Question

Your local bank is offering a new type of retirement savings account. An initial deposit is made to the account when it is opened. This money and any accumulated interest must be left in the account for 27 years. No additional deposits can be made. On the day the account is opened and on each annual anniversary of the initial deposit, the account balance is reviewed and the following terms apply: If the account balance is less than or equal to $20,000, interest for the next annual period is 7%/year compounded annually. If the account balance is greater than $20,000 but less than or equal to $40,000, interest for the next annual period is 10%/year compounded quarterly. If the account balance is greater than $40,000, interest for the next annual period is 12%/year compounded monthly. You decide to open an account under these terms today with $11,600. How much money will you withdraw when the account is closed 27 years from today?

Answer 1

Final value= $287,663.01

Step-by-step explanation:

Giving the following information:

If the account balance is less than or equal to $20,000, interest for the next annual period is 7% compounded annually.

If the account balance is greater than $20,000 but less than or equal to $40,000, interest for the next annual period is 10%/year compounded quarterly.

If the account balance is greater than $40,000, interest for the next annual period is 12%/year compounded monthly.

You decide to open an account under these terms today with $11,600.

We need to calculate the time required for the initial investment to reach each limit until the 27 years have passed.

We will use the following formula:

n= ln(FV/PV) / ln(1+i)

First, the number of years to reach $20,000

n= ln(20,000/11,600) / ln(1+0.07)

n= 8.05 years

In the firsts 9 years the account will be invested at a 7% interest rate.

FV= PV*(1+i)^n

FV= 11,600*(1.07)^9

FV= $21,326.13

Now, we need to calculate the number of quarters required to reach $40,000.

i= 0.10/4= 0.025

n= ln(40,000/21,326.13) / ln(1.025)

n= 25.4 quarters

n= 7 years= 28 quarters

FV= 21,326.13*(1.025^28)

FV= $42,577.51

Finally, the 16 years left at a 12% interest rate compounded monthly.

n= 16*12= 192

i= 0.12/12= 0.01

FV= 42,577.51*(1.01^192)

FV= $287,663.01