My initial investment is $ 100, and I earn 1 % interest per day. I can opt for any number of compoundings per day (if twice per day, then the interest rate per compounding period is 0.5 % , and so on), but I have to pay $ 0.01 each time my interest is compounded. After 365 days I will close the account.What would this equation look like, and how should I include this to maximize my total deposit? How to generalize and figure out a good or optimal maximization?

Question

My initial investment is   $  100, and I earn   1  % interest per day. I can opt for any number of compoundings per day (if twice per day, then the interest rate per compounding period is   0.5  %  , and so on), but I have to pay   $  0.01 each time my interest is compounded. After 365 days I will close the account.What would this equation look like, and how should I include this to maximize my total deposit? How to generalize and figure out a good or optimal maximization?

Marlee Guerra · Accepted Answer

Let m be the number of compound periods per day.Then       T          n      +      1        =      T    n        (    1    +          0.01      m        )    −  0.01      (  n  ≥  0  )  .This is a first-order linear recurrence relation; derive the closed-form expression for       T    n  .The task is to finally determine the positive integer m at which       T          365      m       is greatest (among the positive integer values of   m).(Alternatively, the answer is easy to determine by simply varying   m in the above equation and observing the effect.)Additionally, the closed-form formula above for       T    n   also reveals that1.   m  =  100    ⟹   you will collect exactly   $  100 when closing the account;2.   100  &amp;lt;  m  &amp;lt;  103    ⟹   you will collect less than your   $  100 principal when closing the account;3.  m  ≥  103    ⟹   you will be owing money when closing the account.

My initial investment is <mi mathvariant="normal">$ 100 , and I earn 1

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