# applications of derivatives

Bank A is offering you an investment account with 2.1% annual simple rate. This means that if you deposit \$1000 then after t years you will get \$1000 + 21t back (t can be fractional). Bank B is offering 2% annual rate compounded continuously. You would like to invest your savings so that you get maximum return in 5 years.

a) Is it better to deposit your money in Bank A or Bank B?

b) Can you increase the amount of interest you get if you first deposit money in one bank, wait some time, and then move them to the other bank? What is the best strategy? What is the maximum amount of interest that you can earn?

(Remark: actually, the best strategy is to put your money in Bank A, come the next day and withdraw them with the interest earned and put them back again the same day. If you do this every day you will turn 2.1% simple rate into 2.1% compounded (almost) continuously. But certainly Bank A will not like this, so let’s assume that both of these saving accounts are a one time offer).

First, your "Remark" is incorrect. If the interest was compounded annually, you would get 2.1% more money if you left the money in there for 1 year. If you only left it in for 1 day, you would get 2.1% / 365.

Second, with the premise of a real bank, this is not a calculus problem, but rather, simple arithmetic. If bank B was compounding monthly, then the balance at time T would be:

B (balance) = \$1000 * (1+i)^n, WHERE n = number of time periods.

Bank A @ 2.1% annual compounding, i = 2.1% / 1, B @ 5 years = 1,109.50

Bank A @ 2.1% monthly compounding, i = 2.1% / 12, B @ 5 years = 1,110.61

Bank B @ 2.0% monthly compounding, i = 2% / 12 = .001666, B @ 5 years = 1,105.08

Bank B @ 2.0% daily compounding, i = 2% / 365 = 5.479 E -5, B @ 5 years = 1,105.17

a) Bank A is the better choice, but just barely. b) no, you would not earn interest while traversing from one bank to the other.