How to Estimate Expected Shortfall When Probabilities Are Known with Interval or Fuzzy Uncertainty

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1 How to Estimate Exected Shortfall When Probabilities Are Known with Interval or Fuzzy Uncertainty Christian Servin Information Technology Deartment El Paso Community College El Paso, TX 7995, USA Hung T. Nguyen Deartment of Mathematical Science New Mexico State University Las Cruces, NM 88003, USA and Faculty of Economics Chiang Mai University, Thailand Vladik Kreinovich Deartment of Comuter Science University of Texas at El Paso 500 W. University El Paso, Texas 7996, USA Abstract To gauge the risk corresonding to a ossible disaster, it is imortant to know both the robability of this disaster and the exected damage caused by such otential disaster ( exected shortfall ). Both these measures of risk are easy to estimate in the ideal case, when we know the exact robabilities of different disaster strengths. In ractice, however, we usually only have a artial information about these robabilities: we may have an interval (or, more generally, fuzzy) uncertainty about these robabilities. In this aer, we show how to efficiently estimate the exected shortfall under such interval and/or fuzzy uncertainty. I. FORMULATION OF THE PROBLEM How to gauge risk. In the ideal world, there should be no risk: all engineering designs should be 00% reliable. To achieve such reliability, civil engineers use the record of historic floods, tsunamis, hurricanes, earthquakes, and other natural disasters to estimate the largest ossible strength of such a disaster, and design the buildings, bridges, and other structures so that they can withstand such disasters. The historic exerience shows, however, that there is always a ossibility that the disaster strength S exceeds the estimated threshold s 0 : this was the reason why the hurricane Katrina devastated New Orleans, why in 20, Fukushima nuclear ower station in Jaan was destroyed by an unusually high tsunami, etc. Since we cannot have a threshold s 0 that would guarantee that the disaster strength never exceeds s 0, the next best thing is to select a threshold s 0 for which the robability of exceeding s 0 does not exceed a given small number 0, i.e., for which, with robability = 0, we have S s 0. The choice of the threshold robability 0 deends on the situation. For examle: for manned sace flights, NASA selected 0 = 0 3 ; smaller values were not feasible because of high uncertainty associated with sace flights; on the other extreme, for reliability of a cell forming a comuter memory, we need 0 0 9, because otherwise, if we allow 0 0 9, at least one of the billions of cells will always go wrong. In addition to knowing the threshold s 0, it is also desirable to also know how much damage will come, on average, if this threshold is exceeded. For each ossible value S of the corresonding disaster strength, we can estimate the corresonding damage X; the stronger the disaster, the larger the damage. Let x denote the damage corresonding to the threshold value s 0 ; then, the condition S s 0 is equivalent to X x. In these terms, the robability that the disaster strength exceeds the threshold s 0 is equal to Prob(X > x ). In addition to this robability, it is desirable to also know the conditional exectation of the damage under the condition that the disaster strength exceeds the threshold x, i.e., the value ES = E[X X x ]. The corresonding conditional exectation is known as exected shortfall. These two values: the threshold x, and the exected shortfall ES, is how we gauge the risk. Similar two measures are used in finance to describe the risk that an investment would result in a big loss; see, e.g., [4]. How to estimate the exected shortfall in the ideal case, when we know the robability distribution describing damage. In the ideal case, we know the robability distribution that describes ossible values of the damage X. A usual way to describe a robability distribution is by describing its cumulative distribution function (cdf) F (x) = Prob(X x); see, e.g., [6]. In terms of cdf, the robability of exceeding the threshold value x is simly equal to F (x ). Thus, we have F (x ) = 0 and hence, F (x ) = 0 =. For each robability, the value x for which F (x ) = is known as the -th quantile. For examle: for = 0.5, we get the median;

2 for = 0.25 and = 0.75, we get quartiles, etc. In mathematical terms, the function that mas to x is an inverse function to the cdf F (x). The conditional exectation can then be comuted as the ratio x ES = x df (x). () In ractice, we only have artial information about the robabilities. In ractice, we rarely know the exact values of all the robabilities, we only have artial information about these robabilities. This may mean that, instead of the exact values F (x) corresonding to different values x, we only know an interval [F (x), F (x)] that contains the actual (unknown) value F (x). Such situation when, for each d, we only know the corresonding intervals, is known as a robability box or, for short, a -box; see, e.g., [], [2]. Even more generally, for each x, we may have several intervals [F (x), F (x)] corresonding to different degrees of certainty α [0, ], i.e., in effect, a fuzzy number; see, e.g., [3], [5], [7]. How to gauge risk under such an uncertainty? For different distributions F (x) [F (x), F (x)] within a given -box, we get different values of quantiles x for which F (x ) =. One can easily check that: the smallest value x corresonds to the largest values F (x) of the cdf; while the largest value x corresonds to the smallest values F (x) of the cdf. Thus, ossible values of the quantile x form an interval [x, x ] in which F (x ) = F (x ) =. Such quantile intervals are often useful when we erform comutations with -boxes; see, e.g., [], [2]. We can use this idea to handle the case when we have a fuzzy-valued function F(x), if we take into account the known fact that for all ossible comutation y = f(x,..., x n ) with fuzzy numbers, the alha-cut α y = {y : µ(y) α} of the result is equal to the range f( α x,..., α x n ) = {f(x,..., x n ) : x α x,..., x n α x n (α)} of the values f(x,..., x n ) when each x i belongs to the corresonding α-cut α x i = {x i : µ i (x i ) α}; see, e.g., [3], [5]. Thus, to find the α-cut of the quantile x, it is sufficient to comute the interval [x, x ] in situation when each F (x) belongs to the corresonding α-cut of the fuzzy number F(x). Such straightforward comutation is ossible because the deendence of the robability = F (x ) on the unknown function F (x) is monotonic (namely, increasing), and that each cdf F (x) is also an increasing function. So, if we increase the values of the function F (x), then for newly increased function F (x), we will have F (x ) > F (x ) =. Thus, to find the value x for which F (x ) =, we need to decrease the value x : x < x. So, to find the range of ossible values [x, x ] of the quantile x, we do not need to enumerate all ossible functions F (x) from the -box [F (x), F (x)] because of monotonicity, we know that: the smallest value x of the quantile x is attained when the cdf F (x) attains its largest ossible value, i.e., when F (x) = F (x) for all x; and the largest value x of the quantile x is attained when the cdf F (x) attains is smallest ossible value, i.e., when F (x) = F (x) for all x. For exected shortfall, different robability distributions from the -box, in general, lead to different values. We would like to find the range of ossible values of the exected shortfall. Estimating this range, however, is not a very straightforward task: when we increase F (x), it is not a riori clear whether the corresonding integral increases or decreases. Thus, we cannot immediately come u with a simle exression for the range of the exected shortfall. It is therefore necessary to come u with efficient algorithms for estimating the range of the exected shortfall under -box (interval) uncertainty. Similarly, for the fuzzy case, we need to be able to transform the fuzzy-valued cdf F(x) into a fuzzy value for the resulting exected shortfall. What we do in this aer. In this aer, we rovide the efficient algorithms for comuting the exected shortfall under interval (-box) and fuzzy uncertainty. To be more recise, we roduce an algorithm for comuting the exected shortfall under interval (-box) uncertainty. Based on what we mentioned earlier about fuzzy comutations, to find the α-cut of the exected shortfall, it is sufficient to comute the range of ossible values of ES of the exected shortfall in situation when each F (x) belongs to the corresonding α-cut of the fuzzy number F(x). In other words, from the algorithmic viewoint, the roblem of comuting the exected shortfall under fuzzy uncertainty can be indeed reduced to the case of interval (-box) uncertainty. II. ANALYSIS OF THE PROBLEM Let us find an equivalent exression for ES. To find the range of ossible values of exected shortfall ES, let us find an equivalent exression for ES that would simlify the comutation of this range. where Thus: According to formula (), we have ES = I, (3) I = x df (x). (4) x

3 the exected shortfall ES attains its smallest ossible value ES when the integral I attains its smallest ossible value I, and the exected shortfall ES attains its largest ossible value ES when the integral I attains its largest ossible value I. The integral I has an infinite uer bound. This integral can be thus reresented as a limit of integrals I T with a finite uer bound T when T : where Thus, for very large T, we have I I T. I = lim T I T, (5) I T = x df (x). (6) x The integral I T can be integrated by art: I T = x F (x) T x F (x) dx = x T F (T ) x F (x ) F (x) dx. (7) x For large T, we have F (T ) ractically equal to so T F (T ) = T and lim F (T ) =, T I T = T x F (x ) x F (x) dx. (8) By inition of a quantile x, we have F (x ) =, so I T = T x x F (x) dx. (9) When does the integral I T attain its largest and smallest ossible values? According to our analysis: the exected shortfall ES attains its largest ossible value when the value of the integral I T (corresonding to a very large value T ) is the largest ossible, and the exected shortfall ES attains its smallest ossible value when the value of the integral I T (corresonding to a very large value T ) is the smallest ossible. Let us thus analyze when the integral I T attains its largest and smallest ossible values. When does the integral I T attain its largest ossible value? Let us start with the largest ossible value. Different cdfs F (x) from the given -box result, in general, in different values of the integral I T. Let x max be the value corresonding to the cdf F max (x) for which this integral is the largest ossible. This means, in articular, that among all cdfs F (x) with the same value of the -th quantile x max (i.e., for which F (x max ) = ), this articular cdf F max (x) leads to the largest ossible value of the integral I T. One can easily see, from the exression (9), that the integral I T is a decreasing function of the values F (x). Thus, this integral is the largest when all the values F (x) are the smallest. What limitations on the values F (x) do we have? We have a limitation F (x) F (x) F (x) coming from the fact that we only consider cdfs from a given -box [F (x), F (x)]. We also have a limitation F (x) F (x ) =, which, for values x x, comes from the requirement F (x ) = and from the fact that each cdf is an increasing function of x. These constraints F (x) F (x) F (x) and F (x) can be equivalently described by a single constraint max(f (x), ) F (x) F (x). (0) Thus, the smallest ossible values of F (x) corresond to When F (x), we have F (x) = max(f (x), ). () max(f (x), ) = F (x) and hence F (x) = F (x). As we described earlier, the equality F (x) = is equivalent to x = x, thus the condition F (x) is equivalent to x x. On the other hand, when F (x) <, i.e., equivalently, when x < x, we have F (x) =. Thus, x max F (x) dx = x x max dx + x F (x) dx = (x x max ) + F (x) dx. (2) x Thus, the exression (9) takes the form I T = T x max (x x max ) x F (x) dx. (3) The two terms x max and (x x max ) can be easily combined into a single term x, so I T = T x x F (x) dx. (4) Since x is the quantile corresonding to the lower endoint F (x) of the -box, we can therefore conclude that the exression (4) is the value of the integral I T corresonding to F (x) = F (x). Thus, the largest value of the integral I T and hence, of the exected shortfall is attained when F (x) = F (x). When does the integral I T attain its smallest ossible value? Let us now consider the smallest ossible value. Let x min be the value corresonding to the cdf F min (x) for which this integral is the largest ossible. This means, in articular, that among all cdfs F (x) with the same value of the -th quantile x min (i.e., for which F (x min ) = ), this articular cdf F min (x) leads to the smallest ossible value of the integral I T.

4 Since the integral I T is a decreasing function of the values F (x), this integral is the smallest when all the values F (x) are the largest. Under the limitations (0), the largest ossible values are F (x) = F (x). Thus, the smallest value of the integral I T and hence, of the exected shortfall ES is attained when F (x) = F (x). III. RESULTING ALGORITHM: CASE OF INTERVAL UNCERTAINTY Problem: reminder. We want to find the range [ES, ES ] of ossible values of the exected shortfall ES when cdf F (x) is in the given -box [F (x), F (x)]. Conclusion. The above analysis leads to the following conclusion: The largest ossible value ES of the exected shortfall ES is attained when F (x) = F (x) for all x. The smallest ossible value ES of the exected shortfall ES is attained when F (x) = F (x) for all x. Thus, we arrive at the following algorithm: Resulting algorithm. First, we comute the exected shortfall ES corresonding to F (x) = F (x), as the ratio x x df (x), where x = (F ) (). This ratio is the uer endoint ES of the desired interval [ES, ES ]. Then, we comute the exected shortfall ES corresonding to F (x) = F (x), as the ratio IV. x x df (x), where x = (F ) (). This ratio is the lower endoint ES of the desired interval [ES, ES ]. RESULTING ALGORITHM: CASE OF FUZZY UNCERTAINTY Problem: reminder. We have a fuzzy-valued cdf F(x). In other words, for every real number x, we know a fuzzy number F(x) that describes the robability Prob(X x). We are also given a robability. We want to comute the corresonding fuzzy shortfall ES. Idea. To comute the desired fuzzy number ES, we comute, for each value α [0, ], the corresonding α-cut α ES. Of course, in ractice, we can only erform comutations with finitely many values α, so usually, we choose the threshold values α for degrees of certainty in accordance with the accuracy with which we can realistically determine these degrees e.g., for values α = 0, α = 0., α = 0.2,..., α = 0.9, and α =.0. As we have mentioned earlier, the interval forming this α-cut can be comuted by solving the interval-uncertainty roblem in which we use the α-cuts of the inuts. In our case, the inuts are the values of the fuzzy cdf F(x). Thus, to comute the desired α-cut α ES, we can solve the corresonding interval-uncertainty roblem in which the inuts are the α-cuts α F(x) of the fuzzy cdf. We already know how to solve the corresonding intervaluncertainty roblem, so we arrive at the following fuzzy-case algorithm. Resulting algorithm. We start with the α-cuts α F(x) = [ α F (x), α F (x)] corresonding to the values of the fuzzy cdf F(x). Our objective is to comute the α-cuts α ES of the exected shortfall ES. For that urose, for each α, we erform the following comutations: First, we comute the exected shortfall ES corresonding to the cdf α F (x), as the ratio α x x d α F (x), where α x = ( α F ) (). This ratio is the lower endoint α ES of the desired α-cut α ES = [ α ES, α ES ]. Then, we comute the exected shortfall ES corresonding to the cdf α F (x), as the ratio α x x d α F (x), where α x = ( α F ) (). This ratio is the uer endoint α ES of the desired α-cut α ES = [ α ES, α ES ]. ACKNOWLEDGMENT This work was suorted in art by the Faculty of Economics of Chiang Mai University and by the National Science Foundation grants HRD and HRD (Cyber- ShARE Center of Excellence) and DUE The authors are thankful to Scott Ferson and Paul Embrechts for valuable discussions, and to the anonymous referees for imortant suggestions. REFERENCES [] S. Ferson, Risk Assessment with Uncertainty Numbers: RiskCalc, CRC Press, Boca Raton, Florida, [2] S. Ferson, V. Kreinovich, W. Oberkamf, and L. Ginzburg, Exerimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty, Sandia National Laboratories, Reort SAND , [3] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Alications, Prentice Hall, Uer Saddle River, New Jersey, 995.

5 [4] A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concets, Techniques, Tools, Princeton University Press, Princeton, New Jersey, 200. [5] H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, Chaman and Hall/CRC, Boca Raton, Florida, [6] D. J. Sheskin, Handbook of Parametric and Nonarametric Statistical Procedures, Chaman & Hall/CRC, Boca Raton, Florida, 20. [7] L. A. Zadeh, Fuzzy sets, Information and Control, 965, Vol. 8,