Generalized quantiles as risk measures
F. Bellini1 , B. Klar2 , A. Müller3 , E. Rosazza Gianin1
1 Dipartimento
di Statistica e Metodi Quantitativi, Università di Milano Bicocca
für Stochastik, Karlsruher Institut für Technologie (KIT), Germany
3 Department of Mathematics, University of Siegen, Germany
2 Institut
XIII International Conference Stochastic Programming,
Bergamo 10/7/2012.
Quantiles as minimizers
It is well known that the quantiles xα∗ (X ) of a random variable X
can be defined as the minimizers of an asymmetric piecewise linear
loss function:
xα∗ (X ) = arg min πα (X , x)
x∈R
with
πα (X , x) := E [α(X − x)+ + (1 − α)(X − x)− ],
for α ∈ (0, 1).
This fundamental property lies at the heart of quantile regression
(see for example Koenker, 2005) and has been exploited by
Rockafellar and Uryasev (2002) for their approach to the
computation of the Conditional Value at Risk (CVaR).
Generalized quantiles
In the statistical literature, several generalizations of the notion of
quantile have been suggested, corresponding to more general loss
functions:
I
expectiles (Newey and Powell, 1987)
I
Lp -quantiles (Chen, 1996)
I
M-quantiles (Breckling and Chambers, 1988)
We will consider general asymmetric loss functions of the type
πα (X , x) := αE Φ1 (X − x)+ + (1 − α) E Φ2 (X − x)− ,
with Φ1 , Φ2 : [0, +∞) → [0, +∞) convex and strictly increasing
satisfying Φi (0) = 0, Φi (1) = 1 and lim Φi (x) = +∞.
x→+∞
Elicitability
Functionals that are defined as the minimizers of a suitable
expected loss are called elicitable in statistical decision theory (see
Gneiting, 2011 and the references therein).
It has been recently suggested (see for example Embrechts and
Hofert, 2013) that elicitability is a very important property of a
risk measure, since it provides a natural methodology to perform
backtesting. As an example, it is very well known how to perform
backtesting of VaR, while backtesting of CVaR (that is not
elicitable) is not equally straightforward.
The connections between elicitability and coherence are also
investigated in Ziegel (2013); elicitable functionals in a dynamic
setting have been considered by Bignozzi (2013).
Luxemburg norm
We recall that the Orlicz space LΦ and the Orlicz heart M Φ
associated to a convex Young function Φ : [0, +∞) → [0, +∞)
satisfying Φ(0) = 0, Φ(1) = 1, lim Φ(x) = +∞ are defined as
x→+∞
Φ
L
MΦ
|X |
< +∞, for some a > 0 ,
: = X :E Φ
a
|X |
: = X :E Φ
< +∞, for every a > 0 .
a
The Luxemburg norm is defined as
|Y |
kY kΦ := inf a > 0 : E Φ
≤1 .
a
and makes both LΦ and M Φ Banach spaces. The case Φ(x) = x p
corresponds to the usual Lp spaces.
First order condition
Theorem
Let X ∈ M Φ1 ∩ M Φ2 , α ∈ (0, 1). Then xα∗ ∈ arg min πα (X , x) if
and only it satisfies the following first order conditions:
i
i
h
h
−
+
−αE 1{X ≥x ∗ } Φ01+ (X − x ∗ ) + (1 − α) E 1{X <x ∗ } Φ02− (X − x ∗ ) ≤ 0
i
i
h
h
−
+
−αE 1{X >x ∗ } Φ01− (X − x ∗ ) + (1 − α) E 1{X ≤x ∗ } Φ02+ (X − x ∗ ) ≥ 0
where Φi− and Φi+ are the left and right derivatives of Φi .
If Φ1 and Φ2 are differentiable and Φ01+ (0) = Φ02+ (0) = 0, then
the f.o.c. becomes simply
αE Φ01 (X − x ∗ )+ = (1 − α) E Φ02 (X − x ∗ )− .
Generalized quantiles as shortfall risk measures
When the f.o.c. is given by an equation, generalized quantiles may
also be defined as the unique solutions of the equation
E [ψ(X − xα∗ )] = 0,
where
ψ(t) :=
−(1 − α)Φ02 (−t)
αΦ01 (t)
t<0
t≥0
is nondecreasing with ψ(0) = 0.
This shows that generalized quantiles can be seen as special cases
of zero utility premium principles, also known as shortfall risk
measures or u-mean certainty equivalents (see Deprez and Gerber,
1985, Föllmer and Schied, 2002, Ben-Tal and Teboulle, 2007).
Properties of generalized quantiles
The following properties are elementary:
I
πα (X , x) is law-invariant, non-negative, convex, and satisfies
lim πα (X , x) = lim πα (X , x) = +∞
x→−∞
I
x→+∞
the infimum of πα (X , x) is always attained; indeed
arg min πα (X , x) := [xα∗− , xα∗+ ]
I
if X is essentially bounded, then
[xα∗− , xα∗+ ] ⊂ [ess inf (X ) , ess sup (X )]
I
if α ≤ β, then xα∗− ≤ xβ∗− and xα∗+ ≤ xβ∗+
I
if Φ1 and Φ2 are strictly convex, then xα∗− = xα∗+ .
Properties of generalized quantiles /2
Let Φ1 ,Φ2 : [0, +∞) → [0, +∞) be strictly convex and
differentiable with Φ01+ (0) = Φ02+ (0) = 0. Then:
I
xα∗ (X + h) = xα∗ (X ) + h, for each h ∈ R
I
xα∗ (X ) is positively homogeneous if and only if
Φ1 (x) = Φ2 (x) = x β , with β > 1.
I
if X ≥ Y a.s. or if X ≥st Y , then xα∗ (X ) ≥ xα∗ (Y )
I
xα∗ (X ) is convex if and only if the function ψ : R → R is
convex; it is concave if and only if ψ is concave.
I
as a consequence, xα∗ (X ) is coherent if and only if
Φ1 (x) = Φ2 (x) = x 2 and α ≥ 21 .
Thus expectiles with α ≥ 21 are the only generalized quantiles that
are a coherent risk measure.
Alternative formulations
Before focusing on expectiles, we mention that other choices of the
loss function have been considered in the literature. In connection
with the so called Haezendonck-Goovaerts risk measures, Bellini
and Rosazza Gianin (2012) introduced the so called Orlicz
quantiles, defined as the minimizers of
πα (X , x) := α (X − x)+ Φ + (1 − α) (X − x)− Φ ,
1
2
while Jaworski (2006) considered instead loss functions of the type
πα (X , x) := E Φ1 (X − x)+ + Φ2 (x),
that depend only on the right tail of the distribution.
Alternative formulations /2
Some comparisons between the different formulations:
I
Orlicz quantiles are always positively homogeneous, while
generalized quantiles have this properties only in the case
Φ1 (x) = Φ2 (x) = X p
I
Orlicz quantiles lack the property of monotonicity with respect
to the ≤st order, as it was shown by means of a
counterexample in Bellini and Rosazza Gianin (2012)
I
Jaworski quantiles are monotonic with respect to the ≤st
order and (under additional assumptions) convex, but they do
not satisfy translation invariance or positive homogeneity.
Expectiles
Thus the expectiles, introduced by Newey and Powell (1987) as
eα (X ) = arg min E [α(X − x)+2 + (1 − α)(X − x)−2 ]
x∈R
are the only example of a generalized quantile that is also a
coherent risk measure (for α ≥ 21 ). When α = 12 the expectile
coincides with the mean, while when α ≤ 21 expectiles are
superadditive, in the sense that
eα (X + Y ) ≥ eα (X ) + eα (Y ).
Moreover, we have the following elementary properties:
I
X ≤ Y and P(X < Y ) > 0 ⇒ eα (X ) < eα (Y )
I
eα (X ) = −e1−α (−X ).
First order condition
The first order condition can be written as
Z eα
Z +∞
(1 − α)
F (t)dt = α
F (t)dt
−∞
eα
or equivalently
eα (X ) − E [X ] =
or also
α=
2α − 1
E [(X − eα (X ))+ ]
1−α
E [(X − eα )− ]
,
E [|X − eα |]
that shows that expectiles can be seen as the usual quantiles of a
transformed distribution (Jones, 1994).
Dual representation
Theorem
Let X ∈ L1 , and α ∈ (0, 1). Then

 max E [ϕX ]
ϕ∈Mα
eα (X ) =
 min E [ϕX ]
ϕ∈Mα
if α ≥
if α ≤
1
2
1
2
,
where
ess sup ϕ
∞
Mα = ϕ ∈ L , ϕ ≥ 0 a.s., EP [ϕ] = 1,
≤β ,
ess inf ϕ
n
o
α
with β = max 1−α
, 1−α
. When α ≥ 12 , the optimal scenario is
α
ϕ :=
α1{X >eα } + (1 − α)1{X ≤eα }
.
E [α1{X >eα } + (1 − α)1{X ≤eα } ]
Kusuoka representation
The Kusuoka representation of a coherent risk measure is its
representation as a supremum of additively comonotone risk
measures, that are convex combinations or integrals of CVaR with
different α (see Kusuoka, 2001, Pflug, 2007, Shapiro, 2013).
Theorem
Let X ∈ L1 ,
1
2
≤ α < 1 and β =
α
1−α .
Then
eα (X ) = max {(1 − γ)CVaR β− γ1 + γE [X ]}.
γ∈[ β1 ,1]
β−1
In particular,
eα (X ) ≥
E [X ]
2α
+ 1−
1
2α
CVaRα (X ).
Expectiles are an example of coherent risk measures that are not
additively comonotone.
Comparison between quantiles and expectiles
In the statistical literature it is usually argued that typically
expectiles are closer to the centre of the distribution than the
corresponding quantiles. This is indeed the case for common
distributions as the uniform or the normal, as the following figure
shows:
Uniform
Standard normal
1
2.5
expectiles
quantiles
0.9
2
0.8
1.5
0.7
Quantiles and expectiles curves
Quantiles and expectiles curves
expectiles
quantiles
0.6
0.5
0.4
0.3
1
0.5
0
-0.5
-1
0.2
-1.5
0.1
-2
0
0
0.2
0.4
0.6
α
0.8
1
-2.5
0
0.2
0.4
0.6
α
0.8
1
Comparison between quantiles and expectiles /2
We are not aware of general comparisons results. Koenker (1993)
provided an example of a distribution with infinite variance for
which expectiles eα and quantiles qα coincide for every α. We can
prove the following:
Theorem
Let X be a Pareto-like distribution with tail index β > 1. Then
1
F (eα (X )) ∼ 1 − α ∼ F (qα (X ))
β−1
as α → 1.
If β < 2, then there exists some α0 < 1 such that for all α > α0 it
holds eα (X ) > qα (X ); if β > 2, the reverse ordering applies.
So for high α expectiles are more conservative than the quantiles
for distributions with very heavy tails (infinite variance).
Robustness
In robust statistics, the notion of qualitative robustness of a
statistical functional corresponds essentially to the continuity with
respect to weak convergence. It is generally argued that this notion
is not appropriate for financial risk measures; the mean itself it’s
not robust. Stahl et al. (2012) suggest that a better notion of
robustness might be continuity with respect to the Wasserstein
distance, defined as
dW (F , G ) := inf{E [|X − Y |] : X ∼ F , Y ∼ G }.
or equivalently
Z
∞
Z
|F (t) − G (t)|dt =
dW (F , G ) =
−∞
0
1
|F −1 (α) − G −1 (α)|dα.
Robustness /2
As it is well known, convergence in the Wasserstein distance is
stronger than weak convergence, indeed
dW (Xn , X ) → 0 ⇐⇒ Xn → X in distribution and E [Xn ] → E [X ].
Thus convergence in the Wasserstein metric is a particular type of
the so called ψ-weak convergence with ψ(x) = |x|. We can prove
that expectiles are Lipschitz with respect to the Wasserstein metric:
Theorem
For all X , Y ∈ L1 and all α ∈ (0, 1) it holds that
α
; 1−α
|eα (X ) − eα (Y )| ≤ βdW (X , Y ), where β = max{ 1−α
α }.
Isotonicity results for generalized quantiles
A natural question is that there exist other generalized quantiles
that are subadditive, or at least convex.
Such a statistical functional would be an example of a convex law
invariant risk measure, so from the results of Bauerle and Muller
(2006), under weak technical additional conditions, it should be
isotonic with respect to the convex order.
In Bellini (2012) the general problem of isotonicity of generalized
quantiles with respect to several stochastic orderings have been
studied, by using a purely order-theoretic comparative static
approach, in the spirit of Topkis (1978) and Milgrom and Shannon
(1994).
The basic idea
In the defining minimization problem
xα∗ (X ) = arg min πα (X , x)
x∈R
we consider X as a parameter, belonging to the partially ordered
set (L, -), where the role of the partial order - will be played in
turn by ≤st , ≤cx and ≤icx .
The theory of monotone comparative statics provides necessary
and sufficient conditions on the function πα (X , x) for increasing
optimal solutions, that is in order to have that
X - Y ⇒ xα∗ (X ) ≤ xα∗ (Y ).
Submodularity and single crossing condition
Topkis (1978) showed that a sufficient condition for increasing
optimal solutions in our situation is the submodularity of the
function πα (X , x), that is
πα (X , y ) − πα (X , x) ≥ πα (Y , y ) − πα (Y , x),
for y ≥ x and Y X . Later, Milgrom and Shannon (1984) proved
that a necessary condition for increasing optimal solutions is the so
called single crossing condition, defined as follows:
i) πα (Y , y ) − πα (Y , x) > 0 ⇒ πα (X , y ) − πα (X , x) > 0
ii) πα (Y , y ) − πα (Y , x) ≥ 0 ⇒ πα (X , y ) − πα (X , x) ≥ 0,
for y ≥ x and Y X .
Isotonicity results
By applying these techniques, we can prove the following (Bellini,
2012):
Theorem
Let πα (X , x) := αE [φ(X − x)+ ] + (1 − α)E [φ(X − x)− ] with
X ∈ M Φ and α ∈ (0, 1).
a) If Φ(x) = x 2 and α ≥ 21 , then X ≤cx Y ⇒ xα∗ (X ) ≤ xα∗ (Y ); if
instead α ≤ 21 then X ≤cx Y ⇒ xα∗ (X ) ≥ xα∗ (Y )
b) If Φ(x) = x s with s odd, then X ≤s−cx Y ⇒ xα (X )∗ ≤ xα (Y )∗
b’) If Φ(x) = x s with s even and α ≥ 21 , then
X ≤s−cx Y ⇒ xα∗ (X ) ≤ xα∗ (Y ); if instead α ≤
X ≤s−cx Y ⇒ xα∗ (X ) ≥ xα∗ (Y )
1
2
then
c) Let Φ ∈ C 2 [0, +∞). If xα∗ (X ) is isotonic with respect to the ≤cx
order, then Φ00 (x) = k.
Some references
Bellini, F., Klar, B., Müller, A., Rosazza Gianin, E. Generalized
quantiles as risk measures, downloadable from SSRN
Bellini, F., Rosazza Gianin, E., 2012. Haezendonck-Goovaerts
risk measure and Orlicz quantiles, Insurance: Mathematics and
economics, vol. 51, Issue 1, pp. 107-114
Bellini, F., 2012. Isotonicity results for generalized quantiles,
Statistics and Probability Letters, vol. 82, pp. 2017-2024
Embrechts, P., Hofert, M., 2013. Statistics and Quantitative
Risk Management for Banking and Insurance, ETH Zurich.
Gneiting, T., 2011. Making and evaluating point forecasts.
Journal of the Anerican Statistical Association, 106(494),
746-762.