Estimating value at risk and conditional tail expectation for extreme and aggregate risks
EEstimating value at risk and conditional tail expectation for extremeand aggregate risks
Suman ThapaSchool of Mathematics and StatisticsCarleton University, Ottawa, ON Canada K1S 5B6Yiqiang Q. ZhaoSchool of Mathematics and StatisticsCarleton University, Ottawa, ON Canada K1S 5B6January, 2021
Abstract
In this paper, we investigate risk measures such as value at risk (VaR) and the conditional tailexpectation (CTE) of the extreme (maximum and minimum) and the aggregate (total) of two depen-dent risks. In finance, insurance and the other fields, when people invest their money in two or moredependent or independent markets, it is very important to know the extreme and total risk beforethe investment. To find these risk measures for dependent cases is quite challenging, which has notbeen reported in the literature to the best of our knowledge. We use the FGM copula for modellingthe dependence as it is relatively simple for computational purposes and has empirical successes. Themarginal of the risks are considered as exponential and pareto, separately, for the case of extreme riskand as exponential for the case of the total risk. The effect of the degree of dependency on the VaRand CTE of the extreme and total risks is analyzed. We also make comparisons for the dependentand independent risks. Moreover, we propose a new risk measure called median of tail (MoT) andinvestigate MoT for the extreme and aggregate dependent risks.
Keywords:
Dependent risk measures; median of tail; extreme risks; aggregate risk.
In this paper, we derive the joint risk measures and extreme and aggregate values of dependent riskmeasures. In finance and insurance, several risk measures, such as value at risk (VaR), conditionaltail expectation (CTE), the distorted risk measure, the copula distorted risk measures, among possibleothers, are considered, of which VaR and the CTE arae the most popular ones.Let X be a nonnegative random variable with distribution function F X ( x ) , which represents the riskor claim for an insurance company, or a loss of a portfolio. Let F − X be the left continuous inverse of F X , called the quantile function. For every α ∈ [0 , X ( α ) and definedby VaR X ( α ) = inf( x : F X ( x ) ≥ α ) . a r X i v : . [ q -f i n . R M ] J a n n actuarial science, it is also known as the quantile risk measure. VaR is often specified with aconfidence level, say α = 90% or 95% or 99%. Hence, VaR X ( α ) represents the loss such that theprobability distribution of VaR X ( α ) will not exceed α .The conditional tail expectation (CTE), or the expected shortfall of X given that X >
VaR X ( α ),denoted by CTE X ( α ) is defined byCTE X ( α ) = E ( X | X >
VaR X ( α )) . Both VaR and CTE are important measures for the right tail risk, which are most often studiedin insurance and financial investment. CTE satisfies all required properties of a coherent risk measureby [1]. So, CTE is more preferable than VaR in many applications. However, VaR could be better foroptimizing portfolios when good tail models are not available.In practical applications of probability and statistics, the results of an experiment are often describedby more that one random vector that form a multivariate random variable. For example, a person caninvest his/her income in more than one market. Let a multivariate random variable X = ( X , X , ..., X n )be a risk vector, where X i ( i = 1 , , ...., n ) denotes the risk or loss in the sub-portfolio i . If people investtheir money in different markets, say X , X , .., X n and if the markets are not independent, we needto analyze dependent random variables. Copulas are the functions that describe dependencies amongvariables and provide a way to create distributions to model dependent multivariate data. Using copu-las, one can construct a multivariate distribution by specifying marginal distributions.The joint risk measure of X is defined by P ( X ≤ x , X ≤ x ,..., X n ≤ x ) and the extreme risksin a portfolio are X (1) = min( X , X ,..., X n ) and X ( n ) = max( X , X , ..., X n ) consisting of the n sub-portfolios. The extreme risks and the aggregate risk are very important and popular. We study thecases when the risks in the portfolio follow the exponential and pareto distributions. Specifically, weconsider two subportfolios, say X and X of the portfolio, say X . Since FGM copula is very popularand easier to use, we apply FGM copula [2] to analyze the dependency between the two risks.We derive VaR and CTE for extreme and aggregate risks of two dependent risks. That is, for X = ( X , X ), we calculate VaR and CTE for the extreme risks ( X (1) = min( X , X ) and X (2) =max( X , X )) and the aggregate risk ( X = X + X ) when X and X are dependent.There are some papers in the literature in which copula was used in financial risks. Brahim, Fatahand Djabrane, in their paper [3], derived a new risk measure called the copula conditional tail expecta-tion that measures the conditional expectation given that the two dependent losses exceed their valueat risks. Heilpern, in his paper [4], used copula to investigate the sum of dependent random variables.The risk measures, value at risk and the expected shortfall of such sums were calculated in his paper.Cai and Li, in their paper [5], defined the conditional tail expectation of the aggregated risk and theextreme risk for multivariate phase type distributions, but they did not use copula theory. In otherpapers related to this field, Hardy introduced risk measures in actuarial applications in paper [6] andBrazauskas, Jones, Puri and Zitikis estimated the conditional tail expectation with actuarial applica-tions in their paper [7]. Based on the literature review in this field, it is interesting to estimate value atrisk and the conditional tail expectation of the extreme risk and the aggregate risk of two dependentrisks explicitly by using copula. This is a completely new method to define dependent risk measuresthat follow the exponential and pareto distributions.2e propose a new risk measure called median of tail (MoT). In some cases, VaR is the worst riskmeasure in the given confidence interval and it is not coherent. The conditional tail expectation (CTE)is useful and coherent. Sometimes, in the absence of good tail models, CTE cannot be a useful measure.Sarykalin, Serraino and Uryasev [8] tried to explain strong and weak features of these risk measuresand illustrate of them with several examples. Yamai and Yoshiba [9] compared and analyzed value atrisk and the conditional tail expectation under market stress. When VaR and CTE are not suitable,median of tail (MoT) can be used.The median is less affected by outliers and skewed data than the mean and is usually a preferredmeasure of central tendency when the distribution is not symmetrical. Mean and median are approxi-mately close in many distributions. In a skewed distribution, the outliers in the tail pull the mean awayfrom the center towards the longer tail. In that case, the median better represents the central tendencyfor the distribution. In this paper, we derive MoT for the extreme and the aggregate of two dependentrisks in the portfolio and generalize the result in section 4.The rest of the paper is organized as follows. In section 2, we estimate VaR and CTE for the extremeand the aggregate of dependent risks which include VaR and CTE for the exponential distribution insection 2.1, VaR and CTE for extreme risks of the exponential distribution in section 2.2, VaR and CTEfor the pareto distribution in section 2.3, VaR and CTE for extreme risks of the pareto distribution insection 2.4 and VaR and CTE for the aggregate risk of the exponential distribution in section 2.5. Insection 3, we estimate median of tail (MoT) for the extreme and the aggregate of dependent risks whichinclude MoT for the exponential and pareto distributions in section 3.1, the MoT for extreme risks ofthe exponential distribution in section 3.2, MoT for extreme risks of the pareto distribution in section3.3, MoT for the aggregate risk of the exponential distribution in section 3.4 and final conclusions aremade in section 4. In this section, we estimate value at risk (VaR) and the conditional tail expectation (CTE) for theextreme and the aggregate of two dependent risks. The distribution of the risk measures is consideredas exponential and pareto.Let X be an exponential random variable with the cumulative distribution function F X ( x ) = P ( X ≤ x ) = 1 − e − λx , x ≥ . Then, VaR X ( α ) is defined as VaR X ( α ) = inf( x : F X ( x ) ≥ α ) , where α ∈ [0 , F X ( x ) =1 − e − λx = α gives x = − λ ln(1 − α ).So, VaR X ( α ) = − λ ln(1 − α ) and the CTE is given byCTE X ( α ) = E ( X | X >
VaR X ( α )) = 11 − F X (VaR X ( α )) (cid:90) ∞ VaR X ( α ) xdF X ( x ) , where F X is the distribution function of X . Since X is continuous, F X (VaR X ( α )) = α. Therefore, CTE X ( α ) = 11 − α (cid:90) ∞ VaR X ( α ) xdF X ( x ) = 11 − α (cid:90) ∞ VaR X ( α ) xλe − λx dx.
3f VaR X ( α ) = Q α , then CTE X ( α ) = λQ α e − λQ α + e − λQ α λ (1 − α ) , where Q α = − ln(1 − α ) λ .Therefore, CTE X ( α ) = 1 λ + V aR X ( α ) . Let X be a Pareto random variable with the distribution function F X ( x ) = P ( X ≤ x ) = 1 − (cid:16) x o x (cid:17) γ , x o ≥ , γ > , x ≥ x o . Then, VaR X ( α ) is defined as VaR X ( α ) = inf( x : F X ( x ) ≥ α ) , where α ∈ [0 , . We have that, F X ( x ) = 1 − (cid:16) x o x (cid:17) γ = α gives x = x o (1 − α ) − γ . So, VaR X ( α ) = Q α = x o (1 − α ) − γ and CTE is given byCTE X ( α ) = E ( X | X >
VaR X ( α )) = 11 − α (cid:90) ∞ VaR X ( α ) xdF X ( x ) , where F X is the distribution function of X . Since the probability density function of X is f X ( x ) = γx γo x − γ − , CTE X ( α ) = 11 − α (cid:90) ∞ Q α x.γ.x γo .x − γ − dx = γx γo − α (cid:20) x − γ − γ (cid:21) ∞ Q α = γx γo Q − γα ( γ − − α ) . Using Q α = x o (1 − α ) − γ , CTE X ( α ) = γx o (1 − α ) γ ( γ −
1) = γγ − X ( α ) . Let X = ( X , X ) be a risk vector where X and X denote the risks in the subportfolio of X . Then, X (1) = min( X , X ) and X (2) = max( X , X ) are the extreme risks in a portfolio.Let X and X be exp( λ ) and exp( λ ) distributions. Then, the distribution function of X i ( i = 1 ,
2) isgiven by P ( X i ≤ x ) = F X i ( x ) = 1 − e − λ i x , i = 1 , . The distribution function of X (1) = min( X , X ) is given by P ( X (1) ≤ x ) = 1 − P (cid:0) min( X , X ) > x (cid:1) = 1 − P ( X > x, X > x ) . Case (i)
When X and X are independent, we have P ( X (1) ≤ x ) = 1 − P ( X > x, X > x ) = 1 − e − λ x · e − λ x = 1 − e − ( λ + λ ) x . So, X (1) ∼ exp ( λ + λ ) . Therefore,VaR X (1) ( α ) = − − ln(1 − α ) λ + λ , CTE X (1) ( α ) = 1 λ + λ + V aR X (1) ( α ) = 1 − ln(1 − α ) λ + λ . X (2) = max( X , X ) is given by P ( X (2) ≤ x ) = P (cid:0) max( X , X ) ≤ x (cid:1) = P ( X ≤ x, X ≤ x ) = P ( X ≤ x ) P ( X ≤ x )= (1 − e − λ x )(1 − e − λ x ) = 1 − e − λ x − e − λ x + e − ( λ + λ ) x , and its density function is given by f X (2) = λ e − λ x + λ e − λ x − ( λ + λ ) e − ( λ + λ ) x . If VaR X (2) ( α ) = Q α , then F X (2) ( x ) = 1 − e − λ Q α − e − λ Q α + e − ( λ + λ ) Q α = α and CTE X (2) ( α ) = 11 − α (cid:90) ∞ Q α xdF X ( x ) = 11 − α (cid:90) ∞ Q α x (cid:104) λ e − λ x + λ e − λ x − ( λ + λ ) e − ( λ + λ ) x (cid:105) dx. After calculations, we haveCTE X (2) ( α ) = λ Q α e − λ Q α + e − λ Q α λ (1 − α ) + λ Q α e − λ Q α + e − λ Q α λ (1 − α ) − ( λ + λ ) Q α e − ( λ + λ ) Q α + e − ( λ + λ ) Q α ( λ + λ )(1 − α ) . Examples:
If we choose λ = 0 . λ = 0 . α = 0 .
9, thenVaR X (0 .
9) = − ln(1 − . . . , VaR X (0 .
9) = − ln(1 − . . . , CTE X (0 .
9) = 1 − ln(1 − . . . , CTE X (0 .
9) = 1 − ln(1 − . . . , VaR X (1) (0 .
9) = − ln(1 − . . . .
09 and CTE X (2) (0 .
9) = 1 − ln(1 − . . . . For VaR X (2) (0 .
9) = Q . , 1 - e − . Q α − e − . Q α + e − . Q α = 0 .
9. Thus, VaR X (2) (0 .
9) = Q . = 5 .
47. ForCTE X (2) (0 . X (2) (0 .
9) = 7 . . Case (ii)
When X and X are dependent, we use FGM copula C ( u, v ) = uv + θuv (1 − u )(1 − v )where 0 ≤ u, v ≤ − ≤ θ ≤
1. In the next section, we use this copula to analyze the dependencybetween the sub-portfolios.
Consider X (1) = min( X , X ) , u = F X ( x ) and v = F X ( x ), the tail probability distribution of X (1) =min( X , X ) is given by P ( X (1) > x ) = P (min( X , X ) > x ) = P ( X > x, X > x ) , = ¯ C (1 − u, − v ) = 1 − u − v + C ( u, v ) , = 1 − (1 − e − λ x ) − (1 − e − λ x ) + (1 − e − λ x )(1 − e − λ x )+ θ (1 − e − λ x )(1 − e − λ x ) e − λ x e − λ x , = e − ( λ + λ ) x + θe − ( λ + λ ) x (1 − e − λ x − e − λ x + e − ( λ + λ ) x ) , C ( u, v ) = uv + θuv (1 − u )(1 − v ) is used.So, the distribution function of X (1) is given by F X (1) ( x ) = 1 − P ( X (1) > x ) = 1 − e − ( λ + λ ) x − θe − ( λ + λ ) x (cid:2) − e − λ x − e − λ x + e − ( λ + λ ) x (cid:3) and the probability density function of X (1) is given by f X (1) ( x ) = ( λ + λ ) e − ( λ + λ ) x + θ (cid:2) ( λ + λ ) e − ( λ + λ ) x − ( λ + 2 λ ) e − ( λ +2 λ ) x − (2 λ + λ ) e − (2 λ + λ ) x + 2( λ + λ ) e − λ + λ ) x (cid:3) . For VaR( Q α ), we have F X (1) ( x ) = 1 − e − ( λ + λ ) Q α − θe − ( λ + λ ) Q α (cid:2) − e − λ Q α − e − λ Q α + e − ( λ + λ ) Q α (cid:3) = α. If we choose λ = 0 . , λ = 0 . α = 0 .
9, then1 − e − . Q α − θe − . Q α [1 − e − . Q α − e − . Q α + e − . Q α ] = 0 . . Let us take different values of θ (from weak dependency to strong dependency), we have θ X (1) (0 .
9) 2.14 2.22 2.3 2.38 2.45Table 1: Table of VaR X (1) (0 .
9) vs dependencyFor CTE X (1) ( α ),CTE X (1) ( α ) = E [ X (1) | X (1) > Q α ] = 11 − α (cid:90) ∞ Q α xf X (1) ( x ) dx, = 11 − α (cid:104)(cid:16) Q α e − Q α ( λ + λ ) + e − Q α ( λ + λ ) λ + λ (cid:17) + θ (cid:16) Q α e − Q α ( λ + λ ) + e − Q α ( λ + λ ) λ + λ (cid:17) − θ (cid:16) Q α e − Q α ( λ +2 λ ) + e − Q α ( λ +2 λ ) λ + 2 λ (cid:17) − θ (cid:16) Q α e − Q α (2 λ + λ ) + e − Q α (2 λ + λ ) λ + λ (cid:17) + θ (cid:16) Q α e − Q α ( λ + λ ) + e − Q α ( λ + λ ) λ + λ ) (cid:17)(cid:105) . For λ = 0 . λ = 0 . , and α = 0 . , we have,CTE X (1) ( α ) = 10 (cid:104)(cid:16) Q α e − . Q α + Q α e − . Q α . (cid:17) + θ (cid:16) Q α e − . Q α + Q α e − . Q α . (cid:17) − θ (cid:16) Q α e − . Q α + Q α e − . Q α . (cid:17) − θ (cid:16) Q α e − . Q α + Q α e − . Q α . (cid:17) + θ (cid:16) Q α e − . Q α + Q α e − . Q α . (cid:17)(cid:105) . For different values of θ , the values of CTE X (1) (0 .
9) are shown in table 2.We have shown by this example that as the strength of dependency increases, VaR and CTE of theminimum of two risks also increase. Hence, it is better to make the risks X and X in the sub-portfoliosindependent or less dependent so that VaR and CTE of the minimum risks can be smaller than thatfor the dependent case.Line graphs of VaR X (1) (0 .
9) and CTE X (1) (0 .
9) at different dependencies are shown in figure (1).6 X (1) (0 .
9) 3.04 3.17 3.23 3.35 3.44Table 2: Table of CTE X (1) (0 .
9) vs dependencyFigure 1: Line graphs of VaR and CTE vs dependency . . . . . . . Graph of VaR vs dependency
Strength of dependency V a R . . . . Graph of CTE vs dependency
Strength of dependency C T E Consider X (2) = max( X , X ) where X ∼ Exp ( λ ), X ∼ Exp ( λ ), the distribution function of X (2) isgiven by F X (2) ( x ) = P ( X (2) ≤ x ) = P (max( X , X ) ≤ x ) = P ( X ≤ x, X ≤ x ) = C ( u, v )where u = F X ( x ) = 1 − e − λ x and v = F X ( x ) = 1 − e − λ x . Thus, F X (2) ( x ) = uv + θuv (1 − u )(1 − v ) , − ≤ θ ≤ . = (1 − e − λ x )(1 − e − λ x ) + θ (1 − e − λ x )(1 − e − λ x ) .e − λ x .e − λ x = 1 − e − λ x − e − λ x + e − ( λ + λ ) x + θ (cid:2) e − ( λ + λ ) x − e − (2 λ + λ ) x − e − ( λ +2 λ ) x + e − λ + λ ) x (cid:3) . Differentiating it with respect to x , we get the density function of X (2) , f X (2) ( x ) = λ e − λ x + λ e − λ x − ( λ + λ ) e − ( λ + λ ) x + θ [( λ + 2 λ ) e − ( λ +2 λ ) x +(2 λ + λ ) e − (2 λ + λ ) x − ( λ + λ ) e − ( λ + λ ) x − λ + λ ) e − λ + λ ) x ] . X (2) ( α ) = Q α , if we choose λ = 0 . λ = 0 . α = 0 .
9, then1 − e − . Q α − e − . Q α + e − . Q α + θ (cid:2) e − . Q α − e − . Q α − e − . Q α + e − . Q α (cid:3) = 0 . . Taking different values of θ from 0 . . X (2) (0 .
9) = Q . , which are shown in table 3. θ V aR X (2) (0 .
9) 5.46 5.45 5.45 5.44 5.43Table 3: Table of VaR X (2) (0 .
9) vs dependencyFor
CT E X (2) (0 . CT E X (2) ( α ) = E (cid:2) X (2) | X (2) > Q α (cid:3) = 11 − α (cid:90) ∞ Q α xf X (2) ( x ) dx where f X (2) ( x ) is the pdf of X (2) = max( X , X ). CT E X (2) (0 .
9) = 11 − . (cid:90) ∞ Q α (cid:104) . xe − . x + 0 . xe − . x − . xe − . x + θ (cid:0) . xe − . x + 1 . xe − . x − . xe − . x − . xe − . x (cid:1)(cid:105) dx. After calculations, we have
CT E X (2) (0 .
9) = 10 (cid:104)(cid:16) Q α e − . Q α + e − . Q α . (cid:17) + (cid:16) Q α e − . Q α + e − . Q α . (cid:17) − (cid:16) Q α e − . Q α + e − . Q α . (cid:17) + θ (cid:16)(cid:16) Q α e − . Q α + e − . Q α . (cid:17) + (cid:16) Q α e − . Q α + e − . Q α . (cid:17) − (cid:16) Q α e − . Q α + e − . Q α . (cid:17) − (cid:16) Q α e − . Q α + e − . Q α . (cid:17)(cid:17)(cid:105) . For different values of θ , the values of CT E X (2) (0 .
9) are shown in table 4. θ CT E X (2) (0 .
9) 7.369 7.366 7.361 7.356 7.351Table 4: Table of
CT E X (2) (0 .
9) vs dependencyWe have shown by this example that as the strength of dependency increases, VaR and CTE ofthe maximum of two risks X (2) = max( X , X ) do not change significantly. That means that, if weconsider the maximum of our two investments in two different portfolios, it does not matter whetherthe portfolios X and X are dependent or not. 8 .2 VaR and CTE for extreme risks of Pareto distribution Let X and X follow the pareto distributions given respectively by P ( X i ≤ x ) = F X i ( x ) = 1 − (cid:16) x o x (cid:17) γ i ,where i = 1 , x o ≥ , γ > . The extreme risks in a portfolio are X (1) = min( X , X ) and X (2) = max( X , X ) . Then, the distribution function of X (1) = min( X , X ) is given by P ( X (1) ≤ x ) = 1 − P (min( X , X ) > x ) = 1 − P ( X > x, X > x ) . Case (i)
When X and X are independent, for X (1) = min( X , X ), we have P ( X (1) ≤ x ) = 1 − P ( X > x, X > x ) = 1 − (cid:16) x o x (cid:17) γ (cid:16) x o x (cid:17) γ = 1 − (cid:16) x o x (cid:17) γ + γ . So, X (1) ∼ Pareto( γ + γ ). Then, VaR X (1) ( α ) and CTE X (1) ( α ) are given byVaR X (1) ( α ) = x o (1 − α ) − γ γ , CTE X (1) ( α ) = ( γ + γ ) x o (1 − α ) γ γ ( γ + γ − . For X (2) = max( X , X ), the distribution function of X (2) is given by P ( X (2) ≤ x ) = P (cid:0) max( X , X ) ≤ x (cid:1) = P ( X ≤ x, X ≤ x )= (cid:104) − (cid:16) x o x (cid:17) γ (cid:105)(cid:104) − (cid:16) x o x (cid:17) γ (cid:105) = 1 − (cid:16) x o x (cid:17) γ − (cid:16) x o x (cid:17) γ + (cid:16) x o x (cid:17) γ + γ , and its density function is given by f X (2) = γ x γ o x γ +1 + γ x γ o x γ +1 − ( γ + γ ) x ( γ + γ ) o x γ + γ +1 . If VaR X (2) ( x ) = Q α , then 1 − (cid:16) x o x (cid:17) γ − (cid:16) x o x (cid:17) γ + (cid:16) x o x (cid:17) γ + γ = α, andCTE X (2) ( α ) = 11 − α (cid:90) ∞ Q α xf X (2) ( x ) dx, = 11 − α (cid:90) ∞ Q α (cid:34) γ x γ o x γ + γ x γ o x γ − ( γ + γ ) x ( γ + γ ) o x γ + γ (cid:35) dx, = 11 − α (cid:20) γ x γ o ( γ − Q α ) γ − + γ x γ o ( γ − Q α ) γ − +( γ + γ ) x ( γ + γ ) o ( γ + γ − Q α ) γ + γ − (cid:21) . Example:
Assume that x o = 1, γ = 3, γ = 4 and α = 0 . , thenVaR X (0 .
9) = x o (1 − α ) − γ = (1 − . − = 2 . , VaR X (0 .
9) = x o (1 − α ) − γ = (1 − . − = 1 . , CTE X (0 .
9) = γ γ − V aR X (0 .
9) = 33 − × .
154 = 3 . , CTE X (0 .
9) = γ γ − V aR X (0 .
9) = 44 − × .
778 = 2 . , VaR X (1) (0 .
9) = x o (1 − α ) − γ γ = (1 − . − = 1 . , X (2) (0 .
9) satisfies the expression1 − (cid:18) Q α (cid:19) − (cid:18) Q α (cid:19) + (cid:18) Q α (cid:19) = 0 . . This gives, Q α = VaR X (2) (0 .
9) = 2 . . Next, CTE X (1) ( α ) = ( γ + γ ) x o (1 − α ) γ γ ( γ + γ −
1) = 1 . , andCTE X (2) ( α ) = 1 α − (cid:20) γ x γ o ( γ − Q α ) γ − + γ x γ o ( γ − Q α ) γ − − ( γ + γ ) x ( γ + γ ) o ( γ + γ − Q α ) γ + γ − (cid:21) = 3 . . Case (ii)
When X and X are dependent, we use FGM copula given by C ( u, v ) = uv + θuv (1 − u )(1 − v ) , where 0 ≤ u, v ≤
1, and − ≤ θ ≤ For X (1) = min( X , X ), u = F X ( x ) and v = F X ( x ), the survival function of X (1) is given by, P ( X (1) > x ) = P (min( X , X ) > x ) = P ( X > x, X > x ) , = ¯ C (1 − u, − v ) = 1 − u − v + C ( u, v ) , = 1 − u − v + uv + θuv (1 − u )(1 − v ) , = 1 − (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) − (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) + (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) , + θ (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) (cid:16) x o x (cid:17) γ (cid:16) x o x (cid:17) γ , = (cid:16) x o x (cid:17) γ + γ + θ (cid:20)(cid:16) x o x (cid:17) γ + γ − (cid:16) x o x (cid:17) γ + γ − (cid:16) x o x (cid:17) γ +2 γ + (cid:16) x o x (cid:17) γ + γ ) (cid:21) , where FGM copula C ( u, v ) = uv + θuv (1 − u )(1 − v ) is used.So, the distribution function of X (1) is given by F X (1) ( x ) = 1 − (cid:16) x o x (cid:17) γ + γ − θ (cid:20)(cid:16) x o x (cid:17) γ + γ − (cid:16) x o x (cid:17) γ + γ − (cid:16) x o x (cid:17) γ +2 γ + (cid:16) x o x (cid:17) γ + γ ) (cid:21) , and the probability density function of X (1) is given by f X (1) ( x ) = ( γ + γ )( x o ) γ + γ x γ + γ +1 + θ (cid:20) ( γ + γ )( x o ) γ + γ x γ + γ +1 − (2 γ + γ )( x o ) γ + γ x γ + γ +1 − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 + 2( γ + γ )( x o ) γ +2 γ x γ +2 γ +1 (cid:21) . To find the value of VaR X (1) ( α ) = Q α , we have F X (1) ( x ) = 1 − (cid:18) x o Q α (cid:19) γ + γ − θ (cid:20) (cid:18) x o Q α (cid:19) γ + γ − (cid:18) x o Q α (cid:19) γ + γ − (cid:18) x o Q α (cid:19) γ +2 γ + (cid:18) x o Q α (cid:19) γ + γ ) (cid:21) = α.
10f we choose x o = 1, γ = 3, γ = 4 and α = 0 . , then1 − ( Q α ) − − θ (cid:2) ( Q α ) − − ( Q α ) − − ( Q α ) − + ( Q α ) − (cid:3) = 0 . . For the strength of dependency, we consider the dependency from weak to strong which is shown intable 5. For CTE X (1) ( α ), we have θ Q . Q . vs dependencyCTE X (1) ( α ) = E (cid:2) X (1) | X (1) > Q α (cid:3) = 11 − α (cid:90) ∞ Q α xf X (1) ( x ) dx, = 11 − α (cid:90) ∞ Q α (cid:20) ( γ + γ )( x o ) γ + γ x γ + γ + θ (cid:16) ( γ + γ )( x o ) γ + γ x γ + γ − (2 γ + γ )( x o ) γ + γ x γ + γ − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ + 2( γ + γ )( x o ) γ +2 γ x γ +2 γ (cid:17)(cid:21) . Using γ = 3, γ = 4, x o = 1 and α = 0 .
9, we haveCTE X (1) (0 .
9) = 10 (cid:20) Q α ) + θ (cid:16) Q α ) − Q α ) − Q α ) + 1413( Q α ) (cid:17)(cid:21) , where Q α = VaR X (1) ( α ) . The values of the conditional tail expectation at different dependencies areshown in table 6. The line graphs of VaR X (1) (0 .
9) and CTE X (1) (0 .
9) for different positive dependencies θ X (1) (0 .
9) 1.63 1.66 1.69 1.71 1.74Table 6: Table of CTE X (1) (0 .
9) vs dependency.are shown in figure 2.From this example, we conclude that as the strength of dependency increases, the VaR and CTE ofminimum of two risks also increase for the risk of loss having the Pareto distribution.Hence, VaR and CTE of the minimum extreme risks can be smaller if we make risks X and X independent or less dependent. For X (2) = max( X , X ), where X i ∼ Pareto( γ i ), i = 1,2, the distribution function of X (2) is given by F X (2) ( x ) = P (cid:0) X (2) ≤ x (cid:1) = P (max( X , X ) ≤ x ) = P ( X ≤ x, X ≤ x ) = C ( u, v ) , . . . . Graph of VaR vs dependency
Strength of dependency V a R . . . . . . Graph of CTE vs dependency
Strength of dependency C T E where C ( u, v ) = uv + θuv (1 − u )(1 − v ), u = F X ( x ) = 1 − (cid:16) x o x (cid:17) γ and v = F X ( x ) = 1 − (cid:16) x o x (cid:17) γ . Then, F X (2) ( x ) = uv + θuv (1 − u )(1 − v ) , = (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) + θ (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) (cid:104) − (cid:16) x o x (cid:17) γ (cid:105) (cid:16) x o x (cid:17) γ (cid:16) x o x (cid:17) γ = 1 − (cid:16) x o x (cid:17) γ − (cid:16) x o x (cid:17) γ + (cid:16) x o x (cid:17) γ + γ + θ (cid:20)(cid:16) x o x (cid:17) γ + γ − (cid:16) x o x (cid:17) γ + γ − (cid:16) x o x (cid:17) γ +2 γ + (cid:16) x o x (cid:17) γ +2 γ (cid:21) , and the probability density function of X (2) is given by f X (2) ( x ) = γ ( x o ) γ x γ +1 + γ ( x o ) γ x γ +1 − ( γ + γ )( x o ) γ + γ x γ + γ +1 − θ (cid:20) ( γ + γ )( x o ) γ + γ x γ + γ +1 − (2 γ + γ )( x o ) γ + γ x γ + γ +1 − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 + (2 γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 (cid:21) . For VaR X ( α ) = Q α , we choose x o = 1, γ = 3, γ = 4 and α = 0 . . Then, F X (2) (0 .
9) = 1 − ( Q α ) − − ( Q α ) − + ( Q α ) − + θ (cid:2) ( Q α ) − − ( Q α ) − − ( Q α ) − + ( Q α ) − (cid:3) = 0 . . For the different strength of dependencies from weak to strong, the values of Q . are shown in table 7.12 Q . Q . vs dependencyFor CTE X (2) (0 . X (2) ( α ) = 11 − α (cid:90) ∞ Q α xf X (2) ( x ) dx. where f X (2) ( x ) is the pdf of X (2) = max( X , X ). Hence,CTE X (2) ( α ) = 11 − α (cid:90) ∞ Q α x (cid:20) γ ( x o ) γ x γ +1 + γ ( x o ) γ x γ +1 − ( γ + γ )( x o ) γ + γ x γ + γ +1 − θ [ ( γ + γ )( x o ) γ + γ x γ + γ +1 − (2 γ + γ )( x o ) γ + γ x γ + γ +1 − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 + (2 γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 (cid:21) dx. Using x o = 1 , γ = 3 , γ = 4 and α = 0 . , and after integration, we getCTE X (2) (0 .
9) = 10 (cid:20) Q α ) + 43( Q α ) − Q α ) − θ (cid:18) Q α ) − Q α ) − Q α ) + 1413( Q α ) (cid:19)(cid:21) . For different strength of dependency, from weak to strong, the values of CTE X (2) (0 .
9) are shown intable 8. θ X (2) (0 .
9) 3.50 3.49 3.49 3.49 3.49Table 8: Table of CTE X (2) (0 .
9) vs dependencyBy this example, we have shown that when the strength of dependency increases, VaR and CTE ofthe maximum of two risks having the Pareto distribution do not change significantly. It means that VaRand CTE of the maximum of two risks do not depend on the dependency of two risks of sub-portfolios.
Let X = ( X , X ) be a risk vector, where X and X denote risks or losses in sub-portfolios. Let X = X + X be the aggregate or the total risk in a portfolio. We consider, X i ∼ Exp( λ i ), i = 1 , x > . Then, the distribution function of X i is given by F X i ( x ) = P ( X i ≤ x i ) = 1 − e − λ i x i , i = 1 ,
2, and x > , and the distribution function of X = X + X is given by P ( X ≤ x ) = P ( X + X ≤ x ) . ase (i) When X and X are independent, the distribution function and the density function of X = X + X are given by F X ( x ) = 1 + λ λ − λ e − λ x − λ λ − λ e − λ x , and f X ( x ) = λ λ λ − λ ( e − λ x − e − λ x )respectively. X = X + X Let Q α be value at risk of X . Then, F X ( Q α ) = 1 + λ λ − λ e − λ Q α − λ λ − λ e − λ Q α = α and the conditional tail expectation of X is given asCTE X ( α ) = 11 − α (cid:90) ∞ Q α xf X ( x ) dx = 11 − α (cid:90) ∞ Q α x (cid:20) λ λ λ − λ ( e − λ x − e − λ x ) (cid:21) dx = λ λ (1 − α )( λ − λ ) (cid:20) Q α e − λ Q α λ + e − λ Q α λ − Q α e − λ Q α λ − e − λ Q α λ (cid:21) . Example:
If we choose λ = 0 . λ = 0 . α = 0 .
9, then, value at risk ( Q α ) can be found from F X ( Q α ) = 1 + λ λ − λ e − λ Q α − λ λ − λ e − λ Q α = α = 1 + 5 e − . Q α − e − . Q α = 0 . . This gives, Q α = 7 . . Next, The conditional tail expectation of X is given byCTE X ( α ) = λ λ (1 − α )( λ − λ ) (cid:20) Q α e − λ Q α λ + e − λ Q α λ − Q α e − λ Q α λ − e − λ Q α λ (cid:21) CTE X (0 .
9) = 0 . × . − . . − . (cid:20) . e − . × . . e − . × . (0 . − . e − . × . . − e − . × . (0 . (cid:21) = 9 . . Case (ii)
When the risks X and X are dependent, we use FGM copula C ( u, v ) = θuv (1 − u )(1 − v ),0 ≤ u, v ≤ − ≤ θ ≤ u = 1 − e − λ x and v = 1 − e − λ x . Then, the joint CDF of X and X is given by C ( u, v ) = F ( x , x ) = (1 − e − λ x )(1 − e − λ x ) + θ (1 − e − λ x )(1 − e − λ x ) e − λ x e − λ x = 1 − e − λ x − e − λ x + e − λ x − λ x + θ ( e − λ x − λ x − e − λ x − λ x − e − λ x − λ x + e − λ x − λ x ) . and its probability density function is given by f X ,X ( x , x ) = λ λ e − λ x − λ x + θ (cid:16) λ λ e − λ x − λ x − λ λ e − λ x − λ x − λ λ e − λ x − λ x + 4 λ λ e − λ x − λ x (cid:17) . X = X + X is given by f X ( x ) = (cid:90) x f X ,X ( x , x − x ) dx , = (cid:90) x (cid:104) λ λ e − λ x − λ ( x − x ) + θ (cid:16) λ λ e − λ x − λ ( x − x ) − λ λ e − λ x − λ ( x − x ) − λ λ e − λ x − λ ( x − x ) + 4 λ λ e − λ x − λ ( x − x ) (cid:17)(cid:105) dx . After calculations, we get f X ( x ) = λ λ λ − λ ( e − λ x − e − λ x ) + θ (cid:20) λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) + 2 λ λ λ − λ ( e − λ x − e − λ x ) (cid:21) , and the distribution function of X = X + X is given by F X ( x ) = 1 + λ λ − λ e − λ x − λ λ − λ e − λ x + θ (cid:20) λ λ − λ e − λ x − λ λ − λ e − λ x − λ λ − λ e − λ x + 2 λ λ − λ e − λ x − λ λ − λ e − λ x + λ λ − λ e − λ x + λ λ − λ e − λ x − λ λ − λ e − λ x (cid:21) . For value at risk (
V aR ), if we choose λ = 0 .
5, and λ = 0 .
6, the distribution function of X becomes F X ( x ) = 1 + 5 e − . x − e − . x + θ (5 e − . x − e − . x − . e − . x + 1 . e − . x + 2 . e − . x − . e − x + 5 e − . x − e − x ) . For
V aR X ( α ) = x = Q α , we have α = 0 . e − . Q α − e − . Q α + θ (5 e − . Q α − e − . Q α − . e − . Q α + 1 . e − . Q α + 2 . e − . Q α − . e − Q α + 5 e − . Q α − e − Q α ) . We consider the different strengths of dependency from weak to strong. Then, value at risk (VaR) canbe found in table 9. From the table, we can see that as the dependency increases between the two risks θ Q . Q . vs dependencyin a portfolio, their aggregate value at risk also increases.15 .3.2 Conditional tail expectation (CTE) of X = X + X The conditional tail expectation of X = X + X is given by CT E X ( α ) = 11 − α (cid:90) ∞ Q α xf X ( x ) dx, where Q α = V aR X ( α ) . = 11 − α (cid:90) ∞ Q α (cid:20) x λ λ λ − λ ( e − λ x − e − λ x ) + xθ (cid:18) λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) + 2 λ λ λ − λ ( e − λ x − e − λ x ) (cid:19)(cid:21) dx, = λ λ (1 − α )( λ − λ ) (cid:20) Q α e − λ Q α λ + e − λ Q α λ − Q α e − λ Q α λ − e − λ Q α λ (cid:21) + θ λ λ (1 − α )( λ − λ ) (cid:20) Q α e − λ Q α λ + e − λ Q α λ − Q α e − λ Q α λ − e − λ Q α λ (cid:21) − θ λ λ (1 − α )( λ − λ ) (cid:20) Q α e − λ Q α λ + e − λ Q α λ − Q α e − λ Q α λ − e − λ Q α λ (cid:21) − θ λ λ (1 − α )(2 λ − λ ) (cid:20) Q α e − λ Q α λ + e − λ Q α λ − Q α e − λ Q α λ − e − λ Q α λ (cid:21) + θ λ λ (1 − α )( λ − λ ) (cid:20) Q α e − λ Q α λ + e − λ Q α λ − Q α e − λ Q α λ − e − λ Q α λ (cid:21) . Putting λ = 0 . , λ = 0 . α = 0 .
9, we haveCTE X ( α ) = − (cid:20) Q α e − . Q α . e − . Q α . − Q α e − . Q α . − e − . Q α . (cid:21) − θ (cid:20) Q α e − . Q α . e − . Q α . − Q α e − . Q α . − e − . Q α . (cid:21) + 8 . θ (cid:20) Q α e − . Q α . e − . Q α . − Q α e − . Q α . − e − . Q α . (cid:21) − θ (cid:20) Q α e − . Q α . e − . Q α . − Q α e − Q α − e − Q α (cid:21) + 60 θ (cid:20) Q α e − . Q α . e − . Q α . − Q α e − Q α − e − Q α (cid:21) . Taking different strengths of dependency from weak to strong (0.1 to 0.9) and corresponding values atrisk, the values of conditional tail expectation after calculations are shown in a table. Also, the linegraphs of VaR X (0 .
9) and CTE X (0 .
9) for the aggregate risk of two dependent risks at different positivedependencies (from 0 . .
9) are constructed and shown in figure 3. θ X (0 .
9) 9.44 9.58 9.72 9.86 9.99Table 10: Table of CTE X (0 .
9) vs dependencyBy this example, we concluded that when the strength of dependency increases, the CTE of theaggregate risk having exponential distribution increases significantly. That means that both measures,VaR and CTE of the aggregate of two risks, depend on the dependency of two risks of sub-portfolios.16igure 3: Line graphs of Q . and CTE X (0 .
9) vs dependency . . . . . Graph of VaR vs dependency
Strength of dependency V a R . . . . . . Graph of CTE vs dependency
Strength of dependency C T E In actuarial science, finance and insurance, several risk measures, namely, VaR, CTE, the distorted riskmeasures have been used in the literature. We are interested to propose an alternative risk measurewhich is called median of tail (MoT).
Definition 3.1
If measure of risk follows a continuous distribution X with probability density function f ( x ) , distribution function F ( x ) and having value at risk Q α , then median of tail is denoted by MoTand defined to satisfy the relation (cid:82) MQ α f ( x ) dx = 1 − α , or, F ( M ) = F ( Q α ) + 1 − α , with M = MoT. Let the risk function follow the exponential distribution with pdf f ( x ) = λe − λx , x ≥ λ > F ( x ) = 1 − e − λx , x ≥ (cid:82) MQ α f ( x ) dx = 1 − α Q α = − ln(1 − α ) λ . Then, the simple calculation leads toMoT X ( α ) = − λ ln (cid:16) − α (cid:17) . Remarks 1:
For λ = 0 . α = 0 .
95, we get, VaR X (0 .
95) = 5 .
99, MoT X (0 .
95) = 7 .
37, andCTE X (0 .
95) = 7 .
99. 17uppose, the risk function X follows the pareto distribution with pdf f ( x ) = γx o x γ +1 , x o ≥ γ > F ( x ) = 1 − (cid:16) x o x (cid:17) γ . The MoT (M) is given by the equation (cid:82) MQ α f ( x ) dx = 1 − α (cid:104) (1 − α )( x o ) − γ − − α x o (cid:105) − γ . Remarks 2:
For x o = 1, γ = 3,and α = 0 .
9, we get, VaR X (0 .
9) = 2 .
15, MoT X (0 .
9) = 2 .
71, andCTE X (0 .
9) = 3 . Let X and X be exp( λ ) and exp( λ ) distributions. Case (i)
When X and X are independent, from section 3.2, the distribution function of X (1) =min( X , X ) is given by F X (1) = P ( X (1) ≤ x ) = 1 − e − ( λ + λ ) x . Thus, MoT of X (1) = min( X , X ) is given by MoT X (1) ( α ) = − λ + λ ln (cid:16) − α (cid:17) . Next, the distribution function of X (2) = max( X , X ) is given by F X (2) = P ( X (2) ≤ x ) = 1 − e − λ x − e − λ x + e − ( λ + λ ) x and the density function of X (2) = max( X , X ) is given by f X (2) = λ e − λ x + λ e − λ x − ( λ + λ ) e − ( λ + λ ) x . Then, the MoT = M of X (2) = max( X , X ) , where Q α is value at risk, is given by (cid:90) MQ α f X (2) dx = (cid:90) MQ α (cid:104) λ e − λ x + λ e − λ x − ( λ + λ ) e − ( λ + λ ) x (cid:105) dx = 1 − α . After calculations, we get e − λ M + e − λ M − e − ( λ + λ ) M = 1 − α Examples:
If we choose λ = 0 . λ = 0 . α = 0 .
9, thenMoT X (0 .
9) = 5 . , MoT X (0 .
9) = 4 . , MoT X (1) (0 .
9) = 2 .
72 and MoT X (2) (0 .
9) = 6 . . Case (ii)
When X and X are dependent, we use FGM copula C ( u, v ) = uv + θuv (1 − u )(1 − v ) , where 0 ≤ u, v ≤ − ≤ θ ≤
1. In the next subsection, we use this copula to analyze the dependencybetween the subportfolios. 18 .2.1 MoT for minimum of two risks of exponential distribution
For u = F X ( x ) and v = F X ( x ), from subsection 3.2.1, the density function of X (1) = min( X , X ) isgiven by f X (1) ( x ) = ( λ + λ ) e − ( λ + λ ) x + θ (cid:2) ( λ + λ ) e − ( λ + λ ) x − ( λ + 2 λ ) e − ( λ +2 λ ) x − (2 λ + λ ) e − (2 λ + λ ) x + 2( λ + λ ) e − λ + λ ) x (cid:3) . Then, M= MoT of X (1) = min( X , X ) is given by1 − α (cid:90) MQ α f X (1) ( x ) dx = (cid:90) MQ α (cid:104) ( λ + λ ) e − ( λ + λ ) x + θ (cid:2) ( λ + λ ) e − ( λ + λ ) x − ( λ + 2 λ ) e − ( λ +2 λ ) x − (2 λ + λ ) e − (2 λ + λ ) x + 2( λ + λ ) e − λ + λ ) x (cid:3) dx (cid:105) . = − e − ( λ + λ ) M + e − ( λ + λ ) Q α + θ (cid:104) − e − ( λ + λ ) M + e − ( λ + λ ) Q α + e − ( λ +2 λ ) M − e − ( λ +2 λ ) Q α + e − (2 λ + λ ) M − e − (2 λ + λ ) Q α − e − λ + λ ) M + e − λ + λ ) Q α (cid:105) . For λ = 0 . λ = 0 .
6, and α = 0 .
9, we have − e − . M + e − . Q α + θ (cid:104) − e − . M + e − . Q α + e − . M − e − . Q α + e − . M − e − . Q α − e − . M + e − . Q α (cid:105) = 0 . . (3.1)For different measures of dependency ( θ ), and the corresponding values of value at risk ( Q α ) given intable 1, the values of MoT from equation 4.1 are given in table 11. θ X (1) (0 .
9) 2.64 2.88 2.97 3.07 3.14Table 11: Table of MoT X (1) (0 .
9) vs dependency.From the table 11, we can see that when values of θ increase, MoT of the minimum of two risks X and X also increases significantly. From subsection 3.2.2, the density function of X (2) = max( X , X ) is given by f X (2) ( x ) = λ e − λ x + λ e − λ x − ( λ + λ ) e − ( λ + λ ) x + θ [( λ + 2 λ ) e − ( λ +2 λ ) x +(2 λ + λ ) e − (2 λ + λ ) x − ( λ + λ ) e − ( λ + λ ) x − λ + λ ) e − λ + λ ) x ] . X (2) = max( X , X ) is given by1 − α (cid:90) MQ α f X (2) ( x ) dx = (cid:90) MQ α (cid:104) λ e − λ x + λ e − λ x − ( λ + λ ) e − ( λ + λ ) x + θ (cid:16) ( λ + 2 λ ) e − ( λ +2 λ ) x +(2 λ + λ ) e − (2 λ + λ ) x − ( λ + λ ) e − ( λ + λ ) x − λ + λ ) e − λ + λ ) x (cid:17)(cid:105) dx. = − e − λ M + e − λ Q α − e − λ M + e − λ Q α + e − ( λ + λ ) M − e − ( λ + λ ) Q α + θ (cid:104) − e − ( λ +2 λ ) M + e − ( λ +2 λ ) Q α − e − (2 λ + λ ) M + e − (2 λ + λ ) Q α + e − ( λ + λ ) M − e − ( λ + λ ) Q α + e − λ + λ ) M − e − λ + λ ) Q α (cid:105) . For λ = 0 . λ = 0 .
6, and α = 0 .
9, we have − e − . M + e − . Q α − e − . M + e − . Q α + e − . M − e − . Q α + θ (cid:104) − e − . M + e − . Q α − e − . M + e − . Q α + e − . M − e − . Q α + e − . M − e − . Q α (cid:105) = 0 . . (3.2)For different measures of dependency ( θ ), and the corresponding values of value at risk ( Q α ) given intable 2, the values of MoT from equation 4.2 are given in table 12. From the table 12, we can see that θ M oT X (2) (0 .
9) 6.77 6.77 6.78 6.77 6.77Table 12: Table of
M oT X (2) (0 .
9) vs dependency.when values of θ increase, MoT of the maximum of two risks X and X does not change significantly. Let X and X follow the pareto distributions given by P ( X i ≤ x ) = F X i ( x ) = 1 − (cid:16) x o x (cid:17) γ i , where x o ≥ , γ i > i = 1 , . Case (i)
When X and X are independent, from section 3.4, the distribution function of X (1) =min( X , X ) is given by F X (1) = 1 − (cid:16) x o x (cid:17) γ (cid:16) x o x (cid:17) γ = 1 − (cid:16) x o x (cid:17) γ + γ . Thus, MoT of X (1) = min( X , X ) is given byMoT X (1) ( α ) = (cid:104) (1 − α )( x o ) − γ − γ − − α x o (cid:105) − γ γ . Next, the distribution function of X (2) = max( X , X ) is given by F X (2) = (cid:104) − (cid:16) x o x (cid:17) γ (cid:105)(cid:104) − (cid:16) x o x (cid:17) γ (cid:105) = 1 − (cid:16) x o x (cid:17) γ − (cid:16) x o x (cid:17) γ + (cid:16) x o x (cid:17) γ + γ , X (2) = max( X , X ) is given by f X (2) = γ x γ o x γ +1 + γ x γ o x γ +1 − ( γ + γ ) x ( γ + γ ) o x γ + γ +1 . Then, MoT = M of X (2) = max( X , X ) where Q α is value at risk is given by (cid:90) MQ α f X (2) dx = (cid:90) MQ α (cid:104) γ x γ o x γ +1 + γ x γ o x γ +1 − ( γ + γ ) x ( γ + γ ) o x γ + γ +1 (cid:105) dx = 1 − α . = (cid:16) x o Q α (cid:17) γ − (cid:16) x o M (cid:17) γ + (cid:16) x o Q α (cid:17) γ − (cid:16) x o M (cid:17) γ − (cid:16) x o Q α (cid:17) γ + γ + (cid:16) x o M (cid:17) γ + γ . Examples:
For x o = 1, γ = 3, γ = 4 and α = 0 . , thenMoT X (0 .
9) = 2 . , MoT X (0 .
9) = 2 . , MoT X (1) (0 .
9) = 1 .
53 and MoT X (2) (0 .
9) = 2 . . Case (ii)
When X and X are dependent, we use FGM copula C ( u, v ) = uv + θuv (1 − u )(1 − v )where 0 ≤ u, v ≤ − ≤ θ ≤
1. In the next subsection, we use this copula to analyze the dependencybetween the subportfolios.
For u = F X ( x ) and v = F X ( x ), from subsection 3.4.1, the density function of X (1) = min( X , X ) isgiven by f X (1) ( x ) = ( γ + γ )( x o ) γ + γ x γ + γ +1 + θ (cid:20) ( γ + γ )( x o ) γ + γ x γ + γ +1 − (2 γ + γ )( x o ) γ + γ x γ + γ +1 − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 + 2( γ + γ )( x o ) γ +2 γ x γ +2 γ +1 (cid:21) . Then, M= MoT of X (1) = min( X , X ) is given by1 − α (cid:90) MQ α f X (1) ( x ) dx = 1 − α (cid:90) MQ α (cid:104) ( γ + γ )( x o ) γ + γ x γ + γ +1 + θ (cid:16) ( γ + γ )( x o ) γ + γ x γ + γ +1 − (2 γ + γ )( x o ) γ + γ x γ + γ +1 − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 + 2( γ + γ )( x o ) γ +2 γ x γ +2 γ +1 (cid:17)(cid:105) dx. = (cid:16) x o Q α (cid:17) γ + γ − (cid:16) x o M (cid:17) γ + γ + θ (cid:104)(cid:16) x o Q α (cid:17) γ + γ − (cid:16) x o M (cid:17) γ + γ − (cid:16) x o Q α (cid:17) γ + γ + (cid:16) x o M (cid:17) γ + γ − (cid:16) x o Q α (cid:17) γ +2 γ + (cid:16) x o M (cid:17) γ +2 γ + (cid:16) x o Q α (cid:17) γ +2 γ − (cid:16) x o M (cid:17) γ +2 γ (cid:105) . For x o = 1, γ = 3, γ = 4, α = 0 .
9, we have( Q α ) − − ( M ) − + θ (cid:104) ( Q α ) − − ( M ) − − ( Q α ) − + ( M ) − − ( Q α ) − + ( M ) − + ( Q α ) − − ( M ) − (cid:105) = 0 . . (3.3)For different measures of dependency ( θ ), and the corresponding values of value at risk ( Q α ) given intable 3, the values of MoT from equation 4.3 are given in table 13.From table 13, we can see that when values of θ increase, MoT of the minimum of two risks X and X also increases significantly. 21 X (1) (0 .
9) 1.52 1.55 1.58 1.61 1.64Table 13: Table of MoT X (1) (0 .
9) vs dependency.
From subsection 3.4.2, the density function of X (2) = max( X , X ) is given by f X (2) ( x ) = γ ( x o ) γ x γ +1 + γ ( x o ) γ x γ +1 − ( γ + γ )( x o ) γ + γ x γ + γ +1 − θ (cid:20) ( γ + γ )( x o ) γ + γ x γ + γ +1 − (2 γ + γ )( x o ) γ + γ x γ + γ +1 − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 + (2 γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 (cid:21) . Then, M= MoT of X (2) = max( X , X ) is given by1 − α (cid:90) MQ α (cid:20) γ ( x o ) γ x γ +1 + γ ( x o ) γ x γ +1 − ( γ + γ )( x o ) γ + γ x γ + γ +1 − θ (cid:18) ( γ + γ )( x o ) γ + γ x γ + γ +1 − (2 γ + γ )( x o ) γ + γ x γ + γ +1 − ( γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 + (2 γ + 2 γ )( x o ) γ +2 γ x γ +2 γ +1 (cid:19)(cid:21) dx. For x o = 1, γ = 3, γ = 4, α = 0 .
9, we have( Q α ) − − ( M ) − + ( Q α ) − − ( M ) − − ( Q α ) − + ( M ) − − θ (cid:104) ( Q α ) − − ( M ) − − ( Q α ) − +( M ) − − ( Q α ) − + ( M ) − + ( Q α ) − − ( M ) − (cid:105) = 0 . . (3.4)For different strengths of dependency ( θ ), and the corresponding values of value at risk ( Q α ) given intable 3, the values of MoT from equation 4.4 are given in table 14. θ X (2) (0 .
9) 2.98 2.98 2.975 2.97 2.97Table 14: Table of MoT X (2) (0 .
9) vs dependency.From the table 14, we can see that when θ increases, MoT of the minimum of two risks X and X does not change significantly. In this section, we derive MoT for the aggregate risk X = X + X , where the distribution function of X i is given by F X i ( x ) = P ( X i ≤ x i ) = 1 − e − λ i x i , i = 1 ,
2, and x > . ase (i) When X and X are independent, from section 3.5, the distribution function and thedensity function of X = X + X are given by F X ( x ) = 1 + λ λ − λ e − λ x − λ λ − λ e − λ x and f X ( x ) = λ λ λ − λ ( e − λ x − e − λ x )respectively.Then, M = MoT of X = X + X is given by1 − α (cid:90) MQ α (cid:104) λ λ λ − λ ( e − λ x − e − λ x ) (cid:105) dx = λ e − λ M − λ e − λ M − λ e − λ Q α + λ e − λ Q α λ − λ . Example:
If we choose λ = 0 . λ = 0 . α = 0 .
9, then, MoT = 8.71.
Case (ii)
When X and X are dependent, we use FGM copula C ( u, v ) = uv + θuv (1 − u )(1 − v )to analyze the dependency where u = 1 − e − λ x and v = 1 − e − λ x . Then, from subsection 3.5.1, the probability density function of X = X + X is given by f X ( x ) = λ λ λ − λ ( e − λ x − e − λ x ) + θ (cid:20) λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) + 2 λ λ λ − λ ( e − λ x − e − λ x ) (cid:21) . Then, M = MoT of the aggregate risk X is given by= (cid:90) MQ α (cid:20) λ λ λ − λ ( e − λ x − e − λ x ) + θ (cid:16) λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) − λ λ λ − λ ( e − λ x − e − λ x ) + 2 λ λ λ − λ ( e − λ x − e − λ x ) (cid:17)(cid:21) dx. = λ e − λ M − λ e − λ M − λ e − λ Q α + λ e − λ Q α λ − λ + θ (cid:20) λ e − λ M − λ e − λ M − λ e − λ Q α + λ e − λ Q α λ − λ − λ e − λ M − λ e − λ M − λ e − λ Q α + 2 λ e − λ Q α λ − λ − λ e − λ M − λ e − λ M − λ e − λ Q α + λ e − λ Q α λ − λ + λ e − λ M − λ e − λ M − λ e − λ Q α + λ e − λ Q α λ − λ (cid:21) = 1 − α . For λ = 0 . λ = 0 . α = 0 .
9, we get(5 + 7 . θ ) e − . M − (6 + 4 . θ ) e − . M + 4 . θe − . M − . θe − M − (5 + 7 . θ ) e − . Q α + (6 + 4 . θ ) e − . Q α − . θe − . Q α + 7 . θe − Q α − .
05 = 0 . (3.5)For different strengths of dependency ( θ ), and the corresponding values of value at risk ( Q α ) givenin table 9, the values of MoT from equation (3.5) are given in table 15.From table 15, we can see that when θ increases, MoT of the aggregate of two risks X and X alsoincreases significantly. 23 X (0 .
9) 8.78 8.93 9.05 9.20 9.31Table 15: Table of MoT X (0 .
9) vs dependency.
Using FGM copula to capture the dependence between risk measures in the sub-portfolios, we derivedthe explicit expressions of value at risk and the conditional tail expectation (CTE) for the extreme risksand the aggregate risk of the portfolio. Both VaR and the CTE measure the right tail risk, which arefrequently used in the insurance and financial investment. Moreover, we proposed an alternate riskmeasure ”median of tail” (MoT). To evaluate the extreme (maximum and the minimum) of two risks,we considered the cases, where the risks follow the exponential and pareto distributions whereas for theaggregate risk, the risks follow exponential distribution. We have shown, with examples, that as thedependency between two risks in their sub-portfolios increases:(i) VaR, CTE and MoT of the minimum of the two risks and the aggregate risk are also increasing and(ii) VaR, CTE and MoT of the maximum of the two risks are not changing significantly.If people are interested in investing money in two or more different sub-portfolios (areas), it is betterto have the areas independent (or less dependent) so that the risk measures are smaller than that forthe dependent case.
Acknowledgement
The authors would like to acknowledge that this work is supported, in part, by the Natural Sciences andEngineering Research Council of Canada (NSERC) through a Discovery Research Grant, and CarletonUniversity.
AppendixIntroduction to copula
The word copula originally came from the latin word copulare, which means to join together. Inmany cases of modeling, it is important to obtain the joint probability density function between twoor more random variables. Even though the marginals of each of the dependent random variables areknown, their joint distributions cannot, in general, be derived from their marginal distributions. Thefollowing definitions can be found in many references, for example, from the book[10] by Nelsen.
Mathematical definition of copula:
A two dimensional copula is a function C : [0 , × [0 , → [0 , , or C : I → I , satisfing the followingtwo conditions: (i) Boundary conditions: C ( u,
0) = 0 , C (0 , v ) = 0 , C ( u,
1) = u, C (1 , v ) = v for all u, v ∈ [0 , . (ii) u , u , v , v ∈ I such that u ≤ u and v ≤ v ,C ( u , v ) + C ( u , v ) − C ( u , v ) − C ( u , v ) ≥ . ote : If C ( u, v ) is twice differentiable, then the 2-increasing property is equivalent to ∂ C ( u, v ) ∂u∂v ≥ . Farlie-Gumbel-Morgenstern copula:
The Farlie-Gumbel-Morgenstern copula (FGM) is defined by C ( u, v ; θ ) = uv + θuv (1 − u )(1 − v ) , − ≤ θ ≤ . The FGM copula was first proposed by Morgenstern (1956). It is a perturbation of the product copula.This copula [11] is only useful when dependence between the two marginals is modest in magnitude.
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