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Risk management with expectiles
1. Introduction
It is well known that the left and right quantiles x−α and x+α of a random variable X can be defined
through the minimization of an asymmetric, piecewise linear loss function:
[x−α (X ), x+α (X )] = argmin
x∈R
αE[(X − x)+] + (1 − α)E[(X − x)−] for α ∈ (0, 1),
where x+ = max(x, 0) and x− = max(−x, 0); see, for example Koenker (2005). Expectiles eq(X )
have been introduced by Newey and Powell (1987) as the minimizers of an asymmetric quadratic
loss:
eq(X ) = argmin
x∈R
qE[(X − x)2+] + (1 − q)E[(X − x)2−] for q ∈ (0, 1). (1)
When q = 12 , it is well known that eq(X ) = E[X ], thus expectiles can be seen as an asymmetric
generalization of the mean. The term ‘expectiles’ has probably been suggested as a combination
of ‘expectation’ and ‘quantiles’. Expectiles are uniquely identified by the first-order condition
(f.o.c.)
qE[(X − eq(X ))+] = (1 − q)E[(X − eq(X ))−]. (2)
Since Equation (5) is well defined for each X ∈ L1, which is the natural domain of definition of
the expectiles, we take it as the definition of eq(X ). Letting
q(x) := qx+ − (1 − q)x−,
∗Corresponding author. Email: [email protected]
© 2015 Informa UK Limited, trading as Taylor & Francis Group
488 F. Bellini and E. Di Bernardino
we see that Equation (2) can be rewritten as
E[q(X − eq(X ))] = 0.
Hence, expectiles are an example of shortfall risk measures in the sense of Föllmer and
Schied (2002), also known as zero utility premia in the actuarial literature. From this point of
view, they had been considered in Weber (2006) and by Ben Tal and Teboulle (2007), although
the connection with the minimization problem (1) and with the statistical notion of expectiles
emerged only in the more recent literature. In general, a statistical functional that can be defined
as the minimizer of a suitable expected loss function as in Equation (1) is said to be elicitable;
we refer to Gneiting (2011), Bellini and Bignozzi (2013), Ziegel (2014), Embrechts et al. (2014),
Davis (2013) and Acerbi and Szekely (2014) for further information about the elicitability prop-
erty and its financial relevance. See also the discussion in Section 4 on the relationship between
elicitability and backtesting. In this paper we compare expectiles with the more common financial
risk measures, that are Value at Risk (VaRα) and Expected Shortfall (ESα). We define
VaRα(X ) = −x+α (X ) for α ∈ (0, 1),
ESα(X ) = − 1
α
∫ α
0
x+u (X ) du for α ∈ (0, 1].
To be consistent with these sign conventions and to facilitate comparisons, we define, following
Kuan, Yeh, and Hsu (2009), the expectile-VaR (EVaRq) as follows:
EVaRq(X ) = −eq(X ).
EVaRq is the financial risk measure associated with expectiles, in the same way as VaRα is the
financial risk measure associated with the quantiles. For q ≤ 12 , EVaRq is a coherent risk measure,
since it satisfies the well-known axioms introduced by Artzner et al. (1999). Indeed, it is easy to
see that
• EVaRq(X + h) = EVaRq(X ) − h, for h ∈ R (translation invariance),
• X ≤ Y a.s. ⇒ EVaRq(X ) ≥ EVaRq(Y ) (monotonicity),
• EVaRq(λX ) = λEVaRq(X ), for λ ≥ 0 (positive homogeneity) and
• EVaRq(X + Y ) ≤ EVaRq(X ) + EVaRq(Y ) (subadditivity).
Moreover, it has been shown in several papers, albeit starting from different angles, that EVaRq
with q ≤ 12 is the only coherent risk measure that is also elicitable (see Weber 2006; Ben Tal and
Teboulle 2007; Bellini and Bignozzi 2013; Bellini et al. 2014; Delbaen et al. 2014; Ziegel 2014).
We refer the interested reader to these works and to Delbaen (2012) and Delbaen (2013) for the
properties of EVaRq as a coherent risk measure, in particular for its dual representation, Kusuoka
representation and for the identification of the optimal scenario in its dual representation. In order
to better understand the financial meaning of EVaRq, it is interesting to compare its acceptance
set with VaRα and with ESα . Recall that the acceptance set of a translation invariant risk measure
ρ is defined as
Aρ = {X | ρ(X ) ≤ 0},
and that ρ can be recovered by Aρ by the formula
ρ(X ) = inf{m ∈ R | X + m ∈ Aρ}.
The European Journal of Finance 489
We refer to Delbaen (2012), Föllmer and Schied (2011) or Pflug and Romisch (2007) for textbook
treatments. In the case of VaRα ,
AVaRα = {X | P(X < 0) ≤ α};
notice that we can equivalently write
AVaRα =
{
X
∣∣∣∣P(X > 0)P(X ≤ 0) ≥ 1 − αα
}
. (3)
In the case of ESα , we have
AESα =
{
X
∣∣∣∣ 1α
∫ α
0
xu(X ) du ≥ 0
}
.
In the case of EVaRq, the acceptance set can be written as
AEVaRq =
{
X
∣∣∣∣E[X+]
E[X−]
≥ 1 − q
q
}
. (4)
The EVaRq is then the amount of money that should be added to a position in order to have a
prespecified, sufficiently high gain–loss ratio. We recall that the gain–loss ratio or -ratio is a
popular performance measure in portfolio management (see, e.g. Shadwick and Keating 2002)
and is also well known in the literature on no-good-deal valuation in incomplete markets (see,
e.g. Biagini and Pinar 2013 and the references therein). It is sometimes argued that EVaRq is
‘difficult to explain’ to the financial community, but this is probably due to the fact (1) is usually
taken as starting point instead of Equation (4), which has a transparent financial meaning: in the
case of VaRα , a position is acceptable if the ratio of the probability of a gain with respect to
the probability of a loss is sufficiently high (3); in the case of EVaRq, a position is acceptable if
the ratio between the expected value of the gain and the expected value of the loss is sufficiently
high (4). In Section 4, we provide a real-data example for the computation of expectiles by means
of a normal i.i.d. model, an historical method, a Garch(1,1) model with normal innovations and a
Garch(1,1) model with Student t innovations. Choosing q = 0.00145, the magnitude of VaR0.01,
ES0.025 and EVaR0.00145 is closely comparable (see Section 4). In conclusion, we believe that
EVaRq is a perfectly reasonable risk measure, displaying many similarities with VaRα and ESα ,
surely worth of deeper study and practical experimentations by risk managers, regulators and
portfolio managers.
The paper is structured as follows: in Section 2 we review the basic properties of EVaRq, we
discuss the comparison between EVaRq and VaRα and the asymptotic behaviour of eq(X ) for
q → 1. Some results have been independently found by Mao, Ng, and Hu (2015). In Section 3
we provide several examples. In Section 4 we compute expectiles, compare the results with V
aR and with ES and we assess the accuracy of the forecasts by means of the realized loss. Proofs
and auxiliary results are postponed to the appendix.
2. Properties of expectiles
As mentioned in Section 1, we take as definition of expectiles the following equation, valid for
each X ∈ L1:
qE[(X − eq(X ))+] = (1 − q)E[(X − eq(X ))−], (5)
490 F. Bellini and E. Di Bernardino
which can also be written as
q = E[(X − eq(X ))−]
E[|X − eq(X )|] ,
which shows that the expectiles eq(X ) can be seen as the quantiles of a transformed distribution
with distribution function
G(x) := E[(X − eq(X ))−]
E[|X − eq(X )|] ,
as noted by Jones (1994). We collect in the following proposition further properties of expectiles
(see, e.g. Newey and Powell 1987; Bellini et al. 2014) :
Proposition 2.1 Let X ∈ L1 and let eq(X ) be the unique solution of Equation (5). Then
(a) eq(X ) is strictly monotonic in q, for q ∈ (0, 1);
(b) eq(X ) is strictly monotonic in X, in the sense that
X ≥ Y a.s. and P(X > Y ) > 0 → eq(X ) > eq(Y );
(c) eq(−X ) = −e1−q(X );
(d) if X is symmetric with respect to x0, then
eq(X ) + e1−q(X )
2
= x0;
(e) if X has a C1 density, then eq(X ) is a C1 function of q, with
deq(X )
dq
= E[|X − eq(X )|]
(1 − q)F(eq(X )) + qF¯(eq(X ))
.
More refined symmetry properties have been considered in Abdous and Remillard (1995).
2.1 Comparison between expectiles and quantiles
For the most common distributions, expectiles are closer to the centre of the distribution than
the corresponding quantiles. Typically, the quantile and the expectile curve intersect in a unique
point, which corresponds to the centre of symmetry in the case of a symmetric distribution (see
Examples 3.1–3.5 and Figures 1–2).
Koenker (1993) found that for a distribution with quantile function
xα = 2α − 1√
α(1 − α) ,
which is a rescaled Student t distribution with ν = 2, it holds that eα(X ) = xα(X ) for each
α ∈ (0, 1), that is, the quantile and the expectile curve coincide. In Zou (2014) the Koenker’s
argument was generalized to show that for every nondecreasing function q : [0, 1] → [0, 1] with
The European Journal of Finance 491
Expectile vs quantile
Exponential with lambda= 1
