Welcome back to the application module of the course in intuitive introduction to probability, decision making, and in an uncertain world. After looking at two playful applications of probability, today in this lecture I want to talk about very serious application where things went wrong. We're going to look at one of the core problems of the financial crisis that happened in 07 to 09. And this has to do with the structuring of West in finance. Unfortunately, to really hit that point home, I have to introduce a little bit of finance, and it won't be too much. But I need a little bit sort of background so we can understand to what went wrong in the financial crisis. We need to talk about points. What are points? When a company wants to do a big project, a big construction project, or wants to build a huge new factory, that company may need a lot of money. More money than it has in the bank so it needs a loan. Now, one way to get this loan is to get a bank, just like you and I do when we need a loan, we go to a bank and the bank gives us money and we pay the loan back. Companies now have a second way to do this. They can borrow money from the financial markets, where lots of us give this company some money and that's through a bond. So a bond is what's called a debt obligation, so they have a debt obligation to us, the lenders, and that's how they the money and in the end, they give up the money back and along the way we hopefully get some interest. Now, something can go wrong. The company could go bankrupt and then default on its loan, meaning that it does not pay back the loan. So that's very bad for you and me. If we gave some of our money to that company and then they default on us, we're going to be made because they don't give our money back. Now, let's now look at a black and white scenario. No default, we get our money back or default, we get no money back. Now, since we don't know the inner workings of these companies, we or our investors in general rely on so called rating agencies, which give ratings to the sponsor and tells them this is the safe investment or this is sort of a more risky Investments. These companies get so called credit ratings. The best rating is a AAA, AAA, that's the best you can get. Then it goes down all the way to CCC. To simplify the discussion, I'm going to have very good bonds, called investment grade bonds, that are considered quite safe, and risky bonds, sort of the junk bonds. Now, these ratings are very important for the companies for at least two reasons. One, the larger the ratings the safer they are considered, the less interest they need to pay us. We, of course, are more comfortable giving our money to a very safe company than to a risky company. Secondly, some investors, some institutional investors, such as pension funds are only allowed into invest into safe ones. So these ratings are very important. Here is some historical data on these ratings from a ratings agency. We can think of these now. We do concept tool once to get empirical probabilities. AAA both for example, it's very unlikely that they default, probability very close to zero. The same is still true in AA+. And you can see, it's gets gigantic large for really bad ratings such as C. So now we want to understand how these ratings affected the financial crisis, and what happened in the financial crisis. In the financial crisis, or before the crisis actually happened, this risk, these bonds got pooled, in particular for those of you who have followed the crisis there was something called the mortgage backed security where these debts were actually the debts of house buyers who bought house they needed money. And then, lots of these home loans were put together in the sort of funds of these bought. So now I want to show you how the junky bonds, that we put together into a portfolio, we can create some good risks. So here now, let's think about that we have 2 barely investments, 2 risky investments, 2 junk bonds, let's call them X and Y, and they both have a probability of default of 10%, so there's a 10% chance we're going to lose our money, we're not going to be happy about this. And now what people in finance do, and what's at the heart of the financial crisis. They take these junky bonds together and create a new type of asset based on those that's supposedly very safe. And so, make some junky stuff into really safe stuff. And now let's move towards probabilities. We have two investments. Both investments could either lead to a default or no default, now we have actually four combinations in total, the fear of investment, default, no default, characterized by little d or little n. Second investment also a little d or little n, and any pair as possible, dd, both X and Y default, dn, X defaults, Y does not default and so on. And remember we have these probabilities 10% default, 90% no default. What can now happen to a portfolio? If you think about those debts together. When both default no money comes back, 0,0 in the sum, the 2 junk bonds pay 0. One of them does not default, the other one defaults, the payback is 1. And in the best case scenario neither of them defaults, both pay back all the money, we get $1 back on each case, gives us $2. Now, let's assume those are independent. So the question is, how likely are these four possibilities? With independence, we can use the multiplication rule that we learned about in independent probabilities in the first two modules and create this probability table. Here in the probability table, you now see the probability that X and Y defaults. So it's a section probability in the upper left-hand corner of the table equal to the probability of the product of the module probability point one times point one. So there's only a 1% chance that both of them default. You do the similar math and you can fill in the table. So let's now do the formula. We are now taking the money that comes in from X and Y, and we return it to the invests. But we do so in a very special way. We create new investments I and J, an investment grade and a junk bond. And how are those defined? Look at these two tables. On the left, we see the money that comes in from our two original junk bonds. X and Y, 0, 1 or 2. And now, we split this according to the following rule. And this rule is actually written down by lawyers on paper. The rules are following. The first dollar that comes in goes to asset I. Any remaining funds go to asset J, let me repeat this because this is crucial. The first dollar that comes in no matter what, goes to I, any remaining funds go to J. Now, let's look what that means. First case, X and Y both default, nothing comes in. So nothing can go out, I pay 0, J pay 0. The next two scenarios, exactly one dollar comes in, so who gets the first dollar? The rule says S and I. So S and J again gets nothing. Finally, in the good case, $2 come in, both I and J get the dollar. Now, look at the sum of I+J. It's exactly the same SX+Y. So let me emphasize, there is absolutely clean accounting. There's no cheating, no money being hidden, no money being added. Whatever X+Y pay in, are paid out via ING. But boy, are I and J different here? Look at I, I pays in 3 out of the 4 cases. It only does not pay in the default default case, which has a tiny probability of only 1%. So I said he looks pretty good, only a 1% chance of default. That's a very good point. That gets a high rating. It has a 99% chance of paying off. So it has investment grade status even though X and Y, the ingredients in this financial alchemy, are junk bonds. But because of the split up of this legal contract of splitting up the money that comes in, in this peculiar way, I become safe. J, on the other hand, look at J, J only pays in the best case scenario of no default, no default. In all of the other three cases there isn't enough money coming in because either no money is coming in or the money that's coming in goes to asset I. J is super junk or how the pros in finance call it Toxic Junk. So what happened here? We get junk plus junk, X and Y and out of it we get the very safe asset I and we get the toxic junk J. And that's what structure finance is all about. Now, you may have heard or read in the newspaper the term collateralized debt obligations. This is a multi-trillion dollar. And by the way this is not a typo. I'm not talking about millions. Or billions, it's trillion dollar market. These are thousands of billions, and in finance, banks do this with mortgages, car loans, in countries where you have student loans. This is done with student loans, credit card debt. This is a gigantic market where people take group debts together and cut it off in this what's called trenches. Now, I know this was a heavy module. We, I gave you the basic idea of structure finals. How we pull this together and then create newer but different risks level. Remember, junk plus junk gave safe, plus toxic junk. But did you notice a probability in here? We assumed some independence, and that was a crucial mistake the rating agencies made in the financial crisis. They relied on this independence, and that's why this eye looked so golden. In the next lecture, we're going to talk about dependence when that assumption falls apart and then all help breaks loose as we saw in 07. Stay tuned for more. The story continues.