Belief Propagation Algorithm for Portfolio Optimization Problems.

RESEARCH ARTICLE

Belief Propagation Algorithm for Portfolio Optimization Problems Takashi Shinzato1☯*, Muneki Yasuda2☯ 1 Mori Arinori Center for Higher Education and Global Mobility, Hitotsubashi University, Kunitachi, Tokyo, Japan, 2 Graduate School of Science and Engineering, Yamagata University, Yonezawa, Yamagata, Japan ☯ These authors contributed equally to this work. * [email protected]

a11111

Abstract

OPEN ACCESS Citation: Shinzato T, Yasuda M (2015) Belief Propagation Algorithm for Portfolio Optimization Problems. PLoS ONE 10(8): e0134968. doi:10.1371/ journal.pone.0134968 Editor: Yong Deng, Southwest University, CHINA Received: April 17, 2015 Accepted: July 15, 2015

The typical behavior of optimal solutions to portfolio optimization problems with absolute deviation and expected shortfall models using replica analysis was pioneeringly estimated by S. Ciliberti et al. [Eur. Phys. B. 57, 175 (2007)]; however, they have not yet developed an approximate derivation method for finding the optimal portfolio with respect to a given return set. In this study, an approximation algorithm based on belief propagation for the portfolio optimization problem is presented using the Bethe free energy formalism, and the consistency of the numerical experimental results of the proposed algorithm with those of replica analysis is confirmed. Furthermore, the conjecture of H. Konno and H. Yamazaki, that the optimal solutions with the absolute deviation model and with the mean-variance model have the same typical behavior, is verified using replica analysis and the belief propagation algorithm.

Published: August 25, 2015 Copyright: © 2015 Shinzato, Yasuda. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper and its Supporting Information files. Funding: This study was supported by Grant-in-Aid for Young Scientists (B) Research Project Number: 24710169, https://kaken.nii.ac.jp/d/p/24710169.en. html 2. Akita prefectural university, (2014). President project for young scientists 3. Grant-in-Aid for Young Scientists (B) Research Project Number: 15K20999, https://kaken.nii.ac.jp/d/p/15K20999.en.html 4. The National Institute of Informatics, Japan, (2014). Research Project Number: 50, http://www.nii.ac.jp/ kenkyou/files/b/1234/kyoudoukenkyu_ichiran/h26_ ichiran.pdf 5. Japan Institute of Life Insurance, (2015). Research Project Number: 5, http://www.jili.or. jp/pdf/150604.pdf 6. Kyoto university institute of

Introduction Portfolio optimization is one of the most fundamental frameworks of risk diversification management. Its theory was introduced by Markowitz in 1959 and is one of the most important areas being actively investigated in financial engineering [1–3]. In their theoretical research, Ciliberti and Mézard assessed the typical behavior of optimal solutions to portfolio optimization problems, in particular those described by the absolute deviation and expected shortfall models, using replica analysis, one of the most powerful approaches in disordered systems. With this approach, they showed that the phase transitions of these optimal solutions were nontrivial [1]. However, they did not develop an effective algorithm for finding the optimal portfolio with respect to a fixed return set. This requires a rapid algorithm for resolving the optimal portfolio problem with respect to a large enough in-sample set. As a first step in such a research direction, we propose an algorithm based on belief propagation, which is well-known as one of the most prominent algorithms in probabilistic inference, to resolve the microscopic averages of the optimal solution in a feasible amount of time for a fixed return set. We also confirm whether the numerical experimental results of our novel algorithm are consistent with the ones of replica analysis. Furthermore, the conjecture of

PLOS ONE | DOI:10.1371/journal.pone.0134968 August 25, 2015

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Belief Propagation Algorithm for Portfolio Optimization Problems

economic research foundation, (2015). http://www.fd. kier.kyoto-u.ac.jp/grant/grant.html 7. Zengin Foundation for Studies on Economics and Finance, (2015). Research Project Number: 1414, http://www. jade.dti.ne.jp/bankfund/table2014.pdf 8. The institute of statistical mathematics, (2015). Research Project Number: 2068, http://kyodo.ism.ac.jp/public_system/ member/saitaku-kadai.php?n=H27 9. Kampo foundation, (2015). Research Project Number: 2, http://www.kampozaidan.or.jp/CL01_02/185_S1.pdf. Competing Interests: The authors have declared that no competing interests exist.

Konno and Yamazaki, that if the return at each period is independently and identically drawn from the normal probability distribution [2], the optimal portfolio of the mean-variance model is consistent with that of the absolute deviation model, is supported using replica analysis and belief propagation.

Model Setting and Proposed Algorithm Let us define the model setting for our discussion. A portfolio of N assets and the return at period μ are represented by w = {w1, w2, , wN}T 2 RN and xμ = {x1μ, x2μ, , xNμ}T 2 RN, respectively, where wk is the position of asset k, and we assume for simplicity that the mean of the return of asset k in period μ, xkμ, is zero. The notation T indicates matrix transposition. Given a return set for p periods as reference, the problem is to minimize the following cost function (i.e., Hamiltonian) for the portfolio: T p X w xm R pffiffiffiffi ; HðwÞ ¼ ð1Þ N m¼1 2

where R(u) represents a cost function, such as u2 in the mean-variance model and juj in the absolute deviation model, respectively. Furthermore, since the budget is assumed to be finite, the following global constraint is set: N X wk ¼ N:

ð2Þ

k¼1

One of our aims is to develop an effective general algorithm for solving this problem; in particular, our aim is an algorithm that works for all cost functions R(u) and all probability distributions of the returns. As a basis for the proposed algorithm, following examples in statistical mechanics, we set the joint probability of portfolio w used in Eq (1) using finite inverse absolute temperature β as follows: PðwÞ / P0 ðwÞ exp ½bHðwÞ T p Y w xm P01p ðwÞ; / P0 ðwÞg pffiffiffiffi N m¼1

ð3Þ

where g(u) = e−βR(u) is the likelihood function and prior probability P0 ðwÞ / PN

~ exp m for sufficiently large N [4, 5]. Notice that the partition function of this k¼1 wk N posterior probability T p XY w xm Z¼ P01p ðwÞ; P0 ðwÞg pffiffiffiffi ð4Þ N w m¼1 is implicitly ignored in this analysis because intuitively it is possible to evaluate the first- and second-order moments of portfolio wk approximately without the partition function by the following procedure. An arbitrary test probability of portfolio is defined as follows: QðwÞ /

p N Y Y bm ðwÞ b1p k ðwk Þ; m¼1


ð5Þ

k¼1

2 / 10


where the reducibility condition on beliefs bk(wk) and bμ(w), X bm ðwÞ; bk ðwk Þ ¼

ð6Þ

wnwk

must hold and w\wk denotes a subset of w from which wk is excluded. The Kullback-Liebler divergence (KLD) KLðQjPÞ ¼

X QðwÞ QðwÞ log PðwÞ w

ð7Þ

provides a useful guideline for deriving the belief propagation algorithm. However, since it is too complicated to directly assess KLD except in specific graphical models, we here approximate the Bethe free energy denoted as follows: 0 1 FBethe ¼

B C bm ðwÞ B C T C bm ðwÞlogB w x @ A m m¼1 w P0 ðwÞg pffiffiffiffi N N X X bk ðwk Þ ; bk ðwk Þlog þð1 pÞ P0k ðwk Þ k¼1 w p X X

ð8Þ

k

~ k is used. The purpose of this step is to derive the optimal portfolio using where P0k ðwk Þ / emw the beliefs bk(wk) and bμ(w) that minimize the Bethe free energy under the reducibility condihP i Pp PN P tion of Eq (6). By adding the term m¼1 k¼1 wk lkm ðwk Þ wnwk bm ðwÞ bk ðwk Þ to the

right-hand side of Eq (8), it is possible to minimize the Bethe free energy with respect to the beliefs to obtain " # p 1 X l ðw Þ ; ð9Þ bk ðwk Þ / P0k ðwk Þ exp 1 p m¼1 km k " # N X wT x m lkm ðwk Þ : bm ðwÞ / P0 ðwÞg pffiffiffiffi exp N k¼1

ð10Þ

Furthermore, for simplicity, we set l~km ðwk Þ

1 X l ðw Þ þ lkm ðwk Þ; 1 p m¼1 km k p

¼

1 as novel auxiliary functions, and then bk(wk) and bμ(w) can be rewritten using 1p P Pp ¼ m¼1 l~km ðwk Þ and lkm ðwk Þ ¼ nðwmÞ l~kn ðwk Þ as p X bk ðwk Þ / P0k ðwk Þ exp ½ l~km ðwk Þ;

ð11Þ Pp m¼1

lkm ðwk Þ

ð12Þ

m¼1

" # N X X wT xm l~kn ðwk Þ : bm ðwÞ / P0 ðwÞg pffiffiffiffi exp N k¼1 nð6¼mÞ


ð13Þ

3 / 10


Moreover, applying the cumulant generating functions X bk ðwk Þewk yk ; k ðyk Þ ¼ log

ð14Þ

wk

m ðyÞ ¼ log

X

T

bm ðwÞew y ;

ð15Þ

w k ðyk Þ ¼ the first and second moments of wk have the compact forms mwk ¼ @@y k

@ 2 k ðyk Þ @y2k

¼

@ 2 m ðyÞ @y2k

@m ðyÞ @yk

and wwk ¼

at θ = {θ1, , θN}T ! 0. This allows us to disregard the calculation of the parti-

tion function. Then, our proposed algorithm for sufficiently large N comprises the following: ~ mwk ¼ wwk ðhwk þ mÞ;

ð16Þ

1 X hwk ¼ pffiffiffiffi xkm mum þ w~wk mwk ; N m¼1

ð17Þ

p

1X 2 x w ; N m¼1 km um

ð18Þ

1 ; w~wk

ð19Þ

p

w~wk ¼

wwk ¼ @ mum ¼ log @hum

Z

1

Dz gðz 1

qffiffiffiffiffiffi w~um þ hum Þ;

N 1 X xkm mwk w~um mum ; hum ¼ pffiffiffiffi N k¼1

w~um ¼ @2 wum ¼ 2 log @hum

N 1X x2 w ; N k¼1 km wk

Z

1

Dzgðz 1

qffiffiffiffiffiffi w~um þ hum Þ;

ð20Þ

ð21Þ

ð22Þ

ð23Þ

2 ffi ez2 is used. Note that if l~km ðwk Þ is redefined as l~km ðwk Þ ¼ gkm w2k þ h~km wk , where Dz ¼ pdzffiffiffi 2 2p Pp Pp ~ then w~wk ¼ m¼1 gkm and hwk ¼ m¼1 h km [4–7]. In addition, w~wk mwk and w~um mum describe the Onsager reaction terms in the literature of spin glass theory (respectively [8, 9]). Four points should be noticed here. First, the calculation of this procedure is reduced from O(N3) to O(N2). For instance, in the case of the mean-variance model, although we are required to calculate the inverse matrix of the correlation matrix of return set XXT 2 MN×N, where return matrix X = {x1, , xp} 2 MN×p, in order to assess the optimal solution rigorously, it is well-known that this calculation is O(N3). Moreover, fortunately it is found that in the case of the mean-variance model, this algorithm derives the exact optimal solution (see appendix A for details). Second, only Eqs (20) and (23) are dependent on the likelihood function g(u) = ~ is e−βR(u), and the variables of index u are the only model dependent ones. Furthermore, m


4 / 10


determined by Eqs (2) and (16). Third, the randomness of return is not assumed to be sampled from specific distributions. Because it is plausible that the assumption on the Bethe free energy approximation works correctly if the return at each period is asymptotically not correlated with other returns. Lastly, we expect that in the limit as β ! 1, the estimate of the portfolio of asset k, mwk, asymptotically corresponds to the optimal portfolio with respect to the given return set.

Numerical Experimental Results In order to confirm the effectiveness of our method, the numerical experimental results of the proposed algorithm and those of the replica analysis for the case of the Markowitz model are shown in Figs 1 and 2, where xkμ are independently and identically drawn from the normal distribution with mean and variance 0 and 1, respectively. The numerical experimental result of belief propagation is assessed from 102 samples of the number of assets N = 500 and is denoted by error bars and the result of replica analysis is denoted by a solid line. Both findings indicate that the two approaches are consistent with each other.

Fig 1. The reference ratio α = p/N (horizontal axis) versus the quenched overlap parameter q (vertical axis). The numerical experimental results from the proposed algorithm (error bars) are assessed from 102 experiments using N = 500 assets. Comparing with the results of replica analysis (solid line), the effectiveness of proposed algorithm is verified. doi:10.1371/journal.pone.0134968.g001


5 / 10


Fig 2. The reference ratio α (horizontal axis) versus one degree of the cost functionε (vertical axis). This result also indicates that the approximation approach based on probabilistic inference works correctly. doi:10.1371/journal.pone.0134968.g002

With regard to the conjecture of Konno and Yamazaki, the variables in Eqs (20) and (23), in the case of the mean-variance model mum ¼

wum ¼

b h ; 1 þ b~ w um um

ð24Þ

b 1 þ b~ w um

ð25Þ

and the absolute deviation model 1 pffiffiffiffiffiffi hum p ffiffiffiffiffi H b w~um þ w~ B C um 1 C mum ¼ b tanh B log bh þ @ um 2 A; pffiffiffiffiffiffi h H b w~um pumffiffiffiffiffi 0

ð26Þ

w~ um

wum ¼


@mum ; @hum

ð27Þ

6 / 10


are assessed exactly using HðuÞ ¼

R1 u

Dz. Because HðuÞ ’

pffiffiffiffiffiffi 1 u2 2pu e 2 in the case of u 1,

h

and wum ’ w~1um are estimated; that is, this finding indicates that the conjecture of mum ’ w~um um Konno and Yamazaki is valid in part in the sense of the belief propagation approach. See appendices for details.

Summary In conclusion, we have discussed an effective algorithm for finding the optimal solution of the portfolio optimization problem with respect to an arbitrary cost function according to Ciliberti and Mézard [1]. With loss of generality, applying the likelihood function g(u) defined by the cost function R(u) dependent on the risk diversification problem, we proposed a novel approximation derivation method based on one of the most powerful estimation methods in probabilistic inference. In addition, since two types of Onsager reaction terms are derived in Eqs (17) and (21), our algorithm provides the Thouless, Anderson, and Palmer approach rather than the mean-field approximation in the literature of spin glass theory. One advantage of our algorithm is that it rapidly converges by excluding the effect of self-response. In order to confirm the effectiveness of the proposed approach, we have described the case of the mean-variance model. Furthermore, we have shown that the conjecture of Konno and Yamazaki is supported by employing both approaches developed in cross-disciplinary research involving statistical mechanics and information sciences. In future work, we will assess the properties of R(u) and the randomness of return that make solving the portfolio optimization problem using belief propagation possible.

Appendix A. Proof of Exactness We here confirm the exactness of the proposed belief propagation algorithm for the case of the Markowitz model. Our discussion is restricted to α > 1 for simplicity. From Eqs (21), (24), and (25), we obtain mu ¼ pbffiffiNffi X T mw , where mu = {mu1, ,mup}T 2 Rp and mw = {mw1, ,mwN}T ~ ¼ p1ffiffiffi Xmu follows immediately from Eqs (16), (17), and (19), where 2 RN. Furthermore, me N

1

~ ðbXX T Þ e into the constraint N = eT mw e = {1, , 1}T 2 RN. Thus, substituting mw ¼ N m NðXX T Þ1 e

gives the exact optimal solution mw ¼ eT ðXX T Þ1 e.

B. Replica Analysis According to Ciliberti and Mézard, replica symmetry solution of the portfolio optimization problem, where xkμ is independently and identically distributed with N(0,1), is derived as the following extremum: 1 ½ log Zq N q1 1 þ log w ¼ Extrq;w 2w 2 Z 1 Z 1 pffiffiffi pffiffiffi Dy log Dzgðz w þ y qÞg; þa

bf ¼ lim

N!1

1

where Z ¼

P w

P0 ðwÞ

Qp m¼1

g

ð28Þ

1

w T xm p N

ffiffiffi

is the partition function and the notation [ ]q denotes

the quenched average over the return set. Moreover, the quenched overlap parameters become


7 / 10


PN qab ¼ N1 k¼1 wka wkb ¼ w þ q if a = b and q otherwise by employing replica indices a, b = 1, 2, , n and the assumption of replica symmetry. Furthermore, for large N and p, α = p/ N * O(1) remains finite and plays an important role as a control parameter with respect to 1 b 2 phase transition phenomena. If gðuÞ ¼ e2u , then q ¼ 1 1a and χ = (β(α − 1))−1 can be exactly calculated in the case α > 1 and q ! 1, and χ ! 1 otherwise. This analytical finding is also verified in by the following. It is well known that the eigenvalue distribution of the correlation matrix C = N1 XXT in the limit of N ! 1 is asymptotically close to the Marčhenko-Pastur pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 ½ll þ ½lþ lþ þ with l ¼ ð1 aÞ and [u]+ = max{u,0} [10]. law rðlÞ ¼ ½1 a dðlÞ þ 2pl Therefore, q ¼ hl12 ih1li

2

1

and one degree of the cost function ε ¼ limN!1 N1 ½HðwÞq ¼ 12 h1li R1 are obtained straightforwardly using hf ðlÞi ¼ 1 dlrðlÞf ðlÞ. Applying Marčhencko-Pastur pffiffiffiffiffiffiffiffi

2 l l lþ þl 1 a ffiffiffiffiffiffiffiffi 12 ¼ a1 law, that h1li ¼ p and hl12 i ¼ 2pþ p2 12 l1 l1þ ¼ ða1Þ 3 if α > 1 and 4

lþ l

approach infinity otherwise follows directly. This is consistent with the findings of replica analysis [5]. In general, the order parameters are derived as follows: pffiffiffi q ; ð29Þ w¼ aZ q ¼ 1 þ aw2 d;

Z Z¼

0Z B B B DyyB Z B 1 @ 1

ð30Þ

1 pffiffiffi pffiffiffi Dzg ðz w þ y qÞC C C 1 C; 1 C pffiffiffi pffiffiffi A Dzgðz w þ y qÞ 1

0

ð31Þ

1

Z d¼

0Z B B B DyB Z B 1 @ 1

12 p ffiffi ffi p ffiffi ffi Dzg 0 ðz w þ y qÞC C C 1 C: 1 C pffiffiffi pffiffiffi A Dzgðz w þ y qÞ 1

ð32Þ

1

From Eqs (29) and (30), q

¼

1d 1 2 aZ

1

is obtained. In the limit of sufficiently large β of g(u) = e−βjuj, if we assess Z ’

ð33Þ p

ffi

q w

and d ’ wq2

asymptotically, then the conjecture of Konno and Yamazaki is confirmed as correct in the sense of replica analysis.

C. The Conjecture of Konno and Yamazaki This conjecture is related to the assessment of an annealed system in the context of spin glass theory. If the return at period μ, xμ, is independently and identically drawn from N(0, S), wT x

where S 2 MN×N is variance-covariance matrix and w is fixed, the novel variable z ¼ pffiffiNffim is P distributed as N(0, s2(w)) with s2 ðwÞ ¼ N1 wT w 2 R. With respect to fixed w, employing one


8 / 10


degree of the cost function of the annealed optimization problem h P T i R1 w x p εðwÞ ¼ N1 m¼1 R pffiffiNffim ¼ a 1 DuRðusðwÞÞ, which becomes εMV ðwÞ ¼ a2 s2 ðwÞ in the q

ffi j sðwÞ j in the absolute deviation model. case of the mean-variance model and εAD ðwÞ ¼ p2affiffiffi 2p This implies that the optimal portfolios of the annealed situations of the two models are consistent with each other. Note that one degree of the cost function in the case of the annealed portn o v folio problem with the expected shortfall model, εES ðwÞ ¼ min v0 a vg þ H sðwÞ with γ > 1ffi 0 can also be assessed. If sðwÞ pffiffiffi , then this optimal solution is identical to those of the pre2pg

vious mentioned models. This finding, that is, arg min wT e¼N εMV ðwÞ ¼ arg min wT e¼N εAD ðwÞ, is one part of the contributions reported by Konno and Yamazaki. However, they optimistically assumed wMV = wAD with respect to a given return set X without any mathematical proof, using N X N 1 XX ww x x ; 2N m¼1 i¼1 k¼1 i k im km p

wMV ¼ arg Tmin w e¼N

wAD ¼ arg Tmin w e¼N

ð34Þ

p N X 1 X wk xkm : pffiffiffiffi N m¼1 k¼1

ð35Þ

As explained above, arg min wT e¼N εMV ðwÞ ¼ arg min wT e¼N εAD ðwÞ with respect to the annealed optimization problem strictly holds; however, wMV = wAD is not always satisfied. For example, in the simple case of N = p = 2 for the two returns x1 = {a, c}T and x2 = {b, d}T, their assumption wMV = wAD does not hold, except under specific special situations. Although this is apparently contradictory to these obtained findings from both approaches, it is necessary to recognize that the relation wMV = wAD with a fixed return set is equivalent to

the sufficient condition qMV = qAD, where qMV ¼ limN!1 N1 wTMV wMV q and qAD ¼ lim N!1 N1 ½wTAD wAD q are quenched averages of overlap parameters. Moreover, although wMV wT w

MV AD = wAD does not hold in general, it is expected that the inner product jwMV is approximately jjwAD j P 1 1 because N m>v xkm xjv ! 0 in the case of sufficiently large N.

Supporting Information S1 Data. Figure 1 and figure 2 from S1 Data. S1 Data is the result from numerical experiment of S1 File and S2 File. (TXT) S1 File. S1 File is main program of C language. (TXT) S2 File. S2 File is sub program of C language. (TXT)

Acknowledgments We thank I. Kaku, T. Mizuno and Y. Shimazaki for their fruitful comments.


9 / 10


Author Contributions Conceived and designed the experiments: TS. Performed the experiments: TS. Analyzed the data: TS. Contributed reagents/materials/analysis tools: TS MY. Wrote the paper: TS MY.

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Konno H., Yamazaki H., 1991. Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock, Management Science, 37(5), 519–531. doi: 10.1287/mnsc.37.5.519

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Kabashima Y., 2005. Replicated Bethe free energy: A variational principle behind survey propagation, Journal of the Physical Society of Japan, 74(8), 2133–2136. doi: 10.1143/JPSJ.74.2133

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