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 definition with Olivia's case previous videos.

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

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

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

Olivia Skayen, 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 is

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 represent 95% of the distribution.

This is called the 95% VaR level.

This is the level roughly here minus 1.8,

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

So, if we want to use this type of measurement

as notion of downside risk, we can add a constraint

that our portfolio has a maximum value-at-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, some of your graph that we have drawn quite a few times already.

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

So these points are attained 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 portions of this curves,

which represents the efficient frontier.

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

The blue one represents old portfolio level in this

setup that have a value-at-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% VaR 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 line.

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 constraints.

This one is a little bit less tight.

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

that we affix, 10% of 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 intersect the blue line.