[MUSIC] Hello, in this video, we’re going to discuss the notion of risk management applied in a specific example to the problem of portfolio allocation. So, so far we've discussed in the few videos the role of correlation in setting up optimal portfolio location. We've extensively discussed the notion of diversification. The measure of risk that we've used is the standard deviation or variance, a symmetric measure of risk. Which is affected by variation below the mean and variation above the mean in a very similar way. In this video, we're going to ask the question, is mean variance enough? Okay, should we go further? Should we take into account other aspects of the return distribution? The return of the portfolio? And in particular, should we pay particular attention to downside risk? And we're going to see how the portfolio allocation decision is modified when we integrate a constraint on the amount of downside risk that we are willing to take, okay. So to do so in our example, we first need to define a specific measure of downside risk. You've covered a lot of these definitions with previous videos. So the measure of downside risk that we're going to consider in this example is a notion of value at risk. It's a notion that you've discussed already in some previous videos with but let me just briefly remind you what we're talking about here through this illustration. This is a distribution of return depicted in a graph. This is the probability density function, and it describes how likely it is to observe a particular return level. The mean here, just for illustration is set at zero. And, the value at risk it a quantity that measures a maximum level of loss that we are willing to take. So for example here, the shaded area in blue represents 5% of the distribution and the clear area represents 95% of the distribution. This is called the 95% VaR level. This is the level, roughly here, -1.8. The maximum level of loss that will occur, with a probability of, maximum, 5%. So, if we want to these type of measurements as a notion of downside risk, we can add a constraint that our portfolio has a maximum valued risk of a given level. We could, for example, assume that the maximum value at risk that we're willing to take is a loss of 10% of our initial level. So we're going to see, now, how the efficient frontier is modified if we integrate such a value at risk constraint. So here we have the familiar graph that we have drawn quite a few times already. The green and black curve represent the efficient frontier. Okay, so these points are obtained by diversifying a portfolio by combining different assets. In a way that uses the correlation between the asset to reduce the risk level for some given target expected return. We are mainly interested in the green portion of this curve, which represents the efficient frontier. I've added to this graph two dotted lines. The blue one represents all portfolio level in this set up that have a VaR level risk of 10%. You see that there are quite a few of these portfolios. And to each of these portfolio corresponds a level of risk and a level of return. If we want to simultaneously diversify our portfolio we should choose a, portfolio on the green line and simultaneously verify the constraint of value at risk. We should choose a portfolio for the 95% value of minus 10% loss. We should choose a portfolio on the blue dotted line. So to simultaneously be on the green curve and the blue dotted line, we have to choose the intersection of these two lines. So still a point on the efficient frontier but there is only one such point that verifies the constraint of value-at-risk of minus 10% in this example. The red dotted line represents another downside risk constraint. This one is a little bit less tight. Here we are allowing to have losses that exceed the minus 10% threshold that we have fixed, 10% of the time. So this is a 90% value at risk. Now you can see that if we allow larger losses to occur more often, we can choose a portfolio that will generate a larger expected return. We can see that the point on the efficient frontier that intersects with the red line is slightly higher than the point of the efficient frontier that intersects with the blue line. Okay so this intersection of the red and green line is an efficient portfolio on the efficient frontier which has a value of risk constraint of minus 10% verified at the 90% confidence interval, okay? So we can simultaneously diversify our portfolio, use the effect of correlation, minimize risk, and maintain a level of risk that we have considered to be the maximum possible loss we are willing to sustain with a given probability. So we can combine the effective diversification with a notion of risk management of the downside. [MUSIC]