 Generally speaking, it will be impossible to find the exact interest rate algebraically. To find an approximate solution, either to check an answer or as an initial guess for a numerical method, we'll make a few simplifications. First we'll scale our time unit so we begin at t equals 0 and end at t equals 1. Next we'll assume an initial balance of 0 so our initial deposit will be the actual amount at t equals 0. Later on we'll see how we can handle the case where we do have an initial balance. And then finally, we'll use some calculus. Suppose amount c is deposited at time t, where t is between 0, our start, and 1, the end. The interest earned on this amount during the remaining time 1 minus t, well that'll be the difference between the deposit amount and the deposit amount times the accumulation function. If we're assuming our compound interest, which we usually are, then this will be, and here's where our approximation comes in. This exponential expression, from calculus, we have the binomial expansion, and if x is small and n isn't too big, then the higher powers of x are very small, and so we can approximate this power using just the first two terms, 1 plus nx. So applying this approximation gives us, now that's just a single contribution. If we contribute amount ck at times tk, then the total amount of interest is going to be approximately, but since every term has this factor of i, we could remove it from the sum to get, and so the total interest earned from all contributions will be. The actual interest i is the difference between the total of the contribution c1 through ck and the final balance b. In other words, to approximate the dollar weighted yield, we let the actual interest equal the approximate interest and solve. And so if i is the total interest, then the approximate dollar weighted yield rate will be, and again we'll throw in our usual qualifier, don't memorize formulas, understand concepts. Here we're equating the actual interest earned with a linear approximation and solving for the interest rate. For example, suppose you invest 3,000 now, 3,000 in one month, and 3,000 in six months. You close out the account in one year by withdrawing $15,000, find the dollar weighted yield rate, and compare it to the approximate value using the preceding method. So we can let tau equal one year, the investment period, and we could write the equation of value, but we'd be wrong. And remember, no computer ever went to prison for misleading a client. Notice that our time units are months or years, and we have to choose. Since we're probably going to quote an annual interest rate, we should use time in years. So remember we'll want the start to be zero, and the end to be one year, and that means a month from now or six months from now will be. And so the correct equation of value for tau equal to one will be. If we assume our accumulation function has the form one plus i to power t, where t is measured in years, our equation becomes. But the most we can simplify this is by letting x equal one plus i to the 12th, which gives us a 12th degree equation, which has no algebraic solutions. Instead, we'll have to find a numerical solution. So we find, since we made the substitution x equals one plus i to the 12th, then we can solve for i, which works out to be about 86.44 percent. Now in the real world, you'll always have a computing device available that will allow you to find these numerical solutions. But it's also useful to have a way of getting approximate solutions to help you build up what's called a number sense, an intuitive feeling of when an answer is correct, or when it's way off and should be investigated. Without the number sense, you have to accept whatever numbers people throw at you. But if you've developed a number sense, you can start to get an idea of when somebody is telling you the truth, and when somebody's just trying to feed you a load of horse stuff. So let's consider, for a total deposit of $9,000, we were able to withdraw $15,000 in one year. So the actual interest received was $6,000. Now let's consider the interest received on the $3,000 deposit initially. That accrued interest for the full year. And so the interest will be, so we can write the approximate equation. The $6,000 interest is the interest from the first deposit. Now let's consider the interest received on the $3,000 deposit in one month, which will be kept in the account for 1112 years. So the interest would be the difference. And we'll use our binomial approximation for 1 plus i to the 1112s is approximately 1 plus 1112s i. And we find, which forms the next part of our interest, the interest received on the $3,000 deposit in six months kept in the account for six 12 years would be approximately, which gives the remainder of the interest. So notice that every term on the right has a factor of i so we can remove it, then divide both sides by the coefficient, and find that our approximate interest rate is about 83%.