What is coherent risk measures?

Coherent risk measures are a type of risk measurement that satisfy certain properties that make them mathematically and economically sound. These properties include subadditivity, positive homogeneity, translation invariance, and monotonicity.

Subadditivity means that the risk measure of a portfolio is always less than or equal to the sum of the individual risk measures of its components. Positive homogeneity means that the risk measure of a portfolio is proportional to its size. Translation invariance means that the risk measure of a portfolio is not affected by a constant shift in its underlying values. Finally, monotonicity means that the risk measure of a portfolio decreases as the probability of a favorable outcome increases.

Coherent risk measures have several advantages over other risk measures. They are easy to compute, robust, and allow for the comparison of risk across different portfolios. They are also compatible with modern portfolio theory and can be used to optimize portfolio risk.

Some examples of coherent risk measures include Value at Risk (VaR), Expected Shortfall (ES), Conditional Value at Risk (CVaR), and Tail Conditional Expectation (TCE). VaR measures the maximum loss with a certain level of probability, while ES measures the expected loss given that the loss exceeded the VaR level. CVaR is similar to ES, but is more robust and better captures the tail risk of a portfolio. TCE is used to measure the expected loss in the tails of the distribution.