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 29 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: 1. If the account balance is less than or equal to $20,000, interest for the next annual period is 7%/year compounded annually. 2. 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. 3. 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 $10,000. How much money will you withdraw when the account is closed 29 years from today? (Do all calculations to 5 decimal places and round final answer to nearest whole dollar. Tolerance is +/- 5.)

Answer 1

Using Rule of 72, at 7% interest rate it will take 72/7 = 10.28 years to double the investment

Hence at the end of year 11 the value will be = 10000*(1+7%)^11 = 10000*1.07^11 = 10000*2.10485 = $21048.51952

Hence from year 12, 10% per year compounded quarterly interest rate will be applicable.

Interest rate = 10%/4 =2.5%

No of quarters required To reach 40000 from 21048.51952 will be given by 40000 = 21048.51952*(1+2.5%)^n

or, 40000/21048.51952=1.025^n

or, log 1.90037 =n log1.025

or, 0.27884 = n*0.01072

or, n = 0.27884/0.01072

or, n = 26

Hence it will take 26/4 =6.5 or 7 years or 28 quarters to cross 40000

Value at the end of year 18 = 21048.51952*(1+2.5%)^28 = 21048.51952*1.025^28=21048.51952*1.996495=$42023.26437

Years remaining = 29-18 =11 years or 11*12=132 months

Interest rate = 12% per year compounded monthly

Hence monthly interest rate = 12%/12=1%

Value at end of 29 years = 42023.26437*(1+1%)6132 = 42023.26437*1.01^132 = 42023.26437*3.71896=$156282.77883

hence final value is $156283