 We've looked at a number of scenarios where a sequence of investments are made at a sequence of times, and there are three variables, the amounts invested, the typing of the investments, and the interest rate. Then we computed several different quantities of interest. The most important is probably the present value, which considers the equivalent single investment made at t equals zero. Another quantity we considered is the equated time, which considers the equivalent time given a single investment. Note that these correspond to either changing the amount of time or the amount of money. We could also look at the interest rate and get the dollar-weighted yield rate, which considers the equivalent interest rate. For example, suppose you invest 1,000 now, 1,000 in a year, and withdraw 2,500 in two years. Let's find the yield rate that allows this. So again, don't memorize formulas, understand concepts. At interest rate i, the amount deposited initially accrues interest for two years and becomes, while the amount deposited in a year only accrues interest for one year to become. So if we assume that the withdrawal closes out the account, we have the equation of equivalent value, and if we set x equal to 1 plus i, this equation becomes an obvious quadratic with solutions, which gives us a dollar-weighted yield rate of about 15.83%. Now while there are many different financial instruments with more being invented all the time, some outcomes might be impossible to obtain. Other outcomes might be achievable through two or more different means, so it's possible for a financial scenario to have no dollar-weighted yield rate or for it to have multiple rates. The only way to determine which case applies is to solve the problem. So for example, suppose you agree to pay 1,000 now and in two years to be able to obtain 1,500 in one year. Find the dollar-weighted yield rate. So assuming the second payment closes out the account, our equation of value will be where x is equal to 1 plus i. So we solve, and if we're lucky or using a cheap calculator, we'll be told this is undefined. The problem is if we're using a more powerful tool, we might get an answer. And the important thing to remember is no computer was at what? You know, you'd think these interns would be better considering that we're giving them experience instead of a paycheck. Let me get a new intern. Yeah, fine, we'll go with it. Remember, it's your responsibility to correctly interpret your results. So even if your calculator or computing device gives you a solution, it's your responsibility to make sure that that solution corresponds to a real situation. And in this case, this equation has no solution, so there is no dollar-weighted yield rate that makes this possible. Now while you could agree to such a payment plan, we might consider what it means for there to be no solution. So let's think about this. If you deposit 1,000 at the start and withdraw 1,500 one year later, the balance at that point is either positive, negative, or zero. But if it's positive, you'd earn interest on the balance, so you'd actually receive money at the end and not pay out an additional $1,000. Similarly, if the balance is zero, then with interest, you'd have zero at the end of two years, and again there's no reason you'd have to pay more money. So the amount after you withdraw 1,500 must be negative. This means the interest rate is less than 50% since otherwise you'd have 1,500 or more in the account. Now if the interest rate was 0%, you'd have negative $500 after withdrawal, but this amount can't grow to 1,000 with an interest rate of less than 50%. What about a different scenario? Suppose you agree to pay 1,000 now and in two years to be able to obtain 2,500 in one year, find the dollar weighted yield rate. So letting x equal 1 plus i gives us the equation of value, and the quadratic formula gives us solutions, and we have two solutions, both of which are positive. And so we have two different interest rates, 100% and negative 50%. Now we've dealt with positive interest rates before, but what should we make of the negative interest rate? One interpretation is that the deposited money loses value over time. This is actually a common scenario in economics where there's inflation. The present value of a dollar now is greater than the value of the same dollar in a year because everything costs more.