About Six Months Ago I figured something out that was very exciting to me. Let me share it with you.
Suppose you have a Certificate of Deposit that is earning a fixed interest rate for a specific term that is compounding daily. Who cares what the term is. Say you want to calculate the exact balance on a specific day. Well I will tell you how you can do it. Of course this will be assuming that it is not a leap year. For leap year calculations just change all of the 365 values in each formula to 366.
Let's first assign variables:
Let P = (The initial amount), r = (The rate), 365 = (365 days in the year), D = (The number of days that our account has been compounding) and I = (The Interest that we will get) and W = (The initial amount + the Interest we get)
The formulas are as follows:

APY (Annual Percentage Yield) = (1+ r/365)^(365) - 1

RATE = ln(APY+1) = [ln(1+r/365)^(365)] = 365*ln(1+r/365) = r

I = P*(1+r/365)^(D) - P So W = (P+I) = P*(1+r/365)^D

D = [ln(W)-ln(P)] / [ln(1+r/365)] where W = (P+I)

So how does this work? Say that we want to know how much interest we would get after 10 days of daily compounding at a 7% rate, with an initial deposit of $100,000.00 to our Certificate of Deposit. Then
I = P*(1+r/365)^(D) - P
So I = $100,000.00*(1+.07/365)^(10) - $100,000.00 = $191.95

Now what if we wanted to know how many days it would take to double our money
Doubling our money would mean that we would have to earn $100,000.00 in interest.
This means we want to know how many days will it take with an initial deposit of $100,000.00 at daily compounding at a rate of 7% to grow to be $200,000.00
So we have D = ln(W)-ln(P)/ln(1+r/365) so we have
ln($200,000.00)-ln($100,000.00)/[ln(1+.07/365)] = 3614.61 days. This means that it would take 3614.61 days to double our money at a rate of 7%

We know that the annual percentage yield is the actual rate of return on an initial deposit that has been compounding at a fixed rate for a full year.

But what if we want to calculate the actual rate of return on any given day after we make the initial deposit?
Well that is easy too. We can take calculate the interest up until that day, and then divide it by our principle account. For example we calculated that we after 10 days on a deposit of $100,000.00 we received 191.95 in interest. Therefore if we wanted to calculate the actual rate of return we would be = to interest / principle = 191.95/100,000 = .0019195 or .19% As you can see although we may be getting 7% as our rate, this rate is actually a yearly rate, in other words, our rate of return actually starts at 0%