# Doubly truncated moment risk measures for elliptical distributions

In this paper, we define doubly truncated moment (DTM), doubly truncated skewness (DTS) and kurtosis (DTK). We derive DTM formulae for elliptical family, with emphasis on normal, student-t, logistic, Laplace and Pearson type VII distributions. We also present explicit formulas of the DTE (doubly truncated expectation), DTV (doubly truncated variance), DTS and DTK for those distributions. As illustrative example, DTEs, DTVs, DTSs and DTKs of three industry segments' (Banks, Insurance, Financial and Credit Service) stock return in London stock exchange are discussed.

## Authors

• 6 publications
• 15 publications
12/10/2021

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### Moments and random number generation for the truncated elliptical family of distributions

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### Right-truncated Archimedean and related copulas

The copulas of random vectors with standard uniform univariate margins t...
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### Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution

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05/23/2019

### Randomized Reference Classifier with Gaussian Distribution and Soft Confusion Matrix Applied to the Improving Weak Classifiers

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### Truncated Variational Expectation Maximization

We derive a novel variational expectation maximization approach based on...
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## 1 Introduction

Landsman et al. (2016b) defined a new tail conditional moment (TCM) risk measure for a random variable

:

 TCMq(Xn)=E[(X−TCEq(X))n|X>xq], (1)

where

 TCEq(X)=E(X|X>xq)

is tail conditional expectation (TCE) of , is

-th quantile and

. Furthermore, they also defined novel types of tail conditional skewness and kurtosis (TCS and TCK):

 TCSq(X)=E[(X−TCEq(X))3|X>xq]TV3/2q(X) (2)

and

 TCKq(X)=E[(X−TCEq(X))4|X>xq]TV2q(X)−3, (3)

where

 TVq(X)=E[(X−TCEX(xq))2|X>xq]

is tail variance (TV) of . Since Landsman et al. (2016b) has been derived formulae of TCM for elliptical and log-elliptical distributions, and has been presented TCS and TCK for those distributions, Eini and Khaloozadeh (2021) generalized those results to generalized skew-elliptical, Zuo and Yin (2021a) extended them to some shifted distributions.

Recently, Roozegar et al. (2020) derived explicit expressions of the first two moments for doubly truncated multivariate normal mean-variance mixture distributions. Zuo and Yin (2021b) defined multivariate doubly truncated expectation and covariance risk measures, and derived formulas of multivariate doubly truncated expectation (MDTE) and covariance (MDTCov) of elliptical distributions. As special cases of MDTE and MDTCov risk measures, authors also defined doubly truncated expectation (DTE) and variance (DTV) risk measures for a random variable as follows, respectively:

 DTE(p,q)(X)=E(X|xp

and

 DTV(p,q)(X)=E[(X−DTE(p,q)(X))2|xp

where is -th quantile, and .

Inspired by those work, we define doubly truncated moment (DTM), and also define doubly truncated skewness (DTS) and kurtosis (DTK). Moreover, we derive doubly truncated moments (DTM) for elliptical family, and also give explicit expressions of DTE, DTV, DTS and DTK for this family and it’s several special cases, such as normal, student-, logistic, Laplace and Pearson type VII distributions. As illustrative example, we discuss DTEs, DTVs, DTSs and DTKs of three industry segments’ (Banks, Insurance, Financial and Credit Service) stock return in London stock exchange.

The rest of the paper is organized as follows. Section 2 defines several doubly truncated risk measures. Section 3 introduces elliptical family and it’s properties. In Section 4, we present -th doubly truncated moments (TCM) for elliptical distributions, and derive explicit expressions of DTV, DTS and DTK for this family. Special cases are given in Section 5. We give illustrative example in Section 6. Finally, in Section 7, is the concluding remarks.

## 2 Doubly truncated risk measures

We define doubly truncated moment (DTM) risk measure of a random variable as follows:

 DTM(p,q)(Xn)=E[(X−DTE(p,q)(X))n|xp

where is as in (4), is -th quantile, and .
When , the doubly truncated moment (DTM) is reduced to tail conditional moment (TCM); When and , the doubly truncated moment (DTM) is reduced to central moment. Further, we define doubly truncated skewness (DTS) and kurtosis (DTK) risk measures:

 DTS(p,q)(X)=E[(X−DTE(p,q)(X))3|xp

and

 DTK(p,q)(X)=E[(X−DTE(p,q)(X))4|xp

where is as in (5).
When , the doubly truncated skewness (DTS) is reduced to tail conditional skewness (TCS), and the doubly truncated kurtosis (DTK) is reduced to tail conditional kurtosis (TCK); When and , the doubly truncated skewness (DTS) is reduced to skewness, and the doubly truncated kurtosis (DTK) is reduced to kurtosis.

Note that Molchanov and Cascos (2016), Cai et al. (2017) and Shushi and Yao (2020) proposed set risk measures (defined as a map from subset of possible outcomes of losses to some measure-valued space , i.e., ) are mathematically abstract and are very complicated when dealing with risks. However, tail conditional moment risk measures are relatively simpler than that of the set risk measures and can be derived explicitly, important for actuarial users (see Landsman et al., 2016a). In addition to these advantages of tail conditional moment, doubly truncated moment risk measures are more flexible than tail conditional moment risk measures. In other words, according to different needs, we can choose different .

## 3 Elliptical distributions

A random variable is said to have an elliptically symmetric distribution (see Landsman and Valdez, 2003)

 fX(x):=c1σg1{12(x−μσ)}, x∈R, (9)

where is a location parameter, is a scale parameter, , , is the density generator of , and denoted by . The density generator satisfies the condition

 ∫∞0s−1/2g1(s)ds<∞, (10)

and the normalizing constant is given by

 c1 =Γ(1/2)(2π)1/2[∫∞0s−1/2g1(s)ds]−1

We define a sequence of cumulative generators :

 ¯¯¯¯G(1)(u)=∫∞ug1(s)ds (11)

and

 ¯¯¯¯G(k)(u)=∫∞u¯¯¯¯G(k−1)(s)ds, k≥2. (12)

The normalizing constants are given by

 c∗(k)=1√2[∫∞0s−1/2¯¯¯¯G(k)(s)ds]−1. (13)

The density generators satisfy the condition

 ∫∞0s−1/2¯¯¯¯G(k)(s)ds<∞, k=1,2⋯,n. (14)

## 4 N-th doubly truncated moment

In this section, we present -th doubly truncated moment (DTM) of elliptical distributions, and also present DTV, DTS and DTK of elliptical distributions.

To derive -th DTM of elliptical distributions, we define a new truncated distribution function as follows (see Zuo and Yin, 2021b):

 FZ(a,b)=∫bafZ(z)dz,

where is pdf of random variable .

Firstly, we give following lemma.

###### Lemma 1

Let . Assume it satisfies conditions (10) and (14). Then

 E[Xn|xp

where

 L1=c1[ξi−1p¯¯¯¯G1(12ξ2p)−ξi−1q¯¯¯¯G1(12ξ2q)]FY(ξp,ξq),
 L2=∫ξqξpyi−2c∗(1)¯¯¯¯G(1)(12y2)dyFY(ξp,ξq),

, , and .

Using definition, we have

 E[Xn|xp

Applying the transformation , and using the Binomial Theorem, we obtain

 E[Xn|xp

Therefore,

 E[Xn|xp

as required.

Now we establish the formula of DTM for elliptical distributions.

###### Theorem 1

Suppose that , which satisfies conditions (10) and (14). Then

 DTM(p,q)(Xn)=(−1)nDTEn(p,q)(X)+(−1)n−1nDTEn(p,q)(X) +n∑k=2(nk)(−DTE(p,q)(X))n−k[μk+kμk−1σDTE(p,q)(Y)] +n∑k=2k∑i=2(nk)(ki)(−DTE(p,q)(X))n−kμk−iσi[L1+(i−1)c1c∗(1)L2], n≥2, (16)

where

 DTE(p,q)(X)=μ+σc1[¯¯¯¯G1(12ξ2p)−¯¯¯¯G1(12ξ2q)]FY(ξp,ξq), (17)

, , and are the same as those in Lemma 1.

Using the Binomial Theorem and basic algebraic calculations, we have

 DTM(p,q)(Xn) =E[(X−DTE(p,q)(X))n|xp

By (45) of Zuo and Yin (2021b), we obtain that is as in (17).
Then, using Lemma 1, we obtain Eq.(1), as required.
When , the -th tail conditional moment (TCM) for elliptical distribution is given by

 TCMp(Xn)=(−1)nTCEnp(X)+(−1)n−1nTCEnp(X) +n∑k=2(nk)(−TCEp(X))n−k[μk+kμk−1σTCEp(Y)] +n∑k=2k∑i=2(nk)(ki)(−TCEp(X))n−kμk−iσi[L1+(i−1)c1c∗(1)L2], n≥2, (18)

where

 TCEp(X)=μ+σc1¯¯¯¯G1(12ξ2p)¯¯¯¯FY(ξp), (19)
 L1=c1ξi−1p¯¯¯¯G1(12ξ2p)¯¯¯¯FY(ξp),
 L2=∫∞ξpyi−2c∗(1)¯¯¯¯G(1)(12y2)dy¯¯¯¯FY(ξp),

and denotes tail function of .
Note that (4) is the result of Theorem 1 in Landsman et al. (2016b).
Letting and in Theorem 1, the -th central moment (CM) for elliptical distribution leads to

 CM(Xn)= (−1)nμn+(−1)n−1nμn+n∑k=2(nk)(−μ)n−kμk +n∑k=2k∑i=2(nk)(ki)(−1)n−kμn−iσi(i−1)c1c∗(1)L2, n≥2, (20)

where

 L2=∫∞−∞yi−2c∗(1)¯¯¯¯G(1)(12y2)dy.

Now, we give explicit expressions of DTV, DTS and DTK for elliptical distributions.

###### Corollary 1

Under conditions of Theorem 1, we have

 DTV(p,q)(X)=−DTE2(p,q)(X)+μ2+2μσDTE(p,q)(Y)+σ2(L1+c1c∗(1)L2), (21)

where

 L1=c1[ξp¯¯¯¯G(1)(12ξ2p)−ξq¯¯¯¯G(1)(12ξ2q)]FY(ξp,ξq), L2=FY(1)(ξp,ξq)FY(ξp,ξq),

where , and are the same as those in Theorem 1. In addition, .

Note that (21) coincides with the result of (65) in Zuo and Yin (2021b). When , (21) is the result of (1.7) in Furman and Landsman (2006).

###### Corollary 2

Under conditions of Theorem 1, we have

 DTS(p,q)(X) =DTV−3/2(p,q)(X){3∑k=2(3k)[−DTE(p,q)(X)]3−k[μk+kμk−1σDTE(p,q)(Y)] (22)

where

 L∗2 =c1[¯¯¯¯G(2)(12ξ2p)−¯¯¯¯G(2)(12ξ2q)]FY(ξp,ξq),

where , and are the same as those in Corollary 1.

###### Corollary 3

Under conditions of Theorem 1, we have

 DTK(p,q)(X) =DTV−2(p,q)(X){−3DTE4(p,q)(X)+6[μ−DTE(p,q)(X)]2σ2(L1+c1c∗(1)L2) +4∑k=2(4k)(−DTE(p,q)(X))4−k[μk+kμk−1σDTE(p,q)(Y)] (23)

where

 L∗∗1=c1[ξ3p¯¯¯¯G(1)(12ξ2p)−ξ3q¯¯¯¯G(1)(12ξ2q)]FY(ξp,ξq),
 L∗∗2 =c1[ξp¯¯¯¯G(2)(12ξ2p)−ξq¯¯¯¯G(2)(12ξ2q)]FY(ξp,ξq)+c1c∗(2)FY(2)(ξp,ξq)FY(ξp,ξq).

Here , and are the same as those in Corollary 2. In addition, .

Note that (2) and (3) coincide with the results of (3.22) and (3.24) in Landsman et al. (2016b) as , repectively.

## 5 Special cases

In the following, we present DTV, DTS and DTK for several special members of univariate elliptical distributions, such as normal, student-, logistic, Laplace and Pearson type VII distributions.
(Normal distribution) Let . In this case, the density generators are expressed:

 g1(u)=¯¯¯¯G(1)(u)=¯¯¯¯G(2)(u)=exp{−u},

and the normalizing constants are written as:

 c1=c∗(1)=c∗(2)=(2π)−12.

Then

 DTV(p,q)(X)=−DTE2(p,q)(X)+μ2+2μσDTE(p,q)(Y)+σ2(L1+1),
 DTS(p,q)(X) =DTV−3/2(p,q)(X){3∑k=2(3k)[−DTE(p,q)(X)]3−k[μk+kμk−1σDTE(p,q)(Y)] +2DTE3(p,q)(X)+3[μ−DTE(p,q)(X)]σ2(L1+1)+σ3(L∗1+2L∗2)},
 DTK(p,q)(X) =DTV−2(p,q)(X){−3DTE4(p,q)(X)+6[μ−DTE(p,q)(X)]2σ2(L1+1) +4∑k=2(4k)(−DTE(p,q)(X))4−k[μk+kμk−1σDTE(p,q)(Y)]

where

 DTE(p,q)(X)=μ+σϕ(ξp)−ϕ(ξq)FY(ξp,ξq),
 L1=ξpϕ(ξp)−ξqϕ(ξq)FY(ξp,ξq), L∗1=ξ2pϕ(ξp)−ξ2qϕ(ξq)FY(ξp,ξq),
 L∗2=ϕ(ξp)−ϕ(ξq)FY(ξp,ξq), L∗∗1=ξ3pϕ(ξp)−ξ3qϕ(ξq)FY(ξp,ξq),

, , and . In addition, is pdf of -dimensional standard normal distribution.
(Student- distribution). Let In this case, the density generators are expressed (for details see Zuo et al., 2021):

 g1(u)=(1+2um)−(m+1)/2,
 ¯¯¯¯G(1)(u)=mm−1(1+2um)−(m−1)/2

and

 ¯¯¯¯G(2)(u)=m2(m−1)(m−3)(1+2um)−(m−3)/2.

The normalizing constants are written as:

 c1=Γ((m+1)/2)Γ(m/2)(mπ)12,
 c∗(1) =(m−1)Γ(1/2)(2π)1/2m[∫∞0u1/2−1(1+2tm)−(m−1)/2du]−1 =(m−1)m3/2B(12, m−22), if m>2

and

 c∗(2) =(m−1)(m−3)Γ(1/2)(2π)1/2m2[∫∞0u1/2−1(1+2tm)−(m−3)/2du]−1 =(m−1)(m−3)m5/2B(12, m−42), if m>4,

where and are Gamma function and Beta function, respectively. Then

 DTV(p,q)(X) =−DTE2(p,q)(X)+μ2+2μσDTE(p,q)(Y)+σ2(L1+mm−2L2), m>2,
 DTS(p,q)(X) =DTV−3/2(p,q)(X){3∑k=2(3k)(−DTE(p,q)(X))3−k[μk+kμk−1σDTE(p,q)(Y)] +2DTE3(p,q)(X)+3[μ−DTE(p,q)(X)]σ2(L1+mm−2L2)+σ3(L∗1+2L∗2)}, m>2,
 DTK(p,q)(X) =DTV−2(p,q)(X){−3DTE4(p,q)(X)+6[μ−DTE(p,q)(X)]2σ2(L1+mm−2L2) +4∑k=2(4k)Ck4(−DTE(p,q)(X))4−k[μk+kμk−1σDTE(p,q)(Y)]

where

 L2 =FY(1)(ξp,ξq)FY(ξp,ξq),
 L∗1=Γ((m+1)/2)√m[ξ2p(1+ξ2pm)−(m−1)/2−ξ2q(1+ξ2qm)−(m−1)/2]Γ(m/2)(m−1)√πFY(ξp,ξq),
 L∗2 =Γ((m+1)/2)m3/2[(1+ξ2pm)−(m−3)/2−(1+ξ2qm)−(m−3)/2]Γ(m/2)(m−1)(m−3)√πFY(ξp,ξq),
 L∗∗1=Γ((m+1)/2)√m[ξ3p(1+ξ2pm)−(m−1)/2−ξ3q(1+ξ2qm)−(m−1)/2]Γ(m/2)(m−1<