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.