Problem calculating percentage compound interest

Question

I have two questions:

1. How can I calculate compound interest when I want to reinvest a percentage of the sum of my principal and interest periodically?

   To make it clearer,

   Usually I use this formula for calculating compound interest: $M = P( 1 + i )^n$.

   This time I want to compound only a percentage of the sum of the principal and interest

   EDIT

   This is what I am looking at:

   I invest $1000$ for a month, at the end of the month I get $1100$ (principal plus interest), and I want to re-invest $60%$ of the total amount for $n$ periods.

2. I want to compound the interest on the principal daily till the investment matures, i.e. if am investing for one year, I want the daily interest to be compounded to the principal.

Answer 1

For 2, you just regard each day as a separate term. So the interest rate is the rate per day and you have 365 (or 366 or 360!?) terms. For 365, if the interest rate were 10%/year, it would be 10/365$\approx $0.0274%/day, and the final value would be $1.000274^{365}$ times the original value, or about 1.105 times the original value.

For 1, what do you want to calculate? The final balance after N terms when at the end of each term the principal is reduced? The amount of interest received under those conditions? Probably it is easiest to make a spreadsheet for what you want.

Answer 2

1. For question 1 you just have to redefine the amount you want to invest after the end of every period. If $x$ is the daily interest rate (if, for example, the daily interest rate is 5% then $x=0.05$), then after 30 days (1 month) your initial capital, $c$, has become $$ c_1=(1+x)^{30} c. $$

At that point you only want to invest a percentage of $c_1$, given, say, by $p$ (in your example $p=0.6$). All you do is multiply with $p$: $$ c_{1,\text{inv}}=(1+x)^{30} pc. $$

At the end of the second month you have $$ c_2=(1+x)^{30}c_{1,\text{inv}}=(1+x)^{60}pc. $$ After $n$ months this becomes $$ c_n=(1+x)^{30n}p^{n-1}c. $$

2. For question 2 just take the yearly interest, $X$, divide it by the number of days in the year, and apply it every day. With initial capital $C$ you get, at the end of the year, $$ C_1=\left(1+\frac{X}{365}\right)^{365}C $$