Suppose an individual makes an initial investment of $1100 in an account that earns 7.5%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (so that the account balance will be reduced to $0). (Round your answers to the nearest cent.)
(a) How much is in the account after the last deposit is made?
$ 1
(b) How much was deposited?
$ 2
(c) What is the amount of each withdrawal?
$ 3
(d) What is the total amount withdrawn?
$ 4
(a)The accumulated value=
1100(1+0.075/12)^144+ 100*((1+0.075/12)^144-1)/(0.075/12)=$25942
(b)1100+100*144=15500
(c)The present value of the annuity=
25942=W*(1-(1+0.075/12)^(-60))/(0.075/12)
where W-the amount of each withdrawal
W=519.82
(d)519.82*60=31189.47
Please check
To answer these questions, we need to break down the problem into different steps and calculate the values step by step.
Step 1: Calculate the future value of the initial investment and the monthly contributions after 12 years.
We can use the future value formula for compound interest.
Future Value = P(1 + r/n)^(nt)
Where:
P = Principal (initial investment)
r = Interest rate (in decimal form)
n = Number of compounding periods per year
t = Number of years
Let's calculate the future value of the initial investment and monthly contributions after 12 years.
P = $1100
r = 7.5% = 0.075 (converted to decimal)
n = 12 (since it is compounded monthly)
t = 12 (12 years)
Future Value = $1100(1 + 0.075/12)^(12*12)
Future Value = $1100(1.00625)^(144)
Future Value ≈ $2970.35
So, after the last deposit is made, the account balance is approximately $2970.35.
Step 2: Calculate the total amount of the deposits.
To calculate the total amount of the deposits, we need to multiply the monthly contribution amount by the total number of months.
Monthly contribution = $100
Total number of months = 12 years * 12 months/year = 144 months
Total deposits = Monthly contribution * Total number of months
Total deposits = $100 * 144
Total deposits = $14400
So, the total amount deposited is $14400.
Step 3: Calculate the amount of each withdrawal.
To calculate the amount of each withdrawal, we divide the remaining account balance after 12 years by the total number of months for the withdrawal period.
Remaining account balance = $2970.35 (from Step 1)
Withdrawal period = 5 years * 12 months/year = 60 months
Amount of each withdrawal = Remaining account balance / Withdrawal period
Amount of each withdrawal = $2970.35 / 60
Amount of each withdrawal ≈ $49.51
So, the amount of each withdrawal is approximately $49.51.
Step 4: Calculate the total amount withdrawn.
To calculate the total amount withdrawn, we multiply the amount of each withdrawal by the total number of months for the withdrawal period.
Total amount withdrawn = Amount of each withdrawal * Total number of months
Total amount withdrawn = $49.51 * 60
Total amount withdrawn ≈ $2970.60
So, the total amount withdrawn is approximately $2970.60.
To summarize the answers:
(a) The account balance after the last deposit is made is approximately $2970.35.
(b) The total amount deposited is $14400.
(c) The amount of each withdrawal is approximately $49.51.
(d) The total amount withdrawn is approximately $2970.60.