Local risk attitude

It is reasonable to suppose that decision makers do not worry about all ran-dom event losses in the same way. Decision makers frequently treat differ-ent random evdiffer-ents distinctly (note that some of these evdiffer-ents can represdiffer-ent benefits or affordable losses). While the area under the distortion function evaluates the accumulated distortion performed over the survival distribu-tion funcdistribu-tion, it does not take into account which part of the survival distri-bution function was distorted. Clearly, from the perspective of a manager, distorting the survival probability in the right tail of the random variable linked to losses is not the same as distorting the probability in the left tail.

Additionally, all distortion functions with an area equal to one half would be associated with global risk neutrality, where thei d function is only a par-ticular case.

In Figure5.4it is shown an example in which the size of the area under several distortion functions is the same. Obviously, these distortion func-tions have not associated the same risk attitude. In the case of the distor-tion funcdistor-tion represented by a dotted line, survival probability values in the interval[0,0.5]are overweighted and survival probability values in the inter-val[0.5,1]are underweighted. So, relatively high losses are overrepresented (right tail) and relatively low losses are underrepresented. On the contrary, the distortion function represented by the solid line overweights relatively low losses and underweights high losses. Note that also the area under the diagonal, which is in fact the distortion function of the mathematical expec-tation, is the same.

Therefore, the global vision of risk embedded in a risk measure has to be completed with local information. One option open to us is to define the risk attitude in absolute terms. Anabsolute risk neutral agentis a decision maker that does not distort the survival probability and who, therefore, uses thei d function as the associated distortion function, i.e. g(u)=id(u)=ufor all 0…u…1. Anabsolute risk intolerant agentis associated with a distortion functiongsuch thatg(u)>u, for all0…u…1. And, similarly, anabsolute risk tolerant agent has a distortion functiong such that g(u)<u, for all 0…u…1. This definition of risk attitude is in absolute terms in the sense that the relationship of ordering betweeng(u)andumust be fulfilled in the whole range[0,1]. Note that these considerations lead to a more restrictive

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definition of risk attitude than that provided by theaggregate risk attitude.

The definition of the absolute risk attitude implies that the implicit attitude of an agent is invariant over the range of values. Yet, there are no reasons as to why the agent should have a unique risk attitude across the whole do-main. An agent’s attitude to risk is likely to differ in accordance with the interval of loss under consideration. The risk attitude implicit in frequently used risk measures is not invariant; this is the case, for instance, of theVaRÆ. When using theVaR_Æ, a risk intolerant attitude is associated with the in-terval[1°Æ,1), but a risk tolerant attitude is associated with the interval (0,1°Æ). Thus, an homogeneous risk attitude cannot be linked to theVaRÆ

risk measure throughout the domain.

Let us define a quotient functionQ_g from(0,1]to^R, based on the distor-tion funcdistor-tiong associated with the risk measure, in order to characterize the local vision of risk. Let the functionQ_gbe defined as the quotient be-tween the distortion functiongand the identity function,Q_g(u)=^g(u)_u for all0<u…1. TheQ_g allows the analysis of the agent’s perception of risk at any point in the survival probability distribution. This quotient function provides a function of survival probabilities,u, which describes the distor-tion factor applied bygat eachulevel. The quotientQ_g is a quantifier of

thelocal risk toleranceof the agent at any point. The quotient value repre-sents the relative risk attitude of the decision maker compared to that of an agent with a risk neutral attitude who is confident of the survival probability.

An agent is risk neutral, risk tolerant or risk intolerant at pointuifQ_g(u)is equal to, lower or higher than one, respectively.

A graphical analysis of the quotient function is proposed to investigate the risk attitude of the agent at any point in the survival distribution function when using a certain risk measure. An interesting characteristic is that the quotient function is bounded. Since the quotient function computes the ratio between the distorted survival probability and the survival probability, so ¹_u is the maximum value attainable by this quotient function. In fact, the maximum risk intolerance frontier at the survival value equal tou is achieved whenQ_g(u)=_u¹(upper bound). Note thatQ_gtakes non-negative values. The maximum local risk tolerance frontier is achieved whenQ_g(u) is equal to zero (lower bound). In addition, when the agent does not distort the survival probabilities,Q_g(u)takes value equal to1(local risk neutrality line).

In Figure5.5bounds of theQg are plotted. Upper and lower bounds are represented in the Figure5.5by a solid line. The local risk neutrality line is plotted by a dotted line. An agent’s risk intolerance (tolerance) attitude emerges at pointuwhen the quotient function is bigger (smaller) than one.

As the quotient function is bounded, we can deduce at any distorted survival value how far the value is from the maximum risk intolerance/tolerance.

The evaluation of the local risk appetite pattern of a manager using theVaR_Æ andTVaRÆis investigated. In Figure5.6the quotient functions associated with theVaRÆandTVaRÆare displayed,Q√_ÆandQ∞_Ærespectively.

If we focus our attention on the quotient function associated with theVaR, Q√Æ, it can be seen that a radical risk attitude is implicit in the interval [1°Æ,1), shifting to the opposite extreme in the interval(0,1°Æ). Indeed, a maximum risk intolerance is involved in[1°Æ,1)and a maximum risk toler-ance attitude is involved in(0,1°Æ). Some similarities are found when the quotient function associated with theTVaRis examined,Q∞_Æ. Two ranges involving a different risk attitude are also distinguished. Maximum risk in-tolerance is involved in the interval[1°Æ,1)and a constant (non-boundary) risk intolerance attitude is involved in(0,1°Æ). In that interval, the quo-tient function value is farther to the maximum as closer is to zero the sur-vival probability. Unlike theVaRÆ, an absolute risk intolerance attitude is associated with theTVaRÆbecause the quotient function is larger than one throughout the range(0,1).

1

0 1

Æ Æ

± 1 • 1°Æ

0 1

1°Æ 1

1 1°Æ

1

0 1°Æ 1

Note: Distortion functions only differ in terms of the interval[0,1°Æ). The quo-tient function of the mathematical expectation,Q_id, is represented by a horizontal dotted line.

The quotient function of theGlueVaR^!_Ø,Æ^1,!2 is not plotted because the par-ticular shape for theQ_∑^!1,!2

Ø,Æ depends on the values of the four parameters that define the risk measure. However, as in the former two risk measures,

a maximum risk intolerance is involved in the interval[1°Æ,1)when the quotient function of theGlueVaR^!_Ø,Æ^1,!2 is analyzed. However, more than just one attitude can be involved in the range(0,1°Æ). The high flexibility ofGlueVaR^!_Ø,Æ^1,!2allows multiple attitudes towards risk to be implicit in the range(0,1°Æ), depending on the values of the remaining three parameters, Ø,!₁and!₂.

In short, the quotient functionQ_g can be used to characterize the relative risk behavior of an agent at any point. The value of a quotient function at a particular point depends on the distortion function as well as on the origi-nal survival function. In other words, risk attitude in the quotient function is contemplated in the size of the distortion performed (the numerator) but also in the position in which this distortion is performed (the denominator).

Note that the area under the quotient function provides similar information to that of the area underg, but expressed in terms of risk neutrality. Indeed, the area under the quotient function can be interpreted as the area under a weighted distortion function, where weights(1/u)are given to distorted val-ues, i.e.g(u)·¹_u. Following this interpretation, greater weights are assigned to distortion function values associated with lower survival values. The areas under the quotient function ofVaR_ÆandTVaR_ÆareA°

Q_√Æ¢

=°ln(1°Æ) andA°

Q∞Æ

¢=1°ln(1°Æ), respectively. Similarly, the area under the quo-tient function of theGlueVaR^!_Ø,Æ^1,!2is equal to

A≥ Q_∑^!1,!2

Ø,Æ

¥=!₁

∑ 1+ln

µ1°Æ 1°Ø

∂∏

+!₂°ln(1°Æ).

Evaluating the area under the quotient function may be useful when analyz-ing the aggregate risk behavior in situations in which values of the distortion function are weighted, indicating that risk intolerance is negatively associ-ated with the size of the survival values. Thus, the area under the quotient function can be interpreted as a weighted quantifier of the aggregate risk attitude, where an area equal to one indicates aggregate risk neutrality, an area larger than one indicates aggregate risk intolerance and an area lower than one indicates aggregate risk tolerance.

Example 5.1 (Obtaining risk attitudes of aRVaRrisk measure). Taking into account the arguments presented in this chapter, the following corollaries of equivalence (4.13) proven in Section4.3of Chapter4may be stated:

• The aggregate risk attitude ofRVaR_Æ,Øcan be obtained as the area under the distortion function plotted in Figure4.1:

2°(a+b)°(1°a)

• The local risk attitude ofRVaR_Æ,Øis given by the following quotient func-tion:

A numerical illustration of the results listed before is provided. Let us eval-uate aRVaR_0.1%,5%risk measure. It combinesVaRat the99.9%andTVaR at the95%. Parameters areÆ=0.1%andØ=5%. From expression (5.1) the aggregate risk behavior ofRVaR0.1%,5%is^2°0.2%°5%_{2 =}^194.8%_{2 =}97.4%. From expression (5.2), the specific risk attitude ofRVaR_0.1%,5%is described by the following quotient function:

Q_RVaR(u)=

The equivalentGlueVaRrisk measure toRVaR_0.1%,5%is the one with pa-rameters a=99.9%, a+b°1=99.9%+95%°1=94.9%,h₁=0and h2=1. In other words,

RVaR_0.1%,5%=GlueVaR^0,199.9%,94.9%.

Given that a general expression for the overall or aggregate risk attitude be-hind aGlueVaR^bh_b^1,bh2

Ø,Æb risk measure is the following b

let us check if, using this formula forGlueVaR^0,199.9%,94.9%the97.4%is

5.2 Application of risk assessment in a scenario