In this class we introduce the concept of coherent risk measure.

A measure of risk is said coherent when, in mathematical terms, it possesses 4 important properties: positive homogeneity, translation invariance, sub-additivity and monotonocity.
Naturally, all these properties have an economic/financial meaning, and we will discuss it.

Believe it or not, despite being one of the queen measures in (credit) risk management, Value-at-Risk is not coherent. Not always, at least. In particular, using a simple example, we will show that the VaR is not generally sub-additive (hence not coherent, given that sub-additivity is one of the four properties). It can be, but often it is not.
What are the implications of this? We will see together.

Conversely, Expected Shortfall is always a coherent measure of risk.

In the second part of the lecture, we will deal with back-testing.
Back-testing is a statistical tool that risk managers use to verify the accuracy and the reliability of the estimated Value-at-Risk. It is a procedure based on historical data, and it belongs to the wide class of statistical hypothesis tests.
The basic approach we will discuss relies on the binomial distribution. It may be worth to have a look at it, before starting this class.

Here you can find the slides of this class.
(New version: we have corrected a typo in the value of ES, pages 9-10)

The script is here