This document summarizes Arthur Charpentier's presentation at the Rennes Risk Workshop in April 2015. It discusses extending concepts of risk from univariate to multivariate prospects, including characterizing attitudes to multivariate notions of increasing risk like the Rothschild-Stiglitz mean preserving increase in risk and Quiggin's monotone mean preserving increase in risk. It also generalizes the Bickel-Lehmann dispersion order to multivariate risks and examines its implications for risk sharing.

1. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Local utility and multivariate risk aversion
Arthur Charpentier, Alfred Galichon & Marc Henry
Rennes Risk Workshop, April 2015
to appear in Mathematics of Operations Research
http ://papers.ssrn.com/id=1982293
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2. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Choice between multivariate risky prospects
– In view of violations of expected utility, a vast literature has emerged on other
functional evaluations of risky prospects, particularly Rank Dependent
Preferences (RDU).
– Simultaneously, a literature developed on the characterization of attitudes to
multiple non substitutable risks and multivariate risk premia within the
framework of expected utility.
– Here we characterize attitude to multivariate generalizations of standard
notions of increasing risk with local utility
– Rothschild-Stiglitz mean preserving increase in risk
– Quiggin monotone mean preserving increase in risk
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3. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Expected utility
– Decision maker ranks random vectors X with law invariant functional
Φ(X) = Φ(FX), with FX the cdf of X.
– Utility x → U(x) :
Φ(F) = U(x)dF(x)
Quiggin-Yaari functional
– Distortion x → φ(x) :
Φ(F) = φ(t)F−1
(t)dt =
x
−∞
φ(F(s))ds
U(x,F )
dF(x)
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4. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Aumann & Serrano Riskiness
– Given X, the index of risk R(X) is defined to be the unique positive solution
(if exists) of E[exp(−X/R(X))] = 1
– Index of Riskiness x → R :
R such that exp −
x
R
U(x)
dF(x) = 1
where U is CARA.
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5. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Local utility
– Decision maker ranks random vectors X with law invariant functional
Φ(X) = Φ(FX), with FX the cdf of X.
– Local utility x → U(x; F) is the Fréchet derivative of Φ at F :
Φ(F ) − Φ(F) − U(x; F)[dF (x) − dF(x)] → 0
– If Φ is expected utility, local and global utilities coincide
– If Φ is the Quiggin-Yaari functional Φ(X) = φ(t)F−1
X (t)dt, then local
utility is U(x; F) =
x
−∞
φ(F(z))dz
– If Φ is the Aumann-Serrano index of riskiness, the local utility
cst[1 − exp(−αx)] is CARA
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6. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Rothschild-Stiglitz mean preserving increase in risk
One of the most commonly used stochastic orderings to compare risky prospects
is the mean preserving increase in risk (MPIR or concave ordering). Let X and Y
be two prospects.
Definition :
Y MP IR X if E[u(X)] ≥ E[u(Y )] for all concave utility u.
Characterization :
∗ Y
L
= X + Z with E[Z|X] = 0 (where “
L
=” denotes equality in distribution).
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7. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Local utility and MPIR
– Aversion to MPIR is equivalent to concavity of local utility (Machina 1982 for
the univariate result)
– Still holds for multivariate risks :
– Φ is MPIR averse (Schur concave) if Φ(Y ) ≤ Φ(X) when Y is an MPIR of
X.
– Equivalent to concavity of U(x; F) in x for all distributions F
Proof : Φ Schur concave iff Φ decreasing along all martingales Xt
Φ(Xt+dt) − Φ(Xt) = E [U(Xt+dt; FXt ) − U(Xt; FXt )]
= E Tr D2
U(Xt; FXt )σT
t σt Itô
≤ 0 iff U(x; F) concave
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8. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Two shortcomings of MPIR in the theory of risk sharing
– Arrow-Pratt more risk averse decision makers do not necessarily pay more
(than less risk averse ones) for a mean preserving decrease in risk.
– Ross (Econometrica 1981)
– Partial insurance contracts offering mean preserving reduction in risk can be
Pareto dominated.
– Landsberger and Meilijson (Annals of OR 1994)
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9. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Bickel-Lehmann dispersion order
These shortcomings are not shared by the Bickel-Lehmann dispersion order
(Bickel and Lehmann 1979). Let X and Y have cdfs FX and FY and quantiles
QX = F−1
X and QY = F−1
Y .
Definition :
Y Disp X if QY (s) − QY (s ) ≥ QX(s) − QX(s ).
Characterization (Landsberger-Meilijson) :
Y
L
= X + Z with Z and X comonotonic,
Examples :
– Normal, exponential and uniform families are dispersion ordered by the variance.
– Arrow (1970) stretches of a distribution X → x + α(X − x), α > 1, are more
dispersed.
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10. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Local utility and attitude to mean preserving dispersion
increase (Quiggin’s monotone MPIR)
– Φ is MMPIR averse if and only if
E
U (X; FX)
E[U (X; FX)]
1{X > x} ≤ E [1{X > x}]
– Example : Quiggin-Yaari functional
Φ(X) = φ(t)F−1
X (t)dt
with local utility is
U(x; F) =
x
−∞
φ(FX(z))dz
is MMPIR averse iff density φ(u) is stochastically dominated by the uniform
(called pessimism by Quiggin)
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11. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Risk sharing and dispersion
– More risk averse decision makers will always pay at least as much (as less risk
averse agents) for a decrease in risk if and only if it is Bickel-Lehmann less
dispersed.
– Landsberger and Meilijson (Management Science 1994)
– Unless the uninsured position is Bickel-Lehmann more dispersed than the
insured position, the existing contract can be improved so as to raise the
expected utility of both parties, regardless of their (concave) utility functions.
– Landsberger and Meilijson (Annals of OR 1994)
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12. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Partial insurance
Consider the following insurance contract :
Insuree Insurer
Before Y 0
After X1 X2
– By construction Y = X1 + X2.
– (X1, X2) is Pareto efficient if and only if comonotonic (Landsberger-Meilijson).
– Hence the contract is efficient iff Y = X1 + X2 Disp X1.
We generalize this result to the case of multivariate risks.
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13. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Multivariate extension of Bickel-Lehmann and Monotone
MPIR
Quantiles and comonotonicity feature in the definition and the characterization
of the Bickel-Lehmann dispersion ordering. They seem to rely on the ordering on
the real line.
However we can revisit these notions to provide
– Multivariate notion of comonotonicity
– Multivariate quantile definition
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14. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Revisiting comonotonicity
– X and Y are comonotonic if there exists Z such that X = TX(Z) and
Y = TY (Z), TX, TY increasing functions.
– Example :
– If X(ωi) = xi and Y (ωi) = yi, i = 1 . . . , n, with x1 ≤ . . . ≤ xn and
y1 ≤ . . . ≤ yn, then X and Y are comonotonic.
– By the simple rearrangement inequality,
i=1,...,n
xiyi = max



i=1,...,n
xiyσ(i) : σ permutation



.
– General characterization : X and Y are comonotonic iff
E[XY ] = sup E[X ˜Y ] : ˜Y
L
= Y .
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15. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Revisiting the quantile function
The quantile function of a prospect X is the generalized inverse of the cumulative
distribution function :
u → QX(u) = inf{x : P(X ≤ x) ≥ u}
Equivalent characterizations :
– The quantile function QX of a prospect X is an increasing rearrangement of X,
– The quantile QX(U) of X is the version of X which is comonotonic with the
uniform random variable U on [0, 1].
– QX is the only l.s.c. increasing function such that
E[QX(U)U] = sup{E[ ˜XU]; ˜X
L
= X}.
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16. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Multivariate µ-quantiles and µ-comonotonicity, (Galichon and
Henry, JET 2012)
– The univariate quantile function of a random variable X is the only l.s.c.
increasing function such that
E[QX(U)U] = sup{E[ ˜XU]; ˜X
L
= X}.
– Similarly, the µ-quantile QX is the essentially unique gradient of a l.s.c.
convex function (Brenier, CPAM 1991),
E[ QX(U), U ] = sup{E[ ˜X, U ]; ˜X
L
= X}, for some U
L
= µ.
– X and Y are µ-comonotonic if for some U
L
= µ,
X = QX(U) and Y = QY (U),
namely if X and Y can be simultaneously rearranged relative to a reference
distribution µ.
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17. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Example : Gaussian prospects
Suppose the baseline U is standard normal,
– X ∼ N(0, ΣX), hence X = Σ
1/2
X OXU, with OX orthogonal,
– Y ∼ N(0, ΣY ), hence Y = Σ
1/2
Y OY U, with OY orthogonal,
E[ ˜X, U ] is minimized for ˜X = Σ
1/2
X U, so when OX is the identity. Hence the
generalized quantile of X relative to U is
QX(U) = Σ
1/2
X U.
X and Y are N(0, I)-comonotonic if OX = OY (they have the same orientation).
The correlation is
E[XY T
] = Σ
1/2
X OXOT
Y Σ
1/2
Y = Σ
1/2
X Σ
1/2
Y .
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18. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Computation of generalized quantiles
The optimal transportation map between µ on [0, 1]d
and the empirical
distribution relative to (X1, . . . , Xn) satisfies
– ˆQX(U) ∈ {X1, . . . , Xn}
– µ( ˆQ−1
X ({Xk})) = 1/n, for each k = 1, . . . , n
– ˆQX is the gradient of a convex function V : Rd
→ R.
The solution for the “potential” V is
V (u) = max
k
{ u, Xk − wk},
where w = (w1, . . . , wn) minimizes the convex function
w → V (u)dµ(u) +
n
k=1
wk/n.
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19. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
µ-Bickel-Lehmann dispersion order
With these multivariate extensions of quantiles and comonotonicity, we have the
following multivariate extension of the Bickel-Lehmann dispersion order (Bickel
and Lehmann 1979).
Definition :
(a) Y µDisp X if QY − QX is the gradient of a convex function.
Characterization :
(b) Y µDisp X iff Y
L
= X + Z, where Z and X are µ-comonotonic,
The proof is based on comonotonic additivity of the µ-quantile transform, i.e.,
QX+Z = QX + QZ when X and Z are µ-comonotonic.
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20. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Relation with existing multivariate dispersion orders
Based on univariate characterization (c), Giovagnoli and Wynn (Stat. and Prob.
Letters 1995) propose the strong dispersive order.
– Y SD X iff Y
L
= ψ(X), where ψ is an expansion, i.e., if
ψ(x) − ψ(x ) ≥ x − x .
This is closely related to our µ-Bickel-Lehmann ordering :
– Y SD X iff Y
L
= X + V (X), where V is a convex function.
– The latter is equivalent to Y
L
= X + Z, where X and Z are c-comonotonic
(Puccetti and Scarsini (JMA 2011)),
– It also implies Y µDisp X for some µ, since X and V (X) are
µX-comonotonic (converse not true in general).
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21. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Partial insurance for multivariate risks
Consider the following insurance contract :
Insuree Insurer
Before Y 0
After X1 X2
– By construction Y = X1 + X2.
– (X1, X2) is Pareto efficient if and only if µ-comonotonic
(Carlier-Dana-Galichon JET 2012).
– Hence the contract is efficient iff Y = X1 + X2 Disp X1 (from the
characterization above).
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22. Arthur CHARPENTIER - Rennes Risk Workshop - April, 2015
Multivariate Quiggin-Yaari functional and risk attitude
– Given a baseline U ∼ µ, decision maker evaluates risks with the functional
Φ(X) = E[QX(U) · φ(U)]
(Equivalent to monotonicity relative to stochastic dominance and comonotonic
additivity of Φ - Galichon and Henry JET 2012)
– Aversion to MPIR is equivalent to Φ(U) = −U
– Aversion to MMPIR is equivalent to Φ(X) ≤ Φ(E[X]) (obtains immediately
from the comonotonicity characterization of Bickel-Lehmann dispersion)
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