 Suppose an investment doubles in value at time t equal to 1. So, if you invest c at t equals 0, your equation of value will be removing a factor of a of tau and letting a of t equal 1 plus i gives, which we can solve to find i equal to 1. So, if you get in on the ground floor at t equal to 0, then your dollar-weighted yield rate will be 100% per year. But what if you waited until right before the investment doubled, say t equal 0.999? Well, how would you know that? Well, for example, you might be president of the company and knew an announcement would cause the value to double. And with this information, you could wait to invest until right before the announcement. Of course, this would be illegal. But if you make money, it doesn't matter, right? So, if you invest at t equal 0.999, your equation of value would be removing a factor of a of tau and again letting a of t equal to 1 plus i gives, which we can solve to find our interest rate, which will be percent. The important thing about this example is certain people should go to jail. I mean that the timing of the investor will determine their dollar-weighted yield rate. But an investor might get lucky and invest right before a jump in value. Since the jump in value is a function of the management of the fund, we might want to be able to evaluate that separately. And this leads to the time-weighted yield rate. To understand how that works, let's consider this example. Suppose you deposit $5,000 into an account at t equals 0. At t equals 1, the balance is 6,000, and you deposit an additional 4,000. At t equals 2, the balance is 13,000, and you deposit another 7,000. At t equals 3, the value is 21,000. What was the effective annual interest rate over each year and interpret your results? So note that in the first year, the amount grew from 5,000 to 6,000. So it grew by a factor of 6,005,000. And so the effective interest rate would be 20%. And so we could say that the fund grew by 20% in the first year. In the second year, you began with $6,000 plus $4,000, $10,000. That's the $6,000 balance plus the additional 4,000 deposit. And at the end of the year, you had 13,000. So this time the fund grew by a factor of 13,010,000. And the effective interest rate is 30%. In the third year, you began with 20,000 and ended with 21,000. So the fund grew by a factor of 21,020,000. And the effective interest rate is 5%. To understand what this means, suppose you made an initial deposit of 10,000, or really any other amount that's easy to work with. Then in the first year, this would grow by 20% to 12,000. In the second year, this would grow by 30% to 15,600. And in the third year, this would grow by... So one way to look at this is that if we deposited 10,000 at the start, then after three years, it would have grown to $16,380. So if we want to find an annual interest rate, then our final amount should be 10,000 times 1 plus i to the third. And solving for the interest rate, we find it's about 17.88%. And this suggests the time-weighted yield rate. Suppose our deposits are made at various times, and the balances immediately after the deposits are B1 through Bn. Let the final balance at time T be B. Then the time-weighted yield rate satisfies the product of all of these growth factors to the one-teeth power. And again, don't memorize formulas. Understand concepts. In this case, the time-weighted yield rate is the average of the growth rates when the deposits are treated as part of the balance. And it's important to note you cannot compute the time-weighted growth rate unless you know the intermediate balances. So, for example, suppose you invest 10,000 in an retirement fund. Five years later, you have 20,000, and you invest another 30,000. Ten years later, the fund has a value of 75,000. Let's compare the dollar-weighted yield rate and the time-weighted yield rate and interpret your results. So note that we have transactions and balances at T equals 0, 5, and 15. That's 10 years after 5 years. So the dollar-weighted yield rate satisfies the equation, which we can solve to get an interest rate of 5.68%. So that initial deposit grew to 20,000. So this initial deposit grew by a factor of... After you deposited the additional $30,000, your balance is now 50,000. And this amount grew to 75,000. So this grew by a factor of... Our growth factors are 20,000, 10,000, and 75,000, 50,000. And the basic idea behind the time-weighted yield rate is that the interest corresponded to this product of the growth factors. And so our time-weighted yield rate satisfies the equation. However, this would be for the entire 15-year period. Since this growth took place over 15 years, we'll take the 15th root to find the annual percentage rate. And this gives us about 7.60%. Now, note that the time-weighted yield rate is higher than the dollar-weighted yield rate. And one way to interpret this is that this means that if all deposits had been made at the start, the final amount would have been greater. This isn't always the case. For example, suppose we open an account with 10,000. 5 years later, the balance is 11,000, and you deposit an additional 4,000. 5 years after that, the balance is 16,000, and you deposit an additional 8,000. 5 years later, the balance is 80,000. Compare the dollar-weighted and time-weighted yield rates. So, note that our transactions occur at t equals 0, 5, 10, and 15. And the dollar-weighted yield rate satisfies, which we solve. For the time-weighted yield rate, note that our balance first went from 10 to 11,000. So, it grew by a factor of 11,000, 10,000. Next, after the deposit of 4,000, the balance became 15,000, and it went from 15,000 to 16,000. So, it grew by a factor of 16,000, 15,000. The next deposit of 8,000 made the balance 24,000, and it grew by 80,000, corresponding to a factor of 80,000, 24,000. So, the time-weighted yield rate will find that product of the growth factors, and since this growth took place over 15 years, we'll take the 15th root and solve to find the interest rate 9.52%.