Spectral Measures of Uncertain Risk

Transcription

1 Spectral Measures of Uncertain Risk Jin Peng, Shengguo Li Institute of Uncertain Systems, Huanggang Normal University, Hubei 438, China Abstract: A key issue in uncertain risk analysis is how to provide risk measures. In this article we present the spectral measure of risk in an uncertain environment which is described by uncertain variable. Firstly, we introduce the concept of spectral measure of risk and harmonious measure of risk based on uncertainty theory. Then, some fundamental properties of the proposed spectral measure of risk are investigated. It is proved that the proposed spectral measure of risk is a harmonious measure of risk. Finally, some numerical examples are illustrated. Keywords: risk analysis, uncertainty theory, uncertain variable, spectral measure of risk, harmonious measure of risk 1 Introduction Risk analysis in an uncertain environment is a very powerful tool for decision making under uncertainty. A key issue in risk analysis is how to provide the risk measures. There has been a great momentum in research on risk measures in recent years. For recent reviews on risk measure we refer the reader to 6, 7, 14, 17, 25, 26. As we know, the simplest way to measure risk in the early era of risk analysis is to calculate the variance and standard deviation of the profit and loss distribution. Value-at-risk (VaR) is the most commonly used measure of market risk 8, 13, 15, 21. However, VaR measure has been subjected to criticism because of mathematical shortcomings (lack of subadditivity and convexity). A related concept, tail value-atrisk (TVaR), also known as expected shortfall (ES) or conditional Value-at-risk (CVaR) in some contexts 22, 24, 28, has proved to be superior in these respects. In 4, 5, Artzner et al. propose a set of axioms to characterize a class of financially meaningful risk measures, called coherent risk measures. A measure of risk is called coherent if it is monotonous, positively homogeneous, translation invariant, and subadditive. Many researchers pay attention to the coherence property of risk measure 1, 12, 16, 27. Acerbi 1 has investigated the coherence of spectral risk measures. Spectral measures of risk are attractive risk measures as they allow the user to obtain risk measures that reflect Proceedings of the First International Conference on Uncertainty Theory, Urumchi, China, August 11-19, 21, pp their weight function or risk-aversion functions. The question of how spectral measures of risk should be produced in decision problems has been extensively discussed in the past few decades, see the articles 2, 3, 9, 11. In economics and finance, risk is customarily measured by means of probabilistic methods. As is well known, probability theory often requires a lot of historical data. However, decision-making problems in real life are not always in these situations. Generally, in risk analysis with little or no relevant historical data, expert judgment is required. Probabilistic risk analysis is subject to a number of severe problems that make its use as measures of risk very questionable. Randomness is only one type of various uncertainties. Liu 18, 2 has found the axiomatic uncertainty theory which identifies general uncertainty as being distinct from probabilistic uncertainty. In some context, risk is a synonym for uncertainty. This motivates uncertain risk analysis a new area of risk analysis which appears to be distinct from the traditional probabilistic risk analysis. Recently, the axiomatic concepts of coherent and convex risk measures have been developed and extended under uncertainty, see 22. We presuppose throughout this paper that risks can be quantified by way of an uncertain variable which can be used to describe uncertain (positive) profit or (negative) loss. In general, we regard a risk measure as a mapping from a set of real valued uncertain variables representing risks to the real line. Given some known risk measures under uncertainty, how to generate a new risk measure? Spectral risk measures is one kind of these methods. More importantly, risk perceptions and rationality in measures of risk is fundamental. We naturally hope the spectral measure of risk is of better properties than the original one in some aspects. The purpose of this paper is to show the spectral measure of risk in an uncertain environment which is described by uncertain variable. The rest of this paper is organized as follows: Section 2 presents some preliminary concepts and results selected from uncertainty theory. In Section 3, the spectral measure of risk of uncertain variable is introduced. Section 4 deals with some properties of the spectral measure of risk. Some computing examples are illustrated in Section 5. The last section contains some concluding remarks.

2 116 JIN PENG AND SHENGGUO LI 2 Preliminaries In this section, we present some definitions and results 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 18 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 18). An uncertain variable can be characterized by its uncertainty distribution Φ : R, 1, which is defined by Liu 18 as follows M { γ Γ (γ) x }. (1) Peng and Iwamura 23 have proved that a function Φ : R, 1 is uncertainty distribution if and only if it is an increasing function except Φ(x) and Φ(x) 1. Let be an uncertain variable with uncertainty distribution Φ. Then the inverse function is called the inverse uncertainty distribution of. An uncertainty distribution Φ is called regular if its inverse function (α) exists and is unique for each α (, 1). The expected value of uncertain variable is defined by Liu 18 as E = + M{ r}dr M{ r}dr (2) provided that at least one of the two integrals is finite. As a useful representation of expected value, it has been proved by Liu 19 that E = (α)dα (3) where is the generalized inverse of the uncertainty distribution Φ of uncertain variable. Let be an uncertain variable with finite expected value µ = E. The variance of is defined as V = E ( µ) 2. And the standard deviation of is defined as σ = V which is previously used as a simple risk measure. Liu 18 introduced the independence concept of uncertain variables. The uncertain variables 1, 2,, m are independent if and only if { m } M { i B i } = min 1 i m M { i B i } (4) for any Borel sets B 1, B 2,, B m of R. Consider an uncertain variable which might be seen as the uncertain profit and loss of an investment or portfolio by a fixed time horizon. Positive values of are regarded as profits, while negative values of are regarded as losses. Peng 22 introduced the concepts of VaR and TVaR of uncertain variable in the sense of the following definitions. Definition 1 Let be an uncertain variable and α (, 1 be a chosen risk confidence level. Then the VaR of can be expressed by VaR (α) inf { x M{ x} α } (α) (5) where (α) denotes the generalized inverse of the uncertainty distribution Φ of. Definition 2 Let be an uncertain variable and α (, 1) be the risk confidence level. Then the TVaR of is the function TVaR : (, 1 R such that TVaR (α) = 1 α VaR 1 α 3 Concept of Spectral Measures of Risk. (6) In this section, we deal with spectral measures of risk, a class of measures based on integrals of the inverse distribution of uncertain variable (which may denote portfolio return). Definition 3 Let be an uncertain variable with uncertainty distribution Φ (x), and w :, 1 R be a function. The spectral measure of risk is defined as M w (α)dα (7) where the function w is called the weight function for convenience. Definition 4 A weight function w :, 1 R is called a risk spectrum or risk aversion function if it satisfies the following conditions 1) Nonnegativity: w(y) for all y, 1; 2) Normality: w = 1; 3) Monotonicity: w(y 1 ) w(y 2 ) for all y 1 y 2 1. Remark 1 The spectral measure of risk can be thought of as a particular uncertainty-weighted average of the inverse distribution (α) of uncertain variable.

3 SPECTRAL MEASURES OF UNCERTAIN RISK 117 Remark 2 It should be noticed that if, then (α) and then M w. That is to say, non-negative uncertain variable means no risk in the sense of spectral measure. Example 1 Different weight functions produce different spectral measures of risk. Several kinds of weight functions are listed as follows. 1) Power weight function: w(y) = λ(1 y) λ 1 with λ > 1; 2) Proportional weight function: w(y) = 1 µ y 1 µ 1 with µ > 1; 3) Exponential weight function: w(y) = c exp{ cy} 1 exp{ c} with coefficient c >. Example 2 Setting w 1 one gets M w E which the negative expected value is trivially a spectral measure of risk. Example 3 The TVaR in Definition 2 is indeed a particular case of spectral measure. In fact, taking the weight function otherwise, we exactly obtain TVaR (α) 1 α. Example 4 The VaR in Definition 1 is also a typical example of spectral measure. In fact, taking the weight function, which is the so called Dirac delta function, = δ(y α), we reproduce that δ(y α) (α) = VaR (α). (,1) α What mathematical properties a meaningful risk should have? Definition 5 Consider a set U of real-valued uncertain variables. A function r : U R is said to be a harmonious measure of risk if it satisfies the following properties (A 1 ) Monotonicity: If, U are two uncertain variables and, then r() r(). (A 2 ) Positive homogeneity: If U is an uncertain variable and c >, then r(c) = cr(). (A 3 ) Translation invariance: If U and b R is a real number, then r( + b) = r() b. (A 4 ) Independent additivity: If,, + U, and and are independent uncertain variables, then r( + ) = r() + r(). A natural question then arises here: is the spectral measures of risk harmonious? Or under what conditions is it a harmonious measure? We will answer these questions in the coming section. 4 Properties of Spectral Measures of Risk The purpose of this section is to show some properties of the spectral measures of risk. First of all, we exhibit two frequently-used lemmas given by Liu 2. Lemma 1 Let be an uncertain variable with uncertainty distribution Φ, and let f be a strictly increasing function. Then f() is an uncertain variable with inverse uncertainty distribution f() (α) = f(φ 1 (α)), α (, 1). (8) Lemma 2 Let 1, 2,, n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f : R n R is a strictly increasing function, then = f( 1, 2,, n ) (9) is an uncertain variable with inverse uncertainty distribution (α) = f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)), α (, 1). (1) Remark 3 A real-valued function f(x 1, x 2,, x n ) is said to be strictly increasing if f(x 1, x 2,, x n ) < f(x 1, x 2,, x n) (11) whenever x i x i for i = 1, 2,, n and x j < x j for at least one index j. Theorem 3 Let be an uncertain variable with uncertainty distribution Φ(x), and w :, 1 R be a risk spectrum. Then the spectral measure of risk is defined by M w is a harmonious measure of risk. (α)dα (12) Proof. It is needed to prove that M w satisfies the following axioms (A 1 ) Monotonicity: If, are two uncertain variables and, then M w M w. (A 2 ) Independent additivity: If and are independent uncertain variables, then M w + = M w + M w. (A 3 ) Positive homogeneity: If is an uncertain variable and c >, then M w c = cm w. (A 4 ) Translation invariance: If is an uncertain variable and b R is a real number, then M w + b = M w b. Let us travel the above properties step by step. (1) Since w is a nonnegative function and implies (α) Φ 1(α), it is easy to see that M w = M w. (α)dα (α)dα

4 118 JIN PENG AND SHENGGUO LI (2) Since and are independent uncertain variables, we see from Lemma 1 that Thus we have + M w + (α) + Φ 1(α). + (α)dα ( = M w + M w. (3) Because c >, we have M w c = cm w. (α) + Φ 1(α))dα c (α)dα c (α)dα (4) From the definition of the spectral measure of risk, it is clear that M w + b = M w b. The proof is accomplished. +b (α)dα ( (α) + b)dα Besides the above harmonious property, there are still other desirable properties below that are worthwhile discussing. Theorem 4 Let be an uncertain variable with uncertainty distribution Φ(x), and w :, 1 R be a weight function. If f(x) is a strictly increasing function on R, then the spectral measure of risk f() is M w f() f ( (α) ) dα. (13) Proof. Under the hypothesis, it follows from Lemma 1 that ( ) f() (α) = f (α). The result follows immediately. Theorem 5 (Law-Invariance) Let and be uncertain variables with uncertainty distributions Φ (x) and Φ (x) respectively, and w :, 1 R be a weight function. If Φ (x) = Φ (x) on R, then the spectral measure of risk M w = M w. Proof. It is an easy consequence of the Definition 3. Theorem 6 Let be an uncertain variable with uncertainty distribution Φ(x), and w :, 1 R be a weight function. If f(x) is a strictly decreasing function on R, then the spectral measure of risk f() is M w f() f ( (1 α) ) dα. (14) Proof. Under the hypothesis, it is easy to check that ( ) f() (α) = f (1 α). The result follows immediately. Especially, it is important to keep in mind that M w = M w. Theorem 7 Let w :, 1 R be a weight function and λ 1, λ 2,, λ n, 1 be nonnegative numbers with n λ i = 1. Assume that 1, 2,, n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. Then spectral measure of risk of the convex combination is = λ i i M w n λ i i = λ i M w i. (15) Proof. Notice that the inverse uncertainty distribution of the convex combination is Thus we have (α) = n M w n λ i i λ i i (α), α (, 1). (16) = The result is proved. λ i λ i M w i. λ i i (α)dα i (α)dα In fact, the convexity property of the spectral measure of risk follows from the positive homogeneity and independent additivity. Theorem 8 Let w :, 1 R be a weight function. Suppose that and are independent uncertain variables with uncertainty distributions Φ (x) and Φ (x), respectively. Then we

5 SPECTRAL MEASURES OF UNCERTAIN RISK 119 have the following results related to the spectral measure of risk M w M w M w M w (α) Φ 1 (α) Φ 1 (α) dα, (17) (α) dα, (18) (α)φ 1 (α) dα, (19) Φ 1 (α) (α) dα. (2) Proof. Note that the following inverse uncertainty distribution expressions / Φ 1 (α) = (α) (α). (α) Φ 1(α), (α) Φ 1(α), (α)φ 1 (α), Thus the results follow directly from the definition of spectral measure of risk. 5 Numerical Examples Now we investigate the spectral measure of risk of some common types of uncertain variable. Example 5 Consider a linear uncertain variable = L(a, b), where a and b are real numbers with < a < b. The uncertainty distribution of the profit and loss uncertain variable is, if x a x a b a, if a x b 1, if x b and its inverse uncertainty distribution is (y) = a(1 y) + by. We obtain the spectral measure of risk expressed as M w L(a, b) = a (y 1)w b Taking the weight function otherwise, yw. we get the spectral measure of risk as follows M w L(a, b) = a b 2 α a. If another weight function w = 1 2 is selected, then y it is easy to see that w is a non-negative, non-increasing, right-continuous integrable function defined on, 1 and w = 1. Thus we have M w L(a, b) = 2a b. 3 Example 6 Consider a zigzag uncertain variable = Z(a, b, c), where a, b, c are real numbers with a < b < c. The uncertainty distribution of the profit and loss uncertain variable is, if x a x a 2(b a), if a x b 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 < α 1, the inverse distribution of the zigzag uncertain variable can be analytically expressed as { a + 2(b a)y, if y.5 (y) = 2b c + 2(c b)y, if y >.5. Taking the weight function otherwise and recalling the definition M w w(y), we get the spectral measure of risk represented as M w Z(a, b, c) (a b)α a, if α.5 = (b c)α + (c 2b) + 2b a c α 1, if α >.5. 4 Example 7 Now we consider the normal uncertain variable = N (e, σ) with uncertainty distribution ( ( )) 1 π(e x) 1 + exp, x R (21) 3σ

6 12 JIN PENG AND SHENGGUO LI where e and σ are real numbers with σ >. The inverse uncertainty distribution of normal uncertain variable = N (e, σ) is (α) = e + σ 3 π ln α 1 α. (22) Setting the weight function otherwise. In this case, we obtain the spectral measure of risk represented as M w N (e, σ) 1 α 1 α w(y) e + σ 3 πα (e + σ 3 π ln y 1 y )dy ln (1 y) ln ydy e + σ 3 (α 1) ln (1 α) α ln α. πα Example 8 Let us now consider the lognormal uncertain variable = LOGN (e, σ) with uncertainty distribution denoted by ( ( )) 1 π(e ln x) 1 + exp, x (23) 3σ where e and σ are real numbers with σ >. The inverse uncertainty distribution of lognormal uncertain variable = LOGN (e, σ) is ( ) 3σ/π α (α) = exp(e) 1 α Setting the weight function otherwise. (24) In this situation, we has the following representation of spectral measure of risk M w LOGN (e, σ) 1 α w(y) exp(e) α 6 Conclusions ( ) 3σ/π y dy. 1 y Risk analysis aims at making systematic use of available information to identify hazards and to estimate the risk to individuals or populations, property or the environment. One key factor in risk analysis in uncertain environment is to introduce the reasonable measure of risk. At the same time, risk measure has a close relationship with premium calculation or insurance price, and it has a great potential applications in actuarial science. Traditionally, probabilistic risk analysis often requires a lot of historical data. However, uncertain risk analysis often requires a lot of experts empirical data. The expert judgment is typically appropriate when data are sparse or difficult to obtain or data are too costly to obtain. The philosophy behind spectral measure of risk is that a spectral risk measure is a new risk measure transformed from a traditional VaR measure via some kind of weight function. It is desired that the spectral risk measure is of better properties than the original one in some aspects. In this article we mainly present the spectral measure of risk under uncertain environment. Firstly, we introduce the concept of spectral measure of risk and harmonious measure of risk from the viewpoint of uncertainty theory. Then, some fundamental properties of the proposed spectral measure of risk are investigated. It is proved that the proposed spectral measure of risk is a harmonious measure of risk. Finally, some numerical examples are illustrated. Various classes of risk measures including spectral risk measures can be used in the framework of risk optimization under risk constraints. For example, portfolio optimization with appropriate spectral measures of risk under risk constraints should be considered in the future research work. Acknowledgements This work is supported by the National Natural Science Foundation (Grant No and No ), the Innovation Team Project of Hubei Provincial Department of Education, China.

7 SPECTRAL MEASURES OF UNCERTAIN RISK 121 References 1 Acerbi C, Spectral measures of risk: a coherent representation of subjective risk, Journal of Banking and Finance, Vol. 26, No. 7, , Acerbi C, Simonetti P, Portfolio optimization with spectral measures of risk, Adam A, Houkari M, Laurent J-P, Spectral risk measures and portfolio selection, Journal of Banking & Finance, Vol. 32, No. 9, , Artzner P, Delbaen F, Eber JM, and Heath D, Thinking Coherently, Risk, Vol. 1, No. 11, 68-71, Artzner P, Delbaen F, Eber JM, and Heath D, Coherent measures of risk, Mathematical Finance, Vol. 9, No. 3, , Brachinger H W, Weber M, Risk as a primitive: A survey of measures of perceived risk, Operations Research Spectrum, Vol. 19, No. 3, , Cheng S, Liu Y, and Wang S, Progress in Risk Measurement, Advanced Modelling and Optimization, Vol. 6, No. 1, 1-2, Choudhry M, An Introduction to Value-at-Risk, 4th Edition, John Wiley, Cotter J, Dowd K, Extreme spectral risk measures: An application to futures clearinghouse margin requirements, Journal of Banking & Finance, Vol. 3, No. 12, , Dhaene J, Laeven RJA, Vanduffel S, Darkiewicz G, and Goovaerts M J, Can a Coherent Risk Measure be too Subadditive? Journal of Risk and Insurance, Vol. 75, , Dowd K, John Cotter J, and Sorwar G, Spectral risk measures: properties and limitations, Journal of Financial Services Research, Vol. 34, No. 1, 61-75, Fabozzi F J, Tunaru R, On risk management problems related to a coherence property, Quantitative Finance, Vol. 6, No. 1, Gourieroux C, Laurent JP, and Scaillet O, Sensitivity analysis of values at risk, Journal of Empirical Finance, Vol. 7, No. 3, , Jaeger C, Renn O, Rosa E, and Webler T, Risk, Uncertainty, and Rational Action, Earhscan Publications, London, Jorion P, Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed., McGraw- Hill, New York, Kusuoka S, On law invariant coherent risk measures, in: Advances in Mathematical Economics, Vol. 3, 83-95, Springer, Tokyo, Kaplanski G, Kroll Y, VaR risk measures vs traditional risk measures: An analysis and survey, Journal of Risk, Vol. 4, No. 3, 1-28, Liu B, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, Liu B, Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems, Vol.3, No.1, 3-1, Liu B, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, Morgan J P, RiskMetricsTM- -Technical Document, Fourth Edition, Morgan Guaranty Trust Companies, Inc. New York, Peng J, Value at Risk and Tail Value at Risk in Uncertain Environment. Proceedings of the Eighth International Conference on Information and Management Sciences, Kuming & Banna, China, July 2-28, pp , Peng Z, Iwamura K, A sufficient and necessary condition of uncertainty distribution Rockafeller RT, Uryasev S, Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, Vol. 26, No. 7, , Szegö G, Risk Measures for the 21st Century, New York: John Wiley and Sons, Szegö G, Measures of risk, European Journal of Operational Research, Vol. 163, No. 1, 5-19, Weber S, Distribution-invariant risk measures, information, and dynamic consistency. Mathematical Finance, Vol. 16, , Yamai Y, Toshinao Y, Value-at-risk versus expected shortfall: A practical perspective, Journal of Banking & Finance, Vol. 29, No. 4, , 25.