Appl. Math. Inf. Sci. 6, No. 2, 233-238 (2012) 233

Applied Mathematics & Information Sciences An International Journal

°c 2012 NSP Natural Sciences Publishing Cor.

Coherent Risk Measure Based on Relative Entropy

Zheng Chengli1 and Chen Yan2 1 School of Economics, Huazhong Normal University, Wuhan, 430079, China 2 School of Mathematics & Statistics, Huazhong Normal University, Wuhan, 430079, China

Received: Jul 8, 2011; Revised Oct. 4, 2011; Accepted Oct. 6, 2011 Published online: 1 May 2012

Abstract: This article proposes a new coherent risk measure called iso-entropic risk measure, which is based on relative entropy under the theory framework of Artzner et al.(1999). It is pointed that this measure is just the negative expectation of the risk portfolio position under the probability measure through Esscher transformation. This iso-entropic risk measure is not a 0-1 risk measure and very smooth in contrast with another important coherent risk measure AV @R (Average Value at Risk). And it is a little larger than AV @R at the same level, namely it is has more prudence. So it maybe a better coherent risk measure.

Keywords: Risk Management, Iso-entropic Risk Measure, Coherent Risk Measure, Relative Entropy, Calculus of Variations.

1. Introduction risk measures has rapidly been evolving; it already occu- pies a considerable part of the modern financial mathemat- How to measure the risk of the uncertainty in the future ics. Some of these papers are [1,4,5,7,8,10,11,14,16–18, value of a position is one of the basic tasks in finance. The 20–22,30]. Excellent reviews on the theory of coherent most well-known and widely used in practice methods to risk measure are given in [12]. Recently, the most fash- this task are variance and subsequent V @R(Value at Risk). ionable coherent risk measure is AV @R (also called Con- In the finance context the standard deviation of continuous ditional Value at Risk, Expected Shortfall,or Tail Value at growth rates usually is called volatility. However, both of Risk). As compared to V @R, it measures not only the them have serious drawbacks. One important drawback for probability of loss but its severity as well. Kusuoka (2001)[20] variance is that it is not monotonic: a better gamble, i.e., a proved that AV @R is the smallest law invariant coher- gamble with higher gains and lower losses, may well have ent risk measure that dominates V @R. It is seemly that a higher variance and thus be wrongly viewed as having AV @R might be the most important subclass of coherent a higher riskiness. About V @R, it takes into account only risk measures. However, its disadvantage is that it depends the quantile of the distribution without caring about what only on the tail of the distribution, i.e. it is a 0-1 risk mea- is happening to the left and to the right of the quantile. sure, so it is not smooth[6]. And it is concerned only with the probability of the loss and does not care about the size of the loss. However, it Our article here propose a new coherent risk measure is obvious that the size of loss should be taken into ac- based on relative entropy, which is obtained under the the count(Cherny and Madan, 2008)[6]. Further criticism of theory framework of coherent risk measure from Artzner variance and V @R can be found in Artzner et al.(1997)[2] et al.(1999)[3]. We call this new risk measure iso-entropic as well as in numerous discussions in financial journals. risk measure. It is pointed that this risk measure is not 0-1 At the final of last century, a new very promising method risk measure, so it is a smooth one. And, we prove that at to quantify risk was proposed in the landmark paper by the same level, the iso-entropic risk measure is more large Artzner, Delbaen, Eber and Heath(1999)[3]. They intro- than AV @R for the same position or portfolio. duced the notion of coherent risk measure, and gave the ax- The remainder of the article is organized as follows. ioms for the measure. And later, the coherent risk measure Section 2 give a brief introduction for monetary, convex was extended to the class of convex risk measures in[11, and coherent risk measures; Section 3 propose the iso- 14,15]. Since their seminal work, the theory of coherent entropic risk measure based on relative entropy. Section 4

∗ Corresponding author: e-mail: [email protected]

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compare our iso-entropic risk measure to several important it acceptable. Note that we work with discounted quanti- risk measures in detail. And at last, Section 5 concludes. ties; cf.[3,19] for a discussion of forward risk measures and interest rate ambiguity. Definition 2.4. Acceptance set associated to a risk mea- 2. Acceptance set, monetary, convex, and sure: the acceptance set associated to a risk measure ρis the A coherent risk measures set denoted by ρ and defined by Aρ = inf {X ∈ G |ρ (X) ≤ 0} . Here, We introduce some concepts related to coherent risk Definition 2.5. A measure of risk ρis called a monetary measures. More detail see Follmer¨ and Schied(2008)[13] risk measure if ρ (0) is finite and if ρ satisfies the following and Artzner et al.(1999)[3]. conditions for all X,Y ∈ G. In financial theory, the uncertainty of value for a po- Monotonicity: If X ≤ Y , then ρ (X) ≥ ρ (Y ). sition (a asset or a portfolio) in the future is usually de- Translation invariance: If c ∈ R, then ρ (X + c) = scribed by a random variable X : Ω → R on a probability ρ (X) − c. space (Ω, F,P ), where Ω is a fixed set of scenarios. For The financial meaning of monotonicity is clear: the instance, X can be the (discounted) value of the portfolio downside risk of a position is reduced if the payoff profile or some economic capital. The goal of risk measure is to is increased. Translation invariance is also called cash in- determine a number ρ (X) that quantifies the risk and can variance. This is motivated by the interpretation of ρ (X) serve as a capital requirement, i.e., as the minimal amount as a capital requirement, i.e., ρ (X) is the amount which of capital which, if added to the position and invested in a should be raised in order to make X acceptable from the risk-free manner, makes the position acceptable. The fol- point of view of a investor/supervisor, as Definition 2.3. lowing axiomatic approach to such risk measures was ini- Thus, if the risk-free amount c is appropriately added to tiated in the coherent case by[2,3] and later extended to the the position or to the economic capital, then the capital re- class of convex risk measures in [11,14,15]. In the sequel, quirement is reduced by the same amount. G denotes a given linear space of functions X : Ω → R Definition 2.6. A monetary risk measure ρ is called a containing the constants. Let G be the set of all risks, that convex risk measure if ρ satisfies the following conditions is the set of all real valued functions on Ω. Convexity: ρ (λX + (1 − λ) Y ) ≤ λρ (X)+(1 − λ) ρ (Y ), As Artzner et al.[3] point that a first, crude but crucial, for 0 ≤ λ ≤ 1. measurement of the risk of a position will be whether its The axiom of convexity gives a precise meaning to the future value belongs or does not belong to the subset of idea that diversification should not increase the risk. acceptable risks, as decided by a investor or a supervisor. Definition 2.7. A convex risk measure ρ is called a co- For an unacceptable risk (i.e. a position with an unaccept- herent risk measure if ρ satisfies the following conditions able future value) one remedy may be to alter the position. Positive Homogeneity: if λ ≥ 0, then ρ (λX) = λρ (X). Another remedy is to look for some commonly accepted Under the assumption of positive homogeneity, the con- instruments which, added to the current position, make its vexity of a monetary risk measure is equivalent to future value become acceptable to the investor/supervisor. Subadditivity: ρ (X + Y ) ≤ ρ (X) + ρ (Y ). The current cost of getting enough of this or these instru- So, a coherent risk measure must satisfies four axioms: ment(s) is a good candidate for a measure of risk of the monotonicity, translation invariance, positive homogeneity initially unacceptable position. Based on this, a series of and convexity or subadditivity. definitions are given as follows. Definition 2.1. A measure of risk ρ is a mapping from G into R. 3. Coherent risk based on relative entropy Definition 2.2. An acceptance set: We call A a set of final values, expressed in currency, are accepted by one In this setction, one new coherent risk measure is proposed investor/supervisor. based on the given relative entropy. It must be pointed that there are different acceptance Suppose now that G consists of measureable functions sets for different investors/supervisors because they are het- on (Ω, F). According to the basic representation theorem erogeneous when faced with risk assets. There is a corre- proved by Artzner, Delbaen, Eber, and Heath (1999)[3] for spondence between acceptance sets and measures of risk. a finite Ω and by Delbaen (2002)[8] in the general case, Definition 2.3. Risk measure associated to an accep- any coherent risk measure ρ admits a representation of the tance set: the risk measure associated to the acceptance set form A is the mapping from G into R denoted by ρA and defined ρ(X) = − inf EQ[X]to11.51 (1) by Q∈D

ρA (X) = inf {m ∈ R |m + X ∈ A} . with a certain set D of probability measures absolutely continuous with respect to P . Here, we apply relative en- The risk measure is the smallest amount of units of date tropy to define the set D of probability measures. 0 money which invested in the admissible asset, must be added at date 0 to the planned future net worth X to make D = {Q : H (Q |P ) = H} (2)

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h i dQ dQ Where H (Q |P ) = E dP log dP is relative entropy of Where m = −1/m2. And E [•] = EP [•], the subscript Q ¿ P . Relative entropy is also called Kullback–Leibler is omit in the sequel. We can see the fomula (7) is just a divergence or information divergence. From Jensen’s in- Esscher transformation of p (x)[9]. equality, we know that H (Q |P ) ≥ 0. Here, we can inter- Then, applying iso-relative-entropy condition(the first pret the meaning for the formula (1), (2): the probability constraint), m is determined. Denote f (m) = H (Q0 |P )− P is the observation, and Q may be the true one which H. We need to find m to satisfy f (m) = 0. About zero- generates X, but the investor/supervisor don’t knows Q point for function f (m), we have theorem as follows. well, she only knows the ‘distance’ from P to Q, namely Theorem 3.1. If H = 0,then f (m) has only one zero- Kullback–Leibler divergence here. Kullback–Leibler di- point which is at m = 0; and if H > 0,then f (m) have vergence is just a pseudo-distance, because H (Q |P ) 6= two zero-points which are at m ∈ (0, ∞) and m ∈ (−∞, 0), H (P |Q). Now, the investor/supervisor try to find the worst respectively. expectation of a position (a asset or a portfolio) X given The brief proofs is as follows: R the divergence H (Q |P ) = H from P to Q. Because the f (m) = q (x) log q0(x) dx − H · G 0 p(x) ¸ set D of probability measures is induced under the same −mx −mx relative entropy, so we call it iso-entropy induction set of (−mx−log E[e ])e = E E[e−mx] − H probability measures. Apparently, this is a functional extremum problem with = E [g (m, x)] − H equality constraints. Denote dQ = q (x) dx and dP = ¡ ∂g = q0(x) mx2 + x log E [e−mx] p (x) dx, rewrite the problem (1), (2) as follows: ∂m p(x) ¶ −mx −mx E[xe ] −x − (1 − mx + log E [e ]) E[e−mx] R ¡ q0(x) 2 −mx ρ (X) = − inf Gxq (x) dx = p(x) mx + x log E [e ] − x q(x) −mx Z − (1 − mx + log E [e ]) EQ [x]) q (x) h i ³ £ 0¤ ´ s.t. H = q (x) log dx (3) df(m) = E ∂g = m E x2 − (E [x])2 p (x) dm ∂m Q0 Q0 G Z = mσ2 Q0 1 = q (x) dx Because of σ2 > 0, so there is only one extremum point G Q0 for function f (m),and further it is minimum point, which Where the second constraint must be satisfied naturally. is at m = 0. Because of the minmum of f (m)is f (0) = Now, we use calculus of variations to solve the prob- −H ≤ 0, so we can get the theorem. The proof is com- lem. Write functional with Lagrange multipliers as fol- pleted. lows: From theorem 3.1., we have following corollary. Z Z Corollary 3.2. If H = 0,then functional J has only J (q (x)) = xq (x) dx − m q (x) dx 1 one extremum which is at q0 (x) = p (x); and if H > G G −mx 0,then J have two extremums which are q (x) = p (x) e Z 0 E[e−mx] q (x) at m ∈ (0, ∞) and m ∈ (−∞, 0), respectively. −m2 q (x) log dx (4) G p (x) So we have the following proposition about coherent risk measure based on relative entropy. In which m1, m2 are Lagrange multipliers. The calculus Proposition 3.3. Given the relative entropy H (Q |P ) = of variations of functional J is H, the coherent risk measure has the form £ ¤ ∂J (q(x) + αδq(x)) E Xe−mX δJ (q(x)) = |α=0 ρ (X) = −EQ0 [X] = − −mX (8) Z µ ∂α ¶ E [e ] q(x) = x − m − m − m log δq (x) dx (5) In which m satisfy f (m) = 0, and m ≥ 0. 1 2 2 p (x) G d(EQ0 [X]) The reason for m ≥ 0 is because of dm = According to Lemma of calculus of variations, functional −σ2 . J gets extremum at q (x), then δJ (q (x)) = 0, and so Q0 0 0 And· from iso-relative-entropy¸ condition: we get (−mx−log E[e−mx])e−mx E E[e−mx] = H, q0 (x) x − m1 − m2 − m2 log = 0 (6) the coherent risk measure can take another form p (x) £ ¤ H + log E e−mX Utilizing unitary condition of probability(the second con- ρ (X) = (9) m straint), we get In which, m is determined by H uniquely. For convenience, e−mx we call it iso-entropic risk measure for our new coherent q (x) = p (x) (7) ie 0 E [e−mx] risk measure, and denote ρ (X).

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In order to apply our iso-entropic risk measure sim- Delbaen, Eber, and Heath (1999)[3]. It satisfies four ax- ilar like V @R(Value at Risk) or AV@R (Average Value ioms: monotonicity, translation invariance, positive homo- at Risk), we discuss relative entropy further. Entropy can geneity and subadditivity. From formula (15), we know describe the uncertainty, and relative entropy describe the that it depends only on the tail of the distribution, i.e. it is ‘distance’ or divergence of two uncertaintys. Confidence a 0-1 risk measure, so it is not smooth. In contrast with this, 1 ie level has the similar meaning. So we denote H = log λ , λ ∈ our coherent risk measure based on relative entropy ρλ (X) = (0, 1] . Then we can get iso-entropic risk measure at confi- log 1 +log E[e−mX ] λ depends on the whole dietribution, it dence level λ: m £ ¤ is very smooth. Concerning to V@R, AV@R and our risk log 1 + log E e−mX measure ρie (X), we have the following theorem. ρie (X) = λ (10) λ λ m Theorem 4.1. At same level λ ∈ (0, 1], the following formula exists In which, m is determined by λ uniquely. ie Here, λ serves as the risk aversion parameter. We have ρλ (X) ≥ AV @Rλ (X) ≥ V @Rλ (X) (16) ie ie The brief proofs is as follows: ρλ (X) → −essinfω X(ω) andρ1 (X) = −E [X]. λ↓0 AV @Rλ (X) ≥ V @Rλ (X) is apparent. Let us see why ρie (X) ≥ AV @R (X). When X√ has Gaussian distribution, X ∼ N (µ, σ), λ λ then m = 2H/σ. So we have iso-entropic risk measure Pay attention that the solution of Q foro optimum prob- 1 for Gaussian distribution: lem (14) satisfies dQλ/dP = λ 1X≤zλ(X). And the rela- √ tive entropy from P to Q is: ie · λ ¸ ρλ (X) = σ 2H − µ (11) dQλ dQλ 1 h i H (Qλ |P ) = E log = log = H 2 2 dP dP λ Where µ = E [X] , σ = E (X − µ) . © ª 1 So, Qλis one element of D = Q : H (Q |P ) = H = log λ . However, from the result of section 3, we know that the 4. Comparison for several risk measures optimization for − infQ∈D EQ[X] , ie so we get ρλ (X) ≥ AV @Rλ (X). The proof is com- In this section,we compare our coherent risk measure based pleted. In fact, for arbitrary dQ /dP = 1 1 , if on relative entropy with other several important risk mea- £ ¤ λ λ X∈(a,b) sures. E 1X∈(a,b) = λ, The first one is V @R(Value at Risk), which is the most then h i V @R λ ∈ (0, 1] dQλ dQλ 1 fashionable one now. at level , defined for H (Qλ |P ) = E dP log dP = log λ = H. X on a probability space (Ω, F,P ) is Figure 1. is the illustration of these three risk measures

V @Rλ (X) = inf {m ∈ R |P [X + m < 0] ≤ λ} (12) supposed that X has Gaussian distribution, X ∼ N (µ, σ). Pay attention that when λ → 0, all the three measures V @R satisfies monotonicty, translation invariance and pos- → −essinfωX(ω), but when itive homogeneity, but not subadditivity, so it is just a mon- λ↓0 ie etary risk measure, not a convex one. λ = 1, V @Rλ (X) → −esssupωX (ω) ρ1 (X) = λ↓1 The second risk measure is AV@R (Average Value at AV @R1 (X) = −E [X] . Risk). At level λ ∈ (0, 1], AV@R is defined as The last risk measure is entropic risk measure, defined Z λ by 1 £ ¤ AV @Rλ (X) = V @Rα (X)dα (13) log E e−θX λ 0 ρent (X) = , θ¿0 (17) θ AV@R is also called Conditional Value at Risk, Expected The entropic risk measure satisfies three axioms: mono- Shortfall,or Tail Value at Risk. According to Follmer¨ and tonicity, translation invariance, and convexity but not for Schied(2008)[13], there is another definition for AV@R: positive homogeneity, so it is just a convex risk measure, not a coherent one. Compare it with our iso-entropic risk AV @Rλ(X) = − inf EQ[X] Q∈D measure ρie (X) in formula (9), we will find that they are ½ ¾ λ dQ 1 very analogous in form. In fact, they are diffierent. The pa- for D = Q : ≤ (14) rameter θ in entropic risk measure ρent (X) is free, but pa- dP λ ie rameter m in our iso-entropic risk measure ρλ (X) is not So it gets free, it is defined by H and X, namely m = m (H,X). 1 If X has Gaussian distribution, X ∼ N (µ, σ), then AV @Rλ (X) = E [−X |X ≤ zλ (X)] , entropic risk measure has the form: ρent (X) = θ σ2 − λ 2 µ, θ z (X) = inf [x : F (x) ≥ λ] (15) ¿0. It is interesting to mention here that the iso-entropic λ risk measure used in this paper is different from the quan- Apparently, AV@R is coherent risk measure accord- tum entropy which has been used to measure the entangle- ing to the basic representation theorem proved by Artzner, ment between two or more parties [24-30].

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[3] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. Coher- ent measures of risk, Mathematical Finance 9(3), 203-228. (1999) [4] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. andH.Ku,, Coherent multiperiod risk adjusted values and Bellman’s principle, Ann.Oper.Ees. 52, 5-22 (2007). [5] Ben-Tal,A.,and M.Teboulle, An old-new concept of convex risk measures: the optimized certainty equivalent, Mathe- matical Finance 17, No. 3, 449-476 (2007). [6] Cherny,A.S., and D.B., Madan,Coherent measurement of factor risks, preprint, Moscow State University (2008). [7] Cherny,A.S.,Wheight V@R and its properties, Finance and Stochastics, 10, No. 3, 367-393 (2006). [8] Delbaen,F. Coherent risk measures on general probability spaces, In: K.Sandmann,P.Schonbucher(Eds.).Advances¨ in finance and stochastics., Essays in honour of Dieter Sonder- mann, Springer, 2002, 1-37 (2002). [9] ESSCHER, F. On the probability function in the collective theory of risk. Scandinavian Actuarial Journal, 15, 175-195 (1932). Figure 1 An Illustration of three risk measures under differnt [10] Fischer, T. Risk capital allocation by coherent risk measures level of λ X ∼ N (µ, σ) , σ = 0.3, µ = −0.5, from top to ie based on one-sided moments, Insurance: Mathematics and bottom: ρλ (X) , AV @Rλ (X) ,V @Rλ (X), respectively. Economics, 32, No.1., p.135-146 (2003). [11] Follmer,¨ H. and A.Schied,Convex measures of Risk and trading constraints, Finance and Stochastics,6, No.4, 429- 447 (2002). 5. Conclusion [12] Follmer,¨ H. and A.Schied,Stochastics Finance:An introduc- tion in discrete time,2nd Ed., Walter de Gruyter (2004). This article proposes a new coherent risk measure called [13] Follmer,¨ H. and A.Schied,Convex and coherent risk mea- iso-entropic risk measure. Comparing with several current sures,preprint, Humboldt University (2008). important risk measures, it turns out that this new risk mea- [14] Frittelli, M. and E. Rosazza Gianin, Putting order in risk sure has advantage over the others. It is coherent, it is a measures, Journal of Banking and Finance, 26, No.7 1473- smooth measure, it is a more prudent risk measure than 1486 (2002). AV@R. So it maybe a better coherent risk measure in the [15] Heath, D. Back to the future., Plenary lecture, First World future financial market in my opinion. But, this needs to Congress of the Bachelier Finance Society, Paris (2000). be verified in many ways, including pure theoretical and [16] Heath, D. and H. Ku, Pareto equilibria with coherent mea- empirical work. In particular, the problem of capital allo- sures of risk, Mathematical Finance 14(2), 163-172 (2004). cation, the problem of pricing and hedging, optimal port- [17] Jaschke, S. and U.Kuchler,¨ Coherent risk measures and good folio choice and equilibrium and so on, must be studied deal bounds.,Finance and Stochastics, 5, 181-200 (2001). under our new coherent measure in practice. [18] Jouini, E. M.Meddeb and N.Touzi,Vector-valued coherent risk measures.,Finance and Stochastics, 8, 531-552 (2004). [19] Karoui, E., Ravanelli N. and Cash C. Sub-additive Risk mea- Acknowledgement sures and Interest Rate Ambiguity.To appear in Math. Fi- nance (2009). The authors acknowledge the financial support provided [20] Kusuoka, S. On law invariant coherent risk measures, Ad- vances in Mathematical Economics, 3, 83-95 (2001). by National Natural Science Foundation of China (NSFC), [21] Leitner, J. Balayage monotonous riskmeasures, Interna- project No.71171095, and Foundation of Huazhong Nor- tional Journal of Theoretical and Applied Finance, 7, No.7 mal University,project No.2010TS021. The author is grate- 887-900 (2004). ful to the anonymous referee for a careful checking of the [22] Rockafellar, R. T. and S. Uryasev, Conditional Value-At- details and for helpful comments that improved this paper. Risk for general loss distrubitions,Journal of Banking and Finance, 26 No.7, 1443-1471(2002). [23] Tasche, D. Expected Shortfall and beyond, Journal of Bank- References ing and Finance, 26, No.7, 1519-1533 (2002). [24] M. Sebawe Abdalla and L. Thabet, Appl. Math. Inf. Sci. No. [1] Acerbi, C. and D. Tasche, On the coherence of expected 5, 570.(2011). shortfall, Journal of Banking and Finance, 26, No. 7, 1505- [25] A. El-Barakaty, M. Darwish and A.-S. F. Obada, Purity loss 1518 (2002). for a Cooper pair box interacting dispersively with a non- [2] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. Thinking co- classical field under phase damping, Appl. Math. Inf. Sci. 5, herently, Risk, 10, No. 11, 68-71 (1997). 122-131 (2011).

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[26] Z. Ficek, Quantum Entanglement processing with atoms, Zheng Chengli is one as- Appl. Math. Inf. Sci. 3, 375-393 (2009). sociate professor from School [27] Barry C. Sanders and Jeong San Kim, Monogamy and of Economics, Huanzhong Nor- polygamy of entanglement in multipartite quantum systems, mal University,China. He ob- Appl. Math. Inf. Sci. 4, 281-288 (2010). tained his PhD from Beihang [28] Li-Hui Sun, Gao-Xiang Li and Zbigniew Ficek, Continuous University (China).And he has variables approach to entanglement creation and processing, worked as a postdoctoral fel- Appl. Math. Inf. Sci. 4, 315-339 (2010). low in Peking University(China). [29] Barry C. Sanders, Algorithm for Quantum Simulation, Appl. He is an active researcher in Math. Inf. Sci. 3, 117-122 (2009). financial mathmatics,financial [30] H. Eleuch, Quantum trajectories and autocorrelation func- engineering and risk manage- tion in semiconductor microcavity, Appl. Math. Inf. Sci. 3, ment, and has published more than 30 research articles in 185-196 (2009). journals of the related fields.

Chen Yan is a teacher at School of mathmatics and Statis- tics,Economics, Huanzhong Nor- mal University,China. She grad- uated from Yangtze University (China).Her main research in- volves finance management and accounting, and has published more than 10 research articles in journals of the related fields.

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