Some general results on risk budgeting portfolios

Some general results on risk budgeting p ortfolios Claudia F assino a , Pierpaolo Ub erti b a University of Geno a, Via Do de c aneso 35, Geno a, 16146, Italy b University of Milano-Bic o c c a, Via Bic o c c a de gli A r cimb oldi 8, Milan, 20126, Italy Abstract Giv en a reference risk measure, the risk budgeting is the p ortfolio where eac h asset con tributes a predetermined amount to the total risk. W e prop ose a no vel approach—alternativ e to the ones prop osed in the literature—for the calculation of the risk budgeting portfolio. This differen t p ersp ectiv e on the problem has several interesting consequences. F or the calculation of the p ortfolio, we define a Cauch y sequence within the simplex of R n , whose limit corresponds to the risk budgeting p ortfolio. This construction allows for the straightforw ard implemen tation of an efficient algorithm, a voiding the need to solv e auxiliary , equiv alen t optimization problems, whic h ma y b e computationally challenging and hard to interpret in the decision theory con text. W e compare our algorithm with the standard optimization-based metho ds proposed in the literature. F rom a theoretical p oin t of view, starting from the Cauc hy sequence, w e define a function for whic h the risk budgeting p ortfolio is a fixed p oint. Therefore, sufficient conditions for the existence and uniqueness of the fixed p oin t can b e used. The metho dology is dev elop ed for general risk measures and implemented in detail in the case of standard deviation. Keywor ds: Risk Budgeting, Risk P arity, Fixed Poin t 1. In tro duction Risk parity refers to an in v estment strategy in which eac h asset con- tributes equally to the total p ortfolio risk. In its original formulation, the risk parity p ortfolio was in tro duced using the standard deviation of returns as the reference risk measure. The idea of allocating the wealth by equally distributing risk exp osures across assets is simple and intuitiv e. This strat- egy has attracted considerable atten tion from b oth academic researc hers and practitioners, as it offers a useful alternative to classical allo cation mo dels -primarily inspired b y the Marko witz framew ork (see Mark owitz (1952))- whic h struggled to address the challenges revealed b y the 2008 global finan- cial crisis. Standard optimization-based allo cation techniques suffer from sev eral w ell-known shortcomings. These approaches are sub ject to model instabilit y , as p oin ted out in Kan and Zhou (2007), and to numerical instability: ev en infinitesimal v ariations in the mo del’s inputs can lead to a large c hanges in the optimal allo cation, see Best and Grauer (1991), causing the in-sample fron tier to b e a biased estimator of the real frontier, see Kan and Smith (2008). F urthermore, optimization mo dels often tend to concentrate the al- lo cation in a small num b er of assets, see Y anushevsky and Y an ushevsky’s (2015). These dra wbacks are among the principal reasons wh y the equally w eighted p ortfolio and other heuristic allo cation strategies frequently out- p erforms optimization-based approac hes in out-of-sample exp eriments (see, among others, DeMiguel et al. (2009), Y uan and Zhou (2024) and Gelmini and Ub erti (2024)). Risk parity portfolios are attractive b ecause they do not exhibit these limitations. They are only slightly sensitive to parameters uncertain ty and yield numerical stable allo cations that are not concentrated 2 in a few assets. Thus, the practical implemen tation is straigh tforward and transaction costs ha ve negligible impact on performance. Over the past 15 y ears, numerous authors ha ve studied the risk parit y portfolio using differ- en t risk measures. Notable examples include Hallerbac h (1999) for V alue at Risk, Mausser and Romanko (2018) for Exp ected Shortfall, Ararat et al. (2024) for Mean Absolute Deviation, and Bellini et al. (2021) for exp ectiles. Risk budgeting is a natural generalization of risk parity: for a given risk measure, the in vestor decides the risk exp osure contribution of eac h asset and, subsequently , computes the allo cation satisfying the predetermined risk exp osures. Despite its simple definition, no closed-form solution exists for the risk budgeting p ortfolio, making n umerical metho ds necessary for its cal- culation. One cen tral issue concerning risk budgeting p ortfolios is the c hoice of the reference risk measure. Ideally , an inv estor can select an y risk measure and subsequen tly compute the corresp onding risk budgeting p ortfolio. T o this end, it is necessary to calculate the marginal risk con tribution of each asset to the total risk, giv en the cho sen risk measure. A v ery recent pap er pro vides general results on this topic. In Cetingoz et al. (2024), the existence and uniqueness of the risk budgeting p ortfolio for general risk measures are in vestigated. F urther researc h, limited to the class of so-called coherent risk measures (see Artzner et al. (1999)), fo cuses on appro ximating the risk bud- geting p ortfolio using sim ulated data da Costa et al. (2023). Both approaches calculate the risk budgeting p ortfolio by solving an auxiliary optimization problem, which generalizes the optimization framew ork originally prop osed in Maillard et al. (2010) for the risk parity p ortfolio when standard devia- tion serves as the reference risk measure. While Maillard et al. (2010) is a milestone in this context, it lacks a formal mathematical pro of of the exis- tence and uniqueness of the risk parity p ortfolio. The first such pro of for 3 the risk parity p ortfolio under the standard deviation measure can b e found in Spin u (2013). Let us underline that b oth the results on the existence and uniqueness of the risk budgeting p ortfolio and its calculation strongly rely on the definition and solution of an auxiliary optimization problem. This approach, although standard, has some known issues. The optimiza- tion problem do es not hav e an immediate economic in terpretation, so it is difficult to explain why an inv estor should allo cate her/his wealth b y solving that problem. F rom an empirical p oin t of view, the computation of the risk budget allo cation suffers from issues related to numerical optimization. F rom a theoretical p oin t of view, the results on the uniqueness and existence of the risk budgeting p ortfolio entirely rely on the con vexit y of the optimization problem. W e study the risk budgeting p ortfolio from a nov el and alternative p er- sp ectiv e. As often happ ens in mathematics, a differen t p ersp ectiv e on a given problem can unv eil interesting new asp ects. W e start by defining a Cauch y sequence within the simplex of R N . Since the simplex with the Euclidean norm forms a complete metric space, ev ery Cauc hy sequence conv erges to a p oint within this space. By construction, the limit of the sequence corre- sp onds to the risk budgeting p ortfolio. This approach immediately leads to an algorithm for computing the risk budgeting p ortfolio. Through n umerical sim ulations, we show that the algorithm is b oth fast and efficien t compared to standard optimization-based approaches prop osed in the literature. F rom a theoretical standp oin t, we sho w that the Cauch y sequence enables the def- inition of a function on the simplex for whic h the risk budgeting p ortfolio is a fixed p oin t b y construction. Therefore, conditions for the existence and uniqueness of the fixed p oin t can b e used to inv estigate the existence and uniqueness of the risk budgeting p ortfolio. Our approach is introduced in a 4 general framework, indep endently of the choice of a reference risk measure. Subsequen tly , we exploit the metho d to the case of the standard deviation and detail its implemen tation. Our contribution to the literature is m ulti- faceted. Theoretically , w e pro vide sufficien t conditions for the existence and uniqueness of the risk budgeting p ortfolio within the simplex. These con- ditions dep end on the functional form of the marginal risk contribution of eac h asset to the total risk, relativ e to the chosen risk measure. Empirically , w e deriv e an efficient algorithm for calculating the risk budgeting p ortfolio, a voiding the need to solve equiv alent auxiliary optimization problems. The pap er is organized as follo ws: Section 2 presents the main idea, the results for a general risk measure, and the structure of the algorithm. Section 3 applies the metho dology to the case where the reference risk measure is the standard deviation. Section 4 is devoted to the numerical results, while the conclusions are presen ted in Section 5. 2. The general idea for an arbitrary risk measure In this section, we illustrate the general idea b ehind our approac h. Its practical implemen tation requires selecting a risk measure and calculating the marginal risk contribution of each asset with resp ect to that measure. W e do not fo cus on this asp ect here and refer to the pap ers cited in the in- tro duction, which study the risk parit y p ortfolio for sp ecific risk measures. 1 This section is divided into four subsections. The first subsection reviews 1 The pap ers cited in the introduction fo cus on the risk parity p ortfolio, whic h is a sp ecial case of risk budgeting. Nev ertheless, the problem of determining the marginal risk con tribution of an asset with resp ect to a given risk measure is common to b oth risk parity and risk budgeting. 5 the basic assumptions and notation. The second subsection is dedicated to constructing the Cauch y sequence that con verges to the risk budgeting p ort- folio. The third subsection presents sufficient conditions for the existence and uniqueness of the risk budgeting p ortfolio via a fixed-p oint approac h. Finally , the last subsection provides the pseudo-co de for the practical com- putation of the risk budgeting p ortfolio. 2.1. Notation and b asic assumptions W e assume that N risky assets are a v ailable on a market. A p ortfolio is iden tified b y the column v ector x ∈ R N where the i -th entry x i , with i = 1 , . . . , N , is the share of the individual wealth in vested in asset i . The analysis is restricted to long-only p ortfolios: no short p ositions, represented b y negative weigh ts, are allow ed. A p ortfolio x is an element of the simplex S in R N , S = ( x = [ x 1 , . . . , x N ] t ∈ R N : N X i =1 x i = 1 , and x i ≥ 0 , ∀ i = 1 , . . . , N ) . The sup erscript t represents the transp ose op eration. If nothing different is sp ecified, w e alwa ys refer to column vectors. The sym b ols 1 , 0 ∈ R N iden tify the N -dimensional column vectors with unitary and n ull en tries resp ectiv ely; diag ( x ) , with x ∈ R N , is the diagonal matrix of order N where the diagonal elemen ts are the entries of x . Giv en a standard probability space (Ω , F , P ) , ρ : L 0 (Ω , R ) → R is the risk measure, where L 0 (Ω , R ) is the set of R -v alued random v ariables, the mea- surable functions defined on Ω with real v alues. In the presen t framew ork, the random v ariables are the returns of the assets. The risk contribution of asset i to the total risk with resp ect to the risk measure ρ is RC ρ i ( x ) , for 6 x ∈ S . The vector RC ρ ( x ) ∈ R N con tains the risk contributions of the N assets. The risk budget is iden tified by x b ∈ S . This vector is giv en b y the in vestor and represents the ob jective in terms of risk allo cation across the N assets. Consequen tly , x b i is the con tribution of asset i to the p ortfolio risk. Ob viously , if x b i = 1 N for i = 1 , . . . , N , the risk budgeting p ortfolio boils do wn to the risk parit y p ortfolio. The v ector x ∗ ∈ S that pro vides the desired allo cation in terms of risk contributions, i.e. R C ρ ( x ∗ ) / 1 t R C ρ ( x ) = x b , is the risk budgeting p ortfolio. 2 As already underlined in the in tro duction, the existence and uniqueness of x ∗ ∈ S for the so-called risk budgeting compatible risk measures has b een recently studied in Cetingoz et al. (2024). 2.2. The se quenc e that c onver ges to the risk budgeting p ortfolio Limited to the presen t subsection, w e assume that the risk budgeting p ortfolio exists and is unique in S for a giv en risk measure ρ . Each choice of the reference risk measure ρ corresp onds to a sp ecific RC ρ ( x ) . When no confusion arises, we omit x and simply write R C ρ . The formalization of the general idea b ehind the prop osal does not require to sp ecify the functional form of R C ρ . W e will exploit the calculations in detail in section 3 when applying the approac h in the case of the standard deviation. Let us define the functions ∆ : R N → R N and G : R N → R N as ∆( x ) = R C ρ ( x ) − x b 1 t R C ρ ( x ) (1) G ( x ) = x − k ( x )∆( x ) . (2) 2 In general, 1 t R C ρ ( x )  = 1 . Then, to obtain a v ector in the simplex it is necessary to normalize dividing by the sum of the elemen ts of RC ρ . 7 Giv en an initial p oin t x 1 ∈ S , we define the sequences { x n } n =1 , 2 ,... and { ∆( x n ) } n =1 , 2 ,... as          ∆( x n ) = R C ρ ( x n ) − x b 1 t R C ρ ( x n ) x n +1 = x n + k ( x n )∆( x n ) , (3) where the function k ( x ) is opportunely c hosen as describ ed in the follo wing. The functions ∆( x ) and k ( x ) are assumed to b e contin uous on their domain. This hypothesis is fundamental for pro ving the con vergence of { x n } n =1 , 2 ,... . The con tinuit y of ∆ requires that the risk contribution R C ρ is con tinuous. Consequen tly , only risk measures characterized by con tinuous risk contribu- tions are compatible with the risk budgeting problem. 3 Remark 1. W e note that the c ondition x 1 ∈ S do es not guar ante e that the entir e se quenc e { x n } n =1 , 2 ,... r emains in S . It holds that 1 t x n = 1 for al l n ∈ N ; in other wor ds, e ach ve ctor in the se quenc e sums to 1 , but some entries may b e c ome ne gative. T o ensur e that the se quenc e { x n } r emains within the simplex S , it is sufficient to b ound the norm of k ( x n ) . W e wil l explicitly show how to r estrict the se quenc e to the simplex in se ction 3. Assuming the existence of a con tinuous function k , we pro ve that { x n } is a Cauc hy sequence. In the pro of of the theorem, w e use the euclidean norm, ∥ · ∥ 2 . The choice of the norm is not fundamental. 4 3 Giv en the arbitrariness of x b ∈ S , it is eviden t that contin uit y of the risk contribution is necessary for the existence of the risk budgeting p ortfolio. W e underline that the existence and uniqueness of the risk budgeting p ortfolio for a given risk measure do not dep end on the computational framew ork used to determine it. Nev ertheless, different approaches to the problem ma y identify distinct classes of risk measures compatible with risk budgeting. 4 Although the result does not dep end on the specific c hoice of the norm and the corre- sp onding distance, R N with the euclidean norm is a complete metric space. Completeness ensures that every Cauc hy sequence conv erges to a p oint within the space. 8 Theorem 1. L et k ( x n ) such that ∥ ∆( x n ) ∥ 2 ≥ L ∥ ∆( x n +1 ) ∥ 2 , with 0 < L < 1 , and | k ( x n ) | < k for n ∈ N , (i.e. the se quenc e { k ( x n ) } n =1 , 2 ,... is b ounde d), then lim n →∞ ∆( x n ) = 0 and { x n } is a Cauchy se quenc e. Pr o of:. First of all, b y assumption, it exists 0 < L < 1 such that ∥ ∆( x n +1 ) ∥ 2 ≤ L ∥ ∆( x n ) ∥ 2 , then it holds that ∥ ∆( x n ) ∥ 2 ≤ L ∥ ∆( x n − 1 ) ∥ 2 ≤ · · · ≤ L i ∥ ∆( x n − i ) ∥ 2 ≤ · · · ≤ L n ∥ ∆( x 0 ) ∥ 2 (4) and so, since L < 1 , lim n →∞ ∥ ∆( x n ) ∥ 2 = 0 or, equiv alen tly , lim n →∞ ∆( x n ) = 0 . A Cauc hy sequence is such that ∀ ϵ > 0 ∃ n : ∥ x p − x q ∥ 2 < ϵ, ∀ p, q > n . Without loss of generalit y , q = p + j , with j ∈ N . Then ∥ x p − x q ∥ 2 = ∥ x p − x p + j ∥ 2 = ∥ x p − x p +1 + x p +1 − x p + j ∥ 2 ≤ ∥ x p − x p +1 ∥ 2 + ∥ x p +1 − x p + j ∥ 2 ≤ ∥ x p − x p +1 ∥ 2 + ∥ x p +1 − x p +2 ∥ 2 + ∥ x p +2 − x p + j ∥ 2 ≤ · · · ≤ j − 1 X i =0 ∥ x p + i − x p + i +1 ∥ 2 = j − 1 X i =0 ∥ k ( x p + i )∆( x p + i ) ∥ 2 ≤ k j − 1 X i =0 ∥ ∆( x p + i ) ∥ 2 , 9 since w e supp ose that the sequence { k ( x n ) } is b ounded. F rom (4) with n = p + i , it follo ws ∥ ∆( x p + i ) ∥ 2 ≤ L i ∥ ∆( x p ) ∥ 2 and so ∥ x p − x q ∥ 2 ≤ k j − 1 X i =0 ∥ ∆( x p + i ) ∥ 2 ≤ k j − 1 X i =0 L i ∥ ∆( x p ) ∥ 2 < k ∥ ∆( x p ) ∥ 2 ∞ X i =0 L i = k ∥ ∆( x p ) ∥ 2 1 1 − L , b ecause lim n →∞ P n i =0 L i = 1 1 − L . Since the limit of the sequence {∥ ∆( x n ) ∥ 2 } is 0 , given ϵ and γ = ϵ 1 − L k , it exists n such that, if p > n then ∥ ∆( x p ) ∥ 2 < γ . Concluding, for each ϵ it exists n suc h that for p > n and, consequen tly , q > p > n , the following holds ∥ x p − x q ∥ 2 ≤ k ∥ ∆( x p ) ∥ 2 1 1 − L < k γ 1 1 − L = ϵ . □ Remark 2. The or em 1 assumes the existenc e of a function k ( x n ) such that ∥ ∆( x n ) ∥ 2 ≤ L ∥ ∆( x n +1 ) ∥ 2 holds for al l n ∈ N . In se ction 3, we wil l show how to c alculate k ( x n ) in the sp e cific c ase wher e the r efer enc e risk me asur e is the standar d deviation. In pr actic e, given a c onstant 0 < L < 1 , it is sufficient to solve the ine quality ab ove to determine k ( x n ) . The existenc e of at le ast one r e al solution dep ends on the value of L . Within the pr esent fr amework, L plays a r ole similar to that of the Lipschitz c onstant in the c ontext of c ontr action mappings, se e Banach (1922). However, deriving a suitable value of L for al l x ∈ S is chal lenging, as it dep ends on the ge ometry of the pr oblem. F urthermor e, sinc e k ( x n ) dep ends on the functional form R C ρ , it c annot b e c ompute d without r efer enc e to a sp e cific risk me asur e. In the applic ation, we arbitr arily cho ose a value 0 < L < 1 . If L is sufficiently close to 1 , the c ontinuity of ∆ guar ante es the existenc e of at le ast one r e al solution to the ine quality, although the c onver genc e r ate of the se quenc e is slow in this c ase. Conversely, when L is closer to zer o, the c onver genc e r ate is p otential ly faster, but a r e al solution to the ine quality may not exist. F or these r e asons, the choic e of L is crucial. W e wil l pr ovide guidanc e on how to pr o c e e d when the ine quality admits no r e al solutions in R in se ction 3. 10 Since R N with the euclidean distance is a complete metric space, each Cauc hy sequence has a finite limit within the space. Therefore, the sequence { x n } has a finite limit. Theorem 2. Under the assumptions of the or em 4, if lim n →∞ x n = e x for e x ∈ R N , then ∆( e x ) = 0 that is R C ρ ( e x ) = x b 1 t R C ρ ( e x ) . Pr o of:. Thanks to the assumptions on the existence of k ( x n ) (with opp ortune prop erties), see theorem 1, lim n →∞ ∆( x n ) = 0 . The function ∆ is con tinuous and lim n →∞ x n = e x , then it holds that lim n →∞ ∆( x n ) = ∆( e x ) = R C ρ ( e x ) − x b 1 t R C ρ ( e x ) . Concluding, ∆( e x ) = 0 and R C ρ ( e x ) − x b 1 t R C ρ ( e x ) = 0 from whic h the thesis of the theorem. □ Theorem 2 sho ws that the limit of the Cauc hy sequence is the risk budgeting p ortfolio, provided that e x ∈ S . 2.3. Existenc e and uniqueness of the fixe d p oint in the simplex This section presen ts sufficient conditions for the existence and unique- ness of the risk budgeting portfolio in S . By Theorem 2, e x is a fixed p oint of the function G , where k is the con tinuous function defined in the previous subsection. The result follows directly from the observ ation that ∆( e x ) = 0 implies G ( e x ) = e x , since k is a b ounded function. Moreo ver, we observe that the risk budgeting p ortfolio x ∗ is also a fixed p oint of the function G in S , since x b = RC ρ ( x ∗ ) / 1 t RC ρ ( x ∗ ) implies G ( x ∗ ) = x ∗ − k ( x ∗ )  RC ρ ( x ∗ ) − x b 1 t RC ρ ( x ∗ )  = x ∗ − k ( x ∗ ) ( R C ρ ( x ∗ ) − R C ρ ( x ∗ )) = x ∗ . 11 If the limit of { x n } b elongs to S , and if the function G admits a unique fixed p oin t in S , then the limit of { x n } is the risk budgeting p ortfolio. While in the ma jorit y of the empirical applications that we ha ve p erformed -using the standard deviation as the reference risk measure- G b ehav es as a contraction on S ⊂ R N , it is p ossible to build examples where G is not a con traction. Consequen tly , the uniqueness of the fixed p oin t is not guaranteed a priori. T o pro ve that G admits a unique fixed p oin t in S , we refer to Kellogg’s theorem, see Kellogg (1976). Let X b e a real Banac h space with a b ounded con vex op en subset D , and let F : D → D b e a con tinu ous function, whic h is also assumed to b e compact if X is infinite dimensional. Under the assumption that F is differentiable, there is a condition which guaran tees the result. If the only relev an t case is the finite dimensional, the compactness hypothesis can b e omitted. Theorem 3. [ Kel lo gg the or em ] L et F : D → D b e a c omp act c ontinuous map which is c ontinuously F r e chet differ entiable on D . Supp ose that • for e ach x ∈ D , 1 is not an eigenvalue of F ′ , and • for e ach x ∈ ∂ D , x  = F ( x ) . Then F has a unique fixe d p oint. Without loss of generalit y , we assume that the dimension of the reference space -the num ber of assets N - is finite, and w e consider the in terior of S , denoted b y ˚ S = ( x = [ x 1 ; . . . ; x N ] t ∈ R N : N X i =1 x i = 1 , and x i > 0 , ∀ i = 1 , . . . , N ) . Equation (5) establishes the existence of the fixed p oin t of G in S . How ever, the economic in terpretation is particularly imp ortant: the risk budgeting p ortfolio is characterized b y a strictly p ositiv e allo cation in eac h of the N 12 assets. It never concentrates the allo cation in a small num b er of large p osi- tions, and therefore x ∗ ∈ ˚ S . The role of k ( x ) is fundamental in ensuring that the function G satisfies the necessary conditions on S . If the functions k and ∆ are contin uous for all x ∈ S , then G is also contin uous, and k is b ounded on the compact set S . F urthermore, if k is appropriately chosen, the following holds: 1. G ( x ) ∈ S for all x ∈ S ; 2. G satisfies the assumptions of Kellogg’s theorem; 3. it exists a p ositiv e real num b er 0 < L < 1 such that ∥ ∆( x n +1 ) ∥ 2 < L ∥ ∆( x n ) ∥ 2 . F rom 1. and 2. it follo ws that the fixed p oint of G ( x ) in S is unique; from 3. it follows that ∆( x n ) → 0 and, since k ( x ) is b ounded in S , k ( x n )∆( x n ) → 0 . In conclusion, if k is appropriately c hosen, { x n } tends to a finite limit in S whic h is the unique fixed p oin t of G ( x ) , that is lim n →∞ x n = x ∗ . While in the presen t section we simply assume the existence of a function k ( x ) with the desired prop erties, in Section 3 we will sho w how to compute it explicitly . This is due to the fact that the deriv ation of k ( x ) requires the selection of a sp ecific reference risk measure. In our framew ork, the existence and uniqueness of the risk budgeting portfolio are equiv alen t to the existence and uniqueness of the fixed p oint of G . Therefore, to apply Kellogg’s theorem, it is necessary that ∆ b e con tinuous. Remark 3. ( Continuity of ∆( x ) ) If ∆ is a c ontinuous function and { x n } is a Cauchy se quenc e with limit x ∗ , then ∆( x n ) → 0 . Conversely, if ∆ is not c ontinuous, x n → x ∗ do es not imply ∆( x n ) → ∆( x ∗ ) . Nevertheless, if ∆ is c ontinuous in a neighb orho o d of x ∗ , sinc e it definitively b elongs to the neighb orho o d, then lo c al c onver genc e to the fixe d p oint c an stil l b e establishe d. 13 2.4. The algorithm In this section, we present the general structure of the algorithm used to compute the risk budgeting p ortfolio. Although the numerical pro cedure is implicit in the sequence defined in Subsection 2.2, it is useful to pro vide a pseudo-co de representation and highlight certain tec hnical aspects relev an t for implementation. The following presen ts the general pseudo-co de for the practical computation of the risk budgeting p ortfolio. The algorithm op- erates as follows: starting from a p oint x n , the v ector ∆( x n ) iden tifies a direction in the space. The sign of k n (for simplicit y , w e use k n to denote k ( x n ) for the rest of the pap er) determines the orient ation along this di- rection, while the v alue | k n |∥ ∆( x n ) ∥ represents the step length. A suitable restriction on | k n | ensures that the sequence x n remains within the simplex S . Algorithm 1. Define the risk c ontribution R C ρ , the risk budget x b , a p ositive p ar ameter L ∈ (0 , 1) , the toler anc e tol , the initial p oint x 1 ∈ S , the maximum numb er of iter ations maxit Initialize the step c ounter: n := 1 while ∥ ∆( x n ) ∥ 2 > tol & n < maxit do Find k n : ∥ ∆( x n + k n ∆( x n )) ∥ 2 < L ∥ ∆( x n ) ∥ 2 if If x n + k n ∆ x n ∈ S then Up date x n +1 = x n + k n ∆ x n end if if x n + k n ∆ x n ∈ S then Calculate e k n = γ n k n , γ n > 0 : ∥ ∆( x n + e k n ∆ x n ) ∥ 2 < L ∥ ∆( x n ) ∥ 2 & x n + e k n ∆ x n ∈ S Up date x n +1 = x n + e k n ∆ x n end if Up date the c ounter: n = n + 1 end while r eturn x n The inputs of the algorithm are: 14 • RC ρ , the v ector of the marginal risk contributions with resp ect to the reference risk measure ρ ; • x b , the target v ector of risk exp osures; • L , the parameter controlling the conv ergence rate of the sequence; • tol , the solution tolerance (default v alue 10 − 6 ); • x 0 ∈ S , the initial p oin t for the iteration; • maxit , the maximum num b er of iterations (default v alue 1000 ) The application of the algorithm requires only the ev aluation of the risk con tribution v ector R C ρ with respect to the selected risk measure ρ , and the computation of k n at eac h iteration step. In practice, given a con- stan t 0 < L < 1 , k n is determined at eac h step by solving the inequalit y ∥ ∆( x n +1 ) ∥ 2 < L ∥ ∆( x n ) ∥ 2 . If | k n | is sufficien tly small to ensure that the next iterate x n +1 remains in the simplex S , the step is accepted. Otherwise, | k n | is appropriately reduced b efore computing x n +1 . Remark 4. Our metho d c omputes G ( x ) = x + k ( x ) ∗ ∆( x ) by c onstruct- ing the se quenc e x n +1 = G ( x n ) . At e ach step, we lo ok for k ( x n ) such that ∥ ∆( x n +1 ) ∥ 2 < L 2 ∥ ∆( x n ) ∥ 2 , L < 1 . It fol lows that the se quenc e | ∆( x n ) | tends to zer o, and ther efor e x n c onver ges to the fixe d p oint of G . The for- mulas r esemble those of line se ar ch metho ds, se e Sun and Y uan (2006), which in gener al al low to c ompute the unc onstr aine d minimum of a func- tion f : R N → R by me ans of the se quenc e x n +1 = x n + α n p n , α n > 0 wher e p n is a se ar ch dir e ction c oinciding with a desc ent dir e ction, that is, such that ∇ f ( x n ) t p n < 0 , and α n is the step length. In our c ase, sinc e we c an r ewrite the r elation in the form x n +1 = x n + | k ( x n ) | sign( k ( x n ))∆( x n ) , the se ar ch dir e ction is sign( k ( x n ))∆( x n ) and the step length is | k ( x n ) |∥ ∆( x n ) ∥ 2 . However, despite the similarity, the metho d is substantial ly differ ent. In- de e d, letting f ( x ) = ∥ ∆( x ) ∥ 2 2 , the line se ar ch metho d applie d to f c omputes, at e ach step, α n that achieves or appr oximates the minimum of the function 15 h ( α ) = f ( x n + αp n ) , with p n a desc ent dir e ction, wher e as our algorithm c om- putes k ( x n ) in or der to r e duc e the value h ( k ) = ∥ ∆( x n + k ∆( x n )) ∥ 2 , that is, we only r e quir e that h ( k ) < Lh (0) , with L < 1 . It is essential that L < 1 . Inde e d, simply implementing at the n -th step a se ar ch for k ( x n ) such that f ( x n + k ( x n )∆( x n )) < f ( x n ) is not an optimal choic e, sinc e the algorithm may stagnate; that is, the r e ductions in the values of the function f may b e ne gligible. 3. The case of the standard deviation In this section, we apply our approach by selecting the standard devia- tion as the reference risk measure. In this case, the vector of marginal risk con tributions is given b y R C ρ ( x ) = diag ( x ) V x , where V denotes the co v ari- ance matrix. 5 A ccordingly , by exploiting the expression for the marginal risk con tributions, the function ∆ : R N → R N b ecomes ∆( x ) = diag ( x ) V x − x b 1 t ( diag ( x ) V x ) , or, equiv alen tly , ∆( x ) = diag ( x ) V x − ( x t V x ) x b . Remark 5. W e note that R C ρ i ( x ) ≥ 0 for al l x ∈ ˚ S . In fact, it is p ossi- ble to find p ortfolios within ˚ S (se e Example 1) for which the mar ginal risk c ontribution of at le ast one asset is ne gative. By c ontinuity of R C ρ , this also implies the existenc e of p ortfolios in ˚ S wher e the mar ginal risk c ontri- bution of a given asset is zer o. R e c al l that ˚ S c onsists of long-only p ortfolios. Intuitively, when numer ous ne gative c orr elations ar e pr esent among assets, the mar ginal risk c ontribution of an asset to the total p ortfolio risk c an turn ne gative. This le ads to the c ounter-intuitive b ehavior wher eby incr e asing the exp osur e to that asset may r esult in a de cr e ase in its mar ginal c ontribution to total risk (se e example 1). R e gar ding the algorithm, in the vast majority of c ases, k n is ne gative, which supp orts the usual e c onomic intuition: if the risk exp osur e of an asset exc e e ds the tar get, then the c orr esp onding al lo c ation 5 The cov ariance matrix is a real, symmetric, p ositiv e definite matrix of order N . 16 should b e r e duc e d, and vic e versa. In some c ases, however, k n may b e c ome p ositive for some x . 6 Example 1. L et us c onsider a market with N = 5 risky assets and the fol lowing c ovarianc e matrix V , which c orr esp onds to the c orr elation matrix C : V =       0 . 1137 − 0 . 0289 0 . 0295 0 . 0279 0 . 0437 − 0 . 0289 0 . 0255 − 0 . 0337 − 0 . 0156 − 0 . 0159 0 . 0295 − 0 . 0337 0 . 1002 0 . 0068 0 . 0427 0 . 0279 − 0 . 0156 0 . 0068 0 . 0281 0 . 0262 0 . 0437 − 0 . 0159 0 . 0427 0 . 0262 0 . 0884       C =       1 . 0000 − 0 . 5367 0 . 2764 0 . 4936 0 . 4359 − 0 . 5367 1 . 0000 − 0 . 6667 − 0 . 5828 − 0 . 3349 0 . 2764 − 0 . 6667 1 . 0000 0 . 1282 0 . 4537 0 . 4936 − 0 . 5828 0 . 1282 1 . 0000 0 . 5257 0 . 4359 − 0 . 3349 0 . 4537 0 . 5257 1 . 0000       The eigenvalues of V ar e 0 . 0031; 0 . 0215; 0 . 0515; 0 . 0802; 0 . 1996 . Given the p ortfolio x = [0 . 0932; 0 . 0495; 0 . 0215; 0 . 5212; 0 . 3147] t the c orr esp onding mar ginal risk c ontributions ar e given by R C ( x ) = diag ( x ) V x t = [0 . 0036; − 0 . 0008; 0 . 0004; 0 . 0130; 0 . 0144] t . Due to the ne gative c orr elations b etwe en asset i = 2 and the other assets, asset i = 2 exhibits a ne gative mar ginal risk c ontribution. If the exp osur e to asset i = 2 is incr e ase d of 1% and the exp osur e to asset i = 4 is r e duc e d by the same amount, the up date d p ortfolio and c orr esp onding mar ginal risk c ontributions b e c ome: x = [0 . 0932; 0 . 0595; 0 . 0215; 0 . 5112; 0 . 3147] t R C ( x ) = [0 . 0036; − 0 . 0009; 0 . 0004; 0 . 0125; 0 . 0142] t . This example il lustr ates a r ar e b ehavior in which incr e asing the exp osur e to one asset le ads to a de cr e ase in its mar ginal risk c ontribution to total risk. Now c onsider the p ortfolio x = [0 . 1450; 0 . 4049; 0 . 0298; 0 . 2896; 0 . 1307] t 6 Based on our exp erience, such p eculiar b eha viors tend to arise only when the num b er of assets N is relatively small. When N is large, we hav e not observed such phenomena. W e conjecture that this effect is related to the structure of the cov ariance matrix and its p ositiv e definiteness. 17 with mar ginal risk c ontributions R C ( x ) = [0 . 0028; − 0 . 0006; 0 . 0000; 0 . 0027; 0 . 0027] t . In this c ase, a p ositive exp osur e to asset i = 3 c orr esp onds to a zer o mar ginal risk c ontribution of that asset to the total risk. Giv en an arbitrary initial p oin t x 1 ∈ ˚ S and 0 < L < 1 , the sequences { x n } n =1 , 2 ,... and { ∆ x n } n =1 , 2 ,... are          ∆( x n ) = diag ( x n ) V x n − ( x t n V x n ) x b x n +1 = x n + k n ∆( x n ) , (5) where k n is computed as follo ws. At each step n , the vectors D 1 , D 2 and D 3 are calculated D 1 = diag ( x n ) V ∆( x n ) − ( x t n V ∆( x n )) x b , D 2 = diag (∆( x n )) V x n − ((∆( x n )) t V x n ) x b , D 3 = ∆(∆( x n )) . W e prop ose tw o alternative criteria for selecting the v alue of k n . • If the quartic p olynomial Q ( k n ) has at least one real root, then k n is c hosen such that Q ( k n ) < 0 . Q ( k n ) = k 4 ∥ D 3 ∥ 2 + 2 k 3 ( D 1 + D 2 ) t D 3 + k 2 (2∆( x t n ) D 3 + ∥ D 1 + D 2 ∥ 2 ) + 2 k ∆( x t n )( D 1 + D 2 ) + (1 − L 2 ) ∥ ∆( x n ) ∥ 2 . • Alternatively , when Q ( k ) has no real ro ots, we consider the cubic p oly- nomial C ( k ) , C ( k ) = k 3 ∥ D 3 ∥ 2 + 2 k 2 ( D 1 + D 2 ) t D 3 + k (2∆( x t n ) D 3 + ∥ D 1 + D 2 ∥ 2 ) + 2∆( x t n )( D 1 + D 2 ) . 18 that alwa ys admits at least one real ro ot. Then k n is chosen to verify C ( k n ) < 0 . As sho wn in the follo wing theorem, the condition Q ( k n ) < 0 is equiv alen t to ∥ ∆( x n +1 ) ∥ 2 2 ≤ L 2 ∥ ∆( x n ) || 2 2 while the condition C ( k n ) < 0 is equiv alent to ∥ ∆( x n +1 ) ∥ 2 2 ≤ ∥ ∆( x n ) ∥ 2 2 . The following theorem holds. Theorem 4. If k n is c alculate d as describ e d ab ove and C ( k n ) = 0 is solve d a finite numb er of times, then lim n →∞ ∥ ∆( x n ) ∥ 2 = 0 that is lim n →∞ ∆( x n ) = 0 . Pr o of:. It holds that ∆( x n +1 ) = diag ( x n +1 ) V x n +1 − x b ( x t n +1 V x n +1 ) = diag ( x n + k n ∆( x n )) V ( x n + k n ∆( x n )) − x b  ( x n + k n ∆( x n )) t V ( x n + k n ∆( x n ))  = diag ( x n ) V x n + k n diag ( x n ) V ∆( x n ) + k n diag (∆( x n )) V x n + k 2 n diag (∆( x n )) V (∆( x n )) − x b  x t n V x n  − 2 k n x b x t n V ∆( x n ) − x b k 2 n  ∆( x t n ) V ∆( x n )  = ∆ x n + k n ( D 1 + D 2 ) + k 2 n D 3 Imp osing ∥ ∆( x n +1 ) ∥ 2 ≤ L 2 ∥ ∆( x n ) ∥ 2 ,  ∆( x n ) + k n ( D 1 + D 2 ) + k 2 n D 3  t  ∆( x n ) + k n ( D 1 + D 2 ) + k 2 n D 3  ≤ L 2 ∥ ∆( x n ) ∥ 2 then (1 − L 2 ) ∥ ∆( x n ) ∥ 2 + 2 k n ∆( x n ) t ( D 1 + D 2 ) + k 2 n (2∆( x n ) t D 3 + ∥ D 1 + D 2 ∥ 2 ) +2 k 3 n ( D 1 + D 2 ) t D 3 + k 4 n ∥ D 3 ∥ 2 ≤ 0 . 19 If Q ( k ) has at least one real ro ot, let α b e the ro ot with the smallest ab- solute v alue. Since Q (0) > 0 , if α > 0 , then any v alue of k n > α , smaller than an ev entual ro ot larger than α , verifies the inequality . Con versely , if α < 0 , a v alue of k n < α , larger than an ev en tual ro ot smaller than α , v erifies the inequality . The v alue of k n p ermits to calculate x n +1 suc h that ∥ ∆( x n +1 ) ∥ 2 ≤ L ∥ ∆( x n ) ∥ 2 . If Q ( k ) has non real ro ots, then w e require ∥ ∆( x n +1 ) ∥ 2 ≤ ∥ ∆( x n ) ∥ 2 , whic h is equiv alen t to the following inequality k n (2∆( x n ) t ( D 1 + D 2 ) + k n (2∆( x n ) t D 3 + ∥ D 1 + D 2 ∥ 2 ) +2 k 2 n ( D 1 + D 2 ) t D 3 + k 3 n ∥ D 3 ∥ 2 ) ≤ 0 . A cubic p olynomial alwa ys has at least one real ro ot. Therefore, it is p ossible to calculate k n suc h that ∥ ∆( x n +1 ) ∥ 2 ≤ ∥ ∆( x n ) ∥ 2 . 7 If the real ro ot of the cubic p olynomial is calculated a finite n umber of times, it holds that ∥ ∆( x n +1 ) ∥ 2 ≤ L m ∥ ∆( x n ) ∥ 2 , where m ≤ n , with lim n →∞ m = ∞ . Since L < 1 , the first result holds. Ob viously , since the norm is a contin uous function, lim n →∞ ∆( x n ) = 0 . □ No w w e prov e that, { x n } is a Cauc hy sequence when k n is calculated as describ ed. This result is a sp ecial case of theorem 1, see subsection 2.2. Compared to the general v ersion, the following theorem differs in tw o main asp ects: first, it exploits the specific functional form of the marginal risk con tributions; second, it explicitly incorp orates the calculation of k n within the pro of, rather than assuming the existence of k n . Theorem 5. If k n is c alculate d as describ e d, C ( k n ) = 0 is solve d a finite 7 When solving the cubic inequality , it is necessary to analyze the sign of k n C ( k n ) . F our cases arise dep ending on the sign of k n and the b ehavior of C ( k n ) . 20 numb er of times and | k n | < k , ∀ n , (i.e. the se quenc e { k n } n =1 , 2 ,... is b ounde d), then { x n } is a Cauchy se quenc e. Pr o of:. Without loss of generality , we assume q = p + j , with j ∈ N . Then ∥ x p + j − x p ∥ 2 = ∥ x p + j − x p +1 + x p +1 − x p ∥ 2 ≤ ∥ x p + j − x p +1 ∥ 2 + ∥ x p +1 − x p ∥ 2 ≤ ∥ x p + j − x p +2 ∥ 2 + ∥ x p +2 − x p +1 ∥ 2 + ∥ x p +1 − x p ∥ 2 ≤ · · · ≤ j − 1 X i =0 ∥ x p + i +1 − x p + i ∥ 2 = j − 1 X i =0 ∥ k p + i ∆ x p + i ∥ 2 ≤ k j − 1 X i =0 ∥ ∆( x p + i ) ∥ 2 . By assumption, it exists L < 1 such that ∥ ∆( x n +1 ) ∥ 2 ≤ L ∥ ∆( x n ) ∥ 2 is satis- fied an infinite n umber of times since ∥ ∆( x n +1 ) ∥ 2 = ∥ ∆( x n ) ∥ 2 can happ en in a finite n umber of steps, it holds that ∥ ∆( x n + h ) ∥ 2 ≤ L m h ∥ ∆( x n ) ∥ 2 with m h ≤ h and w e obtain ≤ k j − 1 X i =0 ∥ ∆ x p + i ∥ 2 ≤ k j − 1 X i =0 L m i ∥ ∆ x p ∥ 2 < k ∥ ∆ x p ∥ 2 ∞ X i =0 L m i = k ∥ ∆ x p ∥ 2 1 1 − L , b ecause lim n →∞ P n i =0 L i = 1 1 − L . The limit of the sequence { ∆( x n ) } is 0 , giv en ϵ and γ = ϵ 1 − L k , it exists n suc h that, if p > n then ∥ ∆( x p ) ∥ 2 < γ . Concluding, for each ϵ it exists n suc h that for p > n and, consequen tly , q > p > n , the following holds ∥ x p − x q ∥ 2 ≤ k ∥ ∆( x p ) ∥ 2 1 1 − L < k γ 1 1 − L = ϵ . 21 □ Eac h Cauch y sequence has a finite limit within a complete metric space. Therefore, { x n } has a finite limit in R N . Theorem 6. L et e x ∈ R N such that lim n →∞ x n = e x , then the fol lowing holds diag ( e x ) V e x = x b ( e x t V e x ) . Pr o of:. Thanks to theorem 4, lim n →∞ ∆ x n = 0 . The function ∆( x ) is con tinuous and lim n →∞ x n = e x , then lim n →∞ ∆( x n ) = ∆( e x ) = diag ( e x ) V e x − x b ( e x V e x ) . Concluding, diag ( e x ) V e x − x b ( e x t V e x ) = 0 from whic h we obtain the thesis of the theorem. □ F rom theorem 6, it follows that x ∗ is a fixed p oint of the function G : R N → R N , defined as G ( x ) = x − k ( x )∆( x ) . This result is immediate, since ∆( x ∗ ) = 0 , k ( x ) is b ounded and thus G ( x ∗ ) = x ∗ . How ever, as discussed more generally at the end of subsection 2.2, the fixed p oin t may not b e unique. 3.1. Existenc e and uniqueness of the fixe d p oint in S In subsection 2.3, w e presen ted general results concerning the existence and uniqueness of the fixed p oint. When the standard deviation is adopted as the referring risk measure, an in teresting alternative to Theorem 3 b ecomes a v ailable for pro ving the existence and uniqueness of the fixed p oin t in S . Sp ecifically , the functional form of the marginal risk contributions allo ws the application of general mathematical results developed in the context of rescaling the row and column norms of matrices to improv e their condition n umber (see, among others, O’Leary (2003)). Reformulating theorem 1 and Corollary 1 from O’Leary (2003) within our framework the follo wing result holds. 22 Theorem 7 (Oleary ). Given a p ositive definite squar e r e al matrix V of or der N and a ve ctor x b ∈ R N , it exists a ve ctor y ∈ R N such that diag ( y ) V y = x b . (6) Mor e over, e quation (6) admits 2 N solutions, e ach lo c ate d in a distinct orthant of R N . The follo wing corollary is an immediate consequence of the previous result. Corollary 1. Given a p ositive definite squar e r e al matrix V of or der N and a ve ctor x b ∈ S , it exists a unique fixe d p oint b x of G ( x ) = x − ( diag ( x ) V x − x b ( x t V x ) in S . Pr o of:. F rom theorem 7, it exists a unique b y ∈ R N × N with p ositive entries suc h that diag ( b y ) V b y = x b . Since x b ∈ S , then 1 = 1 t x b = 1 t ( diag ( b y ) V b y ) = b y t V b y and, consequen tly , diag ( b y ) V b y = 1 · x b =  b y t V b y  x b . Let b z = b y P N i =1 b y i = b y ∥ b y ∥ 1 Note that b z ∈ S b ecause the entries are p ositiv e real num b ers with unitary sum. F rom diag ( b y ) V b y =  b y t V b y  x b it holds that diag  b y ∥ b y ∥ 1  V b y ∥ b y ∥ 1 =  b y t ∥ b y ∥ 1 V b y ∥ b y ∥ 1  x b equiv alen tly , diag ( b z ) V b z =  b z t V b z  x b ⇔ ∆( b z ) = 0 . 23 This pro ves that the ro ot of ∆( x ) is unique. Then the fixed p oin t of G ( x ) is unique in S . □ In general, as previously noted, the sequence x n do es not necessarily re- main within the simplex S , unless the step size | k n | is appropriately b ounded. The following theorem provides an explicit condition on | k n | to ensure that the sequence x n remains in the simplex. This result is particularly relev an t for the practical implemen tation of the numerical pro cedure. 8 Theorem 8. If x n ∈ S , then x n +1 = x n + k n ∆( x n ) ∈ S if | k n | ≤        min n 1 − ( x n ) i | ∆( x n ) i | | k n ∆( x n ) i > 0 o min n ( x n ) i | ∆( x n ) i | | k n ∆( x n ) i < 0 o , wher e ( x n ) i and ∆( x n ) i ar e the i th -entries of x n and ∆( x n ) r esp e ctively. Pr o of:. First, 1 t ∆( x n ) = x t n V x n − ( 1 t x b )( x t n V x n ) = x t n V x n − x t n V x n = 0 ; this means that if the en tries of x n ha ve unitary sum, x n +1 has the same prop ert y if 1 t x n +1 = 1 t ( x n + k n ∆( x n )) = 1 , indep enden tly from the v alue of k n . Therefore, it is necessary to prov e that each comp onen t of x n +1 is non-negativ e. F or simplicity , we omit the index n and refer to ∆ for ∆( x ) . W e require 0 ≤ x i + k ∆ i ≤ 1 , i.e. − x 1 ≤ k ∆ i ≤ 1 − x i , where x i and ∆ i are the i th en tries of x and ∆ resp ectively . By assumption, 0 < x 1 < 1 . T w o 8 Based on our empirical exp erience, in the v ast ma jority of cases, the sequence x n con verges to the fixed p oin t within S without an y explicit restriction on | k n | . How ever, in a small num ber of instances, we observed a finite num b er of iterations falling outside of S . In rare situations, when no restriction is imp osed on | k n | , the sequence may con verge to a solution lo cated in a different orthan t. This t ypically o ccurs when an alternative fixed p oin t lies close outside the b order of the simplex. F or this reason, it is useful to enforce the restriction on | k n | and ensure the sequence remains within S . 24 cases are p ossible. case 1 : k ∆ i > 0 , then k ∆ i = | k ∆ i | = | k || ∆ i | . The v alue of | k | needs to satisfy the conditions          | k | ≥ − x i | ∆ i | | k | ≤ 1 − x i | ∆ i | Note that the first inequalit y is alwa ys verified. case 2 : k ∆ i < 0 , then k ∆ i = −| k ∆ i | = −| k || ∆ i | . The v alue of | k | needs to satisfy the conditions          −| k | ≥ − x i | ∆ i | −| k | ≤ 1 − x i | ∆ i | ⇔          | k | ≤ x i | ∆ i | | k | ≥ x i − 1 | ∆ i | In this case, the second inequalit y is alwa ys verified. Concluding, | k | v erifies | k | ≤          min n 1 − x i | ∆ i | | k ∆ i > 0 o min n x i | ∆ i | | k ∆ i < 0 o □ 4. Empirical results In this section, w e presen t a n umerical application to demonstrate the effectiv eness of the prop osed approac h for computing the risk budgeting p ort- folio. The reference risk measure emplo yed is the standard deviation. Ac- cordingly , we implement the fixed p oin t (FP) algorithm in tro duced in general terms in Section 2.4, specifying the marginal risk contributions and the step 25 size k n as outlined in Section 3. The p erformance of the FP algorithm is compared against three standard methods commonly used in the literature. A brief description of these alternativ e approaches is provided b elow. • Optimization 1 (OP1): The risk budgeting p ortfolio is computed by solving the following optimization problem using the MA TLAB func- tion fmincon . Minimize P N i =1 P N j =1  x i ( V x ) i − x b j  2 sub ject to x i ≥ 0 P N i =1 x i = 1 . This optimization problem was originally prop osed in Maillard et al. (2010) for the calculation of the risk parit y p ortfolio with resp ect to the standard deviation. In our case, w e adapt it b y mo difying the ob jective function in order to compute the risk budgeting p ortfolio. • Optimization 2 (OP2): The risk budgeting p ortfolio is computed by solving the following optimization problem using the MA TLAB func- tion fmincon Minimize p ∥ y t V y − x b ∥ 2 sub ject to P N i =1 log y i ≥ c y i ≥ 0 where log denotes the natural loga rithm and c is an arbitrary con- stan t. This optimization problem was originally prop osed in Maillard et al. (2010) as a useful alternative to OP1 for computing the risk parit y p ortfolio. In our case, we slightly mo dified the ob jective func- tion to generalize the problem and derive the risk budgeting p ortfolio. As highligh ted in the introduction, this optimization approac h is also 26 emplo yed in Cetingoz et al. (2024) to establish sufficient conditions for the existence and uniqueness of the risk budgeting portfolio and widely used in the literature. • Non-linear system (NLS): the condition RC ρ ( x ∗ ) P N i =1 RC ρ i ( x ) = x b whic h charac- terizes the risk budgeting p ortfolio for a general risk measure ρ , defines a nonlinear system of equations. In principle, it is alwa ys p ossible to attempt solving this system numerically to obtain the risk budgeting p ortfolio. F or this purp ose, we use the MA TLAB function lsqnonlin . All the MA TLAB built-in functions used in the application are imple- men ted with their default settings. In particular, the maxim um n um b er of iterations is determined b y the standard configuration. Increasing the n umber of iterations ma y lead to a more accurate solution; ho w ever, this t ypically comes at the cost of increased computational time. The differen t algorithms are compared based on tw o criteria: the computational time re- quired to reach the solution and the distance from the effectiv e solution. If x denotes the solution obtained b y a given algorithm, the distance to the true risk budgeting p ortfolio is measured by the ∥ RC ρ ( x ) / 1 t R C ρ ( x ) − x b ∥ 2 . This metric ev aluates how closely the realized risk contributions match the target risk allo cation x b . Since computational time ma y dep end on the p ortfolio size N , we com- pare the p erformance of the algorithms for v arious v alues: N = 5 , 10 , 50 , 100 , 200 . F or each v alue of N , w e generate T = 1000 random exp eriments. In each exp erimen t, a cov ariance matrix V is sampled, along with t wo vectors x 1 and x b in S , representing the initial p oin t and the target risk budget, re- sp ectiv ely . Both vectors are uniformly sampled from the simplex S to stress the numerical efficiency of the algorithms, particularly when one or b oth are 27 near the b oundary of the simplex, whic h can lead to increased computational complexit y . T o further assess the robustness of the algorithms, some co v ari- ance matrices are chosen to b e numerica lly close to a rank-deficien t matrix. F or each triplet ( V , x 1 , x b ) , the four algorithms are applied, and b oth the computational time and the distance from the target solution are recorded. T able 1 summarizes the results. F or each algorithm and each p ortfolio size, w e rep ort the av erage conv ergence time and the mean distance from the true risk budgeting solution, as defined ab ov e. The key observ ation from T able 1 is that the fixed-p oint (FP) algorithm consistently outp erforms the other metho ds, b oth in terms of sp eed and accuracy , across all tested scenarios. T able 1: Comparison of av erage computational time (in seconds) and solution ac- curacy across differen t algorithms (FP , OP1, OP2, NLS) and p ortfolio sizes ( N = 5 , 10 , 50 , 100 , 200) . FP ( L = 0 . 5) OP1 NLS OP2 N = 5 time(s) 0.0007 0.0140 0.0061 0.0158 accuracy 0.0002 0.0897 0.1527 0.3633 N = 10 time(s) 0.0017 0.0228 0.0069 0.0313 accuracy 0.0004 0.0467 0.1506 0.2916 N = 50 time(s) 0.0047 0.1267 0.0363 0.0836 accuracy 0.0012 0.0397 0.0896 0.2058 N = 100 time(s) 0.0100 0.2612 0.1959 0.1089 accuracy 0.0023 0.0488 0.0626 0.1368 N = 200 time(s) 0.0383 1.4118 3.6511 0.2711 accuracy 0.0040 0.0556 0.0458 0.2877 As previously highligh ted, the accuracy of the solutions obtained via OP1, OP2, and NLS can be impro v ed by increasing the maxim um num- 28 b er of iterations. How ev er, this enhancement w ould come at the cost of significan tly longer conv ergence times, further penalizing the efficiency of these pro cedures. It is worth noting that, in addition to outp erforming OP1 and OP2, the NLS algorithm demonstrates comp etitiv e p erformance. As exp ected, computational time increases with the portfolio size N across all algorithms. Nev ertheless, the FP algorithm exhibits the greatest resilience to dimensionalit y , with its performance b eing least affected by the n umber of assets. F urthermore, p ortfolio size do es not app ear to significantly influence the accuracy of the solution for an y of the metho ds. W e now turn our atten tion to the FP algorithm, fo cusing on an applica- tion designed to inv estigate the impact of the parameter L . This parameter is externally specified and, in theory , go verns the conv ergence rate—and thus the computational time—of the algorithm. F or each triad ( V , x 1 , x b ) , we run the FP algorithm while v arying the v alue of L , recording b oth the a verage computational time and the accuracy of the solution. The outcomes are re- p orted in T able 2. An in teresting observ ation emerges: the smallest v alue of L do es not alwa ys corresp ond to the fastest con vergence. This phenomenon is straigh tforward to interpret. When L is to o small, the algorithm frequen tly encoun ters iterations for whic h the asso ciated polynomial of degree 4 has no real solution. As a result, sev eral steps of the algorithm fail to adv ance to ward the solution, thereby slowing the o verall conv ergence. In contrast, when L approaches 1, although eac h step length is relatively small, it reliably mo ves closer to the solution, enhancing ov erall conv ergence efficiency . Considering the go o d p erformance of the FP algorithm relativ e to alter- nativ e pro cedures, along with its low computational cost p er iteration, we recommend selecting v alues of L < 1 close to 1 for practical implementation. 29 T able 2: Comparison of the FP algorithm for differen t v alues of the parameter L = 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 , 0 . 95 , 0 . 99 . FP FP FP FP FP FP L 0.3 0.5 0.7 0.9 0.95 0.99 N = 5 time(s) 0.0009 0.0007 0.0009 0.0024 0.0050 0.0234 accuracy 0.0006 0.0002 0.0001 0.0002 0.0003 0.0002 N = 10 time(s) 0.0014 0.0017 0.0013 0.0033 0.0054 0.0255 accuracy 0.0003 0.0004 0.0004 0.0003 0.0004 0.0004 N = 50 time(s) 0.0075 0.0047 0.0052 0.0108 0.0204 0.1903 accuracy 0.0010 0.0012 0.0013 0.0014 0.0016 0.0018 N = 100 time(s) 0.0207 0.0100 0.0103 0.0306 0.0464 0.2280 accuracy 0.0020 0.0023 0.0027 0.0029 0.0030 0.0030 N = 200 time(s) 0.0747 0.0383 0.0470 0.1183 0.2803 1.2434 accuracy 0.0029 0.0040 0.0051 0.0057 0.0058 0.0059 5. Conclusions This pap er prop oses an alternative approac h to risk budgeting, different from the standard optimization-based metho ds commonly studied in the lit- erature. The adv an tages of this p erspective are manifold. First, it allo ws for the analysis of existence and uniqueness of the risk budgeting p ortfo- lio for a general risk measure by form ulating the problem as the existence and uniqueness of a fixed p oint of a function. Second, the computation of the risk budgeting p ortfolio do es not rely on solving an auxiliary optimiza- tion problem that often lacks clear economic interpretation. Additionally , the optimization-based techniques—used in standard approaches—can b e- come computationally demanding, particularly as the num b er of assets in- 30 creases. In con trast, the proposed metho d naturally leads to an algorithm for calculating the risk budgeting allo cation, which is c haracterized b y fast con vergence and low computational cost, thanks to the simplicity of each it- eration. Imp ortantly , the computational burden do es not scale significantly with portfolio size. 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