0:32

The measure or risk that we've used is the standard deviation of variants,

Â a symmetric measure or risks.

Â Which is affected by variation below the mean and

Â variation above the mean in a very similar way.

Â In this video, we're going to ask the question, is mean variance enough?

Â Okay, should we go further?

Â Should we take into account other aspect of the return distribution?

Â The return of the portfolio?

Â And in particular, should we pay particular attention to downside risk?

Â And we're going to see how the portfolio allocation decision is modified when we

Â integrate a constraint on the amount of downside risk that we are willing to take.

Â 1:16

So, to do so in our example,

Â we first need to define a specific measure of downside risk.

Â You've covered a lot of these definition with Olivia's case previous videos.

Â So the measure of downside risk that we're going to consider in

Â this example is the notion of value-at-risk.

Â It's a notion that you've discussed already in some previous videos with

Â Olivia Skayen, but let me just briefly remind you what we're talking about here

Â through this illustration.

Â This is a distribution of return depicted in a graph.

Â This is the probability density function.

Â And it describes how likely it is to observe a particular return level.

Â The mean here, just for illustration, is set at zero, and the value-at-risk is

Â a quantity that measures a maximum level of loss that we are willing to take.

Â So for example here, the shaded area in blue represents 5% of the distribution,

Â and the clear area represent 95% of the distribution.

Â This is called the 95% VaR level.

Â This is the level roughly here minus 1.8,

Â the maximum level of loss that will occur with a probability of maximum 5%.

Â So, if we want to use this type of measurement

Â as notion of downside risk, we can add a constraint

Â that our portfolio has a maximum value-at-risk of a given level.

Â We could for example, assume that the maximum value-at-risk that

Â we're willing to take is a loss of 10% of our initial level.

Â So we're going to see now how the efficient frontier

Â is modified if we integrate such a value-at-risk constraint.

Â So here,

Â we have the, some of your graph that we have drawn quite a few times already.

Â The green and black curves represent the efficient frontier, okay?

Â So these points are attained by diversifying a portfolio by combining

Â different assets in a way that uses the correlation between

Â the asset to reduce the risk level for some given target expected return.

Â We are mainly interested in the green portions of this curves,

Â which represents the efficient frontier.

Â I've added to this graph two lines, two dotted lines.

Â The blue one represents old portfolio level in this

Â setup that have a value-at-risk of 10%.

Â You see that there are quite a few of these portfolios.

Â And to each of these portfolio corresponds a level of risk and a level of return.

Â If we want to simultaneously diversify our portfolio, we should choose a portfolio

Â on the green line and simultaneously verify the constraint of value-at-risk.

Â We should choose a portfolio for the 95% VaR of minus 10% loss.

Â We should choose a portfolio on the blue dotted line.

Â So to simultaneously be on the green curve and

Â the blue dotted line, we have to choose the intersection of these two line.

Â So still a point on the efficient frontier, but there is only one such point

Â that verifies the constraint of value at risk of minus 10% in this example.

Â The red dotted line represents another downside risk constraints.

Â This one is a little bit less tight.

Â Here, we're allowing to have losses that exceed the minus 10% threshold

Â that we affix, 10% of time, so this is a 90% value-at-risk.

Â Now you can see that if we allow larger losses to occur more often,

Â we can choose a portfolio that will generate a larger expected return.

Â We can see that the point on the efficient frontier that

Â intersects with the red line is slightly higher

Â than the point of the efficient frontier that intersect the blue line.

Â 5:23

So this intersection of the red and green line is an efficient portfolio

Â on the efficient frontier, which has a value-at-risk constraint of minus 10%,

Â verified at the 90% confidence interval.

Â So we can simultaneously diversify our portfolio.

Â Use the effective correlation, minimize risk, and

Â maintain the level of value-at-risk that we have considered to be the maximum

Â possible loss we're willing to sustain with a given probability.

Â So we can combine the effect of diversification with the notion of risk

Â management of the downside risk.

Â [MUSIC]

Â