 In some financial scenarios, several parties are involved in making payments to various individuals. Done correctly, this is legal. To analyze this, we use a bottom-line approach. The source of the money doesn't matter. All that matters is the money paid out or received by an investor. Now, it's useful to imagine that there is some meta-bank that receives all contributions and disperses all payments. So, for example, John, Charles, and Bernie agree on the following set of payments. John will pay Charles $1,000 now. Charles will pay John $500 in a year and Bernie $1,000 in a year. Bernie will pay John $800 in two years. Find the annual yield of all three investors and, importantly, interpret your findings. Now, we might want to make the following observation. Bernie's payment to John at t equals 2 is the final payment, and so we can assume it closes all of our accounts. So, we only need to know what happens up until time t equals 2. So, let's consider. At t equals 0, John paid Charles $1,000, but the destination is irrelevant. At t equals 1, John received $500 from Charles, but the source is irrelevant. And at t equals 2, John received an additional $800 from Bernie, and again, the source is irrelevant. So, we'll view money John pays as positive, he's putting money into this meta-bank, and money John receives as negative, he's withdrawing money from the meta-bank. So, this allows us to write John's equation of value, which will be... Now, this expression 1 plus i shows up all the time, so we'll let x be equal to 1 plus i, which gives us a quadratic equation, and solving that quadratic equation gives us, which has positive solution, and so, because x is equal to 1 plus i, we can solve this for our interest rate, which will be... Now, let's consider Charles. Charles had a transaction at t equals 0 when he received $1,000 from John, and t equals 1 when he paid $500 to John and $1,000 to Bernie. Since he received $1,000, this should be recorded as a negative contribution. It's a withdrawal from the meta-bank. Since he paid $500 plus $1,500, this should be recorded as a positive contribution. And note at this point there are no further transactions that Charles was involved in, so we can assume a $0 transaction at t equals 2, which closes the account. And so this gives Charles's equation of value, and again, letting x be equal to 1 plus i gives us the quadratic equation. Solving gives us x equals 0, which is unrealistic, and x equals 1.5, and so we find our interest rate is 50%. And finally, Bernie had transactions at t equals 1 of $1,000, a payment from Charles, so that's a withdrawal and negative amount, and $800, a payment to John, so his equation of value will be... and again, letting x equals 1 plus i gives for an interest rate of negative 20%. Huh. And it's worth noting that the three rates mean very different things, so consider John's rate of 17.87%. He initially paid $1,000, and again, it's helpful to think about some meta-bank as receiving and dispersing all amounts, so this is a deposit to the meta-bank. He later received $500 in one year and $800 in two years, and this would happen if the meta-bank paid 17.87% annual interest, and in John's case, he got back more than he put in. Bonus. So the interest of 17.87% represents a gain for John. Now, consider Charles's rate of 50%. Charles initially received $1,000 from the meta-bank, but in return, he paid back $500 plus $1,500 a year later. So in Charles's case, he paid more than he got out, so the interest rate of 50% is actually a loss for Charles. And Bernie's case is the most peculiar. Bernie received $1,000 at t equals 1 and paid out $800 at t equals 2, and this is an interest rate of negative 20%, but the reason that the interest rate is negative is that we assume the payout of $800 closed the account. The $1,000 he received shrank in value to $800, but that $200 loss was actually kept by Bernie, so even though the interest rate is negative, it's a gain. And this leads to the following conundrum. John's interest rate is positive and this is good for him. Charles' interest rate is positive, but this is bad for him. Bernie's interest rate is negative, but this is a good thing. And so the important question to ask is, why did this happen? We might look at it as follows. Note that John initially deposited money to the meta-bank, so earning a positive rate of interest means that his deposits became more valuable and he was able to withdraw more than he put in. But Charles received money, so he started with the negative balance at the meta-bank. And a positive rate of interest means that his negative balance grew even more negative, which is bad for him. Bernie also received money, but the negative rate of interest on the negative balance actually reduced the amount, so at the end he owed less. Thank you.