Notes on Cumulative Entropy as a Risk Measure

Saeid Tahmasebi 1  and Hojat Parsa 2
  • 1 Department of Statistics, Persian Gulf University, Bushehr, Iran
  • 2 Department of Economics, Persian Gulf University, Bushehr, Iran
Saeid Tahmasebi and Hojat Parsa

Abstract

Di Crescenzo and Longobardi [Di Crescenzo and Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 2009, 12, 4072–4087] proposed the cumulative entropy (CE) as an alternative to the differential entropy. They presented an estimator of CE using empirical approach. In this paper, we consider a risk measure based on CE and compare it with the standard deviation and the Gini mean difference for some distributions. We also make empirical comparisons of these measures using samples from stock market in members of the Organization for Economic Co-operation and Development (OECD) countries.

1 Introduction and Background

Let X denote the lifetime of a device or a system with probability density function (pdf) f and cumulative distribution function (cdf) F. Then the differential entropy known as Shannon entropy is defined by (see [12])

H(X)=-0+f(x)logf(x)dx,

where, by convention, 0log0=0. Recently, new measures of information are proposed. Replacing the pdf by the survival function F¯=1-F in Shannon entropy, the cumulative residual entropy (CRE) is defined by Rao et al. [11] as

(X)=0+F¯(x)Λ(x)dx,

where Λ(x)=-logF¯(x). Asadi and Zohrevand [1] also considered a dynamic version of the CRE. More properties of CRE in residual lifetime are given by Navarro, del Aguila and Asadi [6]. Some new connections between CRE and the residual lifetime are obtained in [5] using the relevation transform. Sordo and Psarrakos [13] provided comparison results for the CRE of systems and their dynamic versions. In a recent paper by Yang [15], it is shown that CRE is an alternative measure of risk for heavy-tailed distributions with infinite variance. Recently, Psarrakos and Navarro [8] introduced an extension of CRE called generalized cumulative residual entropy (GCRE), related to upper records statistics. Psarrakos and Toomaj [9] applied GCRE to problems related to risk measures. Navarro and Psarrakos [7] proved some interesting characterization results based on GCRE. Toomaj, Sunoj and Navarro [14] obtained an interesting connection between CRE and the Gini mean difference as a dispersion measure alternative to standard deviation. A new information measure similar to CRE has been proposed by Di Crescenzo and Longobardi [3] as

𝒞(X)=0+F(x)Λ~(x)dx,

where Λ~(x)=-logF(x). Note that 0𝒞(X). Properties on CE in past lifetime are given by Di Crescenzo and Longobardi [3] and Navarro, del Aguila and Asadi [6]. Di Crescenzo and Toomaj [4] obtained further results for the CE including stochastic ordering, bounds and characterization results. Motivated by some of the articles mentioned above, in this paper, we aim to investigate some applications of CE as a risk measure analogous to that proposed by Yang [15]. Investors are entering the stock market in order to earn returns or profit, but they are facing a risk of unexpected fluctuation of stock returns. Traditionally, stock market analysts use standard deviation criteria in order to calculate the risk of stock price fluctuation. In this study, we consider a risk measure based on CE and compare it with the standard deviation and the Gini mean difference as applied to the stock market. The paper is organized as follows: Section 2 presents some basic properties of risk measures. We also compare the risk measures σ(X) (standard deviation of X), 𝒞(X) and the Gini mean difference for some distributions. In Section 3, we make empirical comparisons of these risk measures using samples from stock market in OECD countries.

2 Some Properties of CE

In this section, we present some basic properties of risk measures. We also compare the risk measure σ(X), 𝒞(X) and the Gini mean difference for some distributions.

Definition 2.1.

Let X be a random variable describing outcomes of a risky asset, and let 𝒳 be the set of all functions X:Ω. A mapping ρ:𝒳 is called a risk measure if it fulfills some conditions as follows:

  1. (1)Sub-additivity: If X,Y𝒳, then ρ(X+Y)ρ(X)+ρ(Y).
  2. (2)Positive homogeneity: If X𝒳, a0, then ρ(aX)=aρ(X).
  3. (3)Monotonicity: If XY, then ρ(X)ρ(Y).
  4. (4)Convexity: If λ[0,1] and X,Y𝒳, then ρ(λX+(1-λ)Y)λρ(X)+(1-λ)ρ(Y).
  5. (5)Consistency: If X𝒳, b, then ρ(X+b)=ρ(X).

Definition 2.2.

A random variable X is said to be smaller than a random variable Y in decreasing convex order (XdcxY), if 𝔼(ϕ(X))𝔼(ϕ(Y)) for all decreasing convex functions ϕ such that the expectations exist.

Di Crescenzo and Longobardi [3] obtained the following results for CE. These results include linear transformations, various bounds and stochastic order properties.

Proposition 2.3.

Let X be a random variable with cdf F. Furthermore, let Y=aX+b, where a>0 and b0. Then

𝒞(Y)=a𝒞(X).

Proposition 2.4.

For a non-negative absolutely continuous random variable X with CE(X)<, we have

𝒞(X)=𝔼[T(2)(X)],

where

T(2)(X):=-x+logF(z)dz,x0.

Note that T(2)() is a decreasing convex function.

Proposition 2.5.

Let X and Y be two non-negative random variables such that XdcxY. Then

𝒞(X)𝒞(Y).

Proposition 2.6.

For a non-negative absolutely continuous random variable X with E(X)<, we have

𝒞(X)𝔼(X).

Proposition 2.7.

Cumulative entropy satisfies convexity of a risk measure.

Proof.

For all λ[0,1] and X,Y𝒳, according to the sub-additivity of cumulative entropy, we have

𝒞(λX+(1-λ)Y)𝒞(λX)+𝒞((1-λ)Y).

Now, according to the positive homogeneity of cumulative entropy, we get

𝒞(λX)+𝒞((1-λ)Y)=λ𝒞(X)+(1-λ)𝒞(Y).

Hence, recalling (2.1) and (2.2), the result follows. ∎

Definition 2.8.

A celebrated measure of income inequality, the Gini index, is defined as

Gini(X)=DG(X)2𝔼(X),

where DG(X):=0+2F¯(x)(1-F¯(x))dx is the Gini mean difference as a dispersion measure.

Proposition 2.9.

If X is a non-negative absolutely continuous random variable with cdf F, then

𝒞(X)DG*(X),

where DG*(X)=DG(X)2.

Proof.

The proof is similar to that of [3, Proposition 4.3]. ∎

The standard deviation σ(X) of a risk X is a common risk measure in insurance. Ramsay [10] showed that the standard deviation is not an appropriate tool in measuring large insurance risks with long-tailed skewed distributions. In the following examples, we compare the risk measures σ(X), 𝒞(X) and DG*(X) for some well-known distributions. Throughout these examples, we use the ratio

η(X)=𝒞(X)σ(X).

Example 2.1.

If X is a random variable with cdf FX(x)=1-exp(-θx), x>0, then

η(X)=π26-1.

Note that DG*(X)𝒞(X)<σ(X).

Example 2.2.

If X is a random variable with cdf FX(x)=xa, 0<x<a, then

η(X)=32.

Note that DG*(X)𝒞(X)<σ(X).

Example 2.3.

If X has an inverse Weibull distribution with cdf FX(x)=exp{-c/xγ}, x>0, c>0, γ>0, then

η(X)=Γ(1-1γ)γ[Γ(2-1γ)-Γ2(1-1γ)]12.

Note that for γ>3, we have DG*(X)𝒞(X)<σ(X).

Example 2.4.

If X has standard normal distribution with pdf f(x)=12πexp(-x22), -<x<, then

η(X)=𝒞(X)σ(X)=0.90.

Note that DG*(X)𝒞(X)<σ(X).

3 Empirical Study of Standard Deviation and CE in Stock Market

Let X1,X2,,Xm be a random sample of size m from an absolutely continuous cdf F. If

X(1)X(2)X(m)

represent the order statistics of the sample X1,X2,,Xm, then the empirical measure of F is defined for k=1,2,,m-1 by

F^m(x)={0,x<X(1),km,X(k)xX(k+1),1,x>X(k+1).

The empirical measure of 𝒞(X) is defined by Di Crescenzo and Longobardi [3] as

𝒞(F^m)=-k=1m-1Uk+1(km)log(km),

where Uk=X(k)-X(k-1), k=1,2,3,,m and X(0)=0. Note that limm𝒞(F^m)=𝒞(X). An empirical measure of DG*(X) is obtained by Davidson [2] as

D^G*(X)=2m2k=1mX(k)(k-0.5)-X¯.

Investors enters the stock market in order to earn returns or profit; however, they are facing a risk of unexpected fluctuation of stock returns. Returns on a stock are combinations of dividend’s and increase in the stock price. Sometimes, investors in stock market face decrease of stock price and loss parts of their properties. Therefore, the major part of the risk is related to stock price fluctuations. Traditionally, stock market analysts use the standard deviation criterion in order to calculate the risk of stock price fluctuation. In this study, we use the annual data of share price index of OECD countries during 2000–2017 to calculate the changes of price share as a part of return on stock and its fluctuation. OECD members are the following:

Australia, Austria, Belgium, Canada, Chile, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Korea, Luxembourg, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, Slovak Republic, Slovenia, Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States.

Table 1 shows the stock market values of standard deviation σ, skewness, kurtosis, CE and D* from OECD countries. Figures 1 and 2 also show the apparent positive correlation between standard deviation and cumulative entropy, which confirms the presence of similar behavior of these two risk measures. From Figure 3, we can conclude that, when the system of volatility (standard deviation) increases, the cumulative entropy increases as well. Likewise, when the system of volatility (standard deviation) has stabilized, cumulative entropy becomes small. Finally, a linear regression equation between σ and CE of share price index in OECD countries is obtained as 𝒞=0.8106σ. This model has an R2 value 98 %. Similarly, a linear regression equation between σ and D* of share price index in OECD countries is obtained as D*=0.52808σ. This model has an R2 value 99.7 %.

Table 1

Values of σ, skewness, kurtosis, CE and DG*(X) for stock market in OECD countries.

Countriesσ (%)SkewnessKurtosisCE(nats)DG*(X)
AUS0.1255-0.42-0.580.10960.0687
AUT0.2224-0.07-0.100.18920.1199
BEL0.1834-0.29-0.800.15990.1015
CAN0.15010.09-0.270.12590.0822
CHE0.1552-0.29-1.320.13200.0854
CHL0.14610.65-0.630.10970.0795
CZE0.23250.27-0.390.18990.1275
DEU0.1968-0.53-1.460.16410.1057
DNK0.1969-0.55-0.880.17420.1077
ESP0.1776-0.07-1.760.14240.0976
EST0.3047-0.110.220.26120.1638
FIN0.29951.103.070.22890.1519
FRA0.1873-0.19-1.180.15510.1018
GBR0.1200-0.28-1.220.10120.0660
GRC0.28490.34-1.690.21160.1541
HUN0.24060.19-0.580.19730.1330
IRL0.2166-1.271.120.19760.1112
ISL0.3559-1.332.850.32300.1733
ITA0.1911-0.25-1.420.16120.1048
JPN0.20720.21-0.560.16960.1144
KOR0.16500.05-1.330.13630.0908
LUX0.26230.08-1.040.21790.1450
MEX0.18830.65-0.920.13970.1004
NLD0.1839-0.61-1.270.15820.0984
NOR0.22530.14-0.770.18770.1252
NZL0.1109-1.211.990.10110.0562
POL0.2315-0.23-0.940.20050.1288
PRT0.18670.06-1.360.15290.1041
SVK0.29791.825.350.20590.1443
SWE0.2216-0.23-0.610.18950.1210
TUR0.40791.945.380.27180.1964
USA0.1306-0.70-0.310.11850.0709
Figure 1
Figure 1

Linear regression of σ and CE of share price index in OECD countries.

Citation: Stochastics and Quality Control 34, 1; 10.1515/eqc-2018-0019

Figure 2
Figure 2

Linear regression of σ and D of share price index in OECD countries.

Citation: Stochastics and Quality Control 34, 1; 10.1515/eqc-2018-0019

Figure 3
Figure 3

Graphs of σ and CE and D for share price index in OECD countries.

Citation: Stochastics and Quality Control 34, 1; 10.1515/eqc-2018-0019

References

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    M. Asadi and Y. Zohrevand, On the dynamic cumulative residual entropy, J. Statist. Plann. Inference 137 (2007), no. 6, 1931–1941.

    • Crossref
    • Export Citation
  • [2]

    R. Davidson, Reliable inference for the Gini index, J. Econometrics 150 (2009), no. 1, 30–40.

    • Crossref
    • Export Citation
  • [3]

    A. Di Crescenzo and M. Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 (2009), no. 12, 4072–4087.

    • Crossref
    • Export Citation
  • [4]

    A. Di Crescenzo and A. Toomaj, Further results on the generalized cumulative entropy, Kybernetika (Prague) 53 (2017), no. 5, 959–982.

  • [5]

    S. Kapodistria and G. Psarrakos, Some extensions of the residual lifetime and its connection to the cumulative residual entropy, Probab. Engrg. Inform. Sci. 26 (2012), no. 1, 129–146.

    • Crossref
    • Export Citation
  • [6]

    J. Navarro, Y. del Aguila and M. Asadi, Some new results on the cumulative residual entropy, J. Statist. Plann. Inference 140 (2010), no. 1, 310–322.

    • Crossref
    • Export Citation
  • [7]

    J. Navarro and G. Psarrakos, Characterizations based on generalized cumulative residual entropy functions, Comm. Statist. Theory Methods 46 (2017), no. 3, 1247–1260.

    • Crossref
    • Export Citation
  • [8]

    G. Psarrakos and J. Navarro, Generalized cumulative residual entropy and record values, Metrika 76 (2013), no. 5, 623–640.

    • Crossref
    • Export Citation
  • [9]

    G. Psarrakos and A. Toomaj, On the generalized cumulative residual entropy with applications in actuarial science, J. Comput. Appl. Math. 309 (2017), 186–199.

    • Crossref
    • Export Citation
  • [10]

    C. M. Ramsay, Loading gross premiums for risk without using utility theory, Trans. Soc. Actuar. 45 (1993), 305–349.

  • [11]

    M. Rao, Y. Chen, B. C. Vemuri and F. Wang, Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory 50 (2004), no. 6, 1220–1228.

    • Crossref
    • Export Citation
  • [12]

    C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–432.

    • Crossref
    • Export Citation
  • [13]

    M. A. Sordo and G. Psarrakos, Stochastic comparisons of interfailure times under a relevation replacement policy, J. Appl. Probab. 54 (2017), no. 1, 134–145.

    • Crossref
    • Export Citation
  • [14]

    A. Toomaj, S. M. Sunoj and J. Navarro, Some properties of the cumulative residual entropy of coherent and mixed systems, J. Appl. Probab. 54 (2017), no. 2, 379–393.

    • Crossref
    • Export Citation
  • [15]

    L. Yang, Study on cumulative residual entropy and variance as risk measure, Fifth International Conference on Business Intelligence and Financial Engineering, IEEE Press, Piscataway (2012), 210–213.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    M. Asadi and Y. Zohrevand, On the dynamic cumulative residual entropy, J. Statist. Plann. Inference 137 (2007), no. 6, 1931–1941.

    • Crossref
    • Export Citation
  • [2]

    R. Davidson, Reliable inference for the Gini index, J. Econometrics 150 (2009), no. 1, 30–40.

    • Crossref
    • Export Citation
  • [3]

    A. Di Crescenzo and M. Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 (2009), no. 12, 4072–4087.

    • Crossref
    • Export Citation
  • [4]

    A. Di Crescenzo and A. Toomaj, Further results on the generalized cumulative entropy, Kybernetika (Prague) 53 (2017), no. 5, 959–982.

  • [5]

    S. Kapodistria and G. Psarrakos, Some extensions of the residual lifetime and its connection to the cumulative residual entropy, Probab. Engrg. Inform. Sci. 26 (2012), no. 1, 129–146.

    • Crossref
    • Export Citation
  • [6]

    J. Navarro, Y. del Aguila and M. Asadi, Some new results on the cumulative residual entropy, J. Statist. Plann. Inference 140 (2010), no. 1, 310–322.

    • Crossref
    • Export Citation
  • [7]

    J. Navarro and G. Psarrakos, Characterizations based on generalized cumulative residual entropy functions, Comm. Statist. Theory Methods 46 (2017), no. 3, 1247–1260.

    • Crossref
    • Export Citation
  • [8]

    G. Psarrakos and J. Navarro, Generalized cumulative residual entropy and record values, Metrika 76 (2013), no. 5, 623–640.

    • Crossref
    • Export Citation
  • [9]

    G. Psarrakos and A. Toomaj, On the generalized cumulative residual entropy with applications in actuarial science, J. Comput. Appl. Math. 309 (2017), 186–199.

    • Crossref
    • Export Citation
  • [10]

    C. M. Ramsay, Loading gross premiums for risk without using utility theory, Trans. Soc. Actuar. 45 (1993), 305–349.

  • [11]

    M. Rao, Y. Chen, B. C. Vemuri and F. Wang, Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory 50 (2004), no. 6, 1220–1228.

    • Crossref
    • Export Citation
  • [12]

    C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–432.

    • Crossref
    • Export Citation
  • [13]

    M. A. Sordo and G. Psarrakos, Stochastic comparisons of interfailure times under a relevation replacement policy, J. Appl. Probab. 54 (2017), no. 1, 134–145.

    • Crossref
    • Export Citation
  • [14]

    A. Toomaj, S. M. Sunoj and J. Navarro, Some properties of the cumulative residual entropy of coherent and mixed systems, J. Appl. Probab. 54 (2017), no. 2, 379–393.

    • Crossref
    • Export Citation
  • [15]

    L. Yang, Study on cumulative residual entropy and variance as risk measure, Fifth International Conference on Business Intelligence and Financial Engineering, IEEE Press, Piscataway (2012), 210–213.

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    Linear regression of σ and CE of share price index in OECD countries.

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    Linear regression of σ and D of share price index in OECD countries.

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    Graphs of σ and CE and D for share price index in OECD countries.