# Coherent risk measure

In the field of financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function $\varrho$ that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

## Properties

Consider a random outcome $X$ viewed as an element of a linear space $\mathcal{L}$ of measurable functions, defined on an appropriate probability space. A functional $\varrho : \mathcal{L}$$\R \cup \{+\infty\}$ is said to be coherent risk measure for $\mathcal{L}$ if it satisfies the following properties:[1]

Normalized
$\varrho(0) = 0$

That is, the risk of holding no assets is zero.

Monotonicity
$\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} \;\mathrm{and}\; Z_1 \leq Z_2 \; \mathrm{a.s.} ,\; \mathrm{then} \; \varrho(Z_1) \geq \varrho(Z_2)$

That is, if portfolio $Z_2$ always has better values than portfolio $Z_1$ under almost all scenarios then the risk of $Z_2$ should be less than the risk of $Z_1$.[2] E.g. If $Z_1$ is an in the money call option (or otherwise) on a stock, and $Z_2$ is also an in the money call option with a lower strike price.

$\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} ,\; \mathrm{then}\; \varrho(Z_1 + Z_2) \leq \varrho(Z_1) + \varrho(Z_2)$

Indeed, the risk of two portfolios together cannot get any worse than adding the two risks separately: this is the diversification principle.

Positive homogeneity
$\mathrm{If}\; \alpha \ge 0 \; \mathrm{and} \; Z \in \mathcal{L} ,\; \mathrm{then} \; \varrho(\alpha Z) = \alpha \varrho(Z)$

Loosely speaking, if you double your portfolio then you double your risk.

Translation invariance

If $A$ is a deterministic portfolio with guaranteed return $a$ and $Z \in \mathcal{L}$ then

$\varrho(Z + A) = \varrho(Z) - a$

The portofolio $A$ is just adding cash $a$ to your portfolio $Z$. In particular, if $a=\varrho(Z)$ then $\varrho(Z+A)=0$.

### Convex risk measures

The notion of coherence has been subsequently relaxed. Indeed, the notions of Sub-additivity and Positive Homogeneity can be replaced by the notion of convexity:[3]

Convexity
$\text{If }Z_1,Z_2 \in \mathcal{L}\text{ and }\lambda \in [0,1] \text{ then }\varrho(\lambda Z_1 + (1-\lambda) Z_2) \leq \lambda \varrho(Z_1) + (1-\lambda) \varrho(Z_2)$

## General Framework of Wang Transform

Wang transform of the decumulative distribution function

A Wang transform of the decumulative distribution function is an increasing function $g \colon [0,1] \rightarrow [0,1]$ where $g(0)=0$ and $g(1)=1$. [4] This function is called distortion function or Wang transform function.

The dual distortion function is $\tilde{g}(x) = 1 - g(1-x)$.[5][6] Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$, then for any random variable $X$ and any distortion function $g$ we can define a new probability measure $\mathbb{Q}$ such that for any $A \in \mathcal{F}$ it follows that $\mathbb{Q}(A) = g(\mathbb{P}(X \in A)).$ [5]

For any increasing concave Wang transform function, we could define a corresponding premium principle :[4] $\varrho(X)=\int_0^{+\infty}g\left(\bar{F}_X(x)\right) dx$

Coherent risk measure

A coherent risk measure could be defined by a Wang transform of the decumulative distribution function $g$ if on only if $g$ is concave.[4]

## Examples of risk measure

### Value at risk

It is well known that value at risk is not, in general, a coherent risk measure as it does not respect the sub-additivity property. An immediate consequence is that value at risk might discourage diversification.[1] Value at risk is, however, coherent, under the assumption of elliptically distributed losses (e.g. normally distributed) when the portfolio value is a linear function of the asset prices. However, in this case the value at risk becomes equivalent to a mean-variance approach where the risk of a portfolio is measured by the variance of the portfolio's return.

The Wang transform function (distortion function) for the Value at Risk is $g(x)=\mathbf{1}_{x\geq 1-\alpha}$. The non-concavity of $g$ proves the non coherence of this risk measure.

Illustration

As a simple example to demonstrate the non-coherence of value-at-risk consider looking at the VaR of a portfolio at 95% confidence over the next year of two default-able zero coupon bonds that mature in 1 years time denominated in our numeraire currency.

Assume the following:

• The current yield on the two bonds is 0%
• The two bonds are from different issuers
• Each bonds has a 4% probability of defaulting over the next year
• The event of default in either bond is independent of the other
• Upon default the bonds have a recovery rate of 30%

Under these conditions the 95% VaR for holding either of the bonds is 0 since the probability of default is less than 5%. However if we held a portfolio that consisted of 50% of each bond by value then the 95% VaR is 35% since the probability of at least one of the bonds defaulting is 7.84% which exceeds 5%. This violates the sub-additivity property showing that VaR is not a coherent risk measure.

### Average value at risk

The average value at risk (sometimes called expected shortfall or conditional value-at-risk) is a coherent risk measure, even though it is derived from Value at Risk which is not.

### Entropic value at risk

The entropic value at risk is a coherent risk measure.[7]

### Tail value at risk

The tail value at risk (or tail conditional expectation) is a coherent risk measure only when the underlying distribution is continuous.

The Wang transform function (distortion function) for the tail value at risk is $g(x)=\min(\frac{x}{\alpha},1)$. The concavity of $g$ proves the coherence of this risk measure in the case of continuous distribution.

### Proportional Hazard (PH) risk measure

The PH risk measure (or Proportional Hazard Risk measure) transforms the hasard rates $\scriptstyle \left( \lambda(t) = \frac{f(t)}{\bar{F}(t)}\right)$ using a coefficient $\xi$.

The Wang transform function (distortion function) for the PH risk measure is $g_{\alpha}(x) = x^{\xi}$. The concavity of $g$ if $\scriptstyle \xi<\frac{1}{2}$ proves the coherence of this risk measure.

Sample of Wang transform function or distortion function

### g-Entropic risk measures

g-entropic risk measures are a class of information-theoretic coherent risk measures that involve some important cases such as CVaR and EVaR.[7]

### The Wang risk measure

The Wang risk measure is define by the following Wang transform function (distortion function) $g_{\alpha}(x)=\Phi\left[ \Phi^{-1}(x)-\Phi^{-1}(\alpha)\right]$. The coherence of this risk measure is a consequence of the concavity of $g$.

### Entropic risk measure

The entropic risk measure is a convex risk measure which is not coherent. It is related to the exponential utility.

### Superhedging price

The superhedging price is a coherent risk measure.

## Set-valued

In a situation with $\mathbb{R}^d$-valued portfolios such that risk can be measured in $n \leq d$ of the assets, then a set of portfolios is the proper way to depict risk. Set-valued risk measures are useful for markets with transaction costs.[8]

### Properties

A set-valued coherent risk measure is a function $R: L_d^p \rightarrow \mathbb{F}_M$, where $\mathbb{F}_M = \{D \subseteq M: D = cl (D + K_M)\}$ and $K_M = K \cap M$ where $K$ is a constant solvency cone and $M$ is the set of portfolios of the $m$ reference assets. $R$ must have the following properties:[9]

Normalized
$K_M \subseteq R(0) \; \mathrm{and} \; R(0) \cap -\mathrm{int}K_M = \emptyset$
Translative in M
$\forall X \in L_d^p, \forall u \in M: R(X + u1) = R(X) - u$
Monotone
$\forall X_2 - X_1 \in L_d^p(K) \Rightarrow R(X_2) \supseteq R(X_1)$
Sublinear

### Set-valued convex risk measure

If instead of the sublinear property,R is convex, then R is a set-valued convex risk measure.

## Dual representation

A lower semi-continuous convex risk measure $\varrho$ can be represented as

$\varrho(X) = \sup_{Q \in \mathcal{M}(P)} \{E^Q[-X] - \alpha(Q)\}$

such that $\alpha$ is a penalty function and $\mathcal{M}(P)$ is the set of probability measures absolutely continuous with respect to P (the "real world" probability measure), i.e. $\mathcal{M}(P) = \{Q \ll P\}$.

A lower semi-continuous risk measure is coherent if and only if it can be represented as

$\varrho(X) = \sup_{Q \in \mathcal{Q}} E^Q[-X]$

such that $\mathcal{Q} \subseteq \mathcal{M}(P)$.[10]