A person wants to establish an annuity for retirement purposes. He wants to make quarterly deposits for 20 years so that he can then make quarterly withdrawals of $5,000 for 10 years. The annuity earns 7.32% interest compounded quarterly.
A. How much will have to be in the account at the time he retires?
B. how much should be deposited each quarter for 20 years in order to accumulate the required amount?
C. What is the total amount of interest earned during the 30-year period?
A. How much will have to be in the account at the time he retires?
To calculate the amount needed at retirement, we first need to calculate the future value of the quarterly withdrawals of $5,000 for 10 years, compounded quarterly at an interest rate of 7.32%.
We can use the formula for future value of an annuity:
FV = P * ((1+r)^n - 1) / r
Where:
FV = Future Value
P = Payment amount per period
r = Interest rate per period
n = Number of periods
In this case, P = $5,000, r = 7.32%/4 = 1.83%, and n = 10 * 4 = 40 quarters.
FV = $5,000 * ((1 + 0.0183)^40 - 1) / 0.0183
Calculating this value, we find that the future value of the withdrawals is approximately $303,485.84.
B. How much should be deposited each quarter for 20 years in order to accumulate the required amount?
Now that we know the future value needed at retirement, we can calculate the amount to be deposited each quarter for 20 years.
We can use the formula for present value of an annuity:
PV = P * ((1 - (1 + r)^(-n)) / r)
Where:
PV = Present Value (amount deposited)
P = Payment amount per period
r = Interest rate per period
n = Number of periods
In this case, PV is the amount we need to solve for, P = ?, r = 7.32%/4 = 1.83%, and n = 20 * 4 = 80 quarters.
PV = ? * ((1 - (1 + 0.0183)^(-80)) / 0.0183)
To solve for ?, we rearrange the formula:
? = PV / ((1 - (1 + r)^(-n)) / r)
Substituting the known values:
? = $303,485.84 / ((1 - (1 + 0.0183)^(-80)) / 0.0183)
Calculating this value, we find that the quarterly deposits needed for 20 years is approximately $1,516.01.
C. What is the total amount of interest earned during the 30-year period?
To calculate the total interest earned, we can subtract the total deposits made from the final value at the end of the 30-year period.
First, we calculate the total deposits made over 20 years:
Total deposits = $1,516.01 * 80
Next, we calculate the total interest earned:
Total interest earned = $303,485.84 - Total deposits
Calculating this value, we find that the total amount of interest earned during the 30-year period is approximately $79,043.16.
To solve this problem, we can use the formulas and concepts related to annuities and compound interest.
Here's how you can approach each part of the question:
A. To find the amount that needs to be in the account at the time of retirement, use the formula for the future value of an ordinary annuity:
FV = PMT * [(1 + r)^n - 1] / r
Where:
FV is the future value (the amount in the account at retirement),
PMT is the payment made each quarter ($5,000),
r is the interest rate per compounding period (7.32% / 4 = 1.83% = 0.0183),
and n is the number of compounding periods (10 years * 4 quarters/year = 40).
Plug in these values into the formula to calculate:
FV = $5,000 * [(1 + 0.0183)^40 - 1] / 0.0183
Calculate this expression to find the required amount.
B. To determine the quarterly deposits needed to accumulate the required amount, you can use the formula for the present value of an ordinary annuity:
PV = PMT * [1 - (1 + r)^(-n)] / r
Now we need to solve for PMT, which represents the payment to make each quarter for 20 years.
PV (present value) is the amount he wants to accumulate, which is the amount required at retirement (as calculated in Part A).
r and n remain the same.
Plug in the values into the formula and solve for PMT.
C. To calculate the total amount of interest earned during the 30-year period, subtract the total deposits made from the total amount withdrawn. Total deposits can be calculated by multiplying the quarterly deposit (PMT) by the number of deposits made (20 years * 4 quarters/year = 80 quarters). The total amount withdrawn can be calculated by multiplying the quarterly withdrawal amount ($5,000) by the number of withdrawals made (10 years * 4 quarters/year = 40 quarters).
Once you have these values, subtract the total deposits from the total withdrawals to find the total interest earned.
I hope this helps! Let me know if you have any further questions.