Defining the Expected Shortfall

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En provenance du cours de University of Geneva

Portfolio and Risk Management

749 notes

University of Geneva

Portfolio and Risk Management

749 notes

Cours 3 sur 5 dans Specialization Investment Management

In this course, you will gain an understanding of the theory underlying optimal portfolio construction, the different ways portfolios are actually built in practice and how to measure and manage the risk of such portfolios. You will start by studying how imperfect correlation between assets leads to diversified and optimal portfolios as well as the consequences in terms of asset pricing. Then, you will learn how to shape an investor's profile and build an adequate portfolio by combining strategic and tactical asset allocations. Finally, you will have a more in-depth look at risk: its different facets and the appropriate tools and techniques to measure it, manage it and hedge it. Key speakers from UBS, our corporate partner, will regularly add a practical perspective on these different topics as you progress through the course.

À partir de la leçon

Risk Management

This fourth and final week is dedicated to risk. We will start by looking in more depth at different sources of risk such as illiquidity and currency risk but also at the different tools available to investors to perform risk management. But how should we measure risk? We will see that it may be valuable to go a step beyond standard deviation, the risk measure we used so far, and look at the Value-at-Risk and Expected Shortfall which focus on potential large losses. Finally, we will use the financial instruments at our disposal to hedge market and currency risk.

Defining the Value-at-Risk4:52

Computing the Value-at-Risk5:21

Defining the Expected Shortfall6:34

Computing the Expected Shortfall3:46

Risk management applied to portfolio allocation6:08

Banking regulation & Basel recommendations: How did we get there?7:28

Rencontrer les enseignants

University of Geneva- Tony Berrada
SFI Associate Professor of Finance
Geneva Finance Research Institute
University of Geneva- Ines Chaieb
SFI Associate Professor of Finance
Geneva Finance Research Institute
University of Geneva- Jonas Demaurex
Teaching Assistant
Geneva Finance Research Institute
University of Geneva- Rajna Gibson Brandon
SFI Senior Chaired Professor of Finance and Managing Director of the GFRI
Geneva Finance Research Institute
University of Geneva- Michel Girardin
Lecturer in Macro-Finance - Project Leader for the "Investment Management" specialization
Geneva Finance Research Institute
University of Geneva- Philipp Krueger
SFI Assistant Professor of Finance
Geneva Finance Research Institute
University of Geneva- Kerstin Preuschoff
Associate Professor of Neurofinance and Neuroeconomics
Geneva Finance Research Institute
University of Geneva- Olivier Scaillet
SFI Senior Chaired Professor of Finance and Vice-dean (research) at GSEM
Geneva Finance Research Institute

[MUSIC]

This session will be about the expected shortfall.

So, you can view the expected shortfall as an extension

of the concept of value at risk.

So again, I will look at two questions.

So, the first question is what is the expected shortfall?

And the second question is how can I compute the expected shortfall?

So again the expected shortfall is a quantitative and

synthetic measure of risk and it answer a very simple question.

So, what is the average loss when I

know that my loss will be above the Value-at-Risk?

And if you remember the Value-at-Risk is a quantile of a loss distribution.

So how can I define that informally looking at the graph?

So if you remember the graph of the loss distribution, so

here we have a nice bell shape but it shouldn't necessarily be a bell shape.

And I look at the loss distribution and if you remember,

the value at risk is a level.

So that I have a one person probability to have a loss above that level.

So when I look at the expected shortfall,

what I will do is simply look at the averages of the losses.

When I know that I will have a loss above the value at risk.

So for example if the value at risk is equal to $1 million.

The expected short fall will tell me whether in average the loss, when I

have a loss above $1 million will be equal for example to $50 million or $1 million.

So it will be the average loss when I know that my loss is above my value at risk.

So of course the question that you should ask me is why should we use the expected

shortfall because we have already, at our disposal,

another measure which is called the value at risk?

So in answer your question, what the advantages of the expected shortfall

with respect to the value at risk.

You have in fact two main advantages.

The first main advantage is that the expected shortfall is what is called

subadditive risk measure.

So let us look at the formula defined in subadditivity of the expected shortfall.

As you can see, we have the expected shortfall computed on a1 plus a2.

So it mean that I will have one portfolio made of a1 and

one portfolio made of a2, and I look at the sum.

The other two parts are the sum of the Expected Shortfall computed with only

the port when you made of a1 and only made of the portfolio made of a2.

And as you can see, the Expected Shortfall computed on a1 plus a2 will

be automatically lower than the sum of the two Expected Shortfall.

So, the two is measure computed on the individual portfolios.

So we have already heard about the type of notion which is called diversification.

So if I look at two portfolio, intuitively when I look at the two

portfolios individually, the sum of the two risk when I consider them individually

should be above the risk when I consider them combined.

So when I look at the joint position made up of a1 and a2.

So intuitively, this is a nice notion subadditivity because it's

the translation of the notion of diversification.

So the Expected Shortfall is always subadditive.

This is not necessarily the case for the value at risk.

So the value at risk doesn't ensure that you will have always that

the measure of risk on the sum of two portfolios will be always

lower than the sum of the risk measure computed on the two individual portfolios.

So this is the first advantage of the Expected Shortfall.

Now, what is the second advantage?

And I talked about that a little bit earlier in the video is that

the value at risk just gives you a single point in the PNL distribution.

So the only information that you have is that was a one person

probability level you will have a loss above the value at risk.

So for example above $1 million but you have no clue about whether

that loss will be $1 million, $2 million, or $10 million.

So when you look at the Expected Shortfall, you have a additional

information, which is the average loss, when you have a loss above $1 million.

So the Expected Shortfall give you an additional information.

So, now let us look at how we can compute the Expected Shortfall and

how can we define formally the Expected Shortfall.

So this is again some formula so

the first I'm going to let you see is an expectation.

So you have the expectation of the loss return.

Knowing that the lost return is above the value at risk.

So it is exactly the notion that I show you on the graph.

I look at the value at risk and

I look at the average losses knowing that my loss is above the value at risk.

So this is a conditional expectation.

So I will not enter too much into the detail, but an application of

what is called the biased yuremwhich defines conditions and expectation.

Allows you to rewrite that conditional expectation, so

the expectation of the loss return knowing so it's a conditional expectation.

So knowing something which is knowing that the loss return is above value at risk.

I can rewrite that as a standard expectation.

It will be the expectation of the loss return

multiplied by an indicator function.

And that indicator function tells the value one

if indeed you are above the value at risk and zero otherwise.

So this is the expectation and

you will divide by the probability of the conditioning event.

And here the conditioning event is simply that the loss return

is above the value at risk.

So now if you remember the definition of the value at risk.

By construction, by definition, the value at risk is so that the probability,

the loss return is above the value at risk is equal to 1 minus alpha.

And this gives you the final formula for the Expected Shortfall it will

be the average return multiplied by the integrate of function that

the loss return are above the value at risk divided by one minus five.

[MUSIC]

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