compounding periods in one year.

N is the number of years we plan to take withdrawals)

Answer 1

a]

PV of annuity = P * [1 - (1 + r)-n] / r,

where P = periodic payment. This is $3,000

r = interest rate per period. This is (8%/12)

n = number of periods. This is 20 * 12 = 240

PV of annuity = $3,000 * [1 - (1 + (8%/12))-240] / (8%/12)

PV of annuity = $358,662.88

b]

Future value of annuity = P * [(1 + r)n - 1] / r,

where P = periodic payment. We need to calculate this.

r = periodic rate of interest. This is (8%/12)

n = number of periods. This is 20 * 12 = 240

$358,662.88 = P * [(1 + (8%/12))240 - 1] / (8%/12)

P =  $358,662.88 * (8%/12) / [(1 + (8%/12))240 - 1]

P =  $608.91

c]

This is because of the effect of compound interest. The interest is earned beginning with the first deposit. After the withdrawal period begins, the entire amount is not withdrawn at once, but a part of it is withdrawn monthly. Thus, the remaining amount in the account continues to earn interest until the entire amount is withdrawn. Hence, the amount in part (b) is less than the annuity amount