Maximizing interest question

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?

What do you mean by as many compound per day? do you mean receive one percent interest and compound it any number of times within a day, paying 0.01 each time? — user1001001, Aug 16 '21 at 18:39
Good Question...I can try to answer with an example... lets say that I have 105 deposited...then 24 hours later I can expect 1 percent interest on that 105 but hypothetically I could compound it 12 hours after which would predict .525 instead of waiting 24 hours to get the 1.05.. if that makes sense.. — thestar, Aug 16 '21 at 18:46
the difficulty is that I can compound it as many times as I want per day but at a cost of 0.01 each time — thestar, Aug 16 '21 at 18:48
One approach is to solve a number of intermediate problems, where you are restricted to only "compounding" some fixed number of times. Then you could optimize when those times occur during the 365 day horizon. Start with no compounding, then optimize if only one compounding is allowed, then two, etc. When the net profit stops increasing, you have your answer. But you should show some effort to solve the problem, not so much to answer it yourself but to clarify what the interest compounding scheme really is. — hardmath, Mar 27 '23 at 03:47

ryang · Accepted Answer · 2021-08-18T06:02:10.763

Let $m$ be the number of compound periods per day.

Then $T_{n+1}=T_n\left(1+\frac{0.01}m\right)-0.01\;\;(n\geq0).$

This is a first-order linear recurrence relation; derive the closed-form expression for $T_n$ using the information here.

The task is to finally determine the positive integer $m$ at which $T_{365m}$ 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 that

$m=100\implies$ you will collect exactly $\$100$ when closing the account;
$100<m<103\implies$ you will collect less than your $\$100$ principal when closing the account;
$m\geq103\implies$ you will be owing money when closing the account.

Maximizing interest question

1 Answers1