Welcome back. Let us now talk about two potential extensions of the most basic forms of constant proportion portfolio insurance strategies that we discussed last time. What we are going to do now is we're first going to introduce a specific version of the strategy where we're going to try to protect a floor which is going to be a maximum drawdown floor. The focus and the goal of the strategy is to maintain the maximum drawdown below a certain pressure level. So let's take an example. Let us take look a look at an asset value process and that value process we call it V of t. So that's the value of your asset at time t. What you'd like to have is to maintain a maximum drawdown or let's say 20 percent. So at all costs, you want to avoid a situation in which the value of your assets loses more than 20 percent. How can you accomplish that using CPPI type strategies? Well it's actually fairly simple. What you're going to do is you're going to introduce a process that we're going to call it M of t. M of t is called a running max process in mathematics, and the running max process is a process that keeps track of the maximum value between day zero and day t that has been reached by the underlying asset value process. So M of t is the maximum, but all times S between zero and t of the value V of S. So we record that. Now what we're going to do is we're going to impose as a constraint that the value at any point in time is greater than 80 percent of that maximum value. In other words we are going to impose that a value that we have now is always greater than 80 percent and the maximum value ever reached before starting from the origin of the portfolio implementation of the portfolio strategy. If you do that, that allows you to make sure that the maximum loss you can take with respect to the highest point is 20 percent. More generally speaking, if you call alpha the percentage of the maximum value that you're protecting. So in other words if you're imposing that the value of your portfolio is greater than alpha percent of the maximum value, then this allows you to protect one minus alpha as a maximum drawdown. Let's take a look at an example here. So the blue line is giving you the performance on a historical scenario the performance of a given strategy. Now what we do is we introduce a green line and the green line is exactly the value of that empty process running maximum process. As you can see, the running max process starts at 80 percent. It's 80 percent of the maximum value ever reached. Initial data value is 100. The value of 80 percent time the maximum is 80. Now, as the value increases of your portfolio, the maximum drawdown flow keeps on increasing. Now it keeps on increasing up until a point where you reach some kind of a plateau and then the strategy actually loses ground. Well the strategy loses ground, but the max drawdown floor stays constant. Reflecting that keeps track of 80 percent of the highest value ever reached and not of the current value. Then eventually when the strategy keeps on going back up, the max drawdown floor keeps on increasing again. So the max drawdown floor is as we say in probability theory a never decreasing process, it either increases or stay flat, never decreases. So what you can do it's actually very simple. If you want to protect that maximum drawdown floor, just pick a multiplier and invest in the performance, or in the risky portfolio you invest a multiplier times the distance between the asset value and the floor, which is given in this case by the max drawdown floor. Let us now turn to a second extension which is very different extension in nature, where I would take look at the situation where investors have a floor which is a minimum amount of wealth that they want to protect, but they also have a cap, maximum amount of wealth that it would like to protect. One might wonder why would you need a cap. I mean it sounds like more is always better. It sounds like if you can get more wealth then the asset owner would be happy. That's true in general, but we have to recognize that protecting downside has an opportunity cost. Now by giving up on some of the upside above and beyond the threshold, let's call it a cap, where the investors have literally fit in marginal happiness or utility to go beyond, then you can actually get an opportunity gain and this opportunity gain will cover some or all of the opportunity cost from downside protection. In other words what we're suggesting here is we're suggesting to truncate the asset return distribution by giving up on the downside as we always do when we do portfolio insurance, but also on giving up on the upside. How do we do this in practice when looking at CPPI type strategies? Well, it's actually pretty simple. What you're going to have to do is you're going to have to define a threshold level. Let's call it T of T. That's the value of the threshold level. The threshold level is somewhere between the floor F of T and the cap C of T. The way it works is the following. If your asset value is below the threshold, what you should do is think in terms of CPPI. So in other words, what you should do if the asset value is say between the floor and threshold, the allocation to risky asset should be the multiplier times your asset value minus the floor. Well, that's exactly what CPPI says that we should do. Now, when you are above the floor, you're going to switch the perspective and now you're going to say "Well, I'm doing well, I'm not so much concerned about reaching the floor. My concern is I might hit the cap." Remember that by assumption, you don't want to go beyond the cap. So what you're going to be doing now is you're going to start reducing risk taking when getting closer to the cap. How do you accomplish this in practice? It's actually very simple. When your asset value is anywhere between T of T which is the threshold and C of T which is the cap, you are going to allocate a multiplier again times now cap value minus asset value. So in other words, you're slowing down on the way down and you're slowing down on the way up so as to have a smooth passing on the cap as opposed to crash through the cap just like you had a smooth-pasting on the floor as opposed to crash through the floor. So I guess the only outstanding question is how do you pick the threshold value? Picking the threshold value is actually very simple. It is obtained by what we call a smooth pasting condition. So you can think about what happens at the threshold value. If you're slightly below the threshold value, the prescription is you should allocate m times asset value minus floor value. Now, if you're slightly above the threshold value, you should allocate m times cap value minus asset value. What if you're exactly at the threshold value? Well, those two prescription should coincide. Otherwise, you get skies of clinic. That's what we call a smooth-pasting condition. So what we are writing down is that the allocation should be exactly the same looking up and looking down when you are at the threshold value. That gives you a very simple condition that translates into m times threshold minus asset value equal to m times cap minus threshold value. In this case, if you solve for this equation what you find is the threshold is very simply given by the average value between the floor and the cap, which by the way is extremely intuitive. This strategy is pretty convenient and useful in situations where investors have a goal or target in mind, and they have little incentive to go beyond the target while implementing the strategy would reduce the cost of downside protection. Wrapping up this basic CPPI strategy does a good job by protecting downside while allowing you to generate upside potential. There are a lot of different extensions of the basic CPPI strategy and we've discussed two extremely useful extensions. One is to protect a max draw down level to ensure that the maximum draw down will be staying at a given threshold. The second example is protecting situation with not only a minimum but a maximum wealth level.