There's really not much difference in interest compounded yearly and daily, it's just a question of how much interest is accrued in a certain length of time. They both amount to a finite geometric sequence, and to deal with those:
The interest rate, p, is the ratio of successive terms in the sequence (where p is a number like 1.05 or thereabouts). Then if you start with a certain amount, a, in your account, after n pay periods you have a*p^n in your account.
To find out how interest compounded continuously works, you use limits:
saywe've got "p" interest compounded continuously, where p is a number like .05 or thereabouts. Then if we subdivide the year into m sections, each section will compound p/m interest, and we'll compound it m times. So after the first section of the year we'll have:
a + a*p/m = a(m+p)/m, after the second section we'll have
a(m+p)/m + a(m+p)/m *p/m = a(m+p)^2/m^2, and then
a(m+p)^2/m^2 + a(m+p)^2/m^2 *p/m = a(m+p)^3/m^3, and then
a(m+p)^2/m^3 + a(m+p)^3*m^3 *p/m = a(m+p)^4/m^4, .......
So at the end of the year (i.e. after m sections) we'll have
a(m+p)^m/m^m = a*((m+p)/m)^m in our account.
If we take the limit as m goes to infinity, this will go to a*e^p, where e is the base of the natural logarithm. So it's similar to the non-continuous case.
