Comonotonicity

In probability theory, comonotonicity mainly refers to the perfect positive dependence between the components of a random vector, essentially saying that they can be represented as increasing functions of a single random variable. In two dimensions it is also possible to consider perfect negative dependence, which is called countermonotonicity.

Comonotonicity is also related to the comonotonic additivity of the Choquet integral.[1]

The concept of comonotonicity has applications in financial risk management and actuarial science, see e.g. Dhaene et al. (2002a) and Dhaene et al. (2002b). In particular, the sum of the components $X 1 + X 2 + \cdot \cdot \cdot + X n$ is the riskiest if the joint probability distribution of the random vector $(X 1, X 2, . . . , X n)$ is comonotonic.[2] Furthermore, the $α$ -quantile of the sum equals of the sum of the $α$ -quantiles of its components, hence comonotonic random variables are quantile-additive.[3][4] In practical risk management terms it means that there is minimal (or eventually no) variance reduction from diversification.

For extensions of comonotonicity, see Jouini & Napp (2004) and Puccetti & Scarsini (2010).

Definitions

Comonotonicity of subsets of $R n$

A subset $S$ of $R n$ is called comonotonic[5] (sometimes also nondecreasing[6]) if, for all $(x 1, x 2, . . . , x n)$ and $(y 1, y 2, . . . , y n)$ in $S$ with $x i < y i$ for some $i \in {1, 2, . . . , n$ }, it follows that $x j \leq y j$ for all $j \in {1, 2, . . . , n$ }.

This means that $S$ is a totally ordered set.

Comonotonicity of probability measures on $R n$

Let $μ$ be a probability measure on the $n$ -dimensional Euclidean space $R n$ and let $F$ denote its multivariate cumulative distribution function, that is

F(x_{1},\ldots ,x_{n}):=\mu {\bigl (}\{(y_{1},\ldots ,y_{n})\in {\mathbb {R} }^{n}\mid y_{1}\leq x_{1},\ldots ,y_{n}\leq x_{n}\}{\bigr )},\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n}.

Furthermore, let $F 1, . . . , F n$ denote the cumulative distribution functions of the $n$ one-dimensional marginal distributions of $μ$ , that means

F_{i}(x):=\mu {\bigl (}\{(y_{1},\ldots ,y_{n})\in {\mathbb {R} }^{n}\mid y_{i}\leq x\}{\bigr )},\qquad x\in {\mathbb {R} }

for every $i \in {1, 2, . . . , n$ }. Then $μ$ is called comonotonic, if

F(x_{1},\ldots ,x_{n})=\min _{i\in \{1,\ldots ,n\}}F_{i}(x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n}.

Note that the probability measure $μ$ is comonotonic if and only if its support $S$ is comonotonic according to the above definition.[7]

Comonotonicity of $R n$ -valued random vectors

An $R n$ -valued random vector $X = (X 1, . . . , X n)$ is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means

\Pr(X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})=\min _{i\in \{1,\ldots ,n\}}\Pr(X_{i}\leq x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n}.

Properties

An $R n$ -valued random vector $X = (X 1, . . . , X n)$ is comonotonic if and only if it can be represented as

(X_{1},\ldots ,X_{n})=_{\text{d}}(F_{X_{1}}^{-1}(U),\ldots ,F_{X_{n}}^{-1}(U)),\,

where $= d$ stands for equality in distribution, on the right-hand side are the left-continuous generalized inverses[8] of the cumulative distribution functions $F X 1, . . . , F X n$ , and $U$ is a uniformly distributed random variable on the unit interval. More generally, a random vector is comonotonic if and only if it agrees in distribution with a random vector where all components are non-decreasing functions (or all are non-increasing functions) of the same random variable.[9]

Upper bounds

Upper Fréchet–Hoeffding bound for cumulative distribution functions

Let $X = (X 1, . . . , X n)$ be an $R n$ -valued random vector. Then, for every $i \in {1, 2, . . . , n$ },

\Pr(X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})\leq \Pr(X_{i}\leq x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n},

hence

\Pr(X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})\leq \min _{i\in \{1,\ldots ,n\}}\Pr(X_{i}\leq x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n},

with equality everywhere if and only if $(X 1, . . . , X n)$ is comonotonic.

Upper bound for the covariance

Let $(X, Y)$ be a bivariate random vector such that the expected values of $X$ , $Y$ and the product $X Y$ exist. Let $(X *, Y *)$ be a comonotonic bivariate random vector with the same one-dimensional marginal distributions as $(X, Y)$ .[note 1] Then it follows from Höffding's formula for the covariance[10] and the upper Fréchet–Hoeffding bound that

{\text{Cov}}(X,Y)\leq {\text{Cov}}(X^{*},Y^{*})

and, correspondingly,

\operatorname {E} [XY]\leq \operatorname {E} [X^{*}Y^{*}]

with equality if and only if $(X, Y)$ is comonotonic.[11]

Note that this result generalizes the rearrangement inequality and Chebyshev's sum inequality.

Notes

$(X *, Y *)$ always exists, take for example $(F X -1 (U), F Y -1 (U))$ , see section Properties above.

Citations

(Sriboonchitta et al. 2010, pp. 149–152)
(Kaas et al. 2002, Theorem 6)
(Kaas et al. 2002, Theorem 7)
(McNeil, Frey & Embrechts 2005, Proposition 6.15)
(Kaas et al. 2002, Definition 1)
See (Nelsen 2006, Definition 2.5.1) for the case $n = 2$
See (Nelsen 2006, Theorem 2.5.4) for the case $n = 2$
(McNeil, Frey & Embrechts 2005, Proposition A.3 (properties of the generalized inverse))
(McNeil, Frey & Embrechts 2005, Proposition 5.16 and its proof)
(McNeil, Frey & Embrechts 2005, Lemma 5.24)
(McNeil, Frey & Embrechts 2005, Theorem 5.25(2))

References

Dhaene, Jan; Denuit, Michel; Goovaerts, Marc J.; Vyncke, David (2002a), "The concept of comonotonicity in actuarial science and finance: theory" (PDF), Insurance: Mathematics & Economics, 31 (1): 3–33, doi:10.1016/s0167-6687(02)00134-8, MR 1956509, Zbl 1051.62107, archived from the original (PDF) on 2008-12-09, retrieved 2012-08-28
Dhaene, Jan; Denuit, Michel; Goovaerts, Marc J.; Vyncke, David (2002b), "The concept of comonotonicity in actuarial science and finance: applications" (PDF), Insurance: Mathematics & Economics, 31 (2): 133–161, CiteSeerX 10.1.1.10.789, doi:10.1016/s0167-6687(02)00135-x, MR 1932751, Zbl 1037.62107, archived from the original (PDF) on 2008-12-09, retrieved 2012-08-28
Jouini, Elyès; Napp, Clotilde (2004), "Conditional comonotonicity" (PDF), Decisions in Economics and Finance, 27 (2): 153–166, doi:10.1007/s10203-004-0049-y, ISSN 1593-8883, MR 2104639, Zbl 1063.60002
Kaas, Rob; Dhaene, Jan; Vyncke, David; Goovaerts, Marc J.; Denuit, Michel (2002), "A simple geometric proof that comonotonic risks have the convex-largest sum" (PDF), ASTIN Bulletin, 32 (1): 71–80, doi:10.2143/ast.32.1.1015, MR 1928014, Zbl 1061.62511
McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul (2005), Quantitative Risk Management. Concepts, Techniques and Tools, Princeton Series in Finance, Princeton, NJ: Princeton University Press, ISBN 978-0-691-12255-7, MR 2175089, Zbl 1089.91037
Nelsen, Roger B. (2006), An Introduction to Copulas, Springer Series in Statistics (second ed.), New York: Springer, pp. xiv+269, ISBN 978-0-387-28659-4, MR 2197664, Zbl 1152.62030
Puccetti, Giovanni; Scarsini, Marco (2010), "Multivariate comonotonicity" (PDF), Journal of Multivariate Analysis, 101 (1): 291–304, doi:10.1016/j.jmva.2009.08.003, ISSN 0047-259X, MR 2557634, Zbl 1184.62081
Sriboonchitta, Songsak; Wong, Wing-Keung; Dhompongsa, Sompong; Nguyen, Hung T. (2010), Stochastic Dominance and Applications to Finance, Risk and Economics, Boca Raton, FL: Chapman & Hall/CRC Press, ISBN 978-1-4200-8266-1, MR 2590381, Zbl 1180.91010

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[10] $(X *, Y *)$ always exists, take for example $(F X -1 (U), F Y -1 (U))$ , see section Properties above.

[1] (Sriboonchitta et al. 2010, pp. 149–152)

[2] (Kaas et al. 2002, Theorem 6)

[3] (Kaas et al. 2002, Theorem 7)

[4] (McNeil, Frey & Embrechts 2005, Proposition 6.15)

[5] (Kaas et al. 2002, Definition 1)

[6] See (Nelsen 2006, Definition 2.5.1) for the case $n = 2$

[7] See (Nelsen 2006, Theorem 2.5.4) for the case $n = 2$

[8] (McNeil, Frey & Embrechts 2005, Proposition A.3 (properties of the generalized inverse))

[9] (McNeil, Frey & Embrechts 2005, Proposition 5.16 and its proof)

[11] (McNeil, Frey & Embrechts 2005, Lemma 5.24)

[12] (McNeil, Frey & Embrechts 2005, Theorem 5.25(2))