Kinetic Monte Carlo

2013-10-15T00:50:54Z

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{{Merge|Coherent risk measure|Distortion function|date=January 2014|section= General Framework of Wang Transform}}

{{Technical|date=August 2013}}
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 <math>\varrho</math> that satisfies properties of [[monotonicity]], [[Sub-additive|sub-additivity]], [[homogeneity (statistics)|homogeneity]], and [[translational invariance]].

==Properties==
Consider a random outcome <math> X</math> viewed as an element of a linear space <math> \mathcal{L}</math> of measurable functions, defined on an appropriate probability space. A [[functional (mathematics)|functional]] <math>\varrho : \mathcal{L}</math> → <math>\R \cup \{+\infty\}</math> is said to be coherent risk measure for <math> \mathcal{L}</math> if it satisfies the following properties:<ref name="Artzner">{{cite doi|10.1111/1467-9965.00068}}</ref>

; Normalized
: <math>\varrho(0) = 0</math>
That is, the risk of holding no assets is zero.

; Monotonicity
: <math>\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)</math>
That is, if portfolio <math>Z_2</math> always has better values than portfolio <math>Z_1</math> under [[almost surely|almost all]] scenarios then the risk of <math>Z_2</math> should be less than the risk of <math>Z_1</math>.<ref>{{cite journal|last=Wilmott|first=P.|year=2006|title=Quantitative Finance|publisher=Wiley|edition=2|volume=1|page=342}}</ref> E.g. If <math>Z_1</math> is an in the money call option (or otherwise) on a stock, and <math>Z_2</math> is also an in the money call option with a lower strike price.

; Sub-additivity
: <math>\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} ,\; \mathrm{then}\; \varrho(Z_1 + Z_2) \leq \varrho(Z_1) + \varrho(Z_2)</math>
Indeed, the risk of two portfolios together cannot get any worse than adding the two risks separately: this is the [[Diversification (finance)|diversification]] principle.

; Positive homogeneity
: <math>\mathrm{If}\; \alpha \ge 0 \; \mathrm{and} \; Z \in \mathcal{L} ,\; \mathrm{then} \; \varrho(\alpha Z) = \alpha \varrho(Z)</math>
Loosely speaking, if you double your portfolio then you double your risk.

; Translation invariance
If <math> A</math> is a deterministic portfolio with guaranteed return <math> a</math> and <math> Z \in \mathcal{L}</math> then
: <math>\varrho(Z + A) = \varrho(Z) - a </math>
The portofolio <math> A</math> is just adding cash <math>a</math> to your portfolio <math>Z</math>. In particular, if <math>a=\varrho(Z)</math> then <math>\varrho(Z+A)=0</math>.

===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 [[convex function|convexity]]:<ref>{{cite journal|last=Föllmer|first=H.|last2=Schied|first2=A.|year=2002|title=Convex measures of risk and trading constraints|journal=Finance and Stochastics|volume=6|issue=4|pages=429–447}}</ref>
; Convexity
: <math>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)</math>

== General Framework of Wang Transform ==

;Wang transform of the decumulative distribution function

A Wang transform of the decumulative distribution function is an increasing function <math> g \colon [0,1] \rightarrow [0,1]</math> where <math> g(0)=0</math> and <math> g(1)=1</math>. <ref name="Wang">{{cite journal|last=Wang|first=Shuan|year=1996|title=Premium Calculation by Transforming the Layer Premium Density|journal=ASTIN Bulletin|volume=26|issue=1|pages=71–92}}</ref> This function is called ''distortion function'' or Wang transform function.

The ''dual distortion function'' is <math>\tilde{g}(x) = 1 - g(1-x)</math>.<ref name="PropertiesDRM">{{cite doi|10.1007/s11009-008-9089-z}}</ref><ref name="Wirch">{{cite web|title=Distortion Risk Measures: Coherence and Stochastic Dominance|author=Julia L. Wirch|author2=Mary R. Hardy|url=http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|format=pdf|accessdate=March 10, 2012}}</ref>
Given a [[probability space]] <math>(\Omega,\mathcal{F},\mathbb{P})</math>, then for any [[random variable]] <math>X</math> and any distortion function <math>g</math> we can define a new [[probability measure]] <math>\mathbb{Q}</math> such that for any <math>A \in \mathcal{F}</math> it follows that
<math>\mathbb{Q}(A) = g(\mathbb{P}(X \in A)).</math> <ref name="PropertiesDRM"/>

;Actuarial premium principle

For any increasing concave Wang transform function, we could define a corresponding premium principle :<ref name="Wang"/>
<math> \varrho(X)=\int_0^{+\infty}g\left(\bar{F}_X(x)\right) dx</math>

;Coherent risk measure

A coherent risk measure could be defined by a Wang transform of the decumulative distribution function <math>g</math> if on only if <math>g</math> is concave.<ref name="Wang"/>

==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.<ref name="Artzner"/>
[[Value at risk]] is, however, coherent, under the assumption of [[Elliptical distribution|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 <math> g(x)=\mathbf{1}_{x\geq 1-\alpha}</math>. The non-concavity of <math> g</math> 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 default]]ing 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.<ref name=Ahmadi2>{{cite journal|last=Ahmadi-Javid|first=Amir|title=Entropic value-at-risk: A new coherent risk measure|journal=Journal of Optimization Theory and Applications|year=2012|volume=155|pages=1105–1123|doi=10.1007/s10957-011-9968-2|issue=3}}</ref>

===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 distribution|continuous]].

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

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

The Wang transform function (distortion function) for the PH risk measure is <math> g_{\alpha}(x) = x^{\xi} </math>. The concavity of <math> g</math> if <math>\scriptstyle \xi<\frac{1}{2}</math> proves the coherence of this risk measure.
[[File:Sample_of_Wang_transform_function_or_distortion_function.png|thumb|right|Sample of Wang transform function or distortion function]]

===g-Entropic risk measures===
[[g-entropic risk measure]]s are a class of information-theoretic coherent risk measures that involve some important cases such as CVaR and EVaR.<ref name=Ahmadi2 />

===The Wang risk measure===

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

===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 <math>\mathbb{R}^d</math>-valued portfolios such that risk can be measured in <math>n \leq d</math> 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 cost]]s.<ref>{{cite journal|last=Jouini|first=Elyes|last2=Meddeb|first2=Moncef|last3=Touzi|first3=Nizar|year=2004|title=Vector–valued coherent risk measures|journal=Finance and Stochastics|volume=8|issue=4|pages=531–552}}</ref>

===Properties===
A set-valued coherent risk measure is a function <math>R: L_d^p \rightarrow \mathbb{F}_M</math>, where <math>\mathbb{F}_M = \{D \subseteq M: D = cl (D + K_M)\}</math> and <math>K_M = K \cap M</math> where <math>K</math> is a constant [[solvency cone]] and <math>M</math> is the set of portfolios of the <math>m</math> reference assets. <math>R</math> must have the following properties:<ref>{{cite doi|10.1137/080743494}}</ref>

; Normalized
: <math>K_M \subseteq R(0) \; \mathrm{and} \; R(0) \cap -\mathrm{int}K_M = \emptyset</math>

; Translative in M
: <math>\forall X \in L_d^p, \forall u \in M: R(X + u1) = R(X) - u</math>

; Monotone
: <math>\forall X_2 - X_1 \in L_d^p(K) \Rightarrow R(X_2) \supseteq R(X_1)</math>

; 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 <math>\varrho</math> can be represented as
: <math>\varrho(X) = \sup_{Q \in \mathcal{M}(P)} \{E^Q[-X] - \alpha(Q)\}</math>
such that <math>\alpha</math> is a [[Penalty function (risk)|penalty function]] and <math>\mathcal{M}(P)</math> is the set of probability measures [[absolutely continuous]] with respect to ''P'' (the "real world" [[probability measure]]), i.e. <math>\mathcal{M}(P) = \{Q \ll P\}</math>.

A [[lower semi-continuous]] risk measure is coherent if and only if it can be represented as
: <math>\varrho(X) = \sup_{Q \in \mathcal{Q}} E^Q[-X]</math>
such that <math>\mathcal{Q} \subseteq \mathcal{M}(P)</math>.<ref>{{cite book|first1=Hans|last1=Föllmer|first2=Alexander|last2=Schied|title=Stochastic finance: an introduction in discrete time|publisher=Walter de Gruyter|year=2004|edition=2|isbn=978-3-11-018346-7}}</ref>

==See also==
* [[Risk metric]] - the abstract concept that a risk measure quantifies
* [[RiskMetrics]] - a model for risk management
* [[Spectral risk measure]] - a subset of coherent risk measures
* [[Distortion risk measure]]
* [[Conditional value-at-risk]]
* [[Entropic value at risk]]
* [[Financial risk]]

==References==
{{reflist|30em}}

==External links==
* [http://www.princeton.edu/~dito/riskmeasures/CohConMon/ A list of important papers on coherent and convex risk measures]

[[Category:Actuarial science]]
[[Category:Mathematical finance]]
[[Category:Financial risk]]

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