# Coherent risk measure Canonical: https://web4.xpay.bi/en/wiki/wp-en-5235067/ Language: en Published: 2026-10-26T16:35:39.444Z Updated: 2026-10-26T16:35:39.444Z In the fields of actuarial science and 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 that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. ## Key facts ## Overview In the fields of actuarial science and 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 that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. ## Properties Consider a random outcome X \{\\displaystyle X\} viewed as an element of a linear space L \{\\displaystyle \{\\mathcal \{L\}\}\} of measurable functions, defined on an appropriate probability space. A functional ϱ : L \{\\displaystyle \\varrho :\{\\mathcal \{L\}\}\} → R ∪ \{ + ∞ \} \{\\displaystyle \\mathbb \{R\} \\cup \\\{+\\infty \\\}\} is said to be coherent risk measure for L \{\\displaystyle \{\\mathcal \{L\}\}\} if it satisfies the following properties: ## Normalized ϱ ( 0 ) = 0 \{\\displaystyle \\varrho (0)=0\} That is, the risk when holding no assets is zero. ## Monotonicity I f Z 1 , Z 2 ∈ L a n d Z 1 ≤ Z 2 a . s . , t h e n ϱ ( Z 1 ## Sub-additivity I f Z 1 , Z 2 ∈ L , t h e n ϱ ( Z 1 + Z 2 ) ≤ ϱ ( Z 1 ) + ϱ ( ## Positive homogeneity I f α ≥ 0 a n d Z ∈ L , t h e n ϱ ( α Z ) = α ϱ ( Z ) \{\\displaystyle \\mathrm \{If\} \\;\\alpha \\geq 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. In financial risk management, positive homogeneity implies the risk of a position is proportional to its size. ## Translation invariance If A \{\\displaystyle A\} is a deterministic portfolio with guaranteed return a \{\\displaystyle a\} and Z ∈ L \{\\displaystyle Z\\in \{\\mathcal \{L\}\}\} then ϱ ( Z + A ) = ϱ ( Z ) − a \{\\displaystyle \\varrho (Z+A)=\\varrho (Z)-a\} The portfolio A \{\\displaystyle A\} is just adding cash a \{\\displaystyle a\} to your portfolio ## 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: Convexity If Z 1 , Z 2 ∈ L and λ ∈ \[ 0 , 1 \] then ϱ ( λ Z 1 + ( 1 − λ ) Z 2 ) ≤ λ ϱ ( ## Value at risk It is well known that value at risk is not a coherent risk measure as it does not respect the sub-additivity property. An immediate consequence is that value at risk might discourage diversification. 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 ) = 1 x ≥ 1 − α \{\\displaystyle g(x)=\\mathbf \{1\} \_\{x\\geq 1-\\alpha \}\} . The non-concavity of g \{\\displaystyle 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 bond 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% (= 0.5\*0.7 + 0.5\*0) since the probability of at least one of the bonds defaulting is 7.84% (= 1 - 0.96\*0.96) 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 or A V a R \{\\displaystyle AVaR\} ) is a coherent risk measure, even though it is derived from Value at Risk which is not. The domain can be extended for more general Orlitz Hearts from the more typical Lp spaces. ## Entropic value at risk The entropic value at risk is a coherent risk measure. ## 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 ( x α , 1 ) \{\\displaystyle g(x)=\\min(\{\\frac \{x\}\{\\alpha \}\},1)\} . The concavity of g \{\\displaystyle g\} proves the coherence of this risk measure in the case of continuous distribution. ## Sources - Coherent risk measure · Wikipedia: https://en.wikipedia.org/?curid=5235067 Source: Wikipedia contributors — https://en.wikipedia.org/?curid=5235067&oldid=1375037158 Licence: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation notice: Reformatted for XPay; the source may be abridged, restructured or translated. 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