Coherent risk measure

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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.


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]

\varrho(0) = 0

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

\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[edit]

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]

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[edit]

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]

Actuarial premium principle

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[edit]

Value at risk[edit]

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.


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[edit]

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[edit]

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

Tail value at risk[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

Superhedging price[edit]

The superhedging price is a coherent risk measure.


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]


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]

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
\forall X_2 - X_1 \in L_d^p(K) \Rightarrow R(X_2) \supseteq R(X_1)

Set-valued convex risk measure[edit]

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

Dual representation[edit]

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]

See also[edit]


  1. ^ a b Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D. (1999). "Coherent Measures of Risk". Mathematical Finance 9 (3): 203. doi:10.1111/1467-9965.00068.  edit
  2. ^ Wilmott, P. (2006). "Quantitative Finance" 1 (2 ed.). Wiley. p. 342. 
  3. ^ Föllmer, H.; Schied, A. (2002). "Convex measures of risk and trading constraints". Finance and Stochastics 6 (4): 429–447. doi:10.1007/s007800200072. 
  4. ^ a b c Wang, Shuan (1996). "Premium Calculation by Transforming the Layer Premium Density". ASTIN Bulletin 26 (1): 71–92. doi:10.2143/ast.26.1.563234. 
  5. ^ a b Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability 11 (3): 385. doi:10.1007/s11009-008-9089-z.  edit
  6. ^ Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012. 
  7. ^ a b Ahmadi-Javid, Amir (2012). "Entropic value-at-risk: A new coherent risk measure". Journal of Optimization Theory and Applications 155 (3): 1105–1123. doi:10.1007/s10957-011-9968-2. 
  8. ^ Jouini, Elyes; Meddeb, Moncef; Touzi, Nizar (2004). "Vector–valued coherent risk measures". Finance and Stochastics 8 (4): 531–552. doi:10.1007/s00780-004-0127-6. 
  9. ^ Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk" (pdf). SIAM Journal on Financial Mathematics 1 (1): 66–95. doi:10.1137/080743494. Retrieved August 17, 2012.  edit
  10. ^ Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. ISBN 978-3-11-018346-7. 

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