Acceptability maximization
aa r X i v : . [ q -f i n . M F ] D ec Acceptability Maximization
Gabriela Kov´aˇcov´a ∗ Birgit Rudloff ∗ Igor Cialenco † December 23, 2020
Abstract : The aim of this paper is to study the optimal investment problem by using coher-ent acceptability indices (CAIs) as a tool to measure the portfolio performance.We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates theoriginal problem by a sequence of risk minimization problems. The results are ap-plied to several important CAIs, such as the gain-to-loss ratio, the risk-adjustedreturn on capital and the tail-value-at-risk based CAI. In the second part of thepaper we investigate the acceptability maximization in a discrete time dynamicsetup. Using robust representations of CAIs in terms of a family of dynamiccoherent risk measures (DCRMs), we establish an intriguing dichotomy: if thecorresponding family of DCRMs is recursive (i.e. strongly time consistent) andassuming some recursive structure of the market model, then the acceptabilitymaximization problem reduces to just a one period problem and the maximalacceptability is constant across all states and times. On the other hand, if thefamily of DCRMs is not recursive, which is often the case, then the accept-ability maximization problem ordinarily is a time-inconsistent stochastic controlproblem, similar to the classical mean-variance criteria. To overcome this formof time-inconsistency, we adapt to our setup the set-valued Bellman’s principlerecently proposed in [KR19] applied to two particular dynamic CAIs - the dy-namic risk-adjusted return on capital and the dynamic gain-to-loss ratio. Theobtained theoretical results are illustrated via numerical examples that include,in particular, the computation of the intermediate mean-risk efficient frontiers.
Keywords: acceptability index, acceptability maximization, optimal portfolio, gain-to-lossratio, dynamic performance measures, tail-value-at-risk, set-valued Bellman prin-ciple
MSC2010:
The renowned Sharpe ratio, introduced in [Sha64], besides being one of the best known tools inmeasuring the performance of financial portfolios, also played an important role in developing theportfolio optimization theory. It is well-known that one of the major shortcomings of the Sharpe ratio,as a performance measure, is its lack of monotonicity, i.e. a portfolio with strictly larger future gainsmay have a smaller Sharpe ratio. Over the years, in particular to overcome this drawback, a numberof performance measures were introduced such as the Gini ratio [SY84], the MAD ratio [KY91], the ∗ Vienna University of Economics and Business, Institute for Statistics and Mathematics, Vienna A-1020, AUT, [email protected] and [email protected] . † Department of Applied Mathematics, Illinois Institute of Technology, 10 W 32nd Str, Building RE, Room 220,Chicago, IL 60616, USA, [email protected] +
05] for a comprehensive survey of ratio type performance measures. Thisnaturally raises the question which properties – and, consequently, which performance measures – aredesirable. Cherny and Madan [CM09] took an axiomatic approach to performance measurement, inline with the classical axiomatic approach to risk measurement by Artzner et al. [ADEH99]. In [CM09]the authors introduce the concept of a coherent acceptability index (CAI) as a measure of performancethat satisfies a set of desirable properties – monotonicity, quasi-concavity, scale invariance and theFatou property. It was proved that an unbounded CAI α admits a robust representation in terms ofa family of coherent risk measures (CRMs) ( ρ x ) x ∈ (0 , ∞ ) given by α ( D ) = sup { x ∈ (0 , ∞ ) : ρ x ( D ) ≤ } . (1.1)Equivalently, the representation (1.1) can be stated in terms of a family of acceptance cones, or interms of a family of sets of probability measures. The concept of a coherent acceptability index in adynamic setup was first studied in [BCZ14], and consequently in [BCDK16, BCC15, RGS13, BBN14].The main goal of this paper is to study the optimization problem of the formmax D ∈D α ( D ) , (1.2)where α is a static or dynamic CAI and D is a set of feasible positions. We refer to this problemas acceptability maximization . The obtained results contribute to the rich literature on optimal port-folio selection or optimization of performance, such as classical mean-variance portfolio analysis orSharpe ratio maximization (cf. [AN04, GISW02]). Clearly, acceptability maximization is a practicallyimportant and natural problem to study. Closest to the spirit of our study is [EM14], where theauthors solve an acceptability maximization problem for some specific choices of static CAIs (AIMIN,AIMAX, AIMINMAX and AIMAXMIN) which are represented through families of distortion func-tions (or Choquet integrals). To the best of our knowledge, this is the only work that considersoptimal control problems with CAI criteria. The proposed solution in [EM14] fundamentally relieson the representation of CAIs in terms of distortion functions.In contrast to [EM14], in this work we mostly exploit the representation (1.1), and we considerboth the static and the dynamic case. We start by considering the one-period (static) setup, presentedin Section 2. First, we recall some key definitions and relevant results on CAIs (Section 2.1), andthen we propose a numerical algorithm for solving (1.2) that approximates the maximal acceptabilityby a sequence of risk minimization problems (Section 2.2). In Section 2.3 we apply this algorithmto several important CAIs, such as the gain-to-loss ratio, the risk-adjusted return on capital and thetail-value-at-risk based CAI.Undoubtedly, for many practical purposes, acceptability or performance needs to be measuredin a multi-period setting where the investor dynamically rebalances her portfolio. We study this inSection 3. As was noted in [BCC15] and later systemically addressed in [BCP17, BCP18], the timeconsistency property stays at the heart of the matter when studying dynamic coherent acceptabilityindices (DCAIs) and their robust representations of the form (1.1) in terms of families of dynamiccoherent risk measures (DCRMs) ( ρ xt ) x> t =0 ,...,T . We prove that if for every x > ρ xt ) t =0 ,...,T is recursive (or strongly time consistent as a DCRM), and assuming some recursive structure of theunderlying market model, then the maximal acceptability is constant across all times and states.Thus, in this case, it is enough to solve the corresponding optimization problem only once, at onestate and one period of time; see Section 3.2.1 for details. However, in many relevant examples ofDCAIs, ( ρ xt ) t =0 ,...,T are not strongly time consistent, and, similar to the classical control problems2ith mean-variance criteria, problem (1.2) is time-inconsistent in the sense that the (naive) dynamicprogramming principles does not hold true. To overcome this challenge, we adapt to our setupthe set-valued Bellman’s principle recently proposed in [KR19]. This approach also provides theintermediate mean-risk efficient frontiers, in the spirit of the mean-variance efficient frontier. Withinthis approach, we consider two specific performance measures - the dynamic risk-adjusted return oncapital (Section 3.2.2) and the dynamic gain-to-loss ratio (Section 3.2.3). The majority of the proofsare deferred to the appendices.As mentioned above, this is the first attempt to study stochastic control problems with dynamicCAI criteria. While the static case is now relatively well understood, the acceptability maximizationproblem in a dynamic setup appears to be an interesting research area, with many open problems,primarily due to the time-inconsistent nature of such problems, as argued in this manuscript. Inparticular, it would be far-reaching to develop a Bellman’s principle of optimality for a class of DCAIs,beyond particular examples. In addition, from a practical point of view, it would be important tostudy the acceptability maximization problem for a larger classes of indices, for example not necessarilycoherent ones. The authors plan to treat these problems in future works. In this section, we consider the static setting, before moving to the dynamic one in Section 3. First,we recall the definition of a coherent acceptability index and its connection to coherent risk mea-sures. This serves as our framework for studying the maximization of performance, i.e., acceptabilitymaximization. In Subsection 2.2 we provide a way to solve the acceptability maximization problemthrough a sequence of risk minimizations. At the end of the section, we provide examples of thisapproach. Proofs can be found in Appendix A.
We start by briefly reviewing the notion of a (static) coherent acceptability index (CAI) and itsconnection to (static) coherent risk measures (CRMs), following [CM09]. The concept of acceptabilitywas developed as a methodology to define axiomatically minimal desirable properties of a functionalthat is meant to measure or assess the performance of a financial position or trading portfolio. Asusual, we consider an underlying probability space (Ω , F , P ), and we denote by L ∞ := L ∞ (Ω , F , P )the space of essentially bounded random variables on this space. In what follows, all equalities andinequalities between random variables will be understood in a P almost surely sense. In this section, anelement D ∈ L ∞ can be viewed as a discounted terminal cash flow of a zero-cost self-financed portfolio,or the terminal profit and loss (P&L) of a financial position. The mapping D α ( D ) ∈ [0 , ∞ ] assignsto the portfolio D the degree of its acceptability, with higher values corresponding to more desirablepositions. Definition 2.1. A coherent acceptability index (CAI) is a function α : L ∞ → [0 , ∞ ] satisfyingfor all positions D, D ′ ∈ L ∞ and any level x ∈ (0 , ∞ ) (A1) Monotonicity: if D ≤ D ′ , then α ( D ) ≤ α ( D ′ ) ,(A2) Scale invariance: α ( λD ) = α ( D ) for all λ > ,(A3) Quasi-concavity: if α ( D ) ≥ x and α ( D ′ ) ≥ x , then α ( λD + (1 − λ ) D ′ ) ≥ x for all λ ∈ [0 , , A4) Fatou property: if | D n | ≤ , α ( D n ) ≥ x for all n ≥ , and D n P → D , then α ( D ) ≥ x . In [CM09] four further properties – law invariance, consistency with the second order stochasticdominance, arbitrage consistency and expectation consistency – are discussed. These are not requiredfor the coherent acceptability index, but, as the authors argue, they are desirable. Additionaly,coherent acceptability indices are closely related to coherent risk measures, a concept introducedin [ADEH99].
Definition 2.2. A coherent risk measure (CRM) is a function ρ : L ∞ → R satisfying for any D, D ′ ∈ L ∞ (R1) Monotonicity: if D ≤ D ′ , then ρ ( D ) ≥ ρ ( D ′ ) ,(R2) Positive homogeneity: ρ ( λD ) = λρ ( D ) for all λ > ,(R3) Translation invariance: ρ ( D + k ) = ρ ( D ) − k for all k ∈ R ,(R4) Subadditivity: ρ ( D + D ′ ) ≤ ρ ( D ) + ρ ( D ′ ) ,(R5) Fatou property: if | D n | ≤ , for all n ≥ , and D n P → D , then ρ ( D ) ≤ lim inf n →∞ ρ ( D n ) . A family of coherent risk measures ( ρ x ) x ∈ (0 , ∞ ) is called increasing if x ≥ y > ρ x ( D ) ≥ ρ y ( D ) for any D ∈ L ∞ .As was proved in [CM09], there is a strong connection between CAIs and increasing families ofCRMs. Namely, the following robust representation type result holds true: A map α : L ∞ → [0 , ∞ ],unbounded from above, is a coherent acceptability index if and only if there exists an increasing familyof coherent risk measures ( ρ x ) x ∈ (0 , ∞ ) such that α ( D ) = sup { x ∈ (0 , ∞ ) : ρ x ( D ) ≤ } (2.1)with the convention sup ∅ = 0. Equivalently, the representation (2.1) can be formulated in terms of afamily of acceptance sets, or an increasing family of sets of probability measures associated with dualrepresentations of CRMs (see also Section 2.3). For various degrees of generalization of (2.1) see, forinstance, [MS16, BCDK16, BCC15, RGS13, BBN14]. We fix a set
D ⊂ L ∞ of available or feasible positions. For example, D could be the P&Ls of portfoliosthat satisfy certain trading or other constraints. Our aim is to identify among the available positionsthe ones with the highest degree of acceptability. Namely, we wish to solve the following optimizationproblem, max D ∈D α ( D ) . ( A )We denote the maximal acceptability by α ∗ := sup D ∈D α ( D ) (2.2)4nd the set of maximally acceptable (optimal) portfolios by D ∗ := { D ∈ D : α ( D ) = α ∗ } . Generallyspeaking, a maximally acceptable portfolio may not exist, i.e. the set D ∗ is empty if α ∗ is not attainedas a maximum. For ǫ > ǫ -optimal positions D ǫ := { D ∈ D : α ( D ) ≥ α ∗ − ǫ } . The following Lemma summarizes the properties of these sets.
Lemma 2.3.
1. If the feasible set D is convex, then the sets D ∗ and D ǫ are convex, for any ǫ > .2. The sets D ∗ and D ǫ are nested: D ∗ ⊆ D ǫ ⊆ D ǫ , for any ǫ > ǫ > .3. D ∗ = T ǫ> D ǫ .Proof. The proof is deferred to Appendix A.To solve the maximization problem ( A ), we will use the robust representation (2.1), with ( ρ x ) x ∈ (0 , ∞ ) denoting the corresponding increasing family of CRMs. For a given level x > D , min D ∈D ρ x ( D ) . ( P x )We denote the optimal value of the risk minimization problem ( P x ) by p ( x ) := inf D ∈D ρ x ( D ), and itsoptimal solution by D x ∈ arg min D ∈D ρ x ( D ) , assuming that the infimum p ( x ) is attained. In what followswe make the following standing assumption: Assumption 2.4.
The acceptability index α is unbounded from above, and for every x ∈ (0 , ∞ ) therisk minimization problem ( P x ) attains its minimum.The unboundness from above of α is usually satisfied in all practically important cases, whileattainability of the min in ( P x ) can be guaranteed, for example, by assuming that D is compact.With this at hand, and in view of (2.1), we note that: • if the risk minimization problem ( P x ) has a positive optimal value, then no portfolio in D hasa degree of acceptability above x ; • if the risk minimization problem ( P x ) has a non-positive optimal value, then some portfolio in D has a degree of acceptability of at least x .The next result summarizes the above observations, which are used in developing Algorithm 1that solves the acceptability maximization problem ( A ). Lemma 2.5.
Let Assumption 2.4 hold and let x ∈ (0 , ∞ ) .1. If p ( x ) > , then x ≥ α ∗ .
2. If p ( x ) ≤ , then x ≤ α ∗ .
3. If x < α ∗ , then p ( y ) ≤ for all y ≤ x. . If x > α ∗ , then p ( y ) > for all y ≥ x. Proof.
The proof is given in Appendix A.The main idea of the proposed numerical solution of ( A ) is to approximate the maximal accept-ability by a sequence of risk minimization problems ( P x ) for some appropriately chosen levels - avariation of the bisection method that will find a pair ( x, D ) maximizing the level x while satisfying ρ x ( D ) ≤
0. First, find two levels with opposite signs of the minimal risk p ( x ), namely find a lowerand upper bound on the maximal acceptability α ∗ . Then, iteratively decrease the distance betweenthe two bounds. This replaces one acceptability maximization problem ( A ) with a sequence of riskminimization problems ( P x ). This becomes particularly useful if the acceptability maximization iscomplicated and the risk minimization is easier to solve. Note that if the feasible set D is convex,then the risk minimization becomes a convex optimization problem. Input: initial level x ∈ (0 , ∞ ), max. iterations ¯ M ∈ N of Step 1, tolerance ǫ > Step 1: Find an initial interval x L ≤ α ∗ ≤ x U Set n := 0; x L := 0; x U := ∞ ; ¯ D := null ; while ( x L == 0 or x U == ∞ ) and ( n < ¯ M ) doif The optimal value of (P x n ) is positive then x U := x n , select x n +1 := x U / else x L := x n , select x n +1 := 2 · x L , assign ¯ D := D x n ; end n := n + 1 ; end Step 2: Decrease the length of the interval [ x L , x U ] (via bisection) while ( x U − x L ≥ ǫ ) and ( n < ¯ M ) do Select x := ( x U + x L ) / if The optimal value of ( P x ) is positive then x U := x ; else x L := x, assign ¯ D := D x ; endendOutput: Interval [ x L , x U ] as an approximation to α ∗ , portfolio ¯ D as approximately optimal one Algorithm 1:
Approximating maximal acceptability α ∗ via risk minimizationThe next result summarizes the key features of Algorithm 1. Lemma 2.6.
Suppose that Assumption 2.4 holds, and let x ∈ (0 , ∞ ) be the initial (seed) value, ¯ M ∈ N be the maximal number of iterations (of Step 1) and ǫ > be the tolerance level. Denote α := x · − ¯ M +1 , α := x · ¯ M − .
1. If α ∗ ∈ [0 , α ) , then Algorithm 1 returns α as an upper bound for α ∗ , no acceptable portfolio isfound.2. If α ∗ ∈ ( α, ∞ ] , then Algorithm 1 returns α as a lower bound for α ∗ and a portfolio with (atleast) this degree of acceptability. . If α ∗ ∈ ( α, α ) , then(a) Algorithm 1 returns bounds x L and x U such that x L ≤ α ∗ ≤ x U and x U − x L < ǫ ,(b) Algorithm 1 returns an ǫ -solution, i.e. ¯ D ∈ D ǫ ,(c) Step 2 of Algorithm 1 terminates after at most (cid:6) log x ǫ + ¯ M − (cid:7) iterations.Proof. The proof is postponed to Appendix A.
Remark 2.7.
Several comments are in order:(i) In Step 1, instead of the halving ( x n +1 := x U / x n +1 := 2 · x L ), onecould select any x n +1 < x U , respectively x n +1 > x L . Similarly, in Step 2 one could replace thebisection with any choice of x ∈ ( x L , x U ) . The results of Lemma 2.6 would differ in the interval( α, α ) on which α ∗ is identified and the number of iterations.(ii) The case α ∗ = x is not included in Lemma 2.5. If α ∗ = x , then we can only say that p ( y ) ≤ y < x . The sign of p ( α ∗ ) is not clear, since we do not know if the suprema in (2.2) and (2.1)are attained as maxima. Consequently, in the cases α ∗ ∈ { α, α } we cannot derive the behaviourof Algorithm 1.(iii) Assumption 2.4 allows us to merge the case p ( x ) = 0 with the case p ( x ) <
0. Without it,we would need to distinguish between attained and not attained infimum for p ( x ) = 0. If theinfimum p ( x ) = 0 is attained for some portfolio ˜ D ∈ D , then ρ x ( ˜ D ) = 0 and x is a lower boundon α ∗ . If the infimum is not attained, then for all positions D ∈ D we have ρ x ( D ) > , so x isan upper bound on α ∗ . This distinction would need to be built into Algorithm 1. Alternatively,if p ( x ) = 0, this level x could be discarded and the iteration repeated with some other choice oflevel in the appropriate interval. However, it is not clear how many such repetitions might beneeded.(iv) Instead of specifying the maximal number of iterations ¯ M and the initial value x we coulddirectly specify the interval [ α, α ] in which the optimal α ∗ would be searched. Then, Step 1of the algorithm would need to check the signs of p ( α ) and p ( α ) and terminate immediately if α ∗ lies in the interval [0 , α ) or ( α, ∞ ] . Step 2 would require (cid:6) log α − αǫ (cid:7) iterations to terminate,assuming bisection steps. This might be of interest especially if the risk measure is well definedalso for the limiting cases x = 0 and x = ∞ , see Section 2.3 for an example.Algorithm 1 with tolerance ǫ outputs an ǫ -solution to the acceptability maximization problem, anelement of the set D ǫ . We also know that the sets of ǫ -solutions are nested and intersect in the setof optimal solutions. Therefore, it is natural to ask about the convergence of the algorithm outputas the tolerance ǫ vanishes. As the next result shows, such convergence holds true if the feasible set D is compact. Generally speaking, for a non-compact D it is possible to construct counter-examples,where the ǫ -solutions D ǫ ∈ D ǫ converge to an (infeasible) element outside of a (non-empty) optimalset D ∗ , or diverge. Lemma 2.8.
Let α ∗ ∈ ( α, α ) and suppose that the feasible set D is a compact set w.r.t. the topologyof convergence in probability. Let { D ǫ n } n ∈ N be a sequence of solutions outputed from Algorithm 1 for asequence of tolerances { ǫ n } n ∈ N with lim n →∞ ǫ n = 0 . Then { D ǫ n } n ∈ N has an P -a.s. convergent subsequencewhose limit belongs to the set of optimal solutions D ∗ . Proof.
The proof is given in Appendix A. 7 .3 Numerical Examples
We will illustrate the proposed algorithm with three examples of CAIs: the acceptability index corre-sponding to the tail-value-at-risk (AIT), the gain-to-loss ratio (GLR) and the risk-adjusted return oncapital (RAROC). First, we define these CAIs as well as identify the families of risk measures fromthe robust representation (2.1).1. The TV@R at level q ∈ (0 ,
1) is defined asTV@R q ( D ) = 1 q q Z V@R p ( D ) d p, where V@R p ( D ) := inf { r ∈ R : P ( D + r < ≤ p } is the value-at-risk at level p ∈ (0 , D ) := sup n x ∈ (0 , ∞ ) : TV@R x ( D ) ≤ o . It is easy to show that AIT indeed is a CAI that is also law invariant, consistent with second orderstochastic dominance, and arbitrage and expectation consistent; for more details see [CM09,Section 3.5]. However, one notable drawback of AIT is that it ignores the gains and only takesinto account the tail corresponding to losses.2. The gain-to-loss ratio is a CAI, popular among practitioners, and defined as the ratio of themean and the expectation of the negative tail, namelyGLR( D ) := ( E [ D ]) + E [ D − ] , where D − = max { , − D } , D + = max { D, } and the convention a := + ∞ for all a ≥ e q ( D ) of a random variable D at level q ∈ (0 , q E [( D − e q ( D )) + ] = (1 − q ) E [( D − e q ( D )) − ] , or as a minimizer of an asymmetric quadratic loss, see [BDB15] for more details. The expectile-V@R, EV@R q ( D ) := − e q ( D ) , for q ≤ /
2, is an increasing family of CRMs. One can show, for example by using that A EV@R q = n D | E ( D + ) E ( D − ) ≥ − qq o is the acceptance set of EV@R q , that the representation (2.1) forGLR has the form GLR( D ) := sup n x ∈ (0 , ∞ ) : EV@R x ( D ) ≤ o . Alternatively, one can use the system of supporting kernels corresponding to GLR, as well asthe explicit form of the extreme measures, cf. [CM09, Proposition 2]. It is also clear that for8 >
0, GLR( D ) ≥ x if, and only if, E [ − D ] + x E [ D − ] ≤
0, which can be conveniently used forcomputation purposes. This is not linked to the robust representation (2.1) since the mappings D E [ − D ]+ x E [ D − ] are not CRMs (for instance they are not translation invariant). Finally, weremark that there is another popular version of GLR, defined as GLR( D ) = E [ D + ] E [ D − ] . This versionof GLR is also monotone, scale invariant and has the Fatou property, but lacks quasi-concavity,and thus GLR is not coherent. The two are connected via GLR( D ) = max { GLR( D ) − , } .3. The risk-adjusted return on capital, similar to GLR, is a reward-risk type ratio, formally definedas RAROC( D ) := ( E [ D ]) + ( π ( D )) + , (2.3)where π is a fixed CRM. The corresponding family of CRMs is given by ρ x ( D ) = min (cid:26) π ( D ) ,
11 + x E [ − D ] + x x π ( D ) (cid:27) . For risk measures π satisfying E [ − D ] ≤ π ( D ) this simplifies to ρ x ( D ) = x E [ − D ] + x x π ( D ).For more details see [CM09, Section 3.4]. In our numerical examples we will use RAROC with π = TV@R . , also known as the stable tail-adjusted return ratio (see, for instance, [MRS03]).In our numerical examples below, we maximize the acceptability index over the set of profitsand losses that are possible to attain by investing in the market with d (risky) assets. Without lossof generality, thanks to scale invariance of CAIs, we fix the initial investment to 1. For numericaltractability, we assume that Ω is finite. The (gross or total) returns S j /S j , j = 1 , . . . , d , of these d assets are modeled as a matrix R ∈ R d ×| Ω | . Then, the sets of available profits and losses, with orwithout short-selling, become D = { R T h − T h = 1 } or D = { R T h − T h = 1 , h ≥ } , where = (1 , , . . . , T . So, h corresponds to the trading strategy (the amount invested in eachasset) and D ( h ) = R T h − d = 2 assets and with returns given in Table 1, Panel A. Generally speaking, it isreasonable to select the input parameters such that that ǫ ≤ α . Panel B of Table 1 summarizesthe iterations of the algorithm with the following input parameters: the starting (seed) acceptabilitylevel set to x = 2, the tolerance ǫ = 10 − and the maximal number of iterations ¯ M = 15 of Step1. The algorithm outputs bounds on the maximal acceptability and an ǫ -optimal solution h ǫ ; see thelast two rows of Table 1, Panel B. For each of the three acceptability indices, the optimal portfolioputs more weight on the first asset, with AIT being the most balanced and RAROC being the mostextreme. This is because the first asset carries (in some sense) less risk, although at the cost of lowermean return than the second one. All three considered CAIs are loss based measures, but each in adifferent way. The AIT measures how far and how deep into the tail the losses can go. The optimalposition for AIT balances the return in the second and the third state of the world. The GLR treatsloss directly through the expectation of effective losses (the negative part of the P&L). Thus, thecorresponding optimal position is in the range where the portfolio return is negative in only one stateof the world. Since we are using TV@R . in defining the RAROC, only the worst-case scenario(state of the world) is considered, which is the reason why the corresponding optimal position relies9eavily on the first asset, for which the worst-case loss is lower. We also remark that in this marketmodel, the short-selling constraints do not change the results.For the sake of completeness, we also show the iteration of the modified algorithm outlined inRemark 2.7(iv). We use the fact that for each of the three indices – AIT , GLR , RAROC – thecorresponding risk measures are well-defined for the limiting parameter values x = 0 and x = ∞ . Since the bisection cannot be done on an interval of infinite length, we index the families of riskmeasures by a parameter q = x on a bounded interval [0 , , or, respectively by q = x on[0 , . . Then, the bisection is performed with respect to the parameter q . The iterations for GLR arepresented in Table 2, see the modified algorithm. This modification avoids the risk of failing to find alower or upper bound for a badly chosen starting point x (compare to Table 3). Moreover, zero andinfinite acceptability are often determined after solving two risk minimization problems, instead of¯ M .
On the other hand, one needs to treat the tolerance parameter ǫ carefully: although the bisectionis performed on the parameter q , the termination criterion needs to be set on x in order for the errornot to be distorted (see Table 2). A mixed version of the algorithm is also provided – it switchesto a bisection on the original parameter x , as soon as a finite upper bound is found. The iterationsfor GLR are also given in Table 2, see the mixed algorithm. In addition, we also make the followingslight modification to the algorithm: at each iteration a risk minimization problem P ( x ) is solved,finding its optimal solution D x . If the considered level is found to be a lower bound then the maximalacceptability, then the position D x is used for updating the optimal solution of the acceptabilitymaximization. Otherwise, it is not used at all. One can easily see that given a fixed position D x , alllevels y satisfying ρ y ( D x ) ≤ y such that ρ y ( D x ) = 0, then it can be used to update the lower bound. For theCAIs used in this example such level y can be found without any further optimization. We refer tothis modification of the original algorithm as zero-level version.We also run the proposed algorithm on a more realistic market model, consisting of d = 10 stocksand | Ω | = 1000 states of the world. The return matrix is obtained as draws from a multivariateStudent’s t -distribution. In Table 3 we report the results for various input parameters x , ǫ and ¯ M .The results are intuitively clear, and expected: the distance of the initial guess x from the true α ∗ affects the number of iterations needed to find the upper and lower bound (Step 1). The tolerance ǫ determines the number of iterations in Step 2. We also note that for a badly chosen starting pointthe algorithm can fail to find a lower or an upper bound unless ¯ M is increased. We also note thatthe maximal acceptability differs in a market with and without short-selling, but the impact of theparameters is the same. Similar to the toy model, we list in Table 4 the results for different versionsof the algorithm – original one, modified, mixed and zero-level. We also present the results both withand without short-selling constraints. These results show that neither of the versions of the algorithmis performing strictly better than the others. Similar conclusions were observed for various other setsof parameters. In this section, we consider the acceptability maximization problem in a dynamic setting. We usethe theory of dynamic coherent acceptability indices introduced in [BCZ14] and their link to dynamiccoherent risk measures. We briefly recall the key definitions and results from [BCZ14], and thenfocus on acceptability maximization in the context of optimal investment in a multi-period marketmodel. It turns out that the maximal acceptability is constant in a setting when the determiningfamily of dynamic coherent risk measures is recursive and when the underlying market has a recursive10tructure. We conclude by considering the non-recursive case by focusing on two specific performancemeasures – the dynamic risk-adjusted return on capital and the dynamic gain-to-loss ratio – where weuse the specific structure of the problem to introduce a solution scheme tailored to these performancemeasures.
The concept of a coherent acceptability index was first extended to a dynamic setting in [BCZ14]and consequently studied in [BCDK16, BCC15, RGS13, BBN14]. A dynamic coherent acceptabilityindex (DCAI) is meant to measure the performance of financial positions or instruments over time,accounting for the incoming flow of information. We start by briefly recalling the setup of [BCZ14],where DCAIs are designed to measure the performance of (discounted) cash flows or dividend streamsor unrealized P&Ls. Most of the properties from the static setup are naturally transferred to thedynamic case. An addition is the time consistency property, which stays at the core of financialinterpretations of DCAIs, but is also fundamentally used in establishing the dual representations. Werefer to [BCP17, BCP18] for an in-depth discussion of various forms of time-consistency in decisionmaking, in particular those arising in the theory of dynamic risk and performance measures. Following[BCZ14], we take a discrete and finite state setting by denoting T := { , , . . . , T } for some fixed T ∈ N , and letting (Ω , F , F = ( F s ) s ∈T , P ) be a filtered probability space, with P having full support.We will write E t instead of the conditional expectation given F t . Without loss of generality, we willassume that F is trivial. The dividend streams or unrealized P&Ls will be modeled as F -adaptedreal-valued stochastic process D = { D t } Tt =0 . We will denote by D the set of all such processes, andby L t = L t (Ω , F t , P ) the F t -measurable random variables. As usual, for A ⊂ Ω will A denote theindicator function which is equal to one for ω ∈ A and zero otherwise. Without loss of generality, weassume a zero interest rate, or, view D ∈ D as discounted cash flows. Operations between randomvariables, such as minimum, maximum, product, or sum will be understood ω -wise. Definition 3.1. A dynamic coherent acceptability index (DCAI) is a function α : T × D × Ω → [0 , ∞ ] satisfying for all times t ∈ T , all cash flows D, D ′ ∈ D , all events A ∈ F t , and all randomvariables λ ∈ L t (A1) Adaptiveness: α t ( D ) is F t -measurable,(A2) Independence of the past: if A D s = A D ′ s for all s ≥ t , then A α t ( D ) = A α t ( D ′ ) ,(A3) Monotonicity: if D s ≥ D ′ s for all s ≥ t , then α t ( D ) ≥ α t ( D ′ ) ,(A4) Scale invariance: α t ( λD ) = α t ( D ) for all λ > ,(A5) Quasi-concavity: α t ( λD + (1 − λ ) D ′ ) ≥ min { α t ( D ) , α t ( D ′ ) } for ≤ λ ≤ (A6) Translation invariance: α t ( D + m { t } ) = α t ( D + m { s } ) for any m ∈ L t and s ≥ t ,(A7) Dynamic consistency: if D t ≥ ≥ D ′ t and there exists an m ∈ L t such that α t +1 ( D ) ≥ m ≥ α t +1 ( D ′ ) , then α t ( D ) ≥ m ≥ α t ( D ′ ) .A DCAI α is normalized if for all t ∈ T , ω ∈ Ω , there exist D, D ′ ∈ D such that α t ( D, ω ) = + ∞ and α t ( D ′ , ω ) = 0 . It is right-continuous if lim c → + α t ( D + c { t } , ω ) = α t ( D, ω ) for any t ∈ T , D ∈ D , ω ∈ Ω .
11s in the static case, DCAIs are closely related to dynamic coherent risk measures (DCRMs).
Definition 3.2. A dynamic coherent risk measure (DCRM) is a function ρ : T × D × Ω → R satisfying for all times t ∈ T , all cash flows D, D ′ ∈ D , all states ω ∈ Ω , all events A ∈ F t , and allrandom variables λ ∈ L t (R1) Adaptiveness: ρ t ( D ) is F t -measurable,(R2) Independence of the past: if A D s = A D ′ s for all s ≥ t , then A ρ t ( D ) = A ρ t ( D ′ ) ,(R3) Monotonicity: if D s ≥ D ′ s for all s ≥ t , then ρ t ( D ) ≤ ρ t ( D ′ ) ,(R4) Homogeneity: ρ t ( λD ) = λρ t ( D ) for all λ > ,(R5) Subadditivity: ρ t ( D + D ′ ) ≤ ρ t ( D ) + ρ t ( D ′ ) (R6) Translation invariance: ρ t ( D + m { s } ) = ρ t ( D ) − m for any m ∈ L t and s ≥ t ,(R7) Dynamic consistency: A (cid:18) min ω ∈ A ρ t +1 ( D, ω ) − D t (cid:19) ≤ A ρ t ( D ) ≤ A (cid:18) max ω ∈ A ρ t +1 ( D, ω ) − D t (cid:19) . A family of dynamic coherent risk measures ( ρ x ) x ∈ (0 , ∞ ) is called increasing if x ≥ y > implies ρ xt ( D ) ≥ ρ yt ( D ) for any t ∈ T , D ∈ D . It is left-continuous at x > if lim x → x − ρ xt ( D, ω ) = ρ x t ( D, ω ) for any t ∈ T , D ∈ D , ω ∈ Ω . Originally, DCRMs were introduced in [Rie04], although with a different (stronger) notion oftime-consistency, which will be discussed in Section 3.2.1. As proved in [BCZ14], there is a one-to-one relationship between a DCAI and an increasing family of DCRMs, similar to (2.1). Namely, thefollowing assertions hold true:1. For a normalized dynamic coherent acceptability index α the functions ( ρ x ) x ∈ (0 , ∞ ) defined as ρ xt ( D, ω ) = inf { c ∈ R : α t ( D + c { t } , ω ) ≥ x } , . (3.1)form an increasing, left-continuous family of dynamic coherent risk measures.2. For an increasing family of dynamic coherent risk measures ( ρ x ) x ∈ (0 , ∞ ) a function α defined as α t ( D, ω ) = sup { x ∈ (0 , ∞ ) : ρ xt ( D, ω ) ≤ } (3.2)is a normalized, right-continuous dynamic coherent acceptability index. Moreover, there existsan increasing sequence of sets of probability measures {Q xt } t ∈T ,x ≥ such that ρ xt ( D, ω ) = − sup Q ∈Q xt E Q t " T X s = t D s , x > , t ∈ T . (3.3)The converse implication is also true, under an additional technical property of time consistencyof {Q xt } t ∈T , 12. If α is a normalized, right-continuous dynamic coherent acceptability index, then there existsan increasing, left-continuous family of dynamic coherent risk measures ( ρ x ) x ∈ (0 , ∞ ) , such thatrepresentation (3.2) holds. Vice versa, for an increasing, left-continuous family of dynamiccoherent risk measures ( ρ x ) x ∈ (0 , ∞ ) there exists a normalized, right-continuous dynamic coherentacceptability index α , such that (3.1) holds.Similar to the static case, we are interested in finding the position with highest degree of accept-ability. Given a set of available, or feasible, cash flows D ⊆ D , the problem of interest ismax D ∈D α ( D ) . (3.4)It is straightforward to adapt Algorithm 1 to the dynamic setup to solve the corresponding versionof (3.4) for a given t ∈ T and ω ∈ Ω. However, this approach, generally speaking, is computationallynot feasible. Usually one would aim to establish a recursive set of equations in the form of a dynamicprogramming principle or Bellman’s principle of optimality that would solve (3.4), which will beprovided in the next section for the optimal investment problem.
In this section, we consider the acceptability maximization problem in the context of optimal portfolioselection in a market model with d available assets. We denote by R s +1 = ( R s +1 , . . . , R ds +1 ) the vectorof assets (total or gross) returns between time s and time s + 1, namely, if S js denotes the price of the j -th asset at time s , then R js +1 := S js +1 S js . We assume that R , . . . , R T are independent and identicallydistributed on a probability space (Ω , F , P ), and denote by ( F s ) s ∈T the natural filtration generated bythe process ( R s ) s =1 ,...,T . In addition, we assume that all one step asset returns R js are strictly positive.Note that we implicitly assume that these assets do not pay dividends.We assume that the investor starts with a positive initial wealth V >
0, and invests it in the d avail-able assets by following a self-financing trading strategy, possibly with some additional trading con-straints. A trading strategy is an adapted stochastic process h = ( h s ) s =0 ,...,T − with h s = ( h s , . . . , h ds ),where h is is the monetary amount invested in asset i between time s and s + 1. The portfolio value attime s + 1 arising from the trading strategy h is given by V s +1 ( h ) = R T s +1 h s for any s = 0 , . . . , T − V is H ( V ) := { ( h s ) s =0 ,...,T − | T h s = V s , V s +1 = R T s +1 h s , s = 0 , . . . , T − } . Correspondingly, the set of feasible trading strategies with short-selling constraints is H +0 ( V ) = { ( h s ) s =0 ,...,T − | h ∈ H ( V ) , h js ≥ , s = 0 , . . . , T − , j = 1 , . . . , d } . The time t feasible sets H t ( V t ) and H + t ( V t ) are defined analogously. The next result shows that thefeasible sets are positive homogeneous and recursive. Lemma 3.3.
1. For a positive F t -measurable wealth V t the feasible sets scale as follows H t ( V t ) = V t · H t (1) and H + t ( V t ) = V t · H + t (1) . . The feasible sets are recursive, H t ( V t ) = (cid:8) ( h s ) s = t,...,T − | h t ∈ H t ( V t ) , ( h s ) s = t +1 ,...,T − ∈ H t +1 ( R T t +1 h t ) (cid:9) , H + t ( V t ) = (cid:8) ( h s ) s = t,...,T − | h t ∈ H + t ( V t ) , ( h s ) s = t +1 ,...,T − ∈ H + t +1 ( R T t +1 h t ) (cid:9) , where H t ( V t ) = { h t | T h t = V t } and H + t ( V t ) = { h t ∈ H t ( V t ) | h t ≥ } .Proof. The proof is deferred to Appendix B.Our aim is to find the optimal trading strategy among the feasible ones by maximizing the port-folio’s acceptability as measured by a given DCAI α . We recall that the dynamic setup of [BCZ14]and [Rie04] assumes that the inputs D to a DCAI are (discounted) dividend processes, a setup usu-ally convenient for pricing purposes or assessing the performance or riskiness of some dividend payingsecurities, or random future cash-flows (cf. [AFP12, AP11, BCIR13, BCC15] and references therein).When dealing with optimal investment (i.e. an optimal portfolio selection problem), traditionally andalso more conveniently, one works with the value process or the (discounted) cumulative dividendprocess. Given a portfolio with value process V = ( V s ) s =0 ,...,T , the corresponding dividend stream D = ( D s ) s =0 ,...,T is defined as D s = V s − V s − , s = 1 , . . . , T, and D = 0. Thus, the cumulative P&L up to time t becomes P ts =0 D s = V t − V . We refer the readerto [AP11, BCDK16] for a detailed discussion on use of dividend streams and cumulative dividendstreams within the general theory of assessment indices.We denote by V ( h ) the wealth process generated by the trading strategy h and D ( h ) will standfor the corresponding dividend stream. In addition, for a given dividend stream D = ( D , . . . , D T )we define the time t tail dividend stream as D [ t,T ] := (0 , . . . , , D t , . . . , D T ), and we also put D [ t ] :=(0 , . . . , , D t , , . . . , h ∈H +0 ( V ) α ( D ( h )) ( A ( V ))or the variant thereof, if short-selling is allowed, in which case the feasible set is H ( V ). By property(A2), independence of the past of DCAIs, we have that α t ( D ) = α t ( D [ t,T ] ). This would suggest thatin order to solve A ( V ) we should consider the problemsmax h ∈H + t ( V t ) α t ( D [ t,T ] ( h )) . ( A t ( V t ))One certainly can study ( A t ( V t )), and try to establish a dynamic programming principle for thisstochastic control problem, although, generally speaking, this problem is not time consistent. Moreimportantly, from a practical point of view, including the change in portfolio value from time t − t , namely the term D t = V t − V t − , in the optimization criteria at time t is less desirable. In particular,using (3.3), one would optimize at time t a function that depends on V T − V t − , rather than a functiondepending on total future return V T − V t . With this in mind, we introduce and focus our attentionon a family of auxiliary acceptability maximization problemsmax h ∈H + t ( V t ) α t ( D [ t +1 ,T ] ( h )) , ( ˜ A t ( V t ))14hich are more in line with the setup from the optimal portfolio selection problem. Note that forany trading strategy h the cash flows D ( h ) and D [1 ,T ] ( h ) coincide, and hence at the initial time theauxiliary problem ˜ A ( V ) is the same as the original problem A ( V ). Therefore, solving the auxiliaryfamily of problems, which we will address next, will lead to the solution of the original acceptabilitymaximization problem. ( ρ xt ) t ∈T As we already mentioned, the form of the time consistency property (R7) for DCRMs is tailored forthe robust representation (3.2) of DCAIs with the time consistency property (A7). This form of timeconsistency is weaker than the so-called strong time consistency of risk measures:(R7’)
Strong time consistency: for any
D, D ′ ∈ D and t = 0 , . . . , T −
1, if D t = D ′ t and ρ t +1 ( D ) = ρ t +1 ( D ′ ) , then ρ t ( D ) = ρ t ( D ′ ).Strong time consistency (R7’) is the one usually associated with dynamic risk measures (cf. [Rie04,BCP17]), due to its natural financial interpretation, but also because of its equivalence to:(R7”) Recursiveness: ρ t ( D ) = ρ t (cid:0) − ρ t +1 ( D ) { t +1 } (cid:1) − D t , for any D ∈ D and t = 0 , . . . , T − A t ( V t )) assuming that thecorresponding family of risk measures is strongly time consistent. We work under the market setupof Section 3.2 with short-selling constraints and with an initial value V > ρ x are defined as ρ xt,t +1 ( Z, ω ) := ρ xt ( + { t +1 } Z )( ω ) , t = 0 , . . . , T − , ω ∈ Ω , for any F t +1 -measurable random variable Z , here denotes the zero process. In what follows weassume that the one-step risk measures are identical across all nodes of the multinomial model.Namely, with P t denoting the partition of Ω that generates F t , we assume that for any t, s ∈ T , andany Ω t ∈ P t and Ω s ∈ P s ρ xt,t +1 ( D t +1 , ω ) = ρ xs,s +1 ( D ′ s +1 , ω ′ ) (3.5)for all ω ∈ Ω t , all ω ′ ∈ Ω s and all D, D ′ satisfying Ω t D t +1 ( d ) = Ω s D ′ s +1 and zero otherwise. Aspreviously, we denote the maximal acceptability attainable at the market as α ∗ t ( V t ; ω ) := sup h ∈H + t ( V t ) α t ( D [ t +1 ,T ] ( h ); ω ) . Under the above, what may appear, natural assumptions, we obtain a somehow surprising result:the maximal acceptability α ∗ is constant across wealth level, time and states of the world. Theorem 3.4.
Let α be a normalized right-continuous DCAI and ( ρ x ) x ∈ (0 , ∞ ) be the correspondingfamily of DCRMs. Assume that for each x > the DCRM ρ x is strongly time consistent, and all theone step risk measures ρ xt,t +1 satisfy (3.5) . Then, under the market model assumption of this section,the maximal acceptability α ∗ t is independent of the wealth, time and state, that is, α ∗ t ( V t ; ω ) = α ∗ (1) , for all t ∈ T , ω ∈ Ω and positive V t ∈ L t . roof. The proof is given in Appendix B.In view of Theorem 3.4 the auxiliary acceptability maximization has a constant optimal objectivevalue in time, and since at the initial time the auxiliary and the original problem coincide, we obtainthat it suffices to solve ˜ A T − (1)(¯ ω ) for some ¯ ω ∈ Ω instead of A ( V ). The next result shows how toconstruct the corresponding optimal trading strategy. Theorem 3.5.
Assume that for some ¯ ω ∈ Ω the supremum α ∗ T − (1; ¯ ω ) is attained and denote by h ∗ ∈ R d the corresponding optimal position (given ¯ ω ), h ∗ = arg max h T − ∈ H + T − (1) α T − ( D [ T ] ( h T − ); ¯ ω )(¯ ω ) , where H + T − (1) was defined in Lemma 3.3(2). Let (¯ h s ) s =0 ,...,T − be the trading strategy defined as ¯ h = V · h ∗ , ¯ h s = V s (¯ h s − ) · h ∗ , s = 1 , . . . , T − . Then, the trading strategy ¯ h is an optimal solution of A ( V ) , i.e. α ( D (¯ h )) = α ∗ .Proof. The proof is given in Appendix B.The results of this subsection rely on the properties of the family of risk measures { ( ρ xt ) t =0 ,...,T } x> corresponding to the DCAI α under consideration. Similar to the static case, one may be interestedin the risk minimization problem corresponding to a fixed level x >
0. As may be expected, therecursive setting of this section has direct implications on the minimal achievable risk (infimum of therisk minimization problem). It can be proved that the minimal risk is positively homogeneous and italso has a recursive form. Unlike the maximal acceptability it is not constant, but it maintains thesame sign over all times and states. Furthermore, if an optimal solution (optimal trading strategy)exists, it can be constructed recursively in the spirit of Theorem 3.5.
Using the definition of the static RAROC, the identity (2.3), as well as the representation (3.3), onenaturally defines the dynamic risk adjusted return on capital (dRAROC) as follows:dRAROC t ( D ) = (cid:16) E t (cid:16)P Ts = t D s (cid:17)(cid:17) + (cid:16) π t (cid:16)P Ts = t D s (cid:17)(cid:17) + , D ∈ D , with the convention a = + ∞ , where π is a given dynamic coherent risk measure (not to be confusedwith the family of DCRMs corresponding to an acceptability index).As was shown in [BCZ14, Section 6], dRAROC fulfills the properties (A1)-(A6), but it, in general,fails to satisfy the dynamic consistency property (A7), and therefore, it is not a DCAI. Nevertheless,for some choices of π , dRAROC satisfies some weaker forms of time consistency. In particular, if π isthe dynamic version of TV@R, then the corresponding dRAROC is so-called semi-weakly acceptancetime consistent, but not semi-weakly rejection time consistent; for more details see [BCP17, BCP18].This will be the example we consider in our numerical experiment below. With this in mind, we areinterested in identifying an investment with the highest performance measured by dRAROC, i.e. the16aximization of the dRAROC-performance in the framework of self-financing portfolios introducedin Section 3.2. Furthermore, we focus our attention only on the case of a feasible set with short-sellingconstraints H +0 ( V ), but most of the results can be extended to the case with no trading constraints.Hence, we wish to solve the following optimization problem:max h ∈H +0 ( V ) dRAROC ( D ( h )) . (3.6)As already mentioned, this problem is time-inconsistent (in the sense of optimal control), and in viewof the above it does not fit the framework of Section 3.2.1.We will take the approach of [KR19] to deal with time-inconsistency of (3.6). First we note thatfor a positive level x > ( D ) ≥ x ⇔ min ( π T X s =0 D s ! , − E T X s =0 D s ! + xπ T X s =0 D s !) ≤ . Using this, one could apply the idea of Algorithm 1 to the family of functions ρ x ( · ) = min { π ( · ) , − E ( · )+ xπ ( · ) } for x >
0. In the nutshell, the procedure would consist of the following: First, minimize therisk π ( · ) among the feasible positions, i.e. solve the mean-risk problem with an infinite risk aversion.If the optimal value were negative, an infinite performance measured by dRAROC would be implied.Second, repeatedly minimize − E ( · ) + xπ ( · ) among the feasible positions for various levels x – that is,solve the mean-risk problem for the risk aversion at various levels x . Therefore, the algorithm wouldbe iteratively computing elements of the mean-risk efficient frontier. If the (full) efficient frontier, i.e.the set of all portfolios that are not dominated in terms of their mean and risk, was available instead,the optimal solution of (3.6) could be found simply as the element of the frontier with the highestratio of the mean to the risk. Of course, applying Algorithm 1 is not computationally efficient in thedynamic setup, but it motivates us to compute the efficient frontier, i.e. to consider the bi-objectivemean-risk problem min h ∈H +0 ( V ) (cid:18) − E ( V T ( h ) − V ) π ( V T ( h ) − V ) (cid:19) w.r.t. ≤ R , (3.7)where we used the fact that P Ts =0 D s ( h ) = V T ( h ) − V . This will also overcome the problem of time-inconsistency of (3.6) and thus lead to an efficient way to solve (3.6). As it turns out, problem (3.7) istime consistent in the set-valued sense, i.e., the set-valued Bellman’s principle of optimality recentlyproposed in [KR19] provides a way to solve the mean-risk problem (3.7) recursively, assuming that thedynamic risk measure π is recursive, i.e. strongly time consistent. We emphasis that the recursivenessof π does not imply the recursiveness of the members of the family ( ρ x ) x ∈ (0 , ∞ ) and is a separate propertyfrom the dynamic consistency of the performance measure dRAROC. The set-valued Bellman’sprinciple of optimality of [KR19] also provides the intermediate mean-risk efficient frontiers, namelyit solves the sequence of mean-risk problemsmin h ∈H + t ( V t ) (cid:18) − E t ( V T ( h ) − V t ) π t ( V T ( h ) − V t ) (cid:19) w.r.t. ≤ L t ( R ) , for each time point t = 0 , . . . , T −
1. Note that since dRAROC is equal to the ratio of the mean tothe risk, the element of the frontier of the time-consistent problem (3.7) with the highest ratio is theoptimal solution of the time-inconsistent problem (3.6). The same can be said about the intermediatemean-risk efficient frontiers and (auxiliary) problems max h ∈H + t ( V t ) dRAROC t ( D [ t +1 ,T ] ( h )).17e illustrate this on a dynamic version of the example from Section 2.3. We consider the marketmodel with two assets, and with one-time-step asset returns R it , i = 1 ,
2, having the probability lawgiven in Panel A of Table 1, and we take T = 6. We take the DCRM π to be the recursive dynamicTV@R at significance level 1%. We recall that the dynamic TV@R is defined analogously to thestatic TV@R by replacing V@R with the conditional V@R, which in turn is defined as a conditionalquantile.In Figure 1 we display the mean-risk efficient frontier of problem (3.7), as well as the intermediatefrontiers. The bright green points correspond to the elements with the highest dRAROC.Figure 1: Efficient frontiers (black) of the mean-risk problems and elements with the highest mean-to-risk ratio (green). All frontiers are depicted in the ( ρ, E ) plane for the returns v T − v t with v t = 1.The trading strategy ( h t ) t =0 ,...,T − corresponding to the highest-dRAROC element of the time 0frontier can be recovered from the solution of the mean-risk problem, see [KR19] for details. The mean-risk profiles, and the corresponding values of dRAROC, of this portfolio in the subsequent time pointsare determined by the strategy itself and vary over times and states of the world. They are depicted asyellow triangles for a selected state of the world ω in Figure 2. All of them lie on the efficient frontiers(yellow triangles), but, in general, do not coincide with the highest-dRAROC t element (bright greenpoints). This confirms the time-inconsistency of dRAROC – the strategy optimal from the viewpointof time t = 0 is not dRAROC-maximal at the subsequent time instances.For comparison, we include also a myopic (magenta square) and an inconsistent switching (reddiamond) approach. In the myopic case, the investor at each time solves a one step optimizationproblem, hence looking always only one period ahead and chooses the position that maximizes theRAROC over this one-period horizon. The switching strategy represents a time inconsistent behaviorin the sense that at time t the dRAROC-maximal element of the (time consistent) frontier is selected,the trading strategy ( h ( t ) s ) s = t,...,T − corresponding to it is found, and the position h ( t ) t is taken. At thenext time t + 1 the previously found trading strategy ( h ( t ) s ) s = t +1 ,...,T − is discarded and a new one,18 h ( t +1) s ) s = t +1 ,...,T − corresponding to the dRAROC-maximal element of the t + 1 frontier, is selected.Since each (efficient) trading strategy is discarded after one time period, none of the corresponding(dRAROC-optimal) mean-risk profiles are ever realized. Figure 2 shows the actual means, risks andvalues of dRAROC that these behaviors yield. Clearly, neither the myopic nor the switching give atany time (except at T −
1) the maximal performance. They even lead to portfolios, which are notmean-risk efficient at all, i.e. they do not lie on the frontier.Figure 2: Efficient frontiers for returns over time. The mean-risk profiles and the corresponding valuesof dRAROC are depicted for three trading strategies: the time consistent mean-risk strategy in onestate ω (yellow triangle), the switching strategy (red diamond) and the myopic strategy (magentasquare). The element of the frontier with the highest dRAROC is also depicted at each time (greencircle).Finally, let us look again at the strategy depicted in yellow, namely the strategy that solves (3.6)at time zero. While this stochastic control problem is time-inconsistent, one can ask which objectivedoes the optimal strategy maximize at the intermediate times. Note that the dRAROC -maximalelement of the time 0 frontier corresponds to a nonlinear scalarization (cid:18) − E ( V T ( h ) − V ) ρ ( V T ( h ) − V ) (cid:19) E ( V T ( h ) − V ) ρ ( V T ( h ) − V ) . Thus, we will concentrate on a class of non-linear scalarizations including the one above. Specifically,we consider scalarizations of the time t frontier of the form (cid:18) − E t ( V T ( h ) − V t ) ρ t ( V T ( h ) − V t ) (cid:19) ( E t ( V T ( h ) − V t )) λ t ρ t ( V T ( h ) − V t ) , (3.8)where the mapping is fully determined by the value λ t , which can be interpreted as a non-linear riskaversion parameter. For any given efficient trading strategy, one can compute the value of λ t , such that19he strategy is an optimal solution of a scalar problem with the objective (3.8). This way, a sequenceof λ , . . . , λ T − can be computed for the dRAROC –optimal strategy ( h t ) t =0 ,...,T − (represented on thefrontiers by the yellow mean-risk pairs). Since the frontiers (and the mean-risk profiles) are adapted,also the corresponding scalarization coefficient λ t will be adapted. We computed the corresponding λ t in the given state of the world ω and depicted it also in Figure 2.Thus, the sequence of scalar problems (3.8) is time consistent in the usual sense for the computedrisk aversion parameters λ , . . . , λ T − . As λ = 1 is by construction included, a time zero memberof this time consistent family is the dRAROC –maximization problem. Thus, an investor with adRAROC criteria at time zero and a dRAROC t like criteria, that differs only in a changed riskaversion parameter λ t , where λ t is changing in a certain manner according to the changes in the stockmarket, would behave time consistent in the classical sense. This is in line with the findings aboutthe moving scalarization (a time and state dependent risk aversion parameter) that leads to a timeconsistent problem and a time consistent behaviour of the investor as also discussed in the mean-risk portfolio optimization problem in [KR19] and for other otherwise time inconsistent problemsin [KMZ17]. Similar to dRAROC, the dynamic gain-to-loss ratio (dGLR) is defined asdGLR t ( D ) = (cid:16) E t (cid:16)P Ts = t D s (cid:17)(cid:17) + E t (cid:18)(cid:16)P Ts = t D s (cid:17) − (cid:19) , D ∈ D , (3.9)with the convention a := + ∞ . Unlike dRAROC, dGLR is a normalized and right-continuous DCAI(see [BCZ14, Section 6]). Our aim is to identify among all self-financing portfolios the ones with thehighest dGLR, that is to solve the problemmax h ∈H ( V ) dGLR ( D ( h )) . (3.10)Similar to the static GLR, the family ρ x of DCRM from the robust representation is identified bythe conditional expectiles, and since the conditional expectiles are not strongly time consistent, theresults of Subsection 3.2.1 do not apply here. As was also noted in the static case, instead of thecorresponding family of risk measures one can consider the family − E ( · ) + x E (cid:0) ( · ) − (cid:1) for x >
0. Notethat for any fixed time instance one can view the problem as a static one, and thus one can applyAlgorithm 1, but this would be, as discussed before, computationally infeasible to do for all t ∈ T .Here, with the intention of obtaining a Bellman’s principle of optimality, we take an approach inspiredby the previous subsection and in the spirit of [KR19]. Motivated by the numerator and denominatorof (3.9), we consider the bi-objective mean-loss problemmin h ∈H ( V ) (cid:18) − E ( V T ( h ) − V ) E (cid:0) ( V T ( h ) − V ) − (cid:1)(cid:19) w.r.t. ≤ R . (3.11)By the same argument as in the dRAROC case, the element of the efficient frontier with the high-est ratio corresponds to the portfolio with the highest value of dGLR . The recursive approachof [KR19], unfortunately, can not be applied directly here, due to the lack of translation invarianceof the objective function E t ( X − ) , which makes it impossible to express E (cid:0) ( V T ( h ) − V ) − (cid:1) through20 t (cid:0) ( V T ( h ) − V t ) − (cid:1) . Nevertheless, to solve (3.11), we consider the following sequence of bi-objectiveproblems min h ∈H t ( V t ) (cid:18) − E t ( V T ( h ) − V ) E t (cid:0) ( V T ( h ) − V ) − (cid:1)(cid:19) w.r.t. ≤ R , (3.12)where V is the fixed initial wealth. Problem (3.12) does not have a natural interpretation as amean-loss problem, unless V t = V , however, it does give a recursive solution of (3.11) in terms of theset-valued Bellman’s principle of [KR19].We also note that the computational approach from [KR19] based on scaling arguments is notapplicable here either, and therefore one needs to solve (3.12) for any V t . As the problems (3.12) canbe rewritten as bi-objective linear optimization problems and differ only in the right-hand side of theconstraints, they form a class of parametric bi-objective linear problems with the parameter V t . Wesolved these parametric problems via polyhedral projection (cf. [LW16]).We conclude this section by illustrating the solution to (3.10) in the same market model setupas in Section 3.2.2 and by taking the initial wealth V = 0. Figure 3 contains the efficient frontierat time t = 0 and the highest value of problem (3.10) given by dGLR = 0 .
27. As the intermediatefrontiers are computed for all possible values of V t , we depict for illustration only those frontierscorresponding to V t = V for each time point t . The case of the current wealth V t coinciding with theinitial wealth V would give problem (3.12) the interpretation as the mean-loss problem. Thereforethe corresponding maximal value of dGLR t can be obtained. Since the zero-cost trading strategy canbe scaled, the frontier is naturally a half-line. The highest value of dGLR corresponds to the slopeof the frontier. The optimal trading strategy of (3.10) can be deduced from the solution of (3.12).Thus, an auxiliary, but time-consistent bi-objective problem (3.12) (following a backward recursionby the set-valued Bellman’s principle of optimality) is used to compute the optimal solution of thetime-inconsistent problem (3.10). Acknowledgments
IC acknowledges partial support from the National Science Foundation (US) grant DMS-1907568. ICthanks the Vienna University of Economics and Business for the hospitality during a research visitrelated to the workshop on dynamic multivariate programming in March 2018, where this project hasbeen initiated. The authors thank Christian Diem for helpful discussions in the early stages of theproject.
A Proofs from Section 2
Proof of Lemma 2.3.
The first property follows from the quasi-concavity of α , the second from thedefinition of the sets, for the third consider \ ǫ> D ǫ = { D ∈ D : α ( D ) ≥ α ∗ − ǫ, ∀ ǫ > } = D ∗ . Proof of Lemma 2.5.
1. The positive value of the risk minimization problem p ( x ) > x - that is ρ x ( D ) > D ∈ D . Therefore all21igure 3: Efficient frontiers (black) of the problems (3.12) depicted for wealth V t = 0 . All frontiersare depicted in the ( E t ( V − T ) , E t ( V T )) plane. The corresponding highest value of dGLR (the slope ofthe frontier) is given.portfolios have acceptability at most x - that is α ( D ) ≤ x for all D ∈ D . Consequently, α ∗ ≤ x. We do not obtain a strict inequality as we have no information about the continuity of the riskmeasure in the parameter x.
2. The assumption of attainment of the infimum implies that there exists ˜ D ∈ D such that ρ x ( ˜ D ) ≤ . Then α ( ˜ D ) ≥ x and α ∗ ≥ x.
3. The maximal acceptability α ∗ above x means that there exists some portfolio ˜ D ∈ D with α ( ˜ D ) > x. From the monotonicity of the family of risk measures and (2.1) it follows that ρ y ( ˜ D ) ≤ y ≤ x and therefore p ( y ) ≤ .
4. The maximal acceptability α ∗ below x means that α ( D ) < x for all D ∈ D . Consequently, forall D ∈ D it holds ρ x ( D ) > . Since by Assumption 2.4 the infimum of the risk minimizationproblem is attained, also p ( x ) > . The same follows for all y ≥ x as the family of risk measuresis increasing. Proof of Lemma 2.6.
1. Let α ∗ < α . With the halving update rule, at iteration n (counting from0) the tested value is x n = 2 − n · x and by Lemma 2.5 at each step p ( x n ) > . Therefore after¯ M iterations the algorithm is terminated and no non-zero lower bound on the acceptability isfound. The portfolio ¯ D is never assigned, as no portfolio with a known lower bound on theacceptability is found. 22. Let α ∗ > α . With the doubling update rule, at iteration n (counting from 0) the tested valueis x n = 2 n · x and by Lemma 2.5 at each step p ( x n ) ≤ . Therefore, after ¯ M iterations thealgorithm is terminated and no finite upper bound on the acceptability is found. The optimalsolution D x ¯ M − of the risk minimization problem P ( x ¯ M − ) is outputted as ¯ D . It has a degreeof acceptability of at least α = x ¯ M − .3. (a) According to Lemma 2.5 for α ∗ ∈ ( α, α ) it holds p ( α ) ≤ < p ( α ) , therefore Step 1 ofAlgorithm 1 identifies both lower and upper bound, x L and x U . Lemma 2.5 also guaranteesthat the found values are true bounds, x L ≤ α ∗ ≤ x U . Step 2 continues until the length of theinterval is sufficiently small.(b) The optimal solution to the risk minimization problem P ( x L ) is returned as ¯ D . By As-sumption 2.4 it holds ρ x L ( ¯ D ) ≤ , so α ( ¯ D ) ≥ x L . From part (a) it follows that x L > α ∗ − ǫ, so¯ D ∈ D ǫ . (c) The worst-case scenario for the length of the interval after Step 1 is x L = x · ¯ M − , x U = x · ¯ M − . Since the bisection step decreases the length of the interval by half, after i bisectioniterations the length of the interval would be x · ¯ M − − i . To obtain a length below ǫ we need i > log x ǫ + ¯ M − Proof of Lemma 2.8.
By Lemma 2.6 the algorithm for the tolerance ǫ outputs D ǫ ∈ D ǫ . Since D is compact, there is a subsequence { D ǫ nk } nk ∈ N with a limit, denoted ˜ D , in the feasible set. Thecompactness also implies there exists C < ∞ such that | D | ≤ C for all feasible positions D ∈ D .Then, since | C D ǫ nk | ≤ C D ǫ nk p −→ C ˜ D, scale invariance and the Fatou property of α imply forany fixed δ > ∀ ǫ nk ≤ δ : α ( D ǫ nk ) ≥ α ∗ − δ ⇒ α ( ˜ D ) ≥ α ∗ − δ. Letting δ go to zero, we obtain α ( ˜ D ) ≥ α ∗ . Therefore, ˜ D is an element of the set D ∗ . B Proofs from Section 3
Proof of Lemma 3.3.
For the first part, positive homogeneity follows from the self-financing propertyand the linearity of the portfolio value V s . As for the second part, the recursiveness, we have H t ( V t ) := { ( h s ) s = t,...,T − | T h s = V s , V s +1 = R T s +1 h s , s = t, . . . , T − } = { ( h s ) s = t,...,T − | T h t = V t , V t +1 = R T t +1 h t , T h s = V s , V s +1 = R T s +1 h s , s = t + 1 , . . . , T − } = { ( h s ) s = t,...,T − | h t ∈ H t ( V t ) , ( h s ) s = t +1 ,...,T − ∈ H t +1 ( R T t +1 h t ) } . The result for the set H + t ( V t ) is obtained similarly. Proof of Theorem 3.4.
First we note that α ∗ is scale invariant, i.e. α ∗ t ( λV t ) = α ∗ t ( V t ) for any λ > , λ ∈F t , t ∈ T , which follows immediately from the scale invariance of the DCAI α and Lemma 3.3(1).Thus, it is enough to prove that α ∗ (1) = α ∗ t (1; ω ), for all t ∈ T and ω ∈ Ω, which we will show next.23o prove the second claim we, use two types of sets of risks: the set of risks of one-step-aheaddividends, Q xt ( ω ) := { ρ xt ( D [ t +1] ( h t ); ω ) | h t ∈ H + t (1) } , and the set of risks of feasible portfolios, P xt ( ω ) := { ρ xt ( D [ t +1 ,T ] ( h ); ω ) | h ∈ H + t (1) } . The fact that the asset returns are iid and the assumption that the one-step risk measures are identicalimply that at a given level x the sets of one-step-ahead risks coincide across all times and all states, Q xt ( ω ) = Q xs ( ω ) for all s, t ∈ T and all ω , ω ∈ Ω . (B.1)At time T − x coincide, Q xT − ( ω ) = P xT − ( ω ).The relationship between the acceptability index α and the corresponding family of risk measures( ρ x ) x ∈ (0 , ∞ ) implies the following two equivalence: α ∗ t (1; ω ) ≤ β ⇔ ∀ y > β : P yt ( ω ) ∩ R − = ∅ , and α ∗ t (1; ω ) ≥ β ⇔ ∀ x < β : P xt ( ω ) ∩ R − = ∅ . (B.2)We prove the claim by a backward induction. Let ¯ ω ∈ Ω be an arbitrary state of the world andset α ∗ := α ∗ T − (1; ¯ ω ). In the first step of the induction we prove that α ∗ T − (1; ω ) = α ∗ for all states ω ∈ Ω: Consider a level y > α ∗ . According to (B.2) the set P yT − (¯ ω ) = Q yT − (¯ ω ) contains positiveelements only. Then, (B.1) implies that the same is true for the set P yT − ( ω ) = Q yT − ( ω ), which means α ∗ T − (1; ω ) ≤ α ∗ . Now consider a level x < α ∗ . According to (B.2) the set P xT − (¯ ω ) = Q xT − (¯ ω )contains some non-positive element. By (B.1), the same is true for the set P xT − ( ω ) = Q xT − ( ω ), so α ∗ T − (1; ω ) ≥ α ∗ .The induction hypothesis assumes that α ∗ s (1) ≡ α ∗ for all s > t . For levels y > α ∗ this means thatthe sets P yt +1 ( ω ) and the sets Q yt ( ω ) (via (B.1) and the first step of the induction) contain positiveelements only. For levels x < α ∗ this means that the sets P xt +1 ( ω ) and the sets Q xt ( ω ) (via (B.1)and the first step of the induction) contain some non-positive element. The adaptiveness (R1) andindependence (R2) of the risk measure imply that there exists an element ¯ p ∈ P xt +1 that is non-positivein all states of the world. The same is true also for the set Q xt .Inductive step: The properties of the risk measure imply the following form of the set P xt , P xt = (cid:8) ρ xt (cid:0) − ( V t +1 ( h t ) · p − D t +1 ( h t )) 1 { t +1 } (cid:1) | h t ∈ H + t (1) , p ∈ P xt +1 (cid:9) . Consider a level y > α ∗ . According to the induction hypothesis all p ∈ P yt +1 are positive. Then, byapplying the monotonicity (R3), an arbitrary element of P yt can be bounded by ρ yt (cid:0) − ( V t +1 ( h t ) · p − D t +1 ( h t )) 1 { t +1 } (cid:1) ≥ ρ yt (cid:0) D t +1 ( h t )1 { t +1 } (cid:1) = ρ yt (cid:0) D [ t +1] ( h t ) (cid:1) . The risk ρ yt (cid:0) D [ t +1] ( w t ) (cid:1) is an element of the set Q yt , so by the induction hypothesis it is positive inall states of the world. This shows α ∗ t (1) ≤ α ∗ .Now consider a level x < α ∗ . Consider the elements of P xt of the form (cid:8) ρ xt (cid:0) − ( V t +1 ( h t ) · ¯ p − D t +1 ( h t )) 1 { t +1 } (cid:1) | h t ∈ H + t (1) (cid:9) , p ≤ P xt +1 , whose existence is guaranteed by the induction hy-pothesis. Monotonicity of the risk measure bounds these risks by ρ xt (cid:0) − ( V t +1 ( h t ) · ¯ p − D t +1 ( h t )) 1 { t +1 } (cid:1) ≤ ρ xt (cid:0) D t +1 ( h t )1 { t +1 } (cid:1) = ρ xt (cid:0) D [ t +1] ( h t ) (cid:1) . The risks ρ xt (cid:0) D [ t +1] ( w t ) (cid:1) are elements of the set Q xt , and by the induction hypothesis at least one ofthem is non-positive. Therefore, the set P xt contains at least one non-positive element and α ∗ t (1) ≥ α ∗ . Proof of Theorem 3.5.
Firstly, note that the construction of the trading strategy ¯ h guarantees that itis adapted and feasible. We prove the claim via backward induction by showing that ρ xt ( D [ t +1 ,T ] (¯ h )) ≤ x < α ∗ . This suffices to show that α t ( D [ t +1 ,T ] (¯ h )) ≥ α ∗ . Since α ∗ is a supremum, equality follows.Consider time T −
1. Optimality of the position h ∗ and the positive homogeneity imply that ρ xT − ( D [ T,T ] (¯ h ); ¯ ω ) ≤ x < α ∗ . The iid asset returns and the identical one-step risk measures together with thepositive homogeneity give the same for all states ω ∈ Ω.The induction hypothesis assumes that the risk ρ xt +1 ( D [ t +2 ,T ] (¯ h )) ≤ x < α ∗ . For the inductive step we use the recursiveness of the risk measure to express the time t risk as ρ xt ( D [ t +1 ,T ] (¯ h )) = ρ xt (cid:0) − (cid:0) ρ xt +1 ( D [ t +2 ,T ] (¯ h )) − D t +1 (¯ h ) (cid:1) { t +1 } (cid:1) . At level x < α ∗ the induction hypothesis and the monotonicity provide a bound ρ xt ( D [ t +1 ,T ] (¯ h )) ≤ ρ xt ( D [ t +1] (¯ h )) ≤ . The inequality ρ xt ( ¯ D t +1 (¯ h )) ≤ h corresponding to the scaled position h ∗ . Weconclude α t ( D [ t +1 ,T ] (¯ h )) ≥ α ∗ . References [ADEH99] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk.
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TheReview of Financial Studies , 17(1):63–98, 01 2004.25 a b l e : A l go r i t h m f o r A I T , G L R a nd R A R O C i n a t o y m a r k e t m o d e l. ω ω ω ω A ss e t . . . . A ss e t . . . . P a n e l A : R e t u r n m a t r i x R i n t h e t o y m a r k e t m o d e l (t w oa ss e t s a nd f o u r s t a t e s o f t h e w o r l d ) . A I T G L RR A R O C I t e r x L x U x p ( x ) I t e r x L x U x p ( x ) I t e r x L x U x p ( x ) S t e p S t e p S t e p ∞ + ∞ − ∞ + + ∞ + + . − S t e p . − S t e p − S t e p . . − . + . . − . . + . . + . . + . . . + . . − . . . − . . . + . . . + . . . + . . . + . . . + . . . + . . . − . . . − . . . − . . . − . . . + . . . + . . . − . . . + . . . + . . . − . . . − . . . − . . . − . . . + . . . + . . . + . . . + . . . + . . . − . . . − . . . − . . . − . . . + . . . + . . . . . + . . h ǫ = ( . % , . % ) . . h ǫ = ( . % , . % ) h ǫ = ( . % , . % ) P a n e l B : I t e r a t i o n s o f A l go r i t h m w i t h i npu t p a r a m e t e r s x = , ǫ = − a nd ¯ M = . T h e l a s tt w o r o w s g i v e , r e s p ec t i v e l y , t h e b o und s x L a nd x U o n t h e m a x i m a l a cce p t a b ili t y , a nd a n ε - o p t i m a l p o r t f o li o . a b l e : I t e r a t i o n s o f t h e m o d i fi e d , t h e m i x e d a nd t h eze r o - l e v e l v e r s i o n o f A l go r i t h m f o r G L R i n t h e m a r k e t m o d e l f r o m T a b l e , P a n e l A ( α ∗ = . ) w i t h t h e t o l e r a n ce ǫ = − . I n t h e m o d i fi e d v e r s i o n t h e b i s ec t i o n i s p e r f o r m e d o n t h e p a r a m e t e r q = + x ∈ [ , . ] a f t e r v e r i f y i n g t h e s i g n s o f p ( ) a nd p ( ∞ ) . T h e t e r m i n a t i o n c r i t e r i o n i ss e t o n t h e p a r a m e t e r x t og u a r a n t ee a n ǫ - s o l u t i o n i s o b t a i n e d . W i t h t h e t e r m i n a t i o n c r i t e r i o n o n t h e p a r a m e t e r q t h e a l go r i t h m w o u l dfin i s h a f t e r i t e r a t i o n o f S t e p , h o w e v e r , t h e i n t e r v a l f o r m a x i m a l a cce p t a b ili t y w o u l dh a v e l e n g t h . e - . T h e m i x e d v e r s i o n s w i t c h e s t oa b i s ec t i o n o n t h e p a r a m e t e r x a ss oo n a s a fin i t e upp e r b o und x U i s o b t a i n e d . - t h eze r o - l e v e l v e r s i o n c o m pu t e s a f t e r e a c h i t e r a t i o n t h e l e v e l y f o r w h i c h t h e p o rt f o li o s o l v i n g t h e r i s k m i n i m i z a t i o np r o b l e m h a s ze r o r i s k . T h i s l e v e li s u s e d a s a l o w e r b o und . T h e a l go r i t h m i s r un w i t h i n i t i a l p a r a m e t e r s x = , ǫ = − a nd ¯ M = . M o d i fi e d a l go r i t h m f o r G L R M i x e d a l go r i t h m f o r G L R Z e r o - l e v e l a l go r i t h m f o r G L R I t e r q L q U q x p ( x ) I t e r q L q U q x p ( x ) I t e r x L x U x y p ( x ) S t e p S t e p S t e p ∞ + ∞ + ∞ . − . − . − . ∞ . . + S t e p S t e p S t e p . . − . . − . . . . + . . + . . + . . . . + . . . . + I t e r x L x U x p ( x ) . . . . + . . . . − + . . . . + . . . . − − . . . . + . . . . − . + . . . . + . . . . + . . + . . . . + . . . . + . . − . . . . + . . . . + . . . + . . . . + . . . . − . . . + . . . . + . . . . − . . . − . . . . + . . . . − . . . + . . . . + . . . . + . . . + . . . . + . . . . + . . . − . . . . + . . . . + . . . + . . . . + . . . . − . . . + ( x L , x U ) = ( . , . ) . . . . − . . . − h ǫ = ( . % , . % ) . . . . − . . . + ( x L , x U ) = ( . , . ) . . . + q U − q L = . e - , x U − x L = . e - ( x L , x U ) = ( . , . ) h ǫ = ( . % , . % ) h ǫ = ( . % , . % ) d = 10assets with short-selling constraints. Panel A: AIT, maximal acceptability α ∗ = 25 . x ǫ M Step 1 Step 2 Run timeIter [ x L , x U ] Iter x U − x L (s)2 10 −
15 5 [16 ,
32] 18 6.1e-05 3.7820 10 −
15 2 [20 ,
40] 18 7.6e-05 3.40200 10 −
15 4 [25 ,
50] 18 9.5e-05 3.562 10 −
15 5 [16 ,
32] 31 7.5e-09 6.222 −
15 15 [0 ,
64] no Step 2 1.882 −
30 17 [16 ,
32] 18 6.1e-05 4.672 − −
15 15 [16 , ∞ ] no Step 2 4.612 − −
30 16 [16 ,
32] 18 6.1e-05 7.15
Panel B: GLR, maximal acceptability α ∗ = 279 . x ǫ M Step 1 Step 2 Run timeIter [ x L , x U ] Iter x U − x L (s)2 10 −
15 9 [256 , −
15 5 [160 , −
15 2 [200 , −
15 9 [256 , −
15 15 [0 , −
30 18 [0 . ,
1] 22 6.1e-05 23.912 − −
15 15 [16 , ∞ ] no Step 2 13.412 − −
30 20 [256 , Panel C: RAROC, maximal acceptability α ∗ = 279 . x ǫ M Step 1 Step 2 Run timeIter [ x L , x U ] Iter x U − x L (s)2 10 −
15 2 [2 ,
4] 15 6.1e-05 7.2020 10 −
15 4 [2 . ,
5] 15 7.6e-05 9.41200 10 −
15 8 [1 . , .
13] 14 9.4e-05 12.842 10 −
15 2 [2 ,
4] 28 7.5e-09 11.072 −
15 15 [0 ,
64] no Step 2 10.552 −
30 20 [2 ,
4] 15 6.1e-05 19.592 − −
15 15 [0 . , ∞ ] no Step 2 7.042 − −
30 18 [2 ,
4] 15 6.1e-05 13.4128able 4: A comparison of the different versions of the algorithm in a market with d = 10 assets and | Ω | = 1000 states of the world both with and without short-selling. A tolerance ǫ = 10 − is usedfor all algorithms, the original and zero-level version use x = 2 and ¯ M = 15. Obtaining the finalapproximation [ x L , x U ] is denoted in the table by α ∗ , values are listed to two decimal places. Panel A: AIT, the maximal acceptability with short-selling constraints ( h ≥
0) is α ∗ = 25 . , withoutshort-selling constraints ( h free) it is α ∗ = 25 . Algorithm Step 1 Bisection on q Bisection on x x U − x L Run timeIter [ x L , x U ] Iter [ x L , x U ] Iter [ x L , x U ] (s) h ≥ ,
32] 18 α ∗ , ∞ ] 23 α ∗ , ∞ ] 5 [15 ,
31] 18 α ∗ . , .
84] 18 α ∗ h free Original 5 [16 ,
32] 18 α ∗ , ∞ ] 23 α ∗ , ∞ ] 5 [15 ,
31] 18 α ∗ . , .
89] 18 α ∗ Panel B: GLR, the maximal acceptability with short-selling constraints ( h ≥
0) is α ∗ = 279 . , withoutshort-selling constraints ( h free) it is α ∗ = 288 . Algorithm Step 1 Bisection on q Bisection on x x U − x L Run timeIter [ x L , x U ] Iter [ x L , x U ] Iter [ x L , x U ] (s) h ≥ , α ∗ , ∞ ] 29 α ∗ , ∞ ] 8 [254 , α ∗ . , .
24] 22 α ∗ h free Original 9 [256 , α ∗ , ∞ ] 29 α ∗ , ∞ ] 8 [254 , α ∗ . , .
76] 22 α ∗ Panel C: RAROC, the maximal acceptability with short-selling constraints ( h ≥
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