Value at Risk and Tail Value at Risk in Uncertain Environment

Value at Risk and Tail Value at Risk in Uncertain Environment Jin Peng Institute of Uncertain Systems, Huanggang Normal University, Hubei 438000, China pengjin01@tsinghua.org.cn Abstract: Real-life decisions are usually made in the state of uncertainty or risk. In this article we present the risk measuring techniques value at risk (VaR) and tail value at risk (TVaR) under uncertainty. Firstly, we introduce the VaR concept of uncertain variable based on uncertainty theory and examine its fundamental properties. Then, the TVaR concept is evolved and some fundamental properties of the proposed TVaR are investigated. Finally, uncertain simulation algorithms are designed to calculate the VaR and TVaR. The suggested VaR or TVaR methodology can be widely used as tools of risk analysis in an uncertain environment. Keywords: risk analysis, uncertainty theory, uncertain variable, value at risk, tail value at risk, uncertain simulation 1 Introduction In real life we constantly have to make decisions under uncertainty or risk. However, there is no common agreement (not to mention a lack of an underlying rigorous mathematical theory) on the definition of risk. In general, we regard risk as uncertain profit or loss of a portfolio. It can be positive (gains) as well as negative (losses). A risk measure is a mapping from the uncertain variables representing risks to the real line. It gives a simple number that quantifies the risk exposure in a way that is meaningful for the problem at hand. There has been a great momentum in research on risk measures. Variance and standard deviation have been traditional risk measures in economics and finance since the pioneering work of Markowitz [18]. Value at Risk (VaR) was introduced in 1990s by the leading bankłjp Morgan [20]. The axiomatic definition of coherent risk measures was introduced in the path-breaking paper [2] [3]. Tail value at risk (TVaR)[6], also labeled conditional value at risk (CVaR)[24] or expected shortfall [1] (ES) in the literature, has been proposed as a natural remedy for the deficiencies of VaR, which is not a coherent risk measure in general. The TVaR is a risk measure which is a superior alternative to VaR. Many other kinds of risk measures are also constructed after the introduction of coherent risk measures (see the survey paper [4]). Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20-28, 2009, pp. 787-793. Risk comes from uncertainty. Uncertainty is not a new concept in management, but it has never assumed a prominence as it has today. Risk itself has neither advantages nor disadvantages. There are various types of uncertainty. We should distinguish between two types of uncertainty related to a considered phenomena: objective uncertainty and subjective uncertainty. Randomness is a basic type of objective uncertainty, and probability theory is a branch of mathematics for studying the behavior of random phenomena. Kolmogorov [13] in 1933 developed probability theory in a rigorous way from fundamental axioms. Fuzziness is a basic type of subjective uncertainty initiated by Zadeh [28]. Credibility theory [15] is a branch of mathematics for studying the behavior of fuzzy phenomena. Unfortunately, uncertainty in many decision problems may be neither randomness nor fuzziness. In order to deal with this type of uncertainty, Liu [16] founded an uncertainty theory, which is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. In stochastic environment, risk is defined, just like random uncertainty, in terms of probability theory. In fuzzy environment, risk is defined, just like fuzzy uncertainty, in terms of credibility theory. Likewise, in uncertain environment, risk is defined, just like uncertainty, in terms of uncertainty theory. There have been some studies on VaR risk management. In stochastic case, VaR has become a standard measure since its development by Morgan [20] in 1990s although it has been heavily criticized due to lack of subadditivity ([2], [3]). Considerable amount of research ([5, 8, 9, 11, 12, 23]) was dedicated to development of VaR method of risk management. In fuzzy case, Peng [22] recently presented the credibilistic VaR methods of fuzzy variables against the background of credibility theory. The credibilistic VaR is subadditive under independence consumption. Other fuzzy (even more fuzzy stochastic) risk analysis methodology is described in [14, 29]. Generally, risk analysis incorporates and spans many disciplines. So far, risk analysis in such uncertain environment has seldom been studied. This fact provides a motivation to study the counterparts of value at risk in an uncertain environment. The aim of this paper is to introduce the fundamental concepts, terminology, properties and calculations of TVaR based on uncertainty theory. The paper is organized as follows. Section 2 presents some preliminaries for risk analysis in uncertain environment

788 JIN PENG in terms of uncertainty theory. In Section 3, the VaR concept of uncertain variable is introduced and some properties of VaR are discussed. The TVaR concept of uncertain variable is introduced and some properties of the proposed TVaR are investigated in Section 4. In Section 5, an uncertain simulation algorithm is designed to calculate the TVaR of uncertain variable. Section 6 presents examples. Finally, the conclusions, limitations of this study and directions for future research. 2 Preliminaries In this section, we present some preliminaries for risk analysis in uncertain environment within the framework of uncertainty theory. Let Γ be a nonempty set, and L a σ-algebra over Γ. For any Λ L, Liu [16] presented an axiomatic uncertain measure M{Λ to express the chance that uncertain event Λ occurs. The set function M{ satisfies the following four axioms: (i) (Normality) M{Γ = 1; (ii) (Monotonicity) M{Λ 1 M{Λ 2, whenever Λ 1 Λ 2 ; (iii) (Self-Duality) M{Λ + M{Λ c = 1 for any Λ L; (iv) (Countable Subadditivity) For every countable sequence of events {Λ i, we have M { i Λ i i M{Λ i. The triplet (Γ, L, M) is called an uncertainty space and an uncertain variable is defined as a function from this space to the set of real numbers (Liu [16]). A random variable may be characterized by a probability density function, and a fuzzy variable may be described by a membership function. Naturally, an uncertain variable may be characterized by an identification function (Liu [17]). According to Liu [17], an uncertain variable ξ is said to have a first identification function λ if (i) λ(x) is a nonnegative function on R such that sup(λ(x) + λ(y)) = 1; (1) x y (ii) for any orel set of real numbers, we have sup λ(x), x M{ξ = 1 sup if sup λ(x) < 0.5 x λ(x), if sup λ(x) 0.5. x c x According to Liu [17], an uncertain variable ξ is said to have a second identification function ρ if (i) ρ(x) is a nonnegative function on R such that R (2) ρ(x)dx 1; (3) (ii) for any orel set of real numbers, we have ρ(x)dx, if ρ(x)dx < 0.5 M{ξ = 1 ρ(x)dx, if ρ(x)dx < 0.5 c c 0.5, otherwise. According to Liu [17], an uncertain variable ξ is said to have a third identification function (λ, ρ) if (i) λ(x) is a nonnegative function and ρ(x) is a nonnegative and integrable function on R such that sup λ(x) + ρ(x)dx 0.5 (5) x and/or sup λ(x) + ρ(x)dx 0.5 (6) x c c for any orel set of real numbers; (ii) we have M{ξ = sup x λ(x) + ρ(x)dx, if sup λ(x) + x 1 sup x c λ(x) if sup x c λ(x) + 0.5, otherwise. ρ(x)dx < 0.5 ρ(x)dx, c c ρ(x)dx < 0.5 The expected value of uncertain variable ξ is defined by Liu [16] as + 0 E[ξ] = M{ξ rdr M{ξ rdr (8) 0 provided that at least one of the two integrals is finite. Let ξ be an uncertain variable with finite expected value E[ξ]. The variance of ξ is defined as V [ξ] = E [ (ξ E[ξ]) 2]. And the standard deviation of ξ is defined as σ = V [ξ]. The uncertainty distribution Φ : R [0, 1] of an uncertain variable ξ is defined by Liu [16] as (4) (7) Φ(x) = M { γ Γ ξ(γ) x. (9) Liu [17] introduced the independence concept of uncertain variables. The uncertain variables ξ 1, ξ 2,, ξ m are independent if and only if { m M {ξ i i = min M {ξ i i (10) 1 i m i=1 for any orel sets 1, 2,, m of R.

VALUE AT RISK AND TAIL VALUE AT RISK IN UNCERTAIN ENVIRONMENT 789 3 VaR in an Uncertain Environment The accurate definition of VaR paves the foundation of risk analysis in an uncertain environment. Definition 1 Let ξ be an uncertain variable and α (0, 1] be the risk confidence level. Then the VaR of ξ is the function ξ VaR : (0, 1] R such that ξ VaR (α) = inf { x M{ξ x α. (11) It is worth emphasizing that the value at risk ξ VaR (α) is actually a quantile of the uncertainty distribution Φ of uncertain variable. For a risk confidence level 0 < α 1, we have ξ VaR (α) = inf { x Φ(x) α = Φ 1 (α) (12) where Φ 1 (α) denotes the generalized inverse function of Φ(x). Alternatively, we have ξ VaR (α) = inf { x M{ξ > x 1 α. (13) elow we illustrate the VaR of a triangular uncertain variable. Example 1 Consider a triangular uncertain variable ξ = (a, b, c), whose first identification function is λ(x) = 2(b a), if a x b x c 2(b c), if b x c 0, otherwise. The uncertainty distribution of the triangular uncertain variable ξ is 0, if x a 2(b a), if a x b Φ(x) = x + c 2b 2(c b), if b x c 1, if x c. It is easily calculated that for any given confidence level α with 0 < α 1, we analytically have the following expression of VaR function { a + 2(b a)α, if α 0.5 ξ VaR (α) = 2b c + 2(c b)α, if α > 0.5. What mathematical properties a meaningful risk should have? Now we investigate some properties of VaR measure in uncertain environment defined by Definition 1. Theorem 1 (Positive Homogeneity) Let ξ be an uncertain variable. If c > 0, then (cξ) VaR (α) = cξ VaR (α). Proof. When c > 0, we have (cξ) VaR (α) = inf {r M{cξ r α = c inf {r/c M {ξ r/c α = cξ VaR (α). Theorem 2 (Monotonicity) Let ξ and η be two uncertain variables. If ξ η (where ξ η means ξ(λ) η(λ) for any λ Λ), then ξ VaR (α) η VaR (α). Proof. It is easy to see that ξ VaR (α) = inf {r M{ξ r α inf {r M{η r α = η VaR (α). Theorem 3 (Translation Invariance) Let ξ be an uncertain variable and b is a real number. Then we have (ξ + b) VaR (α) = ξ VaR (α) + b. Proof. From the definition of VaR, it is clear that (ξ + b) VaR (α) = inf {r M{ξ + b r α = inf {r M{ξ r b α = inf {s M{ξ s α + b = ξ VaR (α) + b. More generally, we have the following result. Theorem 4 (Monotonicity Transformation) Let ξ be an uncertain variable. If f(x) is an increasing and rightcontinuous function on R, Then f(ξ) VaR (α) = f(ξ VaR (α)) for any α (0, 1]. Proof. Under the hypothesis, we have f(ξ) VaR (α) = inf { x M{f(ξ) x α = inf { f(f 1 (x)) M{ξ f 1 (x) α = f ( inf { f 1 (x)) M{ξ f 1 (x) α ) = f(ξ VaR (α)). Theorem 5 (Subadditivity under Independence) Let ξ and η be two uncertain variables. If ξ and η are independent uncertain variables, then for any α (0, 1], we have (ξ + η) VaR (α) ξ VaR (α) + η VaR (α). (14) Proof. For any given number ε > 0, since ξ and η are independent uncertain variables, we have M{ξ + η ξ VaR (α) + η VaR (α) + ε M {{ξ ξ VaR (α) + ε/2 {η η VaR (α) + ε/2 =M{ξ ξ VaR (α) + ε/2 M{η η VaR (α) + ε/2 α

790 JIN PENG which yields (ξ + η) VaR (α) ξ VaR (α) + η VaR (α) + ε. Letting ε 0, we obtain (ξ + η) VaR (α) ξ VaR (α) + η VaR (α). This completes the proof. Note that the convexity property of ξ VaR (α) follows from the positive homogeneity and subadditivity under the condition of independence. Theorem 6 (Convexity under Independence) Suppose that ξ and η are independent uncertain variables and α (0, 1]. Then for any λ [0, 1], we have (λξ + (1 λ)η) VaR (α) λξ VaR (α) + (1 λ)η VaR (α). Theorem 7 Let ξ be an uncertain variable. Then ξ VaR (α) is an increasing and left-continuous function of α. Proof. The fact that ξ VaR (α) is an increasing function of α follows from the definition of VaR. Next, the left-continuity of ξ VaR (α) with respect to α can be proved in the following way. Let {α i be an arbitrary sequence of positive numbers such that α i α. Then {ξ VaR (α i ) is an increasing sequence. If the limitation is equal to ξ VaR (α), then the left-continuity is proved. Otherwise, there exists a number z such that ξ VaR (α i ) lim i ξ VaR (α i ) < z < ξ VaR (α). Thus M{ξ z α i for each i. Letting i, we get M{ξ z α. Hence z ξ VaR (α). A contradiction proves the left-continuity of ξ VaR (α) with respect to α. 4 TVaR in an Uncertain Environment In financial mathematics and financial risk management, VaR is a widely used measure of the risk of loss on a specific portfolio of financial assets. A disadvantage of VaR is that it does not give any information about the severity of losses beyond the VaR level. It may happens that the VaRs of two uncertain variable are equal to but one has a thicker tail and is more risky than the other. TVaR accounts for the severity of failure and not only the chance of failure. TVaR is considered to provide a better measure of risk, in that it is afected by the extreme values in the tail (while VaR is not). It is defined as follows. Definition 2 Let ξ be an uncertain variable and α (0, 1) be the risk confidence level. Then the TVaR of ξ is the function ξ TVaR : (0, 1) R such that ξ TVaR (α) = 1 1 α 1 α ξ VaR (β)dβ. (15) Let U denote the set of uncertain variables ξ : Γ R defined on the uncertainty space (Γ, L, M). A risk measure is a mapping from the uncertain variables representing risk assets to the real line. Risk measure can be axiomatized directly in the following way. Definition 3 A mapping µ : U R is called a convex risk measure under independence, if it satisfies the following axioms: (1) convexity under independence: µ(λξ + (1 λ)η) λµ(ξ) + (1 λ)µ(η) for any λ [0, 1] and any independent ξ, η. (2) monotonicity: If ξ η, then µ(ξ) µ(η). (3) translation invariance: if b is constant, then µ(ξ + b) = µ(ξ) + b. Definition 4 A mapping ν : U R is called a coherent risk measure under independence, if it satisfies (1) positive homogeneity: If c > 0, then ν(cξ) = cν(ξ) holds. (2) subadditivity under independence: ν(ξ+η) ν(ξ)+ν(η) for any ξ, η U and any independent ξ, η. (3) monotonicity: If ξ η, then ν(ξ) ν(η). (4) translation invariance: if b is constant, then ν(ξ + b) = ν(ξ) + b. Observe that a risk measure is coherent under independence if and only if it is of positive homogeneitive and convex under independence. What mathematical properties should a meaningful risk have? Now we investigate some fundamental properties of TVaR measure in uncertain environment defined by Definition 2. Theorem 8 (Law Invariance) Let ξ, η be any two uncertain variables. If M{ξ x = M{η x for all x R, then ξ TVaR (α) = η TVaR (α). Proof. An immediate consequence of the Definition 1 and Definition 2. Note that the distributions of ξ and η need not be identical in order to ensure ξ AVaR (α) = η AVaR (α). Theorem 9 (Positive Homogeneity) Let ξ be an uncertain variable. If c > 0, then (cξ) TVaR (α) = cξ TVaR (α). Proof. It follows from the positive homogeneity of VaR that 1 (cξ) TVaR (α) = 1 (cξ) VaR (β)dβ = c 1 ξ VaR (β)dβ = cξ TVaR (α). Theorem 10 (Monotonicity) Let ξ and η be two uncertain variables. If ξ η, then ξ TVaR (α) η TVaR (α). Proof. The result follows immediately the monotonicity of VaR. Theorem 11 (Translation Invariance) Let ξ be an uncertain variable and b is a real number. Then we have (ξ + b) TVaR (α) = ξ TVaR (α) + b.

VALUE AT RISK AND TAIL VALUE AT RISK IN UNCERTAIN ENVIRONMENT 791 Proof. It follows immediately the translation invariance of VaR. More generally, we have the following result. Theorem 12 (Monotonicity Transformation) Let ξ be an uncertain variable. If U(x) is an increasing function on R, Then U(ξ) TVaR (α) = U(ξ TVaR (α)) for any α (0, 1). Proof. From the definition of TVaR and the monotonicity transformation property of VaR in Theorem 4, we can draw the conclusion U(ξ) TVaR (α) = U(ξ TVaR (α)). Theorem 13 (Subadditivity under Independence) Let ξ and η be two uncertain variables. If ξ and η are independent uncertain variables, then for any α (0, 1), we have (ξ + η) TVaR (α) ξ TVaR (α) + η TVaR (α). Proof. y the definition of TVaR and Theorem 5, we have 1 (ξ + η) TVaR (α) = 1 (ξ + η) VaR (β)dβ 1 1 (ξ VaR (α) + η VaR (α))dβ = ξ TVaR (α) + η TVaR (α). The proof is completed. Here we note that the convexity property under independence of ξ TVaR (α) follows from the positive homogeneity and subadditivity under independence. Mathematically speaking, TVaR does have the convexity property under the condition of independence. We state the following result. Theorem 14 (Convexity under Independence) Suppose that ξ and η are independent uncertain variables and α (0, 1). Then for any λ [0, 1], we have (λξ + (1 λ)η) TVaR (α) λξ TVaR (α) + (1 λ)η TVaR (α). Sadly speaking, the TVaR measure is not a a convex risk measure or coherent risk measure because it does not fulfill the axiom of subadditivity in the case of nonindependence. However, it is a convex risk measure or coherent risk measure under independence in the sense of Definition 3 or Definition 4. Theorem 15 Let ξ be an uncertain variable. Then ξ TVaR (α) is an increasing and left-continuous function of α. Proof. y Theorem 7, we know that ξ VaR (α) is an increasing and left-continuous function of α. Then ξ TVaR (α) is well defined and is left-continuous to α. Since dξ TVaR dα = 1 α ξ VaR (α)dβ (1 α)ξ VaR (α) (1 α) 2 0, we find that ξ TVaR (α) is an increasing function of α. 5 Calculating TVaR by Uncertain Simulation The TVaR is a risk measure which is a superior alternative to VaR. The definition of VaR is nonconstructive, it specifies a property VaR must have, but not how to compute VaR. VaR measures can have many applications, such as in risk management, to evaluate the performance of risk takers and for regulatory requirements, and hence it is very important to develop methodologies that provide accurate estimates. The calculation of TVaR is sometimes of computational challenges. In this section we exhibit the uncertain simulation algorithm to calculate TVaR. Let us think about finding VaR at first. Let ξ be an uncertain variable and f be a function. Our object is to find the minimal f such that M { f(ξ) f α. It follows from monotonicity that we may employ bisection search to find the minimal value f such that M { f(ξ) f α. The uncertain simulation algorithm for TVaR is structured as follows. Algorithm for Calculating TVaR: Step 1. Given risk confidence level α and a sufficiently small number ε > 0. Step 2. Set A = 0. Step 3. For any number r, calculate L(r) = M {f(ξ) r. Step 4. Sample point α i = α + i (1 α) from interval M (α, 1) for i = 1, 2,, M. Step 5. Calculate the minimal value r i such that L(r i ) = M {f(ξ) r i α i. Step 5.1 Determine the range interval of L(r i ), for example [a i, b i ]. Step 5.2 Set x = a i + b i. If L(x) α, set a i = x; else 2 set b i = x; Step 5.3 Repeat Step 5.2 till a i b i < ε. Step 5.4 Return r i = x. Step 6. Set A = A + r i α i. Step 7. Repeat the fourth to sixth steps M times. Step 8. Return A = A as the answer. 1 α 6 Example Although many types of uncertain variables have been used to describe risk and uncertainties, one of the simplest uncertain variables triangular uncertain variables are extensively used in the applications because the parameters defining them can be easily specified in linguistic terms. Now we investigate the VaR and TVaR of a triangular uncertain variable.

792 JIN PENG Example 2 Consider a triangular uncertain risk variable ξ = (a, b, c, d), whose first identification function is b a, if a x b 1, if b x c λ(x) = x d c d, if c x d 0, otherwise. The uncertainty distribution of the triangular uncertain risk variable ξ is 0, if x a 2(b a), if a x b 1 Φ(x) = 2, if b x c x + d 2c 2(d c), if c x d 1, if x d. It is easily calculated that for any given confidence level α with 0 < α 1, the VaR can be analytically expressed as { a + 2(b a)α, if α 0.5 ξ VaR (α) = 2c d + 2(d c)α, if α > 0.5. Furthermore, the TVaR function ξ TVaR (α) can be analytically expressed as 1 [(1 2α) (a + b + 2α(b a)) + d + c], 4(1 α) if α 0.5 (16) (1 α)c + αd, if α > 0.5. 7 Conclusions The problem of risk measurement is an old one in statistics, economics and finance. Coping with risk management issues in decision making under uncertainty is a very important and complex problem. The aim of risk management is to identify, measure and control uncertain events, in order to minimize loss, and optimize the return. This paper is a self-contained introduction to the concepts and methodologies of value at risk and tail value at risk, which are new tools for measuring risk in uncertain environment. We explain the concepts of value at risk and tail value at risk, and then examine some fundamental properties of the proposed risk measures. We describe in detail the uncertain simulation algorithms to calculate the risk measures. Finally, we briefly describe some alternative risk measures. The proposed risk measures can be applied in many real problems of risk analysis in an uncertain environment. The TVaR introduced in this paper is a risk measure which is a superior alternative to VaR. It provides a new tool for dealing with risk modelling in an uncertain environment. There are convenient ways for computing and estimating TVaR which allows its application in optimal portfolio problems. TVaR satisfies all axioms of independent coherent risk measures and it is consistent with the preference relations of risk-averse investors. TVaR is a special case of spectral risk measures. To conclude, the framework for risk and risk management in uncertain environment is still in the making, and further research and analysis is needed. Practically, extensive testing or case study of the TVaR framework on real data is needed. Acknowledgements This work is supported by the National Natural Science Foundation (Grant No.70671050), the Major Research Program (Grant No.Z20082701) and the Group Innovation Project of Hubei Provincial Department of Education, China. References [1] Acerbi C, and Tasche D, On the Coherence of Expected Shortfall, Journal of anking and Finance, Vol. 26, 1487-1503, 2002. [2] Artzner P, Delbaen F, Eber JM, and Heath D, Thinking Coherently, Risk, Vol. 10, No. 11, 68-71, 1997. [3] Artzner P, Delbaen F, Eber JM, and Heath D, Coherent measures of risk, Mathematical Finance, Vol. 9, No. 3, 203-228, 1999. [4] Cheng S, Liu Y, and Wang S, Progress in Risk Measurement, Advanced Modelling and Optimization, Vol. 6, No. 1, 1-20, 2004. [5] Choudhry M, An Introduction to Value-at-Risk, 4th Edition, John Wiley, 2006. [6] Desmedt S, and Jean-Francois W, On the Subaddivity of Tail Value at Risk: An Investigation with Copulas, Variance, Vol. 2, No. 2, 231-252, 2008. [7] Dhaene J, Laeven R J A, Vanduffel S, Darkiewicz G, and Goovaerts M J, Can a Coherent Risk Measure be too Subadditive? Journal of Risk and Insurance, Vol. 75, 365-386, 2008. [8] Duffie D, Pan J, An overview of value at risk, Journal of Derivatives, Vol. 4, 7-49, 1997. [9] Gourieroux C, Laurent JP, Scaillet O, Sensitivity analysis of values at risk, Journal of Empirical Finance, Vol. 7, No. 3, 225-245, 2000. [10] Jaeger C, Renn O, Rosa E, and Webler T, Risk, Uncertainty, and Rational Action, Earhscan Publications, London, 2001. [11] Jorion P, Value at Risk: The New enchmark for Managing Financial Risk, 2nd ed., McGraw- Hill, New York, 2001. [12] Kaplanski G & Kroll Y, VaR risk measures vs traditional risk

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