This section illustrates how the above findings can be applied in character-izing underlying risk attitudes. It is devised to highlight situations in which the implicit risk attitude linked to theVaRis unable to detect changes in potential catastrophic losses. We argue that the use of equivalent GlueVaR risk measures can be helpful in overcoming this drawback.
Suppose that theVaRwith a confidence levelÆ=99.5%is required to as-sess the regulatory capital under some regulatory framework. Note that the selection of the confidence level involves a trade-off between protection and competitiveness. The level of the protection could be reduced with low con-fidence levels. An increase in the concon-fidence levels could involve higher eco-nomic reserves and, therefore, the protection would rise; however, this could also affect the competitiveness.
Risk managers may dislike using theVaRas a risk measure, because of the lack of risk-based information it provides on catastrophic losses. Indeed, two firms with marked differences in the sum of their potential losses in ad-verse scenarios may report the same risk value, even though they are not exposed to the same level of risk. As such, their disparities would go un-observed by decision makers. Moreover, the lack of subadditivity may well
constitute another drawback. Alternatives toVaR99.5%that take into ac-count catastrophic losses can be considered by risk managers. Traditional approaches frequently lead to severely higher economic reserves. Managers need to find a risk measure that generates similar economic reserves than VaR99.5%for the overall risk faced by the insurance company and, addition-ally, they would like that the alternative risk measure provides risk-based information on catastrophic losses and that, hopefully, it satisfies appealing subadditivity properties.
5.2.1 Calibration of GlueVaR parameters
Our goal is to find the set of GlueVaR risk measures that return the same risk value that theVaR99.5%in a particular context. So, we need to find the parameter values that define theGlueVaR!Ø,Æ1,!2risk measures. All the steps required in calibratingGlueVaRrisk measures are described here. The crite-rion followed in the calibration procedure is the need to obtain the same risk measure value with theGlueVaRrisk measures as the one obtained with theVaR99.5%. Moreover, the selection of the risk measure is restricted to the subfamily ofGlueVaRcandidates that may satisfy that their distortion function is concave in[0,1°Æ). The strategy for calibrating the parameters is as follows:
• Minimum and maximum admissible values of theÆandØconfidence levels have to be determined,ÆminandØmax.
• Let us assume thatZrandom variable represents the overall risk. A set of d£dconstrained optimization problems is defined at this step:
Pi,j: min
!1,!2|GlueVaR!Ø1,!2
j,Æi(Z)°VaR99.5%(Z)|, subject to
8<
:
0…!1…1, 0…!2,
!1+!2…1 wherei,j =1,...,d,Æi =Æmin+ i°1
d°1(Ømax°Æmin)andØj =Æi+ j°1
d°1(Ømax°Æi). Flexibility rises with the number of partitionsd, as do computational costs. Constraints are fixed to guarantee that the distor-tion funcdistor-tion of the GlueVaR is concave in[0,1°Æi).
• An optimization algorithm should be used to solve this set of problems.
If Pi§,j§represents the problem for which the minimum value of the ob-jective function is reached and(!§1,!§2)is the associated solution, then
aGlueVaR!ا1j§,!,Ƨ2i§ is found with its distortion function concave in[0,1° Æi§)and gives similar risk values to those obtained withVaR99.5%when applied to the overall risk of the company. Pi,j problems may not have solutions. Were this to be the case, then the optimization criteria would have to be revised, including a lowerÆmin, a higherØmaxand/or a largerd.
• More than one GlueVaR solution is frequently found. Alternative com-binations of parameter values return the same objective function value, or a value that differs insignificantly. In this situation, solutions could be ranked in accordance with the underlying risk attitude involved. Here, we propose ranking the solutions based on the value of the area under the distortion function associated with each optimal risk measure. With this goal in mind, degrees of orness are computed for (multiple) optimal GlueVaR!ا1j§,!,Ƨ2i§ solutions. Two particular GlueVaR risk measures among the set of solutions are of special interest:
Lower-limit solution. Selection of the GlueVaR risk measure with the as-sociated minimum area under the distortion function;
Upper-limit solution. Selection of the GlueVaR risk measure with the as-sociated maximum area under the distortion function.
In other words, boundaries of the area size under distortion functions are detected. Optimal GlueVaR risk measures linked to boundaries reflect the extreme risk attitudes of agents when the random variableZ is analyzed.
5.2.2 Data and Results
We are going to use the dataset used in previous chapters. It containsX1, X2andX3. Total claim costs are the sum of the three random variables, Z =X1+X2+X3. So the aggregate risk faced by the insurer is the sum of the three random variablesXi,i=1,2,3. We assume that the insurer uses theVaR99.5%as its risk measure.
Before dealing with the calibration of the GlueVaR risk measures, we first compute theVaR99.5%(Z)and its associated area under its distortion func-tion. The risk measure value is equal toVaR99.5%(Z)=51.05and the area under its distortion function is equal to0.995. Let us now focus on the strat-egy used to calibrate the GlueVaR parameters. The following steps are per-formed to obtain GlueVaR risk measures that are comparable to theVaR99.5%(Z): a) the minimum and maximum values of confidence levels are fixed at90%
and99.9%, i.e.Æmin=90%andØmax=99.9%;
b) the number of partitions is stipulated ind =25, so we deal with625 optimization problems;
c) the empirical distribution function of total claim costs is used for the risk quantification, and, finally;
d) the outcome choice of the GlueVaR solutions are obtained using con-strOptimfunction fromrootSolvelibrary in R.
A more complex calibration problem involving a modified random variable including catastrophic losses can be found inBelles-Samperaet al.[2016c].
We obtained a set of optimal GlueVaR risk measures that give the same risk value as theVaR99.5%in this specific context. Thus,192optimal solutions were found. Once a set of GlueVaR risk measures has been obtained as feasi-ble solutions, the areas under the distortion functions linked to each Glue-VaR were computed to characterize the respective underlying aggregated risk attitude. The boundary values and the associated GlueVaR risk mea-sures were identified. We should emphasize that the maximum area was equal to the area of theVaR99.5%. In fact, the optimalGlueVaR!Ø,Æ1,!2solution with the highest area size was the GlueVaR with parametersÆ=99.5%,Ø= 99.9%and!1=!2=0, and it holds thatGlueVaR0,099.9%,99.5%=VaR99.5%
(see expression (3.4)). In other words, given a certain risk value, theVaRÆ is the GlueVaR risk measure that presents the highest area under the asso-ciated distortion function of all the GlueVaR risk measures that return this value. Recall that the distortion function associated with theVaRÆassigns one to survival values higher than(1°Æ)and zero to the rest.
The minimum area under the distortion function and the associated Glue-VaR risk measure for the original dataset are reported in Table5.1. Informa-tion about the underlying aggregate risk attitude of the agent can be inferred from the minimum area. Table5.1shows that, for this dataset, there exists an optimal GlueVaR risk measure for which the area of the associated distor-tion funcdistor-tion is approximately 0.949. Thus, this GlueVaR risk measure gives the same value as that given byVaR99.5%when applied toZ, but, in aggre-gate terms, it involves a more moderate distortion of the original survival distribution function and, consequently, less aggregate risk intolerance.
The area under the distortion function should be complemented with the examination of the quotient function which allows the relative risk attitude at any point of the survival distribution to be analyzed. The quotient func-tions of risk measures associated with boundary areas in both scenarios are examined. All the quotient functions analyzed are located in the upper
risk-Original dataset
Æ 0.900
Ø 0.999
!1 0.483
!2 1.035·10°6
!3 0.516
Area under the
distortion function 0.949
Note: Parameter values of the associated GlueVaR risk measure equivalent to VaR99.5%and minimum area under the distortion function.
tolerance frontier in the range[0.10,1). For ease of comparison, the quo-tient functions are rescaled and their left-tails are shown only in the range [0,0.10]in Figure5.7.
Notable differences can be observed in the relative risk attitudes locally im-plicit in the left-tail of the quotient functions. Let us first examine the quo-tient function of the GlueVaR risk measure that presents the maximum de-gree of orness (left), which is the same quotient function associated with theVaR99.5%. The agent is most risk intolerant at any point of the interval [0.5%,1)and maximum risk tolerant at(0,0.5%). This means, the quotient function is located in the upper frontier at[0.5%,1)and in the lower frontier at(0,0.5%). When the GlueVaR risk measure that presents the minimum area is analyzed (right), the patterns of the left-tails of the quotient func-tions are undoubtedly different. An interesting finding is that theQg(u)is not located within the boundaries at any point of the interval(0,0.10). This means, the risk intolerant attitude is not maximized in the range[0.5%,0.10) but, on the contrary, the agent is more risk intolerant to catastrophic losses at(0,0.5%)than when usingVaR99.5%.
Note: Maximum (left) and minimum (right) areas under distortion functions of op-timal GlueVaR risk measures. Dashed curve indicates the upper-bound quotient function curve, i.e. 1ufor all0…u…1.