## What are the four properties of a coherent risk measure?

A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

**Which risk measures are coherent?**

A coherent risk measure imposes specific, seemingly innocuous, technical requirements for a risk measure (ρ): 1) ρ [0] = 0 The risk of nothing is zero. This is referred to as normalization. 2) Consider two random outcomes, Z1 and Z2, e.g., the returns from two portfolios or capital assets.

### Why is VaR not considered a coherent measure of risk?

In other words, VaR is not a “coherent” measure of risk. This problem is caused by the fact that VaR is a quantile on the distribution of profit and loss and not an expectation, so that the shape of the tail before and after the VaR probability need not have any bearing on the actual VaR number.

**Is CTE a coherent risk measure?**

Academics have lauded CTE as a “coherent” statistic. Those outside the in- surance industry call it “Tail VaR” or “expected tail loss” (ETL). Actuaries, who have always been suspicious or even hostile to the usage of value at risk (VaR) as a risk measurement stan- dard, have readily embraced CTE.

#### What is monotonicity in coherent risk measure?

Monotonicity means that, if Z1 and Z2 are two losses and Z1 is smaller than Z2, then the value of the risk measure in Z2 is greater than the value of the risk measure in Z1.

**Is standard deviation a coherent risk measure?**

The standard deviation is always coherent. Notice that standard deviation, in finance, is often called volatility.

## Which of the following is a property of a coherent risk metric?

(1999) defined that a risk measure is coherent if it satisfies the following four properties: monotonicity, positive homogeneity, sub-additivity and translation invari- ance.

**Is variance a coherent risk measure?**

Properties VaR follows the properties of monotonicity, positive homogeneity and translation invariance. It is also considered a monetary risk measure because it satisfies monotonicity and translation invariance. The big drawback to using VaR over other risk measures is VaR is not coherent.

### Is VaR a spectral measure?

A spectral risk measure can be regarded as a weighted average of losses (or VaRs) at all possible confidence levels, where the weights chosen reflect not just the probabilities associated with the losses but the strength of the user’s risk-aversion.

**Why is expected shortfall better than VaR?**

A risk measure can be characterised by the weights it assigns to quantiles of the loss distribution. VAR gives a 100% weighting to the Xth quantile and zero to other quantiles. Expected shortfall gives equal weight to all quantiles greater than the Xth quantile and zero weight to all quantiles below the Xth quantile.