### Abstract

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

We propose a new family of risk measures, called GlueVaR, within the class of distortion risk measures. Analytical closed-form expressions are shown for the most frequently used distribution functions in financial and insurance applications. The relationship between GlueVaR, value-at-risk, and tail value-at-risk is explained. Tail subadditivity is investigated and it is shown that some GlueVaR risk measures satisfy this property. An interpretation in terms of risk attitudes is provided and a discussion is given on the applicability in nonfinancial problems such as health, safety, environmental, or catastrophic risk management.

### 1. INTRODUCTION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

Financial and insurance risk management practitioners typically have to deal with two opposing demands: on the one hand, they want business units to achieve or outperform the objectives fixed by the firm's executive committee, yet, on the other, they are responsible for controlling their economic risks. Finding a tradeoff between these two demands is the challenging task that risk managers face on a daily basis. At the same time, they need to decide how risk should be quantified.

Financial and insurance firms are subject to the capital requirements established by regulators' guidelines and directives. These requirements are typically equal to, or proportional to, a risk measure value that determines a minimum cushion of economic liquidity. The selection of such risk measures and tolerance levels is crucial, therefore from the regulators' point of view.

Our aim is to propose a new family of risk measures, which we name GlueVar, and that have analytical closed-form expressions for many statistical distributions that are frequently used in financial and insurance applications. This new family combines the most popular risk measures and considers more than just one parameter to capture managerial and regulatory attitudes toward risk.

Financial institutions and insurance companies prefer to minimize the level of capital reserves required by solvency regulations because they must contend with many restrictions on how this capital can be invested and, as such, the return on their capital reserves is usually lower than that provided by other opportunities. For this reason, companies typically favor regulations that impose risk measures and tolerance levels that are not overly conservative. Managers also prefer simple, straightforward risk measures rather than more complicated alternatives, since they claim that the former are more easily communicated.

From the regulators' perspective, controlling the risk of financial institutions and insurance companies is fundamental in order to protect consumers and investors, which may have conflicting objectives. Strict solvency capital requirements may limit the capacity of firms, but they also reassure consumers and guarantee the position of the financial industry in the economy. Thus, the debate as to what constitutes a suitable risk measure and what represents a suitable tolerance level is interminable, without their apparently having been much investigation as to what might represent an appropriate compromise.

We contend that the GlueVaR family could be useful in helping regulators and practitioners reach a consensus. As we discuss below, the GlueVaR family should enhance the way in which regulatory capital requirements are calculated, as GlueVaR can incorporate more information about agents' attitudes to risk. It is our belief that the incorporation of qualitative information in decision-making tools is essential for risk managers and, as such, the GlueVaR risk measures can play a key role in achieving this goal.

### 2. BACKGROUND AND MOTIVATION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

Value-at-risk (VaR) has been adopted as a standard tool to assess the risk and to calculate capital requirements in the financial industry. Value-at-risk at level α is the α-quantile of a random variable *X* (which we often call loss), that is, VaR, where is the cumulative distribution function (cdf) of *X* and α is the confidence or the tolerance level . However, VaR is known to present a number of pitfalls when applied in practice. A disadvantage when using VaR in the financial context is that the capital requirements for catastrophic losses based on the measure can be underestimated, that is, the necessary reserves in adverse scenarios may well be less than they should be. The underestimation of capital requirements may be aggravated when fat-tailed losses are incorrectly modeled by mild-tailed distributions such as the normal distribution. There are attempts to overcome this kind of model risk when using VaR or, at least, to quantify the risk related to the modeling. [1] A second drawback is that the VaR may fail the subadditivity property. A risk measure is subadditive when the aggregated risk is less than or equal to the sum of individual risks. Subadditvity is an appealing property when aggregating risks in order to preserve the benefits of diversification. VaR is subadditive for elliptically distributed losses.[2] However, the subadditivity of VaR is not granted.[3, 4]

Tail value-at-risk (TVaR) may be interpreted as the mathematical expectation beyond VaR, and is defined as TVaR. The TVaR risk measure does not suffer the two drawbacks discussed above for VaR and, as such, would appear to be a more powerful measure for assessing the actual risks faced by companies and financial institutions. However, TVaR has not been widely accepted by practitioners in the financial and insurance industry. VaR is currently the risk measure contemplated in the European solvency regulation for the insurance sector (Solvency II), and this is also the case of solvency regulation for the banking sector (Basel accords). The TVaR measures average losses in the most adverse cases rather than just the minimum loss, as the VaR does. Therefore, capital reserves based on the TVaR have to be considerably higher than those based on VaR and significant differences in the size of capital reserves can be obtained depending on which risk measure is adopted.

This article is motivated, therefore, by an attempt to respond to the following question. Can a risk measure be devised that would provide a risk assessment that lies somewhere between that offered by the VaR and the TVaR? To this end, we propose a new family of risk measures (GlueVaR), which forms part of a wider class referred to as distortion risk measures. We analyze the subadditivity properties of these GlueVaR risk measures and show that a subfamily of GlueVaR risk measures satisfies tail-subadditivity.

GlueVaR risk measures are defined by means of a four-parameter function. By calibrating the parameters, GlueVaR risk measures can be matched to a wide variety of contexts. Specifically, once a confidence level has been fixed, the new family contains risk measures that lie between those of VaR and TVaR and that may adequately reflect the risk of mild-tailed distributed losses without having to resort to VaR. In certain situations, however, more conservative risk measures even than TVaR may be preferred. We show that these highly conservative risk measures can also be defined by means of the GlueVaR family. We derive analytical closed-form expressions of GlueVaR for commonly used statistical distributions in the financial context. These closed-form expressions should enable practitioners to undertake an effortless transition from the use of VaR and TVaR to GlueVaR.

### 3. DISTORTION RISK MEASURES

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

Consider a probability space and the set of all random variables defined on this space. Any risk measure[5] ρ is a mapping from the set of random variables to the real line , . Distortion risk measures were introduced by Wang [6, 7] and are closely related to the distortion expectation theory. A review on how risk measures can be interpreted from several perspectives is provided in Ref. [8], and a clarifying explanation of the relationship between distortion risk measures and distortion expectation theory is included. A detailed literature review of distortion risk measures is available in Refs. [9] and [10]. There are two key elements to define a distortion risk measure: first, the associated distortion function; and, second, the concept of the Choquet[11] Integral. The distortion function, Choquet Integral, and the distortion risk measure can be defined as follows:

**Distortion function**. Let be a function such that , , and *g* is nondecreasing. Then *g* is called a distortion function.**Choquet Integral**. The (asymmetric) Choquet Integral with respect to a set function μ of a μ-measurable function is denoted as and is equal to , if , where denotes the *survival function* of *X* with respect to μ. Note that Ω denotes a set, which in financial and insurance applications is the sample space of a probability space. A set function μ in this context is a function defined from 2^{Ω} (the set of all subsets of Ω) to . A μ-measurable function *X* is, widely speaking, a function defined on Ω so that expressions like or make sense. See Ref. [12] for more details.**Distortion risk measure**. Let *g* be a distortion function. Consider a random variable *X* and its survival function . Function defined by is called a distortion risk measure.

From the previous definitions, it is straightforward to see that for any random variable *X*, is the Choquet Integral of *X* with respect to the set function , where *P* is the probability function associated with the probability space in which *X* is defined.

The mathematical expectation is a distortion risk measure whose distortion function is the identity function, [9] . Therefore, a straightforward way to interpret a distortion risk measure is as follows: first, the survival function of the random variable is distorted ; second, the mathematical expectation of the distorted random variable is computed. From a theoretical point of view, note that this interpretation fits the discussion that risk may be defined as an expected value in many situations. [13]

VaR and TVaR measures are in fact distortion risk measures. The associated distortion functions of these risk measures are shown in Table 1.

Table 1. VaR and TVaR Distortion FunctionsRisk Measure | Distortion Function |
---|

VaR | |

TVaR | |

For a confidence level . |

Based on the distortion functions shown in Table 1, once α is fixed it can be proved that VaR TVaR for any random variable *X*.

### 4. A NEW FAMILY OF RISK MEASURES: GlueVaR

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

A wide range of risk measures may be defined under this framework. Note that VaR_{α} and TVaR_{α} are particular cases of this new family of risk measures. Namely, for a random variable *X*, VaR and TVaR correspond to distortion functions and , respectively. By establishing suitable conditions on the heights *h*_{1} and *h*_{2}, the GlueVaR family is very flexible. For example, risk managers might like to select α, β, *h*_{1}, and *h*_{2} so that VaR GlueVaR TVaR: this can be achieved by selecting a set of parameters for their associated distortion functions to ensure that for any following Remark 3.1, that is, by forcing condition . An example of such a case is shown in Fig. 1 (left-hand side).

### 6. SUBADDITIVITY IN THE TAIL

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

This section is devoted to an analysis of the properties of the GlueVaR family of risk measures, with special attention to subadditiviy. Our reason for defining these GlueVaR risk measures is a response to the concerns expressed by risk managers regarding the choice of risk measures in the case of regulatory capital requirements. However, an axiomatic approach to define or represent risk measures is more frequent in the literature.[3, 19-26]

In a seminal article,[3] the following set of axioms that a risk measure should satisfy was established: positive homogeneity, translation invariance, monotonicity, and subadditivity. Authors referred to such risk measures as “coherent risk measures.” Distortion risk measures always satisfy the first three properties, but subadditivity is only guaranteed when the distortion function is concave.[12, 27, 28] Therefore, VaR, unlike TVaR, is not coherent. In some situations, coherence of risk measures is a requirement [29] but, nonetheless, some criticisms can be found. [30] Additional properties for distortion risk measures are provided in Refs. [31] and [11].

Subaddtitivity in the whole domain is a strong condition. When dealing with fat tail losses (i.e., low-frequency and large-loss events), risk managers are especially interested in the tail region. Fat right tails have been extensively studied in insurance and finance. [32-37] To the best of our knowledge, however, previous studies of the subadditivity of risk measures in the tail region are scarce.[38, 39] The milder condition of subadditivity in the tail region is investigated here.

When there is no ambiguity as to which confidence level α and random variables *X*, *Y* are taken into account, *tail-subadditivity* is used to refer to this property. If notation is introduced, the integral condition used in the definition can be rewritten, in terms of survival functions, as:

The proof is contained in Appendix C.

Tail-subadditivity is a desirable property because it implies that the benefits of diversification may not be valid in every situation but, at least, they hold in extreme cases.

Note that, in terms of parameters *h*_{1} and *h*_{2}, a GlueVaR risk measure may be tail-subadditive if, and only if, , as a corollary of Theorem 6.1.

### 7. RISK ATTITUDES IN GlueVaR

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

An interesting interpretation in the context of decision making and risk management is that GlueVaR risk measures arise as a linear combination of three possible scenarios. So, two levels of severity can be fixed, namely, α and β, with . Then, the risk can be measured in the highly conservative scenario with TVaR at level β; in the conservative scenario with TVaR at level α; and in the less conservative scenario with VaR at level α.

Each combination of these risk scenarios reflects a specific risk attitude. Therefore, we can say that the combination of these risk scenarios in this context is something that is directly identified by an explicit GlueVaR risk measure. To some extent, these risk attitudes could be related to risk appetite. [40]

From the practitioner's point of view, four parameters must be fixed to define the GlueVar risk measure. The α and β values correspond to the confidence levels used for bad and very bad scenarios, respectively. We could select, for instance, and , which is equivalent to one bad event every 20 years or one bad event every 200 years, respectively. The other two parameters are directly related to the weights given to these scenarios. For instance, we could say that the three components of GlueVaR in Equation (4) are equally important. This would imply , so we could find the corresponding *h*_{1} and *h*_{2} parameters. When and , , these parameters are and .

#### 7.1. Geometrical Discussion on Risk Attitudes

If , risk managers are optimistic regarding the impossibility of the occurrence of the worst-case scenario, and so attach a negative weight to it.

### 8. OTHER NONFINANCIAL APPLICATIONS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

New risk measures based on distortion functions can be valuable outside the scope of finance and insurance. There is a natural bridge from financial applications to any discipline where the choice of a risk measure plays a role for decision making. GlueVaR risk measures can be applied to health, safety, environmental, adversarial risks, or catastrophic risks including terrorism. Health or safety regulations report quantile risks and could be enhanced with GlueVaR, as the latter allows to combine risk measures and risk levels.

A crucial feature is that in GlueVaR risk measures, the decisionmaker sets up two tolerance levels: one for the “bad cases” and another one for the “very bad cases.” Then, weights are set up according to a decisionmaker's or a regulator's risk attitude. If a decisionmaker is very conservative, then he should give all weight to the “very bad case” outcome, whereas a less cautious decisionmaker would assign all weight to a lesser “very bad case” outcome.4 An intermediate decisionmaker could set up a reasonable position, where he could balance a position between VaR versus TVaR, and a tradeoff between a lower tolerance level versus a higher tolerance level.

One good example of difficulties in agreeing upon a suitable risk measure is found in a recent article [43] on the optimal security investments and extreme risks with an application to terrorism risks. These authors focus on both amount and timing of security investment, but they fundamentally model risk based on the principle of expected net benefit of investment in security. An analogous principle is also used [44] to investigate security in information systems and was earlier proposed [45] for the same purpose. The expected net benefit of investment in security is a simple analytical model that maximizes the gain when investing in security procedures. Optimization is straightforward once a level of loss severity and a probability of occurrence are assumed (see section 'BACKGROUND AND MOTIVATION' in Ref. [43]). When addressing terrorism risk in form of intentional attacks on the food sector using chemical, biological, and radionuclear tools, several scenarios for casualties resulting from a terrorist attack are established in Ref. [43], which have a very low probability. The choice of those scenarios is done subjectively, even if there is a technical part that relies on extreme value theory to approximate the tail probabilities. At least two scenarios are necessary to carry out the optimization procedure and their corresponding anticumulative distributions. A GlueVaR risk measure could be used directly in the optimization procedure. Moreover, GlueVaRs could help to compare different contexts, such as terrorist attacks on the food sector versus other sectors, where the number of casualties could be much lower. GlueVaRs could provide a single value for every phenomenon and there would be no need to define interval scenarios, whose choice could potentially be controversial.

Extensions can be found in many other applications. Let us imagine an employer who has to manage worker compensation reserves. In order to assess the risk of being short in reserves in a one-year horizon, the classical procedure is to estimate VaR, which would equal the minimum reserve amount that would be likely to cover the compensations to be paid. If the manager has a tolerance of one in 100 years, that would correspond to a confidence level equal to 99% and VaR at that level should provide the estimate of sufficient reserve. In the GlueVaR framework, the manager could be slightly more cautious. He would recalculate the risk with a higher confidence and raise reserves accordingly, in case he might face a bad scenario. He could fix a confidence level of 99.5% or tolerate shortfall of reserves once in 200 years. The fund manager could select transitional position and assign weights so that a GlueVaR reflects his risk attitude. He could give equal weight to the two scenarios and be equally positioned with regard to the classical risk measure. Not only risk managers, but regulators could leave freedom to fund managers to choose their preferred GlueVaR risk measure to set up their reserves and thus, regulators could inform about the risk measure that is being used by the worker compensation fund managers.

Generalizations and extensions to disaster management could be implemented. A public safety agency could use GlueVaR risk measures to help better plan how many resources to set aside to meet the needs of next year's hurricane season. Let us assume that resources are established proportional to a VaR at certain level α. Similarly, as in the example concerning worker compensation, agencies could allow for a risk attitude, so that there would be two tolerance levels, α and β, and then weights would determine whether the agency would rather have a position in between those two levels, as it is natural to do with GlueVaRs.

Managers of scarce resources with uncertain supply and demand (e.g., strategic petroleum reserve, antibiotic stockpiles, blood bank, etc.) could use the GlueVaR measures instead of classical measures to improve decisions. Quantile-based risk measures are applied to solve a variety of optimization issues in presence of uncertainty in supply and/or demand, such as strategic planning for hospital care services, [46] operational planning of chemical and petrochemical plants,[47, 48] level of capacity in auto industry facilities, [49] water resources management[50, 51] or hydrocarbon supply-chain designs.[52, 53] Many decisions are taken based on an optimization procedure where an expected loss is minimized subject to the risk being lower than an upper bound. Risk could be defined by a GlueVaR rather than by a traditional risk measure. In that respect, GlueVaR would allow for defining the bad and very bad scenarios and would weight them according to risk aversion.

A related area where the application of GlueVaR measures might be useful is the reservoir management decision making. A set of percentiles are commonly computed to subdivide hydrocarbon reserve estimates into categories that describe the probability of extracting a certain volume, such as proven (P1), probable (P2), and possible (P3) reserves. [54-56] Management decisions are taken under these alternative scenarios. For instance, they might represent the bad, average, and good cases to evaluate the impact of additional drillings. A unified evaluation may be performed by means of the GlueVaR measures. These three scenarios may be jointly considered by the selection of tolerance levels and weights for the VaR and TVaR measures in accordance with the management decisionmaker's risk profile.

The choice of a risk measure for regulatory purposes in a matter of strong debate in the financial and the insurance sector, as solvency requirements limit the potential benefits of a firm; however, the discussion is not unique to that sector.

### 9. CONCLUSIONS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. BACKGROUND AND MOTIVATION
- 3. DISTORTION RISK MEASURES
- 4. A NEW FAMILY OF RISK MEASURES: GlueVaR
- 5. LINEAR COMBINATION OF RISK MEASURES
- 6. SUBADDITIVITY IN THE TAIL
- 7. RISK ATTITUDES IN GlueVaR
- 8. OTHER NONFINANCIAL APPLICATIONS
- 9. CONCLUSIONS
- Acknowledgments
- Appendix A
- Appendix B
- Appendix C
- References

The VaR measure is broadly used to calculate financial requirements in finance and insurance. However, the VaR measure shows drawbacks when the risk is assessed in extremely adverse scenarios and whether risks are aggregated. The TVaR measure overcomes previous limitations but capital requirements based on the TVaR measure are often observed as excessively conservative by practitioners. A new family of flexible risk measures, called GlueVar, is proposed in this article as a bridge between both approaches.

We have shown that GlueVaR measures can be expressed as linear combinations of standard risk measures and that, similarly, they can be defined based on a straightforward distortion function. Attractive properties of a GlueVaR risk measure are, therefore, readily derived from the definition of its associated distortion function. This is the case of the tail-subadditivity property defined in this article. Basically, concavity of the distortion function on the subrange assures tail-subadditivity. This milder condition in the distortion function than concavity over the whole range might be a sufficient requisite for risk measures when fat right-tail risks are assessed: the benefits of diversification are attained in adverse scenarios but capital requirements are not excessively high. An advantage of this new approach compared to previous types of VaR measures including different extensions available in the literature is flexibility and simplicity, as it is based on standard measures. We note, however, that we have limited the construction of the GlueVaR measures to the linear combination of two standard measures and two tolerance levels, but this could be extended to accommodate a larger number of elements.

The results provided in this article are directly applicable in the financial industry. Closed-form expressions of GlueVaR risk measures are shown for commonly used distributions in finance and insurance. We encourage regulators and financial and insurance risk managers to seek an equilibrium between their different demands. The two levels of qualitative information that GlueVaR risk measures incorporate (one related to the confidence levels of bad- and worst-case scenarios; the other related to the plausibility of those scenarios) can help achieve this goal. We believe that GlueVaR risk measures can play a leading role in helping to reach a satisfactory consensus.

There is potential for extending the application of GlueVaR measures to nonfinancial disciplines where the choice of a risk measure matters.