Betty Sue sets up a retirement account. For the �first 35 years, she deposits
$500 at the end of each month into an account with an annual interest rate of 3.6%, compounded monthly. Then, she stops making monthly payments and transfers the money into a di�fferent account with an annual interest rate of 4%, compounded quarterly for a period of 10 years. How much money has she saved for retirement at the end of her 45 years if saving?
first 35 years
35 * 12 = 420 months = n
r = .036/12 = .003 monthly interest rate
p= present value of sinking fund
N = deposit each period of 1 month = 500
p = N [ (1+r)^n - 1 ] /r
p = 500 [ (1.003)^420 - 1 ] / .003
p = 419,796.33 after 35 years
now the final 10 years
quarterly for 10 years = 40 periods
interest rate = .04/4 = .01
1.01^40 = 1.48886
times 419 etc = 625,019.54
To calculate how much money Betty Sue has saved for retirement at the end of 45 years, we need to calculate the total amount accumulated in both accounts. We can break down the calculation into two parts: the first 35 years with monthly deposits, and the next 10 years without any deposits.
First, let's calculate the total amount accumulated in the account during the first 35 years with monthly deposits.
Step 1: Calculate the monthly interest rate and the number of periods in the first account.
Since the annual interest rate is 3.6% and the interest is compounded monthly, the monthly interest rate would be 3.6% divided by 12 (months) = 0.003.
The number of periods would be 35 years multiplied by 12 (months) = 420 months.
Step 2: Calculate the amount accumulated during the first 35 years.
We can use the formula for the future value of an ordinary annuity:
FV = P * (((1 + r)^n) - 1) / r
Where:
FV = Future Value
P = Monthly Deposit Amount
r = Monthly Interest Rate
n = Number of Periods
Using the formula, we have:
P = $500
r = 0.003
n = 420
FV = $500 * (((1 + 0.003)^420) - 1) / 0.003
Calculate this expression to get the value of FV, which represents the amount accumulated in the first account after 35 years.
Now, let's calculate the total amount accumulated in the second account during the next 10 years without any deposits.
Step 1: Calculate the quarterly interest rate and the number of periods in the second account.
Since the annual interest rate is 4% and the interest is compounded quarterly, the quarterly interest rate would be 4% divided by 4 (quarters) = 0.01.
The number of periods would be 10 years multiplied by 4 (quarters) = 40 quarters.
Step 2: Calculate the amount accumulated during the next 10 years.
Using the same formula as before:
FV = P * (((1 + r)^n) - 1) / r
Where:
FV = Future Value
P = Initial Amount (amount accumulated in the first account)
r = Quarterly Interest Rate
n = Number of Periods
Now we can replace the values and calculate the expression to get the value of FV, which represents the amount accumulated in the second account after 10 years.
Finally, we add the amounts accumulated in both accounts to get the total amount saved for retirement at the end of 45 years.
Total Amount Saved = Amount accumulated in the first account + Amount accumulated in the second account
Make sure to substitute the actual values calculated in the previous steps to find the final answer, i.e., the total amount saved.