 If all transactions occur between t equals 0 and t equals 1, and deposits c1 through ck are made at times t1 through tk, an approximation is good enough, and i is small enough, then the yield rate can be approximated by... where capital I is the total interest earned. But what if the transactions don't all occur between t equals 0 and t equals 1? That's actually a pretty easy fix, as follows, the interest rate is computed per unit of time. Now, the unit of time could be months, years, millennia, or whatever, but we can still make sure that t equals 0 could correspond to the start and t equals 1 to the end. We can then find the annual interest rate by computing the amount of interest accumulated over one year. For example, suppose we open an account with a $1,000 deposit, then deposit another $2,000 in two months and $4,000 in six months. If the account has $8,000 after nine months, approximate the annual yield rate, and while we're at it, let's find the exact annual rate. So the total contributions are... So the total interest earned is... Since the total length of our investment period is nine months, then the amount deposited at the beginning accrues interest for the entire period. So the amount of interest is... The amount deposited in two months will accrue interest for seven months, which is seven-ninths of the investment period. So the interest earned will be... And we can use our binomial to approximate this as... Similarly, the amount deposited in six months will accrue interest for the remaining three months, which is three-ninths of the investment period. So the interest earned will be... Solving gives us... Or about 25.71% per nine months. To convert the interest rate per nine months to an annual interest rate, we note that one year is 12-ninths of nine months, so a dollar would grow too. So the annual interest rate is around 36%. Now, since this is an approximation, we're going to not worry too much about how closely we'll round it. We'll just round it to the nearest whole percentage. To find the exact value, which will just be a more accurate approximation using the real equation, we note the equation of value will be... And note that we can remove a common factor of A of tau. And if A of t is 1 plus i to the t, then our equation will include reciprocals, 1 plus i raised to the two-twelfths, six-twelfths, and nine-twelfths. So if we let x equal 1 divided by 1 plus i to the one-twelfth, we can simplify our equation, too. And we can find a numerical solution. Now, remember, x was 1 divided by 1 plus i to the one-twelfth. And so we can solve for i. And this time, because we're trying to find the exact annual rate, we should be a little bit more careful in our rounding. And so we'll quote an annual interest rate of 37.12%. So, note that if we actually earned 36% annually, which was our approximation, then our deposits would have a final value of... So our approximation doesn't give the exact value of our final balance, but we worked a lot less hard for it, and it came very close. And so by any account, this would be considered a bargain. Again, it's important to consider what we're using the approximation for. We really should be using this approximation to give us a sense of what the numbers should be, and we should try to be as accurate as possible when solving for the actual values that we're going to quote to clients or our bosses. The approximations are really used to give us some assurance that our computations have been done correctly.