A New Class of Time-Consistent Dynamic Risk Measures and Its Application

Technology and Investment, 2013, 4, 36-41 Published Online February 2013 (http://www.SciRP.org/journal/ti)

A New Class of Time-Consistent Dynamic Risk Measures and its Application

Rui Gao, Zhiping Chen School of Mathematics and statistics, Xi’an Jiaotong University, Xi’an, China Email: [email protected], [email protected]

Received 2012

ABSTRACT We construct a new time consistent dynamic convex cash-subadditive risk measure in this paper. Different from existing measures, both potential loss and volatility of risky objects are considered. Based on a one-period measure that dis- torts financial values, punishes downside risk yet rewards upside potential, a dynamic time consistent version is constructed recursively through a modified translation property. We then establish a portfolio selection model and give its optimal condition.

Keywords: Dynamic Risk Measure; Time-Consistent; Cash-Subadditive; Portfolio Optimization; Stochastic Programming

1. Introduction and its measure, we should consider both the potential loss and volatility into consideration and distinguish Financial activity is teemed with risk, therefore it is between downside risk and upside potential. crucial to construct reasonable risk measures and utilize Generally speaking, an ideal risk measure should them on the optimal portfolio selection. One popular satisfy some properties. In their seminal paper [3], definition of risk is volatility of random return of Artzner, Delbaen, Eber, Heath established an axiomatic portfolio, originated from Markowitz’s prominent notion of coherent risk measures. They proposed that an mean-variance model. Following him, hundreds of ideal risk measure should satisfy four properties: moment-based risk measures were proposed, such as monotonicity, subadditivity, positive homogeneity and Mean Absolute Deviation [1] and Lower Partial Moment translation-invariance. Though it has been accepted by [2]. Another common notion of risk is potential downside many scholars, it is not perfect. For example, positive loss below a certain target. Correspondingly, a number of homogeneity sometimes does not hold because a downside risk measures has been suggested in the financial position’s risk increases in a nonlinear way with literature, such widely used measures as Value-at-Risk its volume due to liquidity risk. Hence, Follmer and (VaR) and Conditional Value-at-Risk (CVaR). In all the Schied [4] replaced subadditivity and positive above financial risk measures, the attention is put on homogeneity by convexity and established a more either the volatility of random return or the potential loss. general concept of convex risk measures. In addition, Nevertheless, this is insufficient. Both these two aspects translation-invariance is questioned in [5] since the should be considered simultaneously. Only concentrating ambiguity on interest rates and is suggested to be on the volatility of return ignores the information on the replaced with cash-subadditivity, which implies that degree of potential loss; while merely emphasizing additional loss of some amount of money is covered by potential loss not only neglects the dispersion of future an additional reserve of the same amount. Hence, we return but also throws away upside data. What’s more, believe that for one-period risk measures it is reasonable when considering the volatility of random return, many to assume monotonicity, convexity and researchers punish both downside risk and upside cash-subadditivity. From the perspective of economics potential. However, the volatility of random return above and finance, an ideal risk measure should reflect certain target implies the potential of gaining much more investor’s risk-averse attitude because risk is always a than expected. A higher upside variability generally subjective notion [6]. indicates a higher possibility to acquire good upside During the recent decade, dynamic risk measure has performance, which is desirable for each rational investor. attracted many researchers and practitioners, of which Hence, upside potential should be rewarded. Based on the most important feature is time consistency describing the analysis above, we believe that when defining risk how risk assessments at different times are interrelated.

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By certain translation property, which corresponds to ured. We use absolute deviation to measure the upside translation-invariance in one-period setting, time potential and downside risk separately. The average consistent risk measures can be completely defined by downside deviation is E[|Xx11−−ααXx≥− ], which conditional risk measures recursively [7]. Thus, an usual −+1 equals to α E[]Xx− . Similarly, the upside potential way to construct time-consistent risk measures is 1−α −+1 establishing a static risk measure first and extending it to is (1 −−α )E[ 1−α Xx ] . Since we punish downside risk dynamic setting by translation property. As illustrated and reward upside potential, the volatility of Xx− 1−α later, existed translation property cannot reflect risk is λα −+11EE[Xx−− ] (1− λ )(1 − α ) − [x − X ] +, where aversion and we will modified it into another version. 11−−αα Bearing in mind the above limitation in existed risk λ reflects the asymmetry between downside and upside. measures, a new class of time consistent dynamic To combine the potential loss and volatility together, a cash-subadditive convex risk measure is constructed. simple way is by linear weight. It is not difficult to show Comparing with existing measures, our new risk measure that to make the weighted sum be monotone, they have to has the following advantages: we take into account the have equal weight. Moreover, taking into risk aversion potential loss and volatility of both downside risk and into consideration, all the financial values aforemen- upside potential, and thus the whole domain distribution tioned, including X and Xx− 1−α , should be distorted by a is utilized, which makes the new measure superior for monotonically decreasing convex function w()· . Such finding robust and stable investment decisions; by property of w indicates that risk-averse investors em- suitably selecting the parameters in the model, our risk phasize more on undesirable situation. For normalization measure can explicitly reflect the investor’s risk attitude; = when the risk measure is applied to portfolio selection condition, we require w(0) 0 . Finally, we define our model, we give its optimality condition, which is useful new risk measure as follows. in determining the stochastic dual dynamic programming Definition 1. Given λ ∈[0,1] , α ∈ (0,1) , the new ∞ method to solve the risk-averse multistage problem. one-period risk measure ρλα, :L ()F →  is defined This paper is organized as follows. Section 2.1 gives as the definition and property of the new one-period risk −+1 ρλα()X=+− wx (−−α )λα E [() wX wx (α )] measure, which is then extended to dynamic setting in ,1 1 (1) −+1 Section 2.2. We apply the risk measure to portfolio −−(1λα )(1 − )E [wx (α ) − wX ( )] selection model in Section 3 and presents our conclusion where x =inf{x ∈ : PX [ ≤≥ x ]α } and w()· is a in Section 4. α monotonically increasing convex continuous function 2. The New Risk Measure and its Properties satisfying normalized condition w(0)= 0 . Rewriting the expectation in Equation (1) in integral 2.1. One-Period Setting form, the measure has an equivalent form which facili- We first consider a one-period framework. Given a tates us to study its property, demonstrating by the fol- probability space (Ω,,PF ) , denote the random cost, lowing proposition: discounted by certain numéraire, of at time T by essen- Proposition 1. For any 0< αλ≤≤ 1, the risk measure tially bounded random variable X in L∞ ()F . A ρλα, defined by (1) can be equivalently written as one-period risk measure is a mapping ρ :L∞ ()F →  . A larger value of ρ implies a riskier cost X . For a ∈  , 1−−λ αλ 1 + ρ = + = λα, ()XE [()] w X w() xu du (2) we denote[]a by max{0,a } . The notation “: ” means 1 −−α αα(1 ) ∫1−α “equal by definition”. For a random variable X , E[]X 1 denotes its expectation; x is the α quantile of X . We Obviously, if λ = 1 , ρ α (= wXx () ) du is α time α 1, ∫1−α u denote the indicator function of set A by 1 . A of TNT ()X defined in [8]; if wx()= x , For reasons demonstrated in the introduction, we pro- α pose here a new type of risk measure that takes into ac- ρλα, ()XX=−+ (1)[] ββE CVaR()α X, where count the potential loss, downside risk, upside potential β=−−(1 λα )(1 ) −1 , is appeared in [9]; further ifα = 1/2, and risk aversion. In order to illustrate the derivation of ρλ= + −++ −− λ− our new risk measure, we look back on the notable λ,1/2 EE[]2[]2(X Xx1/2 1)[] Ex1/2 X is downside risk measure VaR, which is defined the deviation measure suggested in [10]. Therefore, our as VaRαα (Xx ) := 1− . When the random cost X is reduced new risk measure can be regarded as extensions to all by x amount of money, it becomes acceptable in the these risk measures. 1−α The choice of depends on the investor's attitude toward sense that the random cost Xx− α is less than zero up to a risk controls the heavy tails of loss distribution. Typical loss with probabilityα . However, as pointed out before, ++ decreasing convex functions are ββ12[]XX−− [ ] the volatility of such random cost should also be meas-

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β They are less conservative in the sense that they bear a ( ββ12>>0 ), exp(β x )− 1( β > 0 ), x (β ≥ 1) . As for concrete selection of and corresponding parameters, one larger probability of loss. can refer to [8] for a detailed discussion. Stochastic dominance rules are often utilized to The following proposition shows that under certain judge a new risk measure. From the point of view of mild specification, the new risk measure satisfies several utility theory, it is desirable for a risk measure to desirable mathematical properties. preserve second order stochastic dominance (SSD). An equivalent definition of SSD in terms of quantile Proposition 2. For any 0< αλ≤≤ 1 and w()· that is function is the following: XY if and only if differentiable and satisfies 0≤≤w 1' , then the risk SSD p measure defined in Equation (1) is a law-invariant, (x− y ) du ≤ 0, ∀≤ 0 p ≤ 1, ∫0 uu convex, cash-subadditive risk measure. and there is a strict inequality of at least one p . Proof. The first two are direct corollaries of Theorem 1. 0 To prove the cash-subadditivity, we show that Proposition 4. The risk measure ρλα preserves second ρρ(Xm+≤ ) () X + m. Rewriting the new risk , λα,,λα order stochastic dominance. 1 measure as ρφλα = w( x )() u du , where , ∫0 u Proof. It suffices to prove the case when two random  φ():uu= λα /11[0,αα ) () +− ( λ α )/(1 − α )[ ,1] () u, by variables XYSSD whose quantile functions satisfy x≡/ y in any interval (ab,)⊂ [0,1] . Since the quantile ()xm+uu =xm + , we have uu function is right-continuous, there exists only countable 1 ρφλα(X+= m ) w (( x + m ) ) ( u ) du intersection of xu and yu . If there is no intersection, it , ∫0 u 1 follows that x≤ y and thus ρρλα(XY )≤ λα () due = wx(+ m )()φ udu uu ,, ∫0 u to the monotonicity of w()· . Otherwise denote all the 1 =[(w x ) + w′(ξφ ) m] () u du intersection point of x and y by {}u N , 1 ≤N ≤∞, ∫ 0 uu u u nn=1 1 and u ::=0,uk = 1, ( ∈  + ) . Then we have ≤+wx( )()φ ud u m 0 Nk+ ∫0 u N un+1 =ρλα, (X ), + m ρρλα(X )−=λα ( Y ) [ w ( x ) − w ( y )]φ ( u ) du ,,∑∫ u uu n=0 n where ξuu∈−[x mx ,] u is determined by the mean [N /2] u + = 21n [w ( x )− w ( y )]φ ( u ) du value theorem. ∑ ∫u uu n=0 2n

Besides, Equation (1) contains two parameters λ and u22n+ −−φ ∫ [w ( yuu ) w ( x )] ( u ) du α , which can be flexibly reflect investor's attitude u21n+ [N /2] toward risk. λ is a factor linearly adjusting the balance u = ξφ21n+ − between downside risk and upside potential; is the ∑ w'(n )∫ (xuuy ) () u du u2n confidence level that the investors can accept. More n=0 u specifically, we have the following theorem. − ηφ22n+ − w'(n )∫ (y uu x) () udu u21n+

Proposition 3. The coherent risk measure ρλα, is ≤ 0, increasing with respect to λ , decreasing with respect to

α , and continuous with respect to α and λ . where ξn∈[,uu2 nn 21+ ] and ηn∈[,uu21 nn++ 22 ] are determined by the mean value theorem. The last The monotonicity property of ρ with respect to λα, inequality is deduced by the convexity of w()· and the λα/ can be used to reflect the investor’s attitude monotonicity of φ()u . toward risk. Concretely, the increasing property of ρ λα, At the end of this section, we consider the computation with respect to λ indicate that the greater the λ , the and minimization of ρλα, . Let X:= gx (,ω ) be the larger the ρ . Investors who adopt a larger λ treat λα, random cost associated with the decision vector x ∈G, λ X riskier than those who choose a smaller . They representing element of feasible set G, and the random have a stronger tendency to risk aversion because the vector ω , standing for the uncertainties in the market concentrate more on downside risk than on upside which affects the random cost, such as capital gains and potential. When λ = 1, investors only consider downside dividends. Similar to the arguments in [11], we have the ρ risk. On the other hand, the decreasing property of λα, following proposition. with respect to α means that ρλα, with large α Proposition 5. Introducing the following auxiliary should be connected with the less risk-averse investor. function

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1 −−λ λα −+1 ρρtT,(,XXXX t t++ 1 ,,…=+T ) t tT,1 (0,,, XX t …T ) (3) Gλα, ( x ,η )= Ewgx [ ( ( ,ω ))] ++ { ηαE [( wgx ( ( , ω )) − η ) ]}, 11−−αα Condition (3) indicates that the risk of a stochastic then ρ has an equivalent form λα, process XXtT… ),(, from the perspective of time t is the ρη= λα,,(x ) minη∈ Gx λα ( , ). aggregation of its riskless component X t and its risky

Minimization of ρλα, with respect to X ∈X is component ()XXtT+1,…, , and by translation-invariance equivalent to property, the risk of the riskless component X t is its op-

minxx∈GGρηλα, (x )= min(,)η∈× Gx λα , ( , ) . posite of its value. Nevertheless, when taking into ac- Moreover, we have count the interest rate [5] and more importantly, the in- (x∗∗ ,ξ )∈ arg min ⇔ vestor’s risk-averse behavior, it is better to measure risk- (,)x η ∈×G  less object’s risk by a distortion of its value instead of

∗∗ itself, i.e., the cash-invariance and corresponding transla- x∈∈arg minρηλα,, (x ), arg min Gxλα ( , η ). x∈∈G η  tion-invariance should be replaced with ρ()m= wm (),( ρρ X += m ) () X + wm (), m ∈ (4) 2.2. Multi-period setting where w()· is a monotonically decreasing convex conti- In this section we extend our new one-period risk nuous function. The monotonicity of w()· guarantees measure ρλα, to the dynamic setting. Consider a that the smaller the m , the riskier it is, and the convexity Ω T entails risk-aversion. Intuitively speaking, the dynamic filtered probability space (),FF ,(tt ) =1,P , where T −1 risk measure {}ρtT,1 t= induced by conditional risk meas- F1 = ∅ Ω},{ and FT = Ω , and an adapted stochastic T −1 ure sequence {}ρtt=1 should satisfies process X t , t=, … ,1 T , representing discounted random ∞ ρρtt,1++(0,X t 1 )= t ( Xt t +1 ), =…− 1, , T 1, (5) return process. Define the space LLtt=(, Ω F ,)P and which identifies the conditional risk measure ρt with LLtT, = t × L T . The one-period risk measure naturally the dynamic risk measure for two-period process ρ . induces a sequence of one-period conditional risk tt,1+ T −1 According to (4) and (5), translation property (3) is measure {}ρtt=1 :L t→  : modified into for all {}X ∈ L , 1− λ tT1, ρ = t tt()XE [()|] wX F ρ (,XXXwX ,,…= ) () +ρρ( ( XX , …, )), (6) 1−α tTtt, +1 T tt ttTt++1, 1 T t λα− where wt )·( is a monotonically decreasing convex conti- ++−tt inf{ηα−+1E [(wX ( ) η) |]F }, α η∈ ttnuous function. t Based on the above analysis, we define our new A dynamic risk measure for stochastic process dynamic risk measure as follows. XX,,… is a sequence of conditional risk measures tT T Definition 3. Let{}wtt=1 is a sequence of monotonically {}ρ T −1 , where ρ : LL→ assesses the risk of the tT,1 t= tT,, tT t decreasing convex differentiable function satisfying for sequence XXtT,,… from the perspective of time t . Our all1 ≤≤tT, wt (0)= 0 and − ≤≤wt′(1 )0. The new time T −1 consistent dynamic risk measure {}ρ T −1 induced by aim is to construct the dynamic risk measure {}ρtT,1 t= tT,1 t= T −1 one-period risk measure (1) through modified translation based on the conditional risk measure {}ρ . tt=1 property (6) is recursively defined as The key to constructing dynamic risk measures from ρ= + ρρ …= …− (7) one-period ones is the translation property, which arises tT, wt t()X t t ( t++1, T(X t 1 ,, XTT) ),1, ,1 from translation-invariance property of one-period risk As aforementioned, time consistency is the most measure. A static risk measure ρ satisfies important issue of dynamic risk measures. One of the most commonly used versions is introduced in [12]. translation-invariance property if for all m ∈  , T −1 ρρ(Xm+= ) () X + m, which implies cash-invariance, Definition 2. A dynamic risk measure{}ρtT,1 t= is time T i.e., ρ()mm= . This suggests the risk of a riskless cost, consistent if for all1 ≤≤<τθT and all sequences{}X tt=1 , T in terms of potential loss, can be described as the its {}Yt t=1,∈ L tT, present value. Then the corresponding translation X kk=Yk, =…−τθ , , 1 and property in existed papers is for all {}X tT∈ L1, and (8) ρρθ,,T(XX θ ,,…≤ TT )θθ (,,) Y …Y T = 1,…−,1Tt ,

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imply ρρττ,,T(,,Xt …≤ X TT ) (,,) YYt … T. decisions xt satisfies nonanticipativity (implementable) constraints, i.e., x is a function of information available This definition is intuitive since it indicates that if t at the current stage, say the process (r1,,… rt ) . We the future subsequence of sequence X t is at least as good assume there are no short sales or borrowing: xit ≥ 0 for as the subsequence of another sequence Yt and today’s value of X is the same as that of Y , then X is at least all i and t . Our goal is minimizing the risk of the whole t t t process over all the implementable and feasible policies, as good as Yt from the perspective of today. We show that measured by our new dynamic risk our new risk measure satisfies time consistency. measure ρ1,TT(,,XX 1 … ). T −1 We write the dynamic programming equations for Proposition 6. Suppose a dynamic risk measure{}ρtT,1 t= = − … satisfies condition (6)-(7), then it is time consistent if and the multistage problem. At the stage tT1 1,, , a realization of []t :(,=rr 1 … ,) rt is known. We solve the only if for all1 ≤≤<τθT and all{}X tT∈ L1, , problem n ρττ,TT(XXX ,,…… θ ,, ) = (9) = + −1 Vxt(xw t−+11 ):min t (∑ it )V t ( x t ) ρρ(X ,,… Xw , ( X ,,…… X ),0,,0) xt ≥0 τ,T τ θθθ− 1,TT θ i=1 (10) nn = + where wτ ()· satisfies properties stated in Proposition 2. s.t. ∑∑xit, (1 rxit,) it ,1− ii=11= T T where Proof. Suppose sequences {}X tt=1 , {}Yt t=1,∈ L tT satisfy n T −1 the Equation (9), by the monotonicity of {}ρtT,1 t= and VVtt+1(x ) := ρρ tttTT[ V + 1 ( x )], ( x −− 11 )= T()∑ (1 + rx iTT ) −1 i=1 , it follows that wθ By conjugate duality theory (c.f. [9]), we can prove the following optimal condition of problem (11). ρρτ,T(XX τ ,,… θθ−1, , TT ( XX θ ,, …… ),0,,0) . ∗ Proposition 7. x is an optimal solution of (11) if and ≤ρρττ,T(,,YY … θ−1, , θθ TT (,,),0,,0) YY …… t ∗ only if there exists π tt∈ D()x −1 such that If identity (10) holds, then ∗∗ 0∈∂Lxtt(,π t ), where D()xt −1 is the set of optimal ρρ…≤ …. τ,,T(X τ ,, X TT )ττ (,,) YY T solutions of the dual problem

n nn

3. Portfolio Selection Model and Optimal max ππ(1+ r ) x −sup x −−wxV+ ( x), π { t ∑ it it [t∑∑ it t ( itt ) t 1 ]} Condition t i=1 xt ii=11= and L(, x π ) is the Lagrangian Based on the new dynamic risk measure, we establish a tt t n nn multistage portfolio selection model in this section. ππ= ++ +− Lt(,)x t t w t()(∑ x itV t+−1( x t ) t ∑∑ (1rxit ) i,1 t xt ), a Suppose we have initial capital X1 in n assets at i=1 ii= 11= stage 1, each of which has respective net expected return nd the subdifferential of f at x is denoted as ∂fx(). rate rt=(,rr1 t … , nt )at stage t=, … ,2 T , forming a Further, the function Vt is differentiable at xt −1 random process with a known distribution (for example, and ∂=Vx()D () x 11 , where 11 is a n can be determined by Vector Auto Regression model tt−−11 tt dimensional vector whose elements are 1. VAR() p ). We assume this process is stagewise Proof. First, the function V is convex for t=, … ,1 T . independent, i.e., r is independent of r,,… r for t t 11t − Indeed, since w and ρ are convex and increasing, t=, … ,2 T . This assumption is not very realistic but in T T the convexity of V follows the fact that the most cases, we can transform the across stage dependent T process into stagewise independent by adding state composition of increasing convex function is still convex variables to the model (cf. [13]). Suppose further a and that the minimum function preserves convexity. By self-balance model, that is, we reallocate our portfolio at induction and increasing and convex property of ρt , = …− each stage 1, ,1Tt , but without investing convexity of Vtt ,1= T − ,,… 1 is obtained. Next, since additional money during the time period. At each stage t , Vt is finite and continuous with respect to xt −1 , by …= we decide the amount of the n assets xt(,1 t,) xx nt , conjugate duality theory (c.f. Theorem 7.8 in [9]), we satisfying the balance of wealth constraints obtain the result. nn The optimal condition and the subdifferential of V is = + = …− t ∑∑xit (1 rxit ) i,1 t− , 1, ,1Tt . The sequence of ii=11 = very useful practically. For example, it has been pointed

R. GAO, Z. CHEN out in [13] that the complexity of Sample Average Journal of Financial Economics 5, no. 2 (1977): 189-200. Approximation (SAA) method for solving multistage doi:10.1016/0304-405X(77)90017-4 stochastic programming grows exponentially in the [3] Artzner, Philippe, Freddy Delbaen, Jean‐Marc Eber, and number of scenarios and stages. An tractable way to David Heath. "Coherent measures of risk." Mathematical solve the SAA problem approximately is by stochastic finance 9, no. 3 (1999): 203-228. dual dynamic programming (SDDP) method. The doi:10.1111/1467-9965.00068 subdifferential of Vt is critical when deciding the SDDP [4] Föllmer, Hans, and Alexander Schied. "Convex measures algorithm. Indeed, we can modify the SDDP algorithm of risk and trading constraints." Finance and Stochastics 6, no. 4 (2002): 429-447. doi:10.1007/s007800200072 easily in [13] based on ∂Vt . [5] El Karoui, Nicole, and Claudia Ravanelli. "Cash subadditive risk measures and interest rate ambiguity." Mathe- 4. Conclusion matical Finance 19, no. 4 (2009): 561-590. By linearly combining the downside measure and doi:10.1111/j.1467-9965.2009.00380.x dispersion measure which punishes downside risk and [6] Rachev, Svetlozar, Sergio Ortobelli, Stoyan Stoyanov, J. rewards upside potential together, meanwhile distorting FABOZZI FRANK, and Almira Biglova. "Desirable the financial value, this paper proposes a new class of properties of an ideal risk measure in portfolio theory." International Journal of Theoretical and Applied Finance one-period risk measure. The new static measure satisfies 11, no. 01 (2008): 19-54. convexity, cash-subadditivity, preserves second order doi:10.1142/S0219024908004713 stochastic dominance, and can reflect investor’s risk [7] Cheridito, Patrick, Freddy Delbaen, and Michael Kupper. attitude. Based on this static measure, we then construct a "Dynamic monetary risk measures for bounded dis- dynamic time consistent risk measure using a modified crete-time processes." Electronic Journal of Probability translation property. Under this new dynamic measure, 11, no. 3 (2006): 57-106. doi:10.1214/EJP.v11-302 we establish a portfolio selection model whose goal is to [8] Chen Zhiping, Li Yang, Daobao Xu, and Qianhui Hu. minimize the risk of the whole process. By conjugate "Tail nonlinearly transformed risk measure and its appli- duality theory, we derive its optimal condition, which cation." OR Spectrum (2011): 1-44. facilitates us implement the SDDP algorithm for solving doi:10.1007/s00291-011-0271-2 the multistage stochastic program. We only consider [9] Shapiro A., Dentcheva D., Ruszczynski A., Lectures on several theoretical property of our new risk measure, stochastic programming: modeling and theory. Society nevertheless, whether such measure is practically ideal for Industrial Mathematics, Vol.9, 2009. needs some empirical research using realistic data. This doi:10.1137/1.9780898718751 issue is left for future research. [10] Denneberg, Dieter. "Premium Calculation." Astin Bulletin 20, no. 2 (1990): 181-190.doi:10.2143/AST.20.2.2005441 5. Acknowledgements [11] Rockafellar, R. Tyrrell, and Stanislav Uryasev. "Optimi- I am grateful to Professor Alexander Shapiro in Georgia zation of conditional value-at-risk." Journal of risk 2 (2000): 21-42. Institute of Technology for his sincere help on my work. [12] Ruszczyński, Andrzej. "Risk-averse dynamic programming for Markov decision processes." Mathematical pro- REFERENCES gramming 125, no. 2 (2010): 235-261. doi:10.1007/s10107-010-0393-3 [1] Konno, Hiroshi, and Hiroaki Yamazaki. "Mean-absolute deviation portfolio optimization model and its applica- [13] Alexander Shapiro, Wajdi Tekaya, Joari Paulo da Costa, tions to Tokyo stock market." Management science 37, no. Murilo Pereira Soares, "Risk neutral and risk averse Sto- 5 (1991): 519-531. doi:10.1287/mnsc.37.5.519 chastic Dual Dynamic Programming method", European Journal of Operations Research, vol. 224, pp. 375-391, [2] Bawa, Vijay S., and Eric B. Lindenberg. "Capital market 2013. doi:10.1016/j.ejor.2012.08.022 equilibrium in a mean-lower partial moment framework."