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Risk management with expectiles
Fabio Bellini & Elena Di Bernardino
To cite this article: Fabio Bellini & Elena Di Bernardino (2017) Risk management
with expectiles, The European Journal of Finance, 23:6, 487-506, DOI:
10.1080/1351847X.2015.1052150
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Published online: 05 Jun 2015.
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The European Journal of Finance, 2017
Vol. 23, No. 6, 487–506, http://dx.doi.org/10.1080/1351847X.2015.1052150
Risk management with expectiles
Fabio Bellinia and Elena Di Bernardinob∗
aDipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Milan, Italy; bCNAM, Département
IMATH, Laboratoire Cédric EA4629, Paris, France
(Received 30 September 2014; final version received 1 May 2015)
Expectiles (EVaR) are a one-parameter family of coherent risk measures that have been recently suggested
as an alternative to quantiles (VaR) and to expected shortfall (ES). In this work we review their known
properties, we discuss their financial meaning, we compare them with VaR and ES and we study their
asymptotic behaviour, refining some of the results in Bellini et al. [(2014). “Generalized Quantiles as
Risk Measures.” Insurance: Mathematics and Economics, 54:41–48]. Moreover, we present a real-data
example for the computation of expectiles by means of simple Garch(1,1) models and we assess the
accuracy of the forecasts by means of a consistent loss function as suggested by Gneiting [(2011). “Making
and Evaluating Point Forecast.” Journal of the American Statistical Association, 106 (494): 746–762].
Theoretical and numerical results indicate that expectiles are perfectly reasonable alternatives to VaR and
ES risk measures.
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: elena.di_bernardino@cnam.fr
© 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
q
expectiles
quantiles
Expectile vs quantile
Gamma with a= 3, b=1
q
expectiles
quantiles
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
3
2
10
8
6
4
2
00
2
4
6
1
–1
–2
–3
0
Expectile vs quantile
Standard Normal distribution
q
expectiles
quantiles
Expectile vs quantile
Standard Log−Normal distribution
q
expectiles
quantiles
Figure 1. Comparison between expectiles (full line) and quantiles (dashed line) in the Exponential (λ = 1),
Gamma (a = 3, b = 1), Normal (μ = 0, σ = 1) and in the Lognormal (μ = 0, σ = 1) cases.
20
10
−1
0
−2
0
0
30
Expectile vs quantile
Student t with nu= 1.5
q
expectiles
quantiles
−5
0
5
Expectile vs quantile
Student t with nu= 2.5
q
expectiles
quantiles
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Expectile vs quantile
Uniform distribution
q
expectiles
quantiles
0.
0
0.
2
0.
4
0.
6
0.
8
Expectile vs quantile
Beta distribution with a=b=2
q
expectiles
quantiles
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0
Figure 2. Comparison between expectiles (full line) and quantiles (dashed line) in the Student t with ν = 1.5,
ν = 2.5, Uniform and Beta (a = 2 and b = 2) cases.
q(0) = 0, q(1) = 1 and q(α0) = 12 , the distribution with quantile function
xα = −S(α) exp
{∫ α
α0
S(t) dt
}
,
with
S(α) = 2q(α) − 1
2αq(α) − α − q(α) ,
satisfies eq(α)(X ) = xα(X ). Hence by a proper choice of the function q(α) any arbitrary number
of intersections between the quantile curve and the expectile curve is possible.
492 F. Bellini and E. Di Bernardino
2.2 Asymptotic behaviour of high expectiles
When X belongs to the domain of attraction of a Generalized Extreme Value distributions, it
is possible to derive the asymptotic behaviour of eq(X ) for q → 1 by using techniques from
Extreme Value Theory. We refer to Hua and Joe (2011), Tang and Yang (2012) and Mao and
Hu (2012) for similar results for other one-parameter families of coherent risk measures. Some
of the results of this subsection have been independently derived in Mao, Ng, and Hu (2015). We
refer to Appendix A.1 for notations and basic definitions and to Appendix A.2 for proofs.
In Bellini et al. (2014), it was shown that when X ∈ MDA( β), with β > 2, there exists q¯ < 1
such that eq(X ) < xq for each q ∈ (q¯, 1). That is, for a Paretian tail with tail index β > 2, expec-
tiles are ultimately (i.e. for q large enough) less conservative than the corresponding quantiles.
On the contrary, when β < 2 the opposite inequality holds. Our first result is the extension of
this asymptotic comparison between quantiles and expectiles to the Gumbel and Weibull cases,
in which expectiles are ultimately less conservative than quantiles.
Proposition 2.2 Let F ∈ MDA( β), with β > 1. If β > 2 ultimately eq(X ) < xq; if β < 2
ultimately eq(X ) > xq. If F ∈ MDA(β) or F ∈ MDA(), then ultimately eq(X ) < xq.
As an illustration, see Figures 1 and 2 which represent a comparison between expectiles (full
line) and quantiles (dashed line) in the Exponential, Gamma, Normal, Lognormal, Student t,
Uniform and Beta cases.
When X belongs to the domain of attraction of a Fréchet distribution, that is, when X has
a Paretian right tail, then it is possible to provide an explicit expression for the asymptotic
behaviour of eq(X ).
Proposition 2.3 Let F ∈ MDA( β), with F¯(x) ∼ x−βL(x), L ∈ RV 0(+∞) and β > 1. Then,
for q → 1,
eq(X ) ∼ (β − 1)−1/βxq. (6)
Moreover, if for some λ0 > 0 it holds that{
L(λ0t)
L(t)
− 1
}
log L(t) → 0 for t → +∞, (7)
then
eq(X ) ∼ (β − 1)−1.β (1 − q)
−1/β
L−1/β((1 − q)−1/β) .
In particular if L = C ∈ (0, +∞), then
eq(X ) ∼
(
β − 1
C
)−1/β
(1 − q)−1/β . (8)
A similar result can be found in Mao, Ng, and Hu (2015), where also the possible refinements
under a second-order regular variation condition are discussed.
In the case of the Gumbel domain of attraction, the situation is more complicated. For several
common distributions (see Examples 3.2–3.5), it holds that eq(X )/xq → 1, as it happens for ES
and for other one parameter families of coherent risk measures (see, e.g. Tang and Yang 2012;
Mao and Hu 2012). Typically, the asymptotic f.o.c. is a transcendental equation which can be
The European Journal of Finance 493
explicitly solved by means of the so-called Lambert W function (see Appendix A.3). We provide
a general result for the class of the so-called Weibull-type distributions (that must not be con-
fused with the distributions in the Weibull domain of attraction; see Beirlant et al. 1995; Dierckx
et al. 2009 and the references therein).
Proposition 2.4 Let F ∈ MDA(), with the additional requirement that F¯ = exp(−xτ L(x)),
with L ∈ RV 0(+∞) and τ > 0. Then, for q → 1,
eτqL(eq(X )) ∼ − log(1 − q) and log eq(X ) ∼ log xq. (9)
Moreover, if for some λ0 > 0 it holds that{
L(λ0t)
L(t)
− 1
}
log L(t) → 0 for t → +∞, (10)
then
eq(X ) ∼ xq.
Finally, for distributions in the Weibull domain of attraction, we have the following (see also
Mao, Ng, and Hu 2015).
Proposition 2.5 Let F ∈ MDA(β), with F¯(x) ∼ (xˆ − x)βL(xˆ − x), L ∈ RV 0(0) and β > 0.
Then, for q → 1,
(xˆ − eq(X ))β+1L(xˆ − eq(X )) ∼ (β + 1)(xˆ − E[X ])(1 − q).
In the particular case, L(x) = C ∈ (0, +∞), then
xˆ − eq(X ) ∼
[
(xˆ − E[X ])(β + 1)
C
]1/(β+1)
(1 − q)1/(β+1). (11)
The quality of the first-order approximations is graphically assessed in Figures 3 and 4.
0.90 0.92 0.94 0.96 0.98 1.00 0.90 0.92 0.94 0.96 0.98 1.00
0.
65
0.
70
0.
75
0.
80
1.
00
0.
90
0.
95
0.
85
Uniform
q
Expectiles
First order Expectiles
0.
6
0.
7
0.
8
0.
9
1.
0
Beta distribution with a=b=2
q
Expectiles
First order Expectiles
Figure 3. Comparison between expectiles (full line) and the first-order asymptotic approximation (dashed
line) in the Uniform and Beta (a = 2 and b = 2) cases.
494 F. Bellini and E. Di Bernardino
0
5
10
15
20
25
30
Pareto with theta=1 alpha=2
q
Exact
First order approx
0
2
4
6
8
Student t with nu=3
q
Exact
First order approx
0.90 0.92 0.94 0.96 0.98 1.00 0.90 0.92 0.94 0.96 0.98 1.00
Figure 4. Comparison between expectiles (full line) and the first-order asymptotic approximation (dashed
line) in the Pareto (θ = 1, α = 2) and Student t (ν = 3) cases.
3. Examples and illustrations
Example 3.1 (Uniform distribution) Let F(t) = t, t ∈ [0, 1]. Then, the f.o.c. is given by
eq(X ) − 12 =
2q − 1
1 − q
[
eq(X )2 − 2 eq(X ) + 1
2
]
,
that gives the explicit solution
eq(X ) = q −
√
q − q2
2q − 1 .
In this case, C = 1 and β = 1, and indeed
1 − eq(X ) ∼
√
1 − q,
in accordance with Proposition 2.5.
Example 3.2 (Exponential distribution) Let F¯ = exp(−λx), x > 0, λ > 0. Then, the f.o.c. is
given by
eq(X ) − 1
λ
= (2q − 1) exp(−λx)
λ(1 − q)
that can be written as
zez = 2q − 1
(1 − q)e ,
The European Journal of Finance 495
with z = λeq(X ) − 1. Recalling the definition of Lambert’s W function (see Appendix A.3), we
find that
z = W
(
2q − 1
(1 − q)e
)
that gives the exact expression
eq(X ) = 1
λ
{
1 + W
(
2q − 1
(1 − q)e
)}
.
Since W(x) ∼ log(x) for x → +∞ (see Lemma A.1), we find
eq(X ) ∼ 1
λ
{− log(1 − q)} = xq.
Example 3.3 (Logistic distribution) Let F¯(x) = 2/(1 + exp(x)). Then mX (x) ∼ 1 (see Beirlant
and Teugels 1992), and the f.o.c. is
eq(X )(1 + exp(eq(X ))) = 21 − q ,
and asymptotically, for q → 1, we obtain
eq(X ) exp(eq(X )) = 21 − q ,
that as in the exponential case gives
eq(X ) ∼ − log(1 − q) ∼ xq.
Example 3.4 (Weibull distribution) Let F¯(x) = exp(−λ xτ ), with τ > 0. Then mX (x) ∼
x1−τ /λτ , hence asymptotically, the f.o.c. becomes
eq(X ) ∼ eq(X )
1−τ exp(−λeq(X )τ )
λτ(1 − q) ,
or
λeq(X )τ exp λeq(X )τ ∼ 1
τ(1 − q) ,
which from Lemma A.1 gives
eq(X ) ∼
(
− log(1 − q)
λ
)1/τ
= xq.
496 F. Bellini and E. Di Bernardino
Example 3.5 (Normal distribution) If X ∼ N(0, 1), it is well known that
mX (x) ∼ F¯(x)f (x) ∼
1
x
, (12)
where f = F′ is the standard normal density. It follows that
E[(X − x)+] ∼ F¯(x)
x
∼ f (x)
x2
.
Asymptotically, the f.o.c. becomes
eq(X )3 exp
(
eq(X )2
2
)
∼ 1√
2π(1 − q) ,
and from Lemma A.1 it follows that
eq(X ) ∼
√
−2 log(1 − q) ∼ xq.
Example 3.6 (Pareto distribution) Let F¯(t) = (θ/(t + θ))α , with t ≥ 0 and α > 1. Then
E[X ] = θ/(α − 1) and
E[(X − x)+] = θ
α(x + θ)−α+1
α − 1
so the f.o.c. becomes
eq(X ) − θ
α − 1 =
(2q − 1)θα(eq(X ) + θ)−α+1
(1 − q)(α − 1) .
If α = 2 the equation can be solved explicitly; we get
eq(X ) = θ
√q√
1 − q > θ
1 − √1 − q√
1 − q = xq.
It is easy to check that eq(X )/xq → 1, in accordance with Proposition 2.3.
4. Risk management with expectiles
In this section we provide a straightforward computation example of EVaRq using standard
econometric models. We compare four different methods: a purely historical method, a normal
model with fixed volatility, a Garch(1,1) model with normal innovations and a Garch(1,1) model
with Student t innovations. All models are estimated on rolling windows of length N = 500. The
aim of this simple experiment is just to show that EVaRq is a perfectly reasonable alternative
to VaRα and ESα . More sophisticated econometric modelling of expectiles has been pursued in
Taylor (2008), Kuan, Yeh, and Hsu (2009) and De Rossi and Harvey (2009). The first ques-
tion that has to be addressed is the choice of q. In the case of VaRα , it is customary to choose
α = 0.01. In the case of ESα , the latest revisions of the Basel Accords suggest α = 0.025, since
for typical portfolios ES0.025 VaR0.01. The idea is to change the risk measure without changing
too much its resulting value and the corresponding capital requirements. In the case of EVaRq,
there seems to be no natural a priori acceptable reference level of gain–loss ratio, so we suggest
q = 0.00145, which satisfies EVaRq(X ) = VaR0.01(X ) for a normally distributed X. The inter-
ested reader is also referred to Rroji (2013) for empirical investigations on the choice of q. The
The European Journal of Finance 497
Figure 5. Logreturns of the SP500 Index from 2 November 1994 to 31 December 2009.
0 500 1000 1500 2000 2500 3000
−0
.1
5
−0
.1
0
−0
.0
5
0.
00
0.
05
0.
10
0.
15
t
B
ac
kt
es
t V
aR
Normal VaR
Historical VaR
VaR Garch(1,1) with normal error distribution
VaR Garch(1,1) with student t error distribution
Figure 6. VaR0.01 for the SP500 log-return series, evaluated using the four considered methods: a purely
historical method based on rolling windows of length N = 500 (dashed magenta line), a normal model
with fixed volatility (dotted-dashed black line), a Garch(1,1) model with normal innovations (red line) and
a Garch(1,1) model with Student t innovations (blue line).
portfolio under consideration is represented by the SP500 Index from 2 November 1994 to 31
December 2009, which correspond to T = 3818 trading days. The corresponding logarithmic
returns are reported in Figure 5.
The values of VaR0.01 for the four standard econometric considered models are displayed in
Figure 6. The number of violations and the p-values of the binomial test are reported in Table 1.
The estimations of the Garch models have been performed by means of the command
estimate in the Econometrics Toolbox of Matlab R2014b. Forecasting of the conditional
variances has been performed by the function forecast. In a very limited fraction of the
498 F. Bellini and E. Di Bernardino
Table 1. Number of violations of VaR0.01 and p-values of the associated binomial test.
Method Violations of VaR0.01 p-Value
Normal model (i) 85 6.9211e − 13
Historical model (ii) 57 5.4592e − 05
Garch(1,1) model with normal innovations (iii) 61 4.4538e − 06
Garch(1,1) model with Student t innovations (iv) 41 0.3231
3318 estimations performed, there had been some numerical instabilities; the best results have
been obtained with the initializations α1 = 0.01, and β1 = 0.95. In accordance with the litera-
ture, models with normal innovations are not conservative enough and have a too high number
of violations, while the Garch(1,1) model with Student t innovations seems able to capture at
least the right frequency of violations. In the last column of Table 1, we report the p-value of the
binomial test, which rejects the null hypothesis in all cases with the exception of the Garch(1,1)
with t innovations. As it has been extensively discussed in the literature on backtesting VaR (see,
e.g. Campbell (2005) and the references therein), considering only the number of violations may
wrongly assess the accuracy of the forecasting model. Since VaRα minimizes a piecewise linear,
asymmetric scoring function, we consider the associated realized loss given by
LVaR(α) = 1
T
T∑
t=1
L(VaRt(α), rt),
with
L(VaRt(α), rt) =
{
(1 − α)· | rt + VaRt(α) | if rt ≤ −VaRt(α),
α· | rt + VaRt(α) | if rt > −VaRt(α).
The values of the realized loss for the different models reported in the following table confirm
the superiority of the Garch(1,1) model with Student t innovations (see Table 2). If one wishes,
it is possible to use the realized loss to build a formal test of the model. The main difficulty is
that in order to derive the distribution of the realized loss one has to rely on resimulations. We
refer the interested reader to Campbell (2005) for a review of the literature on backtesting with
loss functions. For more sophisticated uses of the realized loss in model selection, we refer to
Bernardi, Catania, and Petrella (2014) and the references therein.
Our next step is to use the same four models to compute EVaR0.00145. The obtained results are
displayed in Figure 7. Notice that for EV aR the notion of ‘violation’ is not meaningful, since its
definition is related to gain–loss ratios. Theoretically, we know that it should hold
E(X + EVaR0.00145)+
E(X + EVaR0.00145)− =
E(X − e0.00145)+
E(X − e0.00145)− =
1 − 0.00145
0.00145
689.
Then a first rough assessment of the accuracy of the different models can be obtained by means
of the realized gain–loss ratios in Table 3.
A low realized gain–loss ratio suggests that the model is not conservative enough. The results
show that the comparatively better model is Garch(1,1) with Student t innovations, although its
gain–loss ratio seems far from the theoretical value of 689. It might be possible to develop a
formal test based on the realized gain–loss ratio, similar to the binomial test for VaR violations,
although it seems that the null distribution of the gain–loss ratio can be determined only through
The European Journal of Finance 499
Table 2. Realized loss functionLVaR(0.01) evaluated using the four
considered methods.
Method LVaR(0.01)
Normal model (i) 5.6071e − 04
Historical model (ii) 4.8657e − 04
Garch(1,1) model with normal innovations (iii) 4.0404e − 04
Garch(1,1) model with Student t innovations (iv) 2.9631e − 04
Normal Expectile
Historical Expectile
Expectile Garch(1,1) with normal error distribution
Expectile Garch(1,1) with student t error distribution
−0
.1
5
−0
.1
0
−0
.0
5
0.
00
0.
05
0.
10
0.
15
B
ac
kt
es
t E
Va
R
0 500 1000 1500 2000 2500 3000
t
Figure 7. EVaR0.00145 for the SP500 log-return series, evaluated using the four considered methods: a purely
historical method based on rolling windows of length N = 500 (dashed magenta line), a normal model with
fixed volatility (dotted-dashed black line), a Garch(1,1) model with normal innovations (red line) and a
Garch(1,1) model with Student t innovations (blue line).
Table 3. Realized gain–loss ratios evaluated using the four consid-
ered methods.
Method Gain–loss ratio
Normal model (i) 91.27
Historical model (ii) 202.88
Garch(1,1) model with normal innovations (iii) 292.87
Garch(1,1) model with Student t innovations (iv) 452.84
resimulations. Moreover, the gain–loss ratio is quite unstable from a numerical point of view,
being the ratio of quantities of different orders of magnitude. For these reasons, we do not further
pursue this approach here.
As we have already done in the case of VaR, we compute the associated realized losses
LEVaR(α) = 1
T
T∑
t=1
L(EVaRt(α), rt),
500 F. Bellini and E. Di Bernardino
where the loss function is given this time by
L(EVaRt(α), rt) =
{
(1 − α) · (rt + EVaRt(α))2 if rt ≤ −EVaRt(α),
α · (rt + EVaRt(α))2 if rt > −EVaRt(α).
The results are reported in Table 4 where the obtained LEVaR(0.00145) is displayed.
As expected, the best model for forecasting expectiles is Garch(1,1) with Student t innova-
tions, although the differences in the realized losses seem quite small. Further analysis and a
test of significativity could be carried out by means of the model selection procedure outlined in
Bernardi, Catania, and Petrella (2014).
In order to assess the magnitude of the realized loss and to show how to use the realized loss
for testing, we generate 10,000 trajectories of the estimated Garch(1,1) model and computed
a bootstrap distribution of the realized loss. The result is graphically displayed and compared
with the actual value in Figure 8. The fraction of resimulated realized losses bigger than the
actual value of 3.9143 × 10−6 is approximately 10%, so the Garch(1,1) model with Student t
innovations is not rejected at 5% significance level.
Finally, in Figure 9 we compare VaR0.01, ES0.025 and EVaR0.00145 computed with the Garch (1,1)
model with Student t innovations. The three risk measures are very similar, since the forecasting
distributions differs only in a single shape parameter, the number of degrees of freedom ν. In
conclusion, we showed that by a proper choice of q and simple econometric models it is not
difficult to forecast EVaR and obtain capital requirements in line with those of VaR and ES.
Table 4. Realized loss function LEVaR(0.00145) evaluated using the
four considered methods.
Method LEVaR(0.00145)
Normal model (i) 9.2109e − 06
Historical model (ii) 5.5134e − 06
Garch(1,1) model with normal innovations (iii) 4.6263e − 06
Garch(1,1) model with Student t innovations (iv) 3.9143e − 06
00
100
200
300
400
500
600
700
800
900
1 2 3 4 5 6 7 8 9
×10-5
Figure 8. Resimulated distribution of the realized loss using 10,000 resimulations of the Garch (1,1) model
with Student t innovations. The vertical line represents the observed realized loss 3.9143 × 10−6.
The European Journal of Finance 501
ES
VaR
EVaR
0 500 1000 1500 2000 2500 3000
t
−0
.1
5
−0
.2
0
−0
.1
0
−0
.0
5
0.
00
0.
05
0.
10
0.
15
Figure 9. VaR0.01, ES0.025 and EVaR0.00145 for the SP500 log-return, evaluated using Garch(1,1) risk model
with Student t error distribution.
Moreover, the elicitability property of expectiles enables the risk manager to adopt a standard
set of techniques from the forecasting literature to assess the accuracy of the model, to compare
between different models and, if necessary, to provide a statistical test of the model by means of
the realized loss.
Acknowledgments
The authors wish to thank the editor and the two anonymous referees whose comments helped to improve a previous
version of this paper.
Disclosure statement
No potential conflict of interest was reported by the authors.
References
Abdous, B., and B. Remillard. 1995. “Relating Quantiles and Expectiles Under Weighted-Symmetry.” Annals of the
Institute of Statistical Mathematics 47 (2): 371–384.
Acerbi, C., and B. Szekely. 2014. “Backtesting Expected Shortfall.” Risk Magazine 2014, 1–6.
Artzner, P., F. Delbaen, J. M. Eber, and D. Heath. 1999. “Coherent Measures of Risk.” Mathematical Finance 9 (3):
203–228.
Beirlant, J., M. Broniatowski, J. Teugels, and P. Vynckier. 1995. “The Mean Residual Life Function at Great Age:
Applications to Tail Estimation.” Journal of Statistical Planning and Inference 45, 21–48.
Beirlant, J., and J. L. Teugels. 1992. “Modeling Large Claims in Non-Life Insurance.” Insurance: Mathematics and
Economics 11 (1): 17–29.
Bellini, F., and V. Bignozzi. 2013. On elicitable risk measures. Preprint, to appear in Quantitative Finance,
http://papers.ssrn.com/sol3/papers.cfm?abstract_id = 2334746.
Bellini, F., B. Klar, A. Müller, and E. Rosazza-Gianin. 2014. “Generalized Quantiles as Risk Measures.” Insurance:
Mathematics and Economics 54, 41–48.
Ben Tal, A., and M. Teboulle. 2007. “An Old New Concept of Convex Risk Measures: the Optimized Certainty
Equivalent.” Mathematical Finance 449–476.
Bernardi, M., L. Catania, and L. Petrella. 2014. Are news important to predict large losses? Preprint, arXiv:1410.6898v2.
Biagini, S., and M.A. Pinar. 2013. “The Best Gain–Loss Ratio is a Poor Performance Measure.” SIAM Journal on
Financial Mathematics 4 (1): 228–242.
Bingham, N., C. Goldie, and J. Teugels. 1989. Regular Variation. Cambridge: Cambridge University Press.
502 F. Bellini and E. Di Bernardino
Campbell, S. D. 2005. “A review of Backtesting and Backtesting Procedures.” Federal Reserve Finance and Economics
Discussion Series. http://www.federalreserve.gov/pubs/feds/2005/200521/200521pap.pdf
Corless, R. M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth. 1996. “On the Lambert W Function.”
Advances in Computational Mathematics 5(1): 329–359.
Davis, M. 2013. Consistency of Risk Measure Estimates. Preprint.
De Bruijn, N. G. 1970. Asymptotic Methods in Analysis. Vol. 4. North-Holland: Courier Dover Publications.
de Haan, L., and A. Ferreira (2006). Extreme Value Theory: An Introduction, Springer Series in Operations Research and
Financial Engineering, New York: Springer.
De Rossi, G., and A. Harvey. 2009. “Quantiles, Expectiles and Splines.” Journal of Econometrics 152, 179–185.
Delbaen, F. 2012. Monetary Utility Functions. Osaka: Osaka University Press.
Delbaen, F. 2013. A remark on the structure of expectiles. Preprint, arXiv:1307.5881v1.
Delbaen, F., F. Bellini, V. Bignozzi, and J. Ziegel. 2014. Risk measures with the CxLS property. Preprint,
arXiv:1411.0426.
Dierckx, G., J. Beirlant, D. De Waal, and A. Guillou. 2009. “A New Estimaton Method for Weibull-Type Tails Based on
the Mean Excess Function.” Journal of Statistical Planning and Inference 139, 1905–1920.
Embrechts, P., C. Klüppelberg, and T. Mikosch. 1997. Modelling Extremal Events for Insurance and Finance. Berlin:
Springer.
Embrechts, P., G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj. 2014. “An Academic Response to Basel 3.5.” Risks
2 (1): 25–48.
Föllmer, H., and A. Schied. 2002. “Convex Measures of Risk and Trading Constraints.” Finance and Stochastics 6,
429–447.
Föllmer, H., and A. Schied. 2011. Stochastic Finance: An introduction in Discrete Time. Berlin: Walter De Gruyter.
Gneiting, T. 2011. “Making and Evaluating Point Forecast.” Journal of the American Statistical Association 106 (494):
746–762.
Hua, L., and H. Joe. 2011. “Second Order Regular Variation and Conditional Tail Expectation of Multiple Risks.”
Insurance: Mathematics and Economics 49, 537–546.
Jones, M. C. 1994. “Expectiles and M-quantiles are Quantiles.” Statistics and Probability Letters 20, 149–153.
Koenker, R. 1993. “When are Expectiles Percentiles (Solution).” Econometric Theory 9, 526–527.
Koenker, R. 2005. Quantile Regression. Cambridge: Cambridge University Press.
Kuan, C.-M., J.-H. Yeh, and Y.-C. Hsu. 2009. “Assessing Value at Risk with CARE, the Conditional Autoregressive
Expectile Models.” Journal of Econometrics 150, 261–270.
Mao, T., and T. Hu. 2012. “Second-Order Properties of the Haezendonck-Goovaerts Risk Measure for Extreme Risks.”
Insurance: Mathematics and Economics 51, 333–343.
Mao, T., K. Ng, and T. Hu. 2015. “Asymptotic Expansions of Generalized Quantiles and Expectiles for
Extreme Risks.” Probability in the Engineering and Informational Sciences. http://journals.cambridge.org/
article_S0269964815000017.
Newey, W., and J. Powell. 1987. “Asymmetric Least Squares Estimation and Testing.” Econometrica 55 (4): 819–847.
Pflug, G., and W. Romisch. 2007. Modeling, Measuring and Managing Risk. Singapore: World Scientific Publishing
Company.
Rroji, E. 2013. “Risk Attribution and Semi-heavy Tailed Distributions.” PhD thesis, University of Milano Bicocca.
Shadwick, W., and C. Keating. 2002. “A Universal Performance Measure.” Journal of Performance Measurement 6
(3, Spring Issue): 59–84.
Tang, Q., and F. Yang. 2012. “On the Haezendonck-Goovaerts risk Measure for Extreme Risks.” Insurance: Mathematics
and Economics 50, 217–227.
Taylor, J. W. 2008. “Estimating Value at Risk and Expected Shortfall using Expectiles.” Journal of Financial
Econometrics 6, 231–252.
Weber, S. 2006. “Distribution-Invariant Risk Measures, Information, and Dynamic Consistency.” Mathematical Finance
16 (2): 419–441.
Yang, F. 2013. “Asymptotics for Risk Measures of Extreme Risks.” PhD thesis, University of Iowa.
Ziegel, J. 2014. Coherency and elicitability. arXiv:1303.1690v3, to appear in Mathematical Finance.
Zou, H. 2014. “Generalizing Koenker’s Distribution.” Journal of Statistical Planning and Inference 148, 123–127.
The European Journal of Finance 503
A. Appendix
A.1. Basic definitions and results from EVT
We always denote with xˆ the right endpoint of F, that is xˆ := sup{x : F(x) < 1}. The possible limiting distributions of
properly normalized maxima of i.i.d. random variables belong to the class of extreme value distributions, that can take
three possible shapes: Fréchet, Weibull and Gumbel.
Definition A.1 (Extreme value distributions)
β(x) =
{
0 x ≤ 0
exp(−x−β) x > 0 , β > 0
β(x) =
{
exp(−(−x)β) x ≤ 0
0 x > 0
, β > 0
(x) = exp(−e−x), x ∈ R.
A unifying parametrization due to Jenkinson and von Mises is the following:
Gγ :=
⎧⎪⎨
⎪⎩
1/γ , γ > 0,
−1/γ , γ < 0,
, γ = 0.
A convenient characterization of the maximum domain of attractions of the distributions Gγ can be given by means
of the notions of regular variation and extended regular variation:
Definition A.2 (Regular variation) Let β ∈ R. A measurable function h : R → R is of regular variation of index β in
x0 if
lim
t→x0
h(tx)
h(t)
= xβ ,
for each x ∈ R. The class of regularly varying functions of index β in x0 is denoted by RVβ(x0). A regularly varying
function with β = 0 is called slowly varying.
Definition A.3 (Extended regular variation) Let γ ∈ R. A measurable function h : R+ → R is of extended regular
variation with index γ at +∞ if there exists a function a : R+ → R+ such that for all x > 0,
lim
t→+∞
h(tx) − h(t)
a(t)
= x
γ − 1
γ
,
where in the case γ = 0 the right-hand side has to be interpreted as log x. The function a(t) is referred as an auxiliary
function for h. The class of extended regularly varying functions at +∞ is denoted by ERVγ (+∞).
When γ = 0, it is easy to see that the notion of extended regular variation boils down to that of regular variation. The
advantage of Definition A.3 is that it enables to give a unified characterization of the maximum domain of attractions of
the three limiting laws.
Let us denote with f ← the left continuous inverse of a nondecreasing function f : R → R, that is,
f ←(y) := inf{x : f (x) ≥ y},
and with U the so-called tail quantile function, given by
U :=
(
1
F¯
)←
, with F¯ = 1 − F.
Then the following result holds.
504 F. Bellini and E. Di Bernardino
Proposition A.1 (Theorem 1.1.6 in de Haan and Ferreira 2006)
F ∈ MDA(Gγ ) if and only if U ∈ ERVγ .
In the case γ = 0, the notion of regular variation is sufficient to completely characterize MDA(Gγ ) as stated by the
following result. The interested reader is also referred to Examples 3.3.7 and 3.3.12 in Embrechts, Klüppelberg, and
Mikosch (1997).
Proposition A.2 (Theorem 3.3.7 and 3.3.12 in Embrechts, Klüppelberg, and Mikosch 1997)
F ∈ MDA( β) if and only if F¯(x) ∼ x−βL(x),
F ∈ MDA(β) if and only if xˆ < +∞ and F¯
(
xˆ − 1
x
)
∼ x−βL(x),
where L is a slowly varying function and β > 0.
When F ∈ MDA(Gγ ), that is when U ∈ ERVγ , we have the following (see Hua and Joe 2011; Mao and Hu 2012;
Tang and Yang 2012):
Proposition A.3 Let F ∈ MDA( β), with β > 1. Then
lim
x→xˆ
E[(X − x)+]
xF¯(x)
= 1
β − 1 .
Let F ∈ MDA(β); then
lim
x→xˆ
E[(X − x)+]
(xˆ − x)F¯(x) =
1
β + 1 .
Let F ∈ MDA(); then
lim
x→xˆ
E[(X − x)+]
mX (x)F¯(x)
= 1,
where mX (t) := E[X − t|X > t] is the mean excess function of X.
A.2. Proofs
(Proof of Proposition 2.2) Let us first write the f.o.c. (5) in the following equivalent from:
eq(X ) − E[x] = 2q − 11 − q E[(X − x)+].
Let us consider separately the three cases. In the Fréchet case F¯ ∈ RV−β(+∞), and since eq(X ) → +∞ for q → 1,
asymptotically, the f.o.c. becomes
eq(X ) ∼ 11 − qE[(X − eq)+],
and from Proposition A.3, we obtain
eq(X ) ∼ eq F¯(eq)
(1 − q)(β − 1) .
Since F¯(xq) ∼ 1 − q, it follows that F¯(eq(X ))/F¯(xq) ∼ β − 1, from which the thesis follows, as it was already shown in
Bellini et al. (2014). In the Gumbel case, we have to distinguish between the two further subcases xˆ = +∞ and xˆ < +∞.
The European Journal of Finance 505
When xˆ = +∞, from Proposition A.3, asymptotically, the f.o.c. becomes
eq(X ) ∼ mX (eq(X ))F¯(eq(X ))1 − q
and since in this case mX (eq(X )) = o(eq(X )) (see Proposition 3.3.24 in Embrechts, Klüppelberg, and Mikosch 1997), it
follows that F¯(eq(X ))/F¯(xq) → +∞. When xˆ < +∞, we have that
eq(X ) − E[X ] ∼ mX (eq(X ))F¯(eq(X ))1 − q
and since mX (eq(X )) = o(xˆ − eq(X )) → 0 (see Proposition 3.3.24 in Embrechts, Klüppelberg, and Mikosch 1997), it
follows that again F¯(eq(X ))/F¯(xq) → +∞. Finally, in the Weibull case from Proposition A.3 asymptotically, the f.o.c.
is given by
xˆ − E[X ] ∼ (xˆ − eq(X ))F¯(eq(X ))
(1 − q)(β + 1) ,
that gives again
F¯(eq(X ))
F¯(xq)
∼ (β + 1)(xˆ − E[X ])
xˆ − eq(X ) → +∞,
from which the thesis follows.
(Proof of Proposition 2.3) In the Fréchet case, the previously derived asymptotic relationship
F¯(eq) ∼ (β − 1)(1 − q),
can also be written as
eq(X ) = F←(1 − (β − 1)(1 − q + o(1))).
By Theorem 1.5.12 in Bingham, Goldie, and Teugels (1989) if F¯ ∈ RV−β(+∞) then F−1(1 − ·) ∈ RV−1/β (0) (see also
Yang 2013). Then it follows that for q → 1
eq(X ) ∼ (β − 1)−1/βxq.
Under condition (7), from Theorem 2.3.3 and Corollary 2.3.4 in Bingham, Goldie, and Teugels (1989) it follows that
xq ∼
(
1 − q
L((1 − q)−1/β )
)−1/β
,
from which we have the thesis.
(Proof of Proposition 2.4) Under the given asymptotic hypothesis on F¯, Beirlant et al. (1995) prove that
mX (x) ∼ x
1−τ
τL(x)
. (A1)
Asymptotically, the f.o.c. becomes
eq ∼ eq(X )
1−τ exp(−eq(X )τ L(eq(X )))
τ (1 − q)L(eq(X )) ,
that by setting z = eτq L(eq) and applying Lemma A.1 gives
eτqL(eq) ∼ − log(1 − q).
Beirlant et al. (1995) showed also that in this case xq = (− log(1 − q))1/τ L∗(− log(1 − p)), for a slowly varying L∗, that
implies log xq ∼ 1/τ log(− log(1 − q)).
Since for a slowly varying L it holds that log L(x) = o(log(x)), we get log eq ∼ 1/τ log(− log(1 − q)) ∼ log xq.
506 F. Bellini and E. Di Bernardino
Under Equation (10), it also hold that
xτq L(xq) ∼ − log(1 − q),
that by asymptotic inversion gives xq ∼ eq(X ). Hence the result.
(Proof of Proposition 2.5) By Corollary A.3, we have that
xˆ − E[X ] ∼ 1
1 − q (xˆ − eq(X ))
F¯(eq(X ))
β + 1 .
Since
F¯(eq(X )) = F¯(xˆ − (xˆ − eq(X ))) ∼ (xˆ − eq(X ))βL(xˆ − eq(X )),
we get the first thesis. When L = C, the result follows by asymptotic inversion.
A.3. Lambert W function
In the following, we recall the basic properties of the Lambert W function, that is defined implicitly by means of the
equation
W(z) exp(W(z)) = z.
When z ∈ R, z > 0, the solution is unique. For example, it is easy to check that W(e) = 1. The Lambert W function
arises in many applied problems, typically in connection with the solution of transcendental equations; for a detailed
review of its properties see, for example, Corless et al. (1996). In the case of a real z, the asymptotic behaviour of W(z)
for z → +∞ is the same as log z. For completeness, we report the argument in a slightly more general form:
Lemma A.1 Let Wα(t) be the unique positive solution of the equation
Wα exp(W) = t, (A2)
with α > 0 and t > 0. Then for t → +∞,
Wα(t) = log t + O(log log t).
Proof The argument in De Bruijn (1970) applies with almost no modifications. Since the function w → wα exp w is
strictly increasing for w > 0, the solution of Equation (A2) is unique, and t > e → W(t) > 1. Rewriting Equation (A2)
as
W = log t − α log W , (A3)
we get that t > e → W < log t → log W < log log t, which inserted in Equation (A3) gives the thesis.