I nternational J ournal V ol .7, N o .3 (2014)
©
of
I ntelligent T echnologies
pp .185-205,
and
A pplied S tatistics
DOI: 10.6148/IJITAS.2014.0703.01
A iriti P ress
Risk Measures with Wang Transforms under Flexible Skew-generalized Settings Weizhong Tian1, Tonghui Wang1,2* and Baokun Li3 1
De partment M athematical Sciences, N ew M exico State U niversity, N ew M exico, U S A 2 College of Science, N orthwest A & F U niversity, X iányáng City, China 3 School of statistics, Southwestern U niversity of F inance and Economics, Chéngdū City, China
ABSTRACT To provide incentive for active risk managements, tail-preserving and coherent distortion risk measures are needed in the actuarial and financial fields. The purpose of this study is to propose extended versions of Wang transform using two different forms of flexible skewgeneralized distribution f unctions and two dif ferent f orms of f lexible skew-generalized t-distributions with normal kernel and Cauchy kernel. We proved that the f lexible skewgeneralized risk measures in Choquet integral form with normal kernel and Cauchy kernel are coherent and degree-two tail-preserving for usual bi-atomic risk distributions. K eywords: Extended Wang-transform; Flexible skew-generalized normal distribution; Flexible skew-generalized t-distribution; Cauchy kernel; Distortion function
1. Introduction Risk measures are used to decide insurance premiums and required capital for a given risk portfolio by examining its downside risk potential. A widely used risk measure for the risk of loss on a specific portfolio of financial assets is the value at risk (VaR), a threshold value (assuming normal markets and no trading in the portfolio) is the given probability level for a given time horizon. Mathematically, the VaR is simply a percentile on the distribution of losses. Despite its universality, several authors have pointed out the two deficiencies of VaR: Lack of subadditivity, and dif f iculty to optimize measure as it may have multiple local minima [5]. From a regulatory perspective Artzner et al. [2] demonstrated that the VaR does not satisfy consistency rules set for a risk-measure. And they proposed an alternative risk measure, called the conditional alue at risk (CVaR). * Corresponding author:
[email protected]
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Although being coherent, the CVaR ref lects only the mean size of losses exceeding the quartile VaR. It ignores the usef ul information in a large part of the loss distribution, and consequently lacks incentive for mitigating losses below the quartile VaR. Moreover, it does not properly adjust for extreme low-frequency and high-severity losses, since it only accounts for the mean shortfall (not higher moments). Distortion risk measures based on Wang transforms overcome the drawbacks of both the VaR and the CVaR, but they do not always provide incentive for risk management because they do not give capital relief in some simple risk distributions, because they, like CVaR, preserves only degree 1 and 0 tail-preserving order [8]. For risk measures and related properties, see [9-11, 14, 15]. To construct a coherent risk measure which preserves higher degree stoploss orders, recently, Li et al. [12] proposed two extended versions of Type I Wang transf orm using two versions of skew-normal distributions. In this paper, we extended results of Li et al. [12] using flexible generalized skew distribution with normal kernel and Cauchy kernel, and then proposed extend versions of both Type I Wang transform and Type II Wang transform. Finally, we proved that the corresponding risk measures are coherent and preserves higher degree stop-loss orders for diatomic lose distributions. This paper is organized as follows. Two distribution risk measures and their properties are discussed in Section 2 and two forms of f lexible skew-generalized distributions together with two forms of skew t-distributions with normal kernel and Cauchy kernel are introduced in Section 3. Extended versions of Wang transforms under f lexible skew-generalized settings and flexible skew-generalized t-settings, together with their distortion functions are studied in Section 4. Results on properties of risk measures with new distortion functions are obtained in Section 5.
2. Two types of distortion risk measures Let (Ω, A, P) be a probability space where Ω is the space of outcomes or states of the world, A is the σ -algebra of subsets of Ω, and P is the probability measure. For a measurable real-valued random variable X : Ω → ℜ, the probability distribution of X is denoted by F X (x) = P(X ≤ x).
2.1 Coherent distortion risk measures In this paper, the random variable X represents a financial loss such that, for ω ∈ Ω, the real number X(ω ) is the realization of a loss or prof it f unction, with X(ω ) ≥ 0 for a loss and X(ω ) < 0 for a profit. Let X be the set of financial losses. A risk measure, ρ (X ), is a functional from the set of losses to the extended nonnegative real numbers described by a map ρ : X → ℜ+ Note that ρ (X ) can be considered as an amount that a company must reserve to face financial loss X, that is ρ (X ) is a
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minimum amount that the company insurance must reserve to pay the damage made by risk X so that ρ (0) = 0. Definition 1 A risk measure is said to be coherent if it satisf ies the f ollowing desirable properties (see, e.g., [1, 2]). (i) Monotonicity: ρ (X ) ≤ ρ (Y ) provided that P(X ≤ Y ) = 1. (ii) Homogeneity: For any c > 0 and X ∈ X, ρ (cX ) = cρ (X ). (iii) Sub-additivity: For any X, Y ∈ X, ρ (X + Y ) ≤ ρ (X ) + ρ (Y ). (iv) Translation invariance: ρ (X + c) ≤ ρ (X ) + c for any X ∈ X and c ∈ ℜ. Definition 2 Let F X (x) be the distribution function of the random variable X ∈ X and ε ∈ (0, 1) be a probability level. (i) The quartile ε of X is called the value at risk at level ε , i.e., V aR(X; ε ) = F X-1 (ε ). (ii) The conditional VaR of level ε , denoted by CV aR( X ; ε ), is def ined as the expected loss given X > CV aR(X; ε ), i.e., CV aR(X; ε ) = E[X - V aR(X; ε ) | X > V aR(X; ε )]. Note that the V aR(X; ε ) is not coherent, but the CVaR is coherent. A distortion f unction is a continuous non-decreasing function g : [0, 1] → [0, 1] with g(0) = 0 and g(1) = 1. Let F X (x) be the distribution function of the risk X, the transform F Xg (x) = g(F(x)) defines a distribution function, which is called the distorted distribution f unction. By Choquet integral, we will focus on the distortion risk measure of g given by
(1)
where SX (x) = 1 - F X (x) is the survival function of X. Wang [16] proved that the risk measure given in (1) is coherent if the distortion function g(x) is a concave function.
2.2 Tail-preserving distortion risk measures A desirable property f or a risk measure is that increased risk should be penalized with an increased measure. A distortion measure ρ g (x) with concave distortion f unction preserves the stop-loss order (see, e.g., [7]). With equal means and variances, a stop-loss order relation between different random variables cannot exist. In this situation, one is interested in distortion measures that preserve higher degree convex orders. Such measures are called tail-preserving distortion measures.
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For any random variable X ∈ X with probability distribution function F X (x), the higher order partial moments Π Xn (x) = E (X - x)+n , n = 0, 1, 2, ..., are said to be the degree n stop-loss transf orms. Note that Π X0 (x) = SX (x) = 1 - F X (x) is te survival function of X. For two random variables X and Y, if Π Xn (x) = ΠYn (x) for all x, the X is said to precede Y in degree n stop-loss transf orm order, denoted by X ≤ (n) slt Y. Also, we say that the X precedes Y in (n + 1) - convex order, denoted by X ≤ (n+1)-cx Y if X ≤ (n) slt Y k k and E[X ] = E[Y ], for all k = 0, 1, ..., n. Definition 3 A risk measure ρ : X → R is said to be a degree n tail-preserving risk measure if it is preserved under the (n + 1) - convex order. That is, if two random variables X and Y \ ∈ X satisfy X ≤ (n+1)-cx Y, then ρ (X ) ≤ ρ (Y ). It is known that a distortion measure ρg (X ) with concave distortion function g(x) preserves (n + 1) - cx order for n = 0, 1 so that it is a tail-preserving risk measure of degree zero and one. From Examples 3.1 and 3.2 of Hürlimann [8], both the CVaR and Wang’s distortion risk measures are not the degree-two tail-preserving coherent risk measures.
3. Flexible skew-generalized distributions with normal and Cauchy kernels The univariate skew-symmetric models have been considered by several authors, see [3, 4, 6, 18]. In the last three decades there has been substantial work in the area of skew normal (SN) and related distributions. The main feature of these models is that a new parameter X, called skewness parameter, is introduced to control skewness and kurtosis. We will introduce this class of distribution by the following lemma. Lemma 1 Let X and Y be two arbitrary continuous independent random variables symmetric about 0, with density f unctions f (x) and g(y), and commutative distribution functions F(x) and G(y), respectively. Then for any λ ∈ ℜ, the function
is a density function. Ma and Genton [13] introduced a flexible class of skew symmetric distributions, given by
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where φ and Φ denote a standard normal density and commutative distribution function, respectively. Here pk is an odd polynomial of order k, and a polynomial order k in R p is defined as a linear combination of terms of the form
, where
, if each term has an odd order then the polynomial is called an odd polynomial. Definition 4 (1) A random variable Z is said to be f lexible skew-generalized normal distributed, denoted by FSGN(λ 1, λ 2, λ 3), if its density function has the form
(2)
where λ 1, λ 3 ∈ R and λ 2 ≥ 0 are constants. In general, the distribution of X = µ + σ Z is denoted by X ~ FSGN( µ , σ 2, λ 1, λ 2, λ 3). (2) A random variable Z is said to be f lexible skew-generalized Cauchy-normal distributed, denoted by FSGN( γ , λ 1, λ 2, λ 3), if its density function has the form
(3)
where λ 1, λ 3 ∈ R, γ > 0 and λ 2 ≥ 0 are constants. In general, the distribution of X = µ + σ Z is denoted by X ~ FSGCN( µ , σ 2, λ 1, λ 2, λ 3). The density f unction, f (x | λ 1, λ 2, λ 3), of f lexible skew-generalized normal distribution with µ = 0, σ = 1 is given in Figure 1. The density f unction, f (x | γ , λ 1, λ 2, λ 3), for f lexible skew-generalized Cauchynormal distribution with µ = 0, σ = 1 is given in Figure 2. Proposition 1 (1) The moment generating function of X ~ FSGN(0, 1, λ 1, λ 2, λ 3) is
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Figure 1. Density curves of f (x | -3, 0, 0) (dotted line), f (x | 0, 0, 0) (solid line) and f (x | -2, 1, 2) (bold line).
X value
Figure 2. Density curves of f (x | 0.5, -1, 1, -1) (dotted line), f (x | 2, -1, 1, -1) (solid line) and f (x | 1, 0, 0, 0) (bold line). where Z ~ N (0, 1). It is easy to check for k = 0, 1, 2, ..., we have
where
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(2) The Characteristic function of X ~ FSGCN(0, 1, γ , λ 1, λ 2, λ 3) is
where . In particular, the mean and variance of X ~ FSGN( µ , σ 2, α , 0, 0) are
and
,
where . Corresponding to Type II Wang transf orm, we need to def ine the skew t-distributions as follows. Definition 5 Let X ~ FSGN( µ , σ 2, λ 1, λ 2, λ 3), and U ~ χ k2 be independent. The random variable is said to have a f lexible skew-generalized t-distribution with normal kernel with k degrees of freedom, location parameters ( µ , σ 2 ) and skewness parameters λ 1, λ 2, λ 3. Thus the f lexible skew-generalized skew-t distribution with normal kernel is defined below.
Remark 1 From Definition 3.2, we can obtain the density of T N from the joint density of X and U, which ST N,k given by, after simplification
where EU ( ⋅ ) is the expected value of ( ⋅ ) with respect to U ∈ (0, ∞). The density curves, fTN,k(t | λ 1, λ 2, λ 3), of flexible skew-generalized t-distribution with normal kernel with µ = 0, σ = 1 is given in Figure 3.
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Figure 3. Density curves of fTN,1(t | -1, 1, -1) (dotted line), fTN,5(t | -1, 1, -1) (dot-dashed line) and fTN,5(t | -1, 1, 2) (bold line). Similar to the definition of flexible skew-generalized t-distribution with normal kernel, we can def ine the f lexible skew-generalized t-distribution with Cauchy kernel, as follows. Definition 6 Let X ~ FSGCN( µ , σ 2, γ , λ 1, λ 2, λ 3), U ~ χ k2 be independent. The random variable is said to have a f lexible skew-generalized t-distribution with Cauchy kernel with k degrees of freedom, location parameters ( µ , σ 2 ) and skewness parameters λ 1, λ 2, λ 3. Thus the flexible skew-generalized t-distribution with Cauchy kernel is defined below:
The density function, fTC ,k(t | λ 1, λ 2, λ 3), of flexible skew-generalized t-distribution with Cauchy kernel with µ = 0, σ = 1 is given in Figure 4. Proposition 2 Let to be a flexible skew-generalized t-distribution with normal kernel with k degrees of freedom and skew parameters λ 1, λ 2, λ 3, then
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Figure 4. Density curves of fTC ,1(t | 1, -1, 1, -1) (dotted line); fTC ,5(t | 1, -1, 1, -1) (dot-dashed line) and fTC ,5(t | 4, -1, 1, 2) (bold line). where
.
4. Generalized versions of the Wang transf orm Motivated by directly extending the Sharpe ratio concept to risks with skewed distributions, Wang [15] proposed the following Wang transform:
(4)
where Φ(z) is the cumulative distribution f unction (cdf ) of the standard normal random variable Z ~ N (0, 1). The Wang transf orm in Equation is a pricing formula that recovers CAPM and Black-Scholes formula under normal asset-return distributions, and is called Type I Wang transform. Following the statistical sampling theory that uses a student-t distribution in place of a normal distribution, Wang [17] yields the following two-factor model:
(5)
where Q(t) is the cdf of with independent Z ~ N (0, 1) and U ~ χ k2. The equation is called the Type II Wang transform.
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It is well known that the two transforms are concave distortion functions, the distortion risk measures are coherent. By Hürlimann [8], the Type I distortion risk measures are not degree-two tail-preserving risk measures; also it could be verified that the Type II distortion risk measures are not degree- two tail-preserving risk measures either. Li et al. [12] have proved that the skew normal risk measures in Choquet integral form are coherent and the degree-two tail-preserving for usual biatomic risk distributions. In many real world applications, the data sets collected are not symmetrically distributed so that normal or student t-distributions may not be the good f it. Specif ically, data sets are skewed to the lef t of skewed to the right in most applications so that the skew normal distributions may be the better choices. In this paper, we will construct extended versions of Wang transforms using distribution functions with flexible generalized skewed densities. Let
(6)
and
(7)
be the distribution functions of flexible skew-generalized normal random variable, and f lexible skew-generalized Cauchy-normal random variable, respectively. Corresponding to Type I Wang’s transf orm, we def ine our extended distortion functions as follows. Definition 7 For x > 0, the flexible skew-generalized normal distributed, and flexible skewgeneralized Cauchy-normal distributed distortion function are defined by
(8)
and
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respectively, where Φ( ⋅ ) is the cdf of N (0, 1), SNΦ( ⋅ ) is the cdf of FSGN( µ , σ 2, λ 1, λ 2, λ 3) with parameters satisfying µ > 0, λ 1 ≤ 0, λ 3 ≤ 0, and λ 2 ≥ 0, and SCΦ( ⋅ ) is the cdf of γ -skew normal distribution FSGCN( µ , σ 2, γ , λ 1, λ 2, λ 3) with parameters satisfying µ > 0, λ 1 ≤ 0, λ 3 ≤ 0, γ > 0 and λ 2 ≥ 0. The following two Figures represent for different distortion functions given in (8) and (9). The flexible skew-generalized normal distortion functions g N (x) with µ = 0, σ = 1 are shown in Figure 5. The flexible skew-generalized Cauchy-normal distortion functions gC (x) with µ = 0, σ = 1 are shown in Figure 6. Remark 2 When λ 1 = λ 3 = 0, this distortion f unction is reduced to the Type I Wang’s transform with λ = µ . Like the previous Wang transforms, we will show that the distortion functions defined above are concave in next section. Corresponding to Type II Wang transform, we def ine distortion f unctions as follows. Definition 8 For x > 0, the f lexible skew-generalized t-distributed distortion function with normal kernel and the f lexible skew-generalized t-distributed distortion f unction with Cauchy kernel are defined by
(10)
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Figure 5. Distortion curves of SNΦ(Φ-1(x) | 0, 1, -1, 1, -1) (dotted line); SNΦ(Φ-1(x) | 0, 1, -2, 1, -1) (dot-dashed line) and SNΦ(Φ-1(x) | 0, 1, -1, 0, 0) (bold line).
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and
(11)
respectively, where SNQ k( ⋅ ) is the cdf of distribution ST C,k( µ , 1, λ 1, λ 2, λ 3) with parameters satisfying µ > 0, λ 1 ≤ 0, λ 3 ≤ 0, and λ 2 ≥ 0, and SCQ k( ⋅ ) is the cdf of the distribution ST C,k( µ , 1, γ , λ 1, λ 2, λ 3) with parameters satisfying µ > 0, λ 1 ≤ 0, λ 3 ≤ 0, γ > 0 and λ 2 ≥ 0. The following two figures represent for different distortion functions given in (10) and (11). The f lexible skew-generalized t-distributed distortion f unction, g T N ,k(x), with normal kernel with µ = 0, σ = 1 are shown in Figure 7. The f lexible skew-generalized t-distributed distortion f unctions, gTC ,k( y), with Cauchy kernel with µ = 0, σ = 1 are shown in Figure 8. Remark 3 Note that when λ 1 = λ 3 = 0, the above distortion function is reduced to Wang Type II transform and thus they are treated as extended Wang Type II transforms.
5. Properties of risk measures with generalized Wang distortion f unctions
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A distortion risk measure can be def ined as the distorted expectation of any non-negative loss random variable X. It is accomplished by using a utility or the distortion function g as follows:
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Figure 6. The distortion curves of SC Φ(Φ-1(x) | 0, 1, -4, -1, 1, -1) (dotted line), SC Φ(Φ-1(x) | 0, 1, 1, -2, 2, -2) (dot-dashed line), and SC Φ(Φ-1(x) | 0, 1, 2, -1, 1, -1) (bold line).
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(12)
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where SX (x) denotes the survival function of X, while g(SX (x)) is referred to as a distorted survival function.
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Figure 7. The distortion curves of SNQ1(Φ-1(x) | 0, 1, -1, 1, -1) (dotted line); SNQ4(Φ-1(x) | 0, 1, -1, 1, -1) (dot-dashed line) and SNQ4(Φ-1(x) | 0, 1, -2, 1, -2) (bold line).
X value
Figure 8. The distortion curves of SCQ1(Φ-1(x) | 0, 1, 1, -1, 1, -1) (dotted line), SCQ4(Φ-1(x) | 0, 1, 2, -1, 1, -1) (dot-dashed line) and SCQ4(Φ-1(x) | 0, 1, 2, -2, 2, -2) (bold line).
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For the gain/loss-distributions, we know that the loss random variable X can take any real value so that the distortion risk measure is defined as in Equation (12). The VaR is not a distortion risk measure because its distortion f unction is discontinuous in this case. Except the fact that properties of the distortion risk measures come from standard results about the Choquet integral, Wirch and Hardy (see [19]) proved that distortion risk measures are sub-additive, i.e.,
if and only if the distortion function g(x) is concave. Thus the concave distortion risk measures are coherent risk measures. In order to show that each of our extended versions of distortion risk measures is coherent, one only need to prove that each of our extended distortion functions in (8) and (9) are concave. Theorem 1 (Coherent risk measures) (1) The distortion risk measure corresponding to the f lexible skew-generalized normal distribution gN (x) given in (8) is coherent and; (2) The distortion risk measure corresponding to the f lexible skew-generalized Cauchy-normal distribution gC (x) in (9) is coherent. Proof To show that gN (x) is coherent, we need to show that gN (x) has negative second derivative everywhere f or X ∈ X . Thus f or (1), we obtain that the f irst order derivative
which is exists and positive for all X ∈ X. The second derivative of gN (x) also exists and is obtained as
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Given λ 1 ≤ 0, λ 3 ≤ 0, λ 2 ≥ 0 and µ < 0, it is easy to check that gN" (x) is always negative. So the distortion risk measure, corresponding to gN (x) in (8), defined by Choquet integral is coherent. Similarly, for (2), we set t = (Φ -1 (x) - µ ) we have
Thus gC' (x) exists for all X ∈ X. Also we have
which exists because of the same reason discussed above. Since
, so we just care bout
. Give
, λ 1 ≤ 0, λ 3
≤ 0, and µ < 0, it is easy to see that gC" (x) is always negative. So the distortion risk measure, corresponding to gC(x) in (9), defined by Choquet integral is coherent. Corollary 1 (Li et al. [11]) The distortion risk measure corresponding to the skew-generalized normal distortion function gN (x) given by λ 2 = λ 3 = 0 in (8) is coherent.
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Theorem 2 (Coherent risk measures) The distortion risk measure corresponding to the f lexible skew-generalized t-distributed with normal kernel given in (10) by λ 2 = λ 3 = 0 is coherent in [0, 1/2]. Proof For the simplicity of our proof, let
so that
Thus
Taking the first order derivative of gTN,k(x) with respect to x, we have
where ω = (Φ -1 (x) - µ ). Similarly, we obtain
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It is easy to see that φ '(z) = (-z)φ (z) and φ '(ωV ) = (-ωV )φ (ωV ). Given λ 1 ≤ 0 and µ < 0, we would like to find the range of x such that gTN,k (x) ≤ 0. Since EV is the integral sign from 0 to ∞ so that gTN,k (x) ≤ 0 if and only if the integrand, say h(x, v) here, is equal or less than 0. Now the first term of the integrand h(x, v) is equal or less than 0 if ω = Φ -1 (x) - µ ≥ 0, which is equivalent to x ≤ 1/2. Note that the second term of the integrand h(x, v) is equal or less than 0 because λ 1 ≤ 0. For the third term of the integrand to be equal or less than 0, we need φ ' = [Φ -1(x)] ≥ 0, which is equivalent to -Φ -1(x) ≥ 0 and hence x ≤ 1/2. From above discussion, we conclude that gTN,k (x) ≤ 0 for all x ∈ [0, 1/2]. When the loss distribution is normal, i.e., X ~ N( µ 0, σ 2 ),
Thus we can prove the following result: Theorem 3 Assume that X ~ N ( µ 0, σ 2 ), the normal loss distribution. Let F X (x) be the cdf of X and S X (x) = 1 - F X (x). (1) For the flexible skew-generalized normal distributed distortion function given in (8),
where SNΦ λ1,λ 2,λ 3(x | µ , σ 2 ) = SNΦ(x | µ , σ 2, λ 1, λ 2, λ 3). (2) For the flexible skew-generalized Cauchy normal distributed distortion function given in (9),
where SCΦ γ ,λ1,λ 2,λ 3(x | µ , σ 2 ) = SCΦ(x | µ , σ 2, γ , λ 1, λ 2, λ 3). Proof Note that
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and
The desired results follow from the facts given by
and
From Theorem 3, we know that, for a normally distributed loss random variable, its distorted loss distribution is skew-normal. Denote the distorted distribution as the distribution of the loss random variable X *, then the risk measures in Equation (12) for both g N (x) is
where
(13)
.
Similarly, the risk measures in Equation (12) for gC (x) (if exists) is
where
(14)
.
Note that when λ 1 = λ 3 = 0 the risk measures are reduced to that of Wang’s Type I distortion function. In the following we will show that the distortion risk measure is also degree-two tail-preserving. To prove this statement, we need the Proposition 4.1 of Hürlimann [8] given as follows.
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Lemma 2 Let g(x) be a continuous and dif f erentiable increasing concave distortion function. The coherent distortion risk measure ρ g(X ) is a degree-two tail-preserving risk measure for the subset of bi-atomic losses if and only if the following condition holds:
(15)
Theorem 4 (1) The distortion risk measure corresponding to the f lexible skew-generalized normal distortion f unction given in (8) is the degree-two tail-preserving risk measure for the bi-atomic losses. (2) The distortion risk measure corresponding to the f lexible skew-generalized Cauchy-normal distortion function given in (9) is the degree-two tail-preserving risk measure for the bi-atomic losses. Proof We prove the part (i) only and the part (ii) can be proved similarly. Set up G(x) = g N (x) - 2xg'N (x) and it is easy to check for λ 1 > 0, λ 3 < 0, and λ 2 > 0, G'(x) < 0 for all x ∈ (0, 1). For x = 1, we can get G(1) = 0, so which means, the inequality holds. Corollary 2 (Li et al. [11]) The distortion risk measure corresponding to λ 2 = 0 and λ 3 = 0 given in (8) is the degree-two tail-preserving risk measure for the bi-atomic losses.
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