Just to put some numbers and give some life to this, let's look at the US company that will receive one million of certain currency, call it A, six months from now. They don't want to be exposed to the exchange risk for that currency, so they want to hedge. They want to hedge since they're going to be long. That currency they are going to go short in Delta units of six-month futures contracts. Let's say there are six-month futures contracts, however not on currency A, but on some correlated currency B. Currency A is some small currency of a small country, that are no futures contracts, maybe. Let's hedge with the futures contract on a correlated currency. Suppose that the exchange rates today are $0.1 per unit of currency A, that's QA. Here, this should be QB really, it's a typo. QB, let's say, $0.2 per one unit of currency B. Which means in a perfect market, the exchange rate of A versus B has to be the ratio of 0.5. One naive answer might be, I'm hedging with futures and currency B and currency B has a double-worth of currency A. Since I'm going to be getting one million units of currency A, maybe I should just short half a million of futures in B. Maybe that's what I have to do because they are now in the ratio 1:2. But this doesn't take possible randomness into account, that change rates will not necessarily stay the same. There is some variance and correlations between these exchange rates. This might not be the correct answer. In fact, let's check what the correct answer is in terms of minimizing variance using our formula. To use our formula, we need to estimate those correlation standard deviations. Let's assume that we have historical data from which we can just do usual statistical estimation. Suppose that we get from historical data that the volatility, the exchange rate of the currency A is 3 percent.03, and of currency B is 2 percent.02. Suppose that we also estimated the correlation to be 0.9 high correlation. Then, well, I had two formulas. Let's go to the original formula. There I need a covariance, so the covariance as I mentioned in the previous slide I wrote it down, is the correlation times the standard deviations. Unless it's 0.9 times 0.02 times 0.03, which is something 0.00054. Then we had our formula, which said that the optimum delta is the covariance over the variance of the second asset. Its covariance is this number 0.00054 over the variance of the second asset that would be this squared, this is the standard deviation so the variance is Sigma squared 0.02 square is 0.0004 and this ratio is 1.35. What does this mean? This means in terms of units of currency A, for each unit of A, the US company should short an amount of currency B equivalent to 1.35 in currency A. Since I have one million units, that would be 1,000,000 times 1.35. But because of the exchange rate, this would be over 2. The value of this in terms of futures contract on currency B, I had to divide by 2 because of the current exchange rate. Therefore that would be 675,000 units of futures from currency B. Then we have a formula for the minimum of variance, which then you could compute. If you just put the numbers into the formula, you get this number here, 0.000171. Now if you don't hedge, then your minimum variance is simply the variance of currency A, which is 0.03 squared, which is 0.0009. The reduction is quite significant. You reduce your risk from 9th to the 4th decimal point to less than 2 in the 4th decimal point. It's a quite a bit of a reduction of risk. I'm not really going to need this elsewhere in this course. But it's a classical futures, static hedging that if you take a course on derivatives and futures are derivatives, this is basic and you should at least hear about it at least once. I think there's going to be a homework problem on this. But that's about it. We're not going to use it other than this. Let me give you one kind of a historical interesting example. Bad things that happened are usually more interesting than good things that are in movies and books are the same thing here. I'm on owning, mostly for telling you bad stories, bad things that happened with using derivatives. This is one from the early '90s. Its a Germany company, Metallgesellschaft. What they did, they sold over the counter privately to their customers, they sold a large number of basically forward contracts to deliver oil at a fixed price some forward price. They promised to deliver oil long term in the future at a pre-specified price to deliver oil, and then they wanted to hedge this. What is a natural hedge? A natural hedge, since they have to deliver oil they should go along into futures contract on oil. However, there are no long-term futures contract on oil labor, at least not for the maturities they needed. They couldn't trade, there were no maturities in the futures contract on oil on exchange that they could hedge with. What they did is so-called rolling the hedge forward. They went short-term futures contract to receive oil, and then when they expired, they just bought new ones. They just kept investing in short-term futures contract on oil, which seems like a natural strategy. However, what happened is the following. The oil price went down quite a bit, which actually is good for their original position. Because that means if they have to deliver oil, they can deliver the oil, it's cheap and it's easy for them to buy it on the market and deliver it. But since the hedging strategy is opposite, this was bad for their hedging strategy, and they were receiving large margin calls. They had to pay money to the margin account in the futures exchange simply because the oil price was going down. Which means in their futures contract they were losing money, they had to pay to the margin account. This is now a problem potentially for two reasons. One reason is if you don't have enough cash reserves, you cannot pay enough to the margin accounts; that's a problem. Or even if you can pay, and this is partly what happened, you might start panicking and you don't like having these really huge negative cash-flows all the time, just paying, and there is also pressure. It was a huge company and a huge contract, there was a political pressure from the media. Media was talking about these huge losses that Metallgesellschaft experienced. They we're really under pressure to do something, and what they did, they negotiated closing out the long-term contracts, but they had to close them out at a huge loss. In fact, this went to court, and there were big names in finance arguing on both sides of the lawsuit. But you could make a case on both sides. You can say, "If they want to hedge, this is a natural strategy, it's not clear what else they could do." On the other hand, if you do this, maybe you should think about that, really what's going to happen if the oil price goes down a lot? You should think about, are you ready to pay large negative cash-flows if the oil price went down in your short-term futures contract? Will you be able to sustain the pressure from the media, from politicians, whoever was involved became a public story, and you are under a lot of pressure to do something? Will you able to sustain this or not? If not, maybe it's not a good idea to hedge like this. It didn't help that the part of the company that was doing the hedge was in US. It was also this attitude that the American traders of the company are doing these crazy things with their crazy hedges destroying the base company in Germany. Anyway, that's a practical issue that we don't talk much about in the our perfect models, but you have to think about what can happen in the worst-case scenario. In this case, you have to worry if the oil price goes up a lot or if it goes down a lot, whether you will be able to ride it out. This was actually dynamic hedging, not just static hedging, they were rolling over the static hedges. There is one more static hedge that I want to talk about, but we're going to do that in the next set of slides.