Study On Haezendonck-Goovaerts Risk Measure

As an important issue on risk manage, risk measure has wide research in economicfield, finance field, insurance field and other fields. There are a lot of methods of riskmeasure: Variance, Semi-Variance, Lower partial moment, G-expectation, etc. In postfinance crisis era, the tail of risk was more and more concerned by risk managers. Plentyof methods on measure tail risk were presented: Value at Risk （VaR）, Tail-Value atRisk （TVaR）, Expected Policyholder Deficit （EPD）, stop-loss premium, Haezendonck-Goovaerts risk measure, and so on.Haezendonck-Goovaerts risk measure is introduced in Haezendonck and Goovaerts（1982）, based on the Orlicz premium principle （see Haezendonck and Goovaerts,1982;Goovaerts et al.,2003）. Haezendonck-Goovaerts risk measure of random variable X atthe level q is defined as follows:where max[X]=inf{y: P （X≥y）=0}, x is the minimizer, which is called Or-licz quantile in Bellini and Rosazza Gianin （2012）. Ï€q[X, x] is the unique solution ofequationx is a threshold, Ï€> x, is a normalized Young function satisfying:[0,+∞)â†'[0,+∞),（0）=0,（1）=1,（+∞）=+∞, it is negative and strictly increasingcontinuous function （more details on see Krasnosel’skii and Rutickii,1961; Rao andRen,1991）. q∈[0,1) is a level, which is related to the risk aversion of insurers. Whenthe Young function （t）=t, Haezendonck-Goovaerts risk measure of random variableX at the level q is T V aRq（X）. Haezendonck-Goovaerts risk measure is the extensionof TVaR. For the complex definition of Haezendonck-Goovaerts risk measure, it wasno popular early in the field of risk measures. It is well known that the studies on Haezendonck-Goovaerts risk measure mostlyfocus on axiom characterization, empirical estimate, asymptotic theory, analysis expres-sion, practical application, and numerical simulation, etc. The early literatures aboutaxiom characterization are mainly Goovaerts et al.（2004a,b）, Bellini and Rosazza Gi-anin （2008a）. There has been recently an increasing interest in Haezendonck-Goovaertsmeasure of risk variable with the dependence structure. Nam et al.（2011） computedHaezendonck-Goovaerts risk measure on the sum of upper comonotonic （see Cheung,2008,2009; Dong et al.,2010; Cheung and Vandufel,2011; Hua and Joe,2012） ran-dom variables. Bellini and Rosazza Gianin （2006,2008b）, Ahn and Shyamalkumar（2011a,b,c）; Tang and Yang （2012）; Mao and Hu （2012） studied the estimation andasymptotic properties of Haezendonck-Goovaerts risk measure. Bellini and Rosazza Gi-anin （2008a,2012） indicated that it was complex to compute Haezendonck-Goovaertsrisk measure, unless special Young function and distribution of risk X. The weightedpower Young function （t）=αt+（1α）t², α∈[0,1] was investigated and simulatedin Bellini and Rosazza Gianin （2006）, but the transparent expression of Haezendonck-Goovaerts risk measure was not attained. The study on the power case （t）=t^r, r≥1can be found in Nam et al.（2011）, and the explicit expression to Haezendonck-Goovaerts risk measure on the sum of upper comonotonic random variables with ex-ponential distribution was derived.In the paper, we firstly consider the characterization and expression of Haezendonck-Goovaerts risk measure in the special cases. Secondly, we define the conditionalHaezendonck-Goovaerts risk measure. Lastly, we generalize the Haezendonck-Goovaertsrisk measure and derive some properties of the generalized Haezendonck-Goovaerts riskmeasures.The primary results of the paper are as follows:The first aim of this work is to compute the Haezendonck-Goovaerts risk measureon the sum of random variables with Pareto and exponential marginal distributionson diferent Young functions, using the inverse function of the distribution functionand the properties of upper comonotonic random vector. Concretely, suppose thatthere are n insurance policies in one account year, the ith claim is Y_i, i=1,, n.Y=（Y₁,..., Y_n） is upper comonotonic random vector, VaR, TVaR, EPD, stop-loospremium, Haezendonck-Goovaerts risk measure of total claim S=Y₁++Y_n arederived.Suppose the distribution of Yiis P areto（I）（σ_i, α）, i=1,2,, n, Haezendonck-Goovaerts measure, corresponding to power Young function （t）=t^r, r≥1, on thesum S is given by the following theorem: Theorem1Letφ（t）=tr,r≥1,Y=（Y1,…,Yn）be upper comonotonic random vector,the distrbution Fi of the ith component Yi be Pareto（Ⅰ）（σi,α）,i1,2,…,n,upper commonolonic threshold be（ai*,…,an*）,S=Y1+…+Yn,then forThe following theorem shows VaR,TVaR,EPD,stop-loos premium of the total claim S:Theorem2Let Y=（Y1,…,Yn）be upper comontonic random vector,Yi Pareto（Ⅰ）（σi,α）,i=1,2,…,n,upper comonotonic threshold be（a1*,…,an*）,S of S at the levcl q: I∫α>1,then the TVaR,EPD of S at the level q are stop-loss premium of S isAssume the distribution of Yi is Pareto（Ⅱ）（λi,α）,i=1,2,…,n,the similar conclusions to theorem1and theorem2can be drawn.If the distribution of Yi is exponential distribution with the parameter λi>0,i=1,2,…,n,then Haezendonck-Goovaerts measure,corresponding to weighted power Young function φ（l）=αl+（1-α）l2,α∈[0,1],on the sum S is given as the following theorem: Theorem3Let Young function φ（t）=αt（1-α）t2, α∈[0,1], Y=（Y1,…,Yn） be upper comonotonic random vector, the distribution of Yi is exponential distribution with the parameter λi>0, i=1,2,…,n, upper comonotonic threshold6e （a1*,…, an*）,The following conclusions are about VaR, TVaR, EPD, stop-loss premium and Haezendonck-Goovaerts risk measure of the sum S at the level q:Remark1If upper comonotonic threshold be （a1*,…,an*）,λ=（∑i=1n1/λi）-then（1） When q∈（F（a1*,…,an*）,1）, VaR, TVaR, EPD of S at the level q are（2） Stop-loss premium of S isThe second aim of this paper is to define the conditional Haezendonck-Goovaerts risk measure for excess of loss insurance and the liability insurance with a maximum coverage. Then we provide the relationships on the insurer’s deduction, assets held by the insurer for avoiding insolvency, confidence level and the upper bound of average loss of policyholders. Concretely, suppose that there are n policies having loss for the insurance company in one accounting year, the ith loss variable is Xi, i=1,…,n, Xi,i=1,…,n are iid, bounded and non-negative random variables. The total loss is∑in=1Xi, then the mean loss is1/n∑in=iXi.=1n the excess of loss insurance, the claim taking place one time at most in a temporal period, denote the insurer’s deduction by x, so the claim amounts paid to the ith policy by the insurance company is （Xi—x）+, the average pay for Xi is the stop-loss premium E[（Xi—x）+]. Let a be the upper bound of average loss. In accordance with the definition of Haezendonck-Goovaerts risk measure, we define the conditional Haezendonck-Goovaerts risk measure of risk Xi at the level q iswhere max[Xi]=inf{y: P（Xi≥y）=0}, x*c is the minimizer of Ï€qc[Xi, x],Ï€qc[Xi,x] is the unique solution of the equation where x is a threshold,x> x,φ is normalized Young function, a>0is a constant, q∈[0,1) is a confidence level.In this paper, assume that Young function φ（t）=t, X1,…,Xn are independent identically exponential distributions with the rate （3, the solution of equation （2） is the assets held by the insurer for the full claim to each policy loss on the aforemen-tioned condition, see for instance Dhaene et al.（2004）. The conditional Haezendonck-Goovaerts risk measure of each loss is the minimum asset value, and the minimizer is the deduction for each policy. To derive these conclusions, given the upper bound a of the policyholders’ mean loss, we calculate the stop-loss premium E[（Xi-x）+] of Xi in the following lemma:Lemma1If X1,…,Xn are independent identically exponential distributions with the rate0, thenFrom the above lemma, for a fixed upper bound a of average loss, we can get the solution of equation （2） for the loss variable Xi as follows:Theorem4If Xi,…,Xn are independent identically exponential distributions with the rate （3, then the solution of equation （2） for the loss Xi is By Theorem4and the definition of the Haezendonck-Goovaerts risk measure, pro-vided the mean loss upper bound a, the conditional optimal Haezendonck-Goovaerts risk measure of the loss Xi is attained in the following corollary:Corollary1Suppose that the losses X1,…, Xn are independent identically expo-nential distributions with the parameter β, then the conditional Haezendonck-Goovaerts risk measure of the loss Xi at the level q isThen we discuss the modifications of x,Ï€qc[Xi,x],Ï€qc[Xi], q with a varied. The re-lation of deduction x and the upper bound a of the average loss is gained asThe relationship of confidence level q and the upper bound a of the mean loss isRemark2（1）πqc [Xi-x] and Ï€qc[Xi] are all monotone increasing with respect to a. It means that in order to avoid insolvency the asset value Ï€qc[Xi,x] and the minimal asset Ï€qc[Xi] will grow up when the average loss upper bound a increases.（2） Xi,…,Xn are independent identically exponential distributions with the rate （3） With a and （3increased,Ï€qc[Xi-x] grows up to Ï€q[Xi,x], and Ï€qc[Xi] increases up to Ï€q[Xi] gradually.（3） q is monotone decreasing with respect to a, i.e., when the upper bound a of the average loss increases, the confidence level q will reduce.Furthermore, to compare the conditional measure with the unconditional measure, we denote the unconditional and conditional confidence level of the loss Xi by q0and q1. We can draw the following conclusions:Corollary2X1…,Xn are independent identically exponential distributions with the parameter β.（1） If the value of assets for Xi is invariant, i.e., Ï€q[Xi,x]=Ï€qc[Xi,x], then （2） If the confidence level is invariant, i.e., q0=q1, thenWe found that the lower upper bound of these applicants’ average loss is, the less assets are needed for each policy. Considering this, the insurer may apply some discount privilege to these policyholders with the lower upper bound of mean loss.For the liability insurance with a maximum coverage, we have the similar results. Assume that for the insurer there are n policies loss occurred in an accounting year, the ith loss variable is Xi, i=1,…,n, the total loss is∑in=1Xi, the mean loss is1/n∑in=1Xi. The insurer’s claim variable to Xi is Yi which equals0if Xi≤x1Yi=x2—x1in case Xi≥x2and Yi=Xi-x1otherwise, where the deduction of the insurance company is X1, the maximum claim is x2—x1.For a predetermined x2, the conditional Haezendonck-Goovaerts risk measure of the loss variable Xi at the level q is where max[Xi]=inf{y: P（Xi≥y）=0}, the conditional Orlicz quantile x1*c is the minimizer of Ï€qc[Xi,x1]. Ï€qc[Xi,x1] is the unique solution of the equation φ is a normalized Young function. Yi is a bounded non-negative random variable, x1is a threshold, Ï€> x1,α>0is a constant, q G [0,1） is a confidence level.Suppose that Young functionφ（t）=t, X1,…,Xn are independent identically exponential distributions with the parameter β. Provided the upper bound a of the policyholders’ average loss, the mean claim from the insurance company can be obtained in the following lemma:Lemma2If X1,…,Xn are independent identically exponential distributions with the rate β, and Yi is the claim size for Xi, then for n>1, nα> x2> x1,By the above-mentioned Lemma, given the upper bound a of the policyholders’ average loss, the solution of equation （3） for the loss Xi is obtained as follows: Theorem5I∫x1,…,Xn are independent identically exponential distributions with the rate β,then for n>1,nα>x2>x1,the solution of equation（3）for the loss Xi at the level q isBy Theorem5and the definition of the Haezendonck-Goovaerts risk measure,pro-vided the mean loss upper bound α,the conditional Haezendonck-Goovaerts risk mea-sure of the loss Xi is attained in the following corollary.Corollary3Suppose that the losses X1,…,Xn are independent identically expo-nential distributions with the parameter β,then the conditional Haezendonck-Goovaerts risk measure of the loss Xi at the level q isMoreover,for n>1,nα>x2>x1,We can receive the relation of the deduction x1and the upper bound α of the mean loss is The relationship of confidence level q and the upper bound α of the average loss isRemark3（1）πqc[Xi,x1]is monotonously increasing with respect to α,that is to say,when the average loss upper limit α grows up,the assets Ï€qc[Xi,x1]held by the insurer for avoiding insolvency have to increase with il.（2）q is monotonously decreasing with respect to α；i.e.ï¼›when the upper limit α of the average loss grows up,the confidence level q will decrease with it.Now We observe the difference of Haezendonck-Goovaerts risk measure and the conditional Haezendonck-Goovaerts risk measure.Denote Haezendonck-Goovaerts mea-sure and confideuce level of Xi by Ï€0and q0,the corresponding conditional Haezendonck-Goovaerts measure and confidence level by Ï€1and q1,respectively.We can obtain the following corollary: Corollary4X1,…,Xn are independent identically exponential distributions with the parameter β. For n>1, nα> x2> x1,（1） if the asset value of the insurer for each policy is invariant, i.e., π°=Ï€1, then（2） if the confidence level is invariant, i.e., q0=q1, thenNext we investigate the conditional Haezendonck-Goovaerts risk measure on the individual risk in the temporal collective risk model. The classical risk process is U（t）=u+ct-S（t）, where U（t） is the capital held by the insurer on time t, u is the initial reserves, c is the premium income in the unit period, S（t）=∑i=1N（t）Xi is Compound Poisson process, N（t） is a Poisson process with the intensity λ, which represents the claim number up to time t. X1,…,XN（t） are independent identically distributions, Xi is non-negative and independent of N（t）, Xi represents the ith claim size, i=1,2,…, N（t）.The case that U（t）<0is called ruin. On one policy year, let t=1, N=N（1）, then the classical risk model is temporal collective risk model. The insurer’s finance safety condition is U（1）=u+c—∑iN=1Xi≥0, i.e.,∑i=1N=Xi≤u+Assume that the insurer applies for the stop-loss reinsurance. We use Haezendonck-Goovaerts risk measure to evaluate the insurer’s individual risk Xi for reserving and solvency purposes. Denote the insurance company’s available capital by γ. In order to avoid insolvency we regard γ as the upper limit of the insurer’s total claims. Therefore, the conditional Haezendonck-Goovaerts risk measure of the individual risk Xi at the level q is where max[Xi]=inf{y:P{Xi≥y）=0}, x*Cl is the minimizer of Ï€qCl [Xi, x]. Ï€gX[Xi,x] is the unique solution of the equation φ is a normalized Young function, x is a threshold indicating the insurance company retention, Ï€> x,∑iN=1Xi is the aggregate claims of the insurer in the year, γ>0is a constant. In the simplest case, we assume that X1,…,Xn are independent identically distributions, and the distribution is exponential with the rate β, Young function φ（t）=t, given the upper limit γ of the insurer’s aggregate claims, for the stop-loss premium E{（Xi—x）+] of Xi, we have the lemma as follows:Lemma3If N is Poisson distribution with the intensity X, Xi,…,Xn are independent identically distributions, the distribution is exponential with the rate β, thenFrom the above-mentioned Lemma, we can receive the solution of the equation （4） for the ith claim Xprovided that Young function φ（t）=t and the upper bound γ of the insurer’s total claims:Theorem6If N is Poisson distribution with the intensity λ, X1,…,XN are independent identically exponential distribution with the rate β, then the solution of the equation （4） for the ith claim Xi at the level q isBy Theorem6and the definition of the conditional Haezendonck-Goovaerts risk measure, for a fixed γ, the conditional Haezendonck-Goovaerts risk measure for the ith claim Xi is obtained in the following corollary:Corollary5If N is Poisson distribution with the intensity X, Xi,…,Xn are independent identically exponential distribution with the rate β, then the conditional Haezendonck-Goovaerts risk measure of the ith claim Xi is Furthermore, we derive the relation of x and γ as follows: The relationship of q and γ is:By these relations, we draw the following conclusions:Remark4（1）πqc1and Ï€qc1[Xi] are monotonously increasing with γ in-creased. Given the retention and confidence level, the capital requirement Ï€qc1[Xi,x] and the lowest capital requirement Ï€qc1[Xi] for the individual risk grow up with the up-per bound of aggregate claims increased.（2） N～Poisson（λ）, X1;…,XN～i.i.d.Exp（β）. As γ and β grow, Ï€c1q[Xi,x] gradually grows up to Ï€q[Xi,x], and Ï€qc1[Xi] grows up to Ï€q[Xi].（3） q decreases monotonously with the increase of γ, when the upper limit γ of the insurer’s total claims increases, given the retention and the capital for individual risk, the confidence level q will decrease.To investigate the difference of two measures, denote Haezendonck-Goovaerts risk measure and confidence level of the individual risk Xi as Ï€0and q0, respectively. Let Ï€2and q2represent separately the conditional Haezendonck-Goovaerts risk measure and confidence level of the individual risk Xi. The following corollary is obtained:Corollary6N is Poisson distribution with the intensity λ, X1,…,XN are exponential and independent identically distributions with the rate β.（1） If the capital requirement keeps invariant, i.e., Ï€0=Ï€2, then（2） If the confidence level keeps invariant, i.e., q0=q2, thenThe last aim of this paper is to about the generalization of Haezendonck-Goovaerts risk measure. Concretely, we firstly transfer the form of premium in the Orlicz equa-tion, replacing the reinsurance premium by the logarithm equivalent form, to generalize Haezendonck-Goovaerts risk measure based on the Orlicz premium principle. Then the generalized Haezendonck-Goovaerts risk measure is derived.Let risk variable X be a continuous type random variable, with the distribution function F（x）,-∞≤min[X]≤max[X]≤+∞; min[X]=sup{y: P（X <y） 0}, max[X]=inf{y: P（X> y）=0}, x be a retention, q G [0,1） be a level. FX-1（q）=inf{x: F（x）≥q} denotes the q quantile of the risk X, φ is a normalized Young function.（X—x）+represents the residual risk, then we have the following theorem on the generalized Orlicz risk measure of risk X at the level q:Theorem7Let x, X, q and φ be as the aforementioned.Then for any x:-∞<x<max[X] and q G [0,1), the equation has a unique solution Ï€1/q[X,x] satisfying the inequalitiesTheorem7shows that Ï€1/q [X, x]> FX-1（q）, that is to say, Ï€1/q [X, x] is （S,q） consistent （see Goovaerts et al.,2004a）, where S is a set of axioms for risk measures. And Ï€1/q [X,x]>x indicate that the measure value is bigger than the retention, therefore, the insurance company will choose reinsurance.The definition of generalized Haezendonck-Goovaerts risk measure is as follows:Definition1Similar to the definition of Haezendonck-Goovaerts risk measure, we call the generalized Haezendonck-Goovaerts risk measure for the variable X at the level q, whereÏ€1/q[X,x] is the unique solution of equation （5）.There are some properties about generalized Haezendonck-Goovaerts risk measure Ï€1/q[X] as the following theories:Theorem8The generalized Haezendonck-Goovaerts risk measure Ï€1/q[X] of ran-dom variable X at the level q satisfiesTheorem8indicates that the generalized Haezendonck-Goovaerts risk measure is （S,q）-consistent.In the following theorems we derive some axiom properties to the generalized Haezendonck-Goovaerts risk measure:Theorem9Ï€1/q[X] is the generalized Haezendonck-Goovaerts risk measure of risk variable X. Let φ be a normalized Young function, q G [0,1) be arbitrarily given, then we have（1） Monotonicity: Ï€1/q[X]<Ï€1/q[Y] if X≤st Y and max[X]=max[Y];（2） Translation Invariance: Ï€1/q[X+a]=Ï€1/q[X]+a for any a;（3） Preservation of convex ordering: Ï€1/q[X]≤1/q[y] if φ is convex, X≤cx Y, and max[X]=max[Y];（4） Subadditivity: Ï€1/q [X+Y]≤π1/q [X]+Ï€1/q [Y] provided that φ is convex.In addition, we have the following theorem:Theorem10Letφi, i=1,2be two non-negative, strictly increasing, and con-tinuous functions on [0,+∞) with φi（t）=t, for0≤t≤1and φi（+∞）=+∞, let Ï€1i/q[X, x] be severally the unique solution of the equation （5） and the corresponding gen-eralized Haezendonck-Goovaerts risk measure on risk X at level q be Ï€1i/q[X], i=1,2.（1） If φ2is convex in φ1（see Goovaerts et al.,2004a）, then Ï€11/q[X,x]≤π12/q[X,x], and Ï€q11[X]≤πq12[X];（2） If φ2is concave in φ1（see Goovaerts et al.,2004a）, then Ï€11/q[X, x]≥π12/q[X, x], and Ï€q11[X]≤πq12[X];From Theorem10, we have a corollary as follows:Corollary7If then（1） Ï€1/q[X] is the smallest generalized Haezendonck-Goovaerts risk measure on strictly increasing convex functions φ satisfying φ{t)=t for0<t <1;（2） Ï€1/q[X] is the largest generalized Haezendonck-Goovaerts risk measure on strictly increasing concave functions φ satisfying φ（t）=t for0<t <1and φ（+∞）=+∞.Finally, we consider another generalization of Haezendonck-Goovaerts risk mea-sure. Concretely, we simultaneously modify the risk and premium in the Orlicz equa-tion, replacing the reinsurance risk and the reinsurance premium by their exponential equivalent forms respectively. We obtain another form of Orlicz premium and the mod-ified Haezendonck-Goovaerts risk measure. The modified equation isThe definition of modified Haezendonck-Goovaerts risk measure is as follows:Definition2Similar to the definition of Haezendonck-Goovaerts risk measure, we call the modified Haezendonck-Goovaerts risk measure for the variable X at the level q,where Ï€_q²[X, x] is the unique solution of equation （6）.Similar to the generalized Haezendonck-Goovaerts risk measure, the modified riskmeasure is （S, q）-consistent, and it has some properties, such as monotonicity, trans-lation invariance, preservation of convex ordering.