レプリカ解析を用いた予算制約・集中投資度制約がある場合の最小投資リスクの理論解析 (確率的環境下における数理モデルの理論と応用)

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(1)24. 数理解析研究所講究録第2044巻 2017年 24-34. レプリカ解析を用いた予算制約集中投資度制約がある場合の最小投資リスクの理論解析一橋大学. 森有礼高等教育国際流動化センター. *. 隆. 新里. Takashi Shinzato Mori Arinori Center for. Higher. Education and Global. Hitotsubashi. Mobility. University. Introduction. 1. Portfolio introduced. problems. optimization problems first appeared. by. to. asset. management. so as. investment into several assets have become well known. issues related to. theory for. Analytical procedures. Markowitz in 1952.. accurately implement. in the. portfolio optimization problems. to. thoroughly. examined. solving portfolio optimization. disperse the risk by diversifying. Over the next few. decades, several. and. recently several. have been. models and the behavior of the minimal investment risk in have been. of diversification investment. addressed,. portfolio optimization problems. using analytical approaches which have been developed and. improved through multidisciplinary collaboration [1−7]. For instance, Ciliberti the minimal investment risk under model. using replica analysis. examined. [3].. quantitatively. Pafka et al.. in detail via. [4].. a. absolute deviation model and. an. in the absolute. the noise. zero. sensitivity of the. discussed the relation between. scenario ratio. (between. temperature limit. optimal portfolio. predicted risk,. an. et al.. analyzed. expected shortfall. [1, 2].. Kondor et al.. for several risk functions. realized. risk, and. true risk. the number of scenarios and the number of. assets). Shinzato derived the statistics minimal investment risk and investment concentration. and showed that the minimal investment risk is attained and the investment concentration constraint is satisfied. (e.g., by. a. portfolio). and that these two statistics have the. property which is frequently used in statistical mechanical informatics analysis et al.. developed. a. faster. algorithm. for. These previous works have 1. Investment risk. (as. independently. and not. cost. function). (like. Shinzato. in the. a. belief. case. that. identically using replica analysis [7].. analyzed. 2. Investment concentration. [5].. solving portfolio optimization problems using. propagation method [6]. Shinzato examined the portfolio optimization problem each asset return rate is distributed. self‐averaging. Herfindahl‐Hirschman. Index).

(2) 25. of the. risk) the. optimal portfolio which. with. As. one. the. of the natural extentions,. expected we. As the first step of. with several constraints.. portfolio optimization. (not. minimize the investment risk. [1−7].. constraint. budget. can. investment. need to examine. analyzing portfolio. optimization problems with several constraints, noting the risk minimization problem,. develop. a. novel. constraints,. analysis.. a. approach. budget. for. solving. constraint and. We will discuss the. portfolio optimization problem. a. constraint. a. on. concentration, using replica. investment. with the constraints of. portfolio optimization problem. we. with two representative. budget. and investment concentration.. 2. Preliminary \bullet. We will revisit stochastic. \bullet. We will introduce the Boltzmann distribution. optimization. so as. clarify. to. target.. our. approach. with respect to the. optimization. problem. .. We will. in order to discuss the. explain replica analysis. portfolio optimization problem. with two constraints. problem from the viewpoint of stochastic optimization. First,. We consider the minimization we assume. that with respect to controal parameter. function. f(w, X). f(w, X). is bounded below. is. prepared, where on w. dom variable X is known. is defined. Then,. w.r. \mathrm{t}.. follows; w^{*}(X). optimal. the. following identity, f(w^{*}(X), X). f(w^{*}(X), X)=\displaystyle \min_{w\in W}f(w, X). ,. E[g(X)]. the. W,. (w, X) f(w, X) ,. =. expectation. as. of. and here the. \geq. .. .. From. f(w, X). From. is obtained.. Then,. follows, E[f(w, X)] \geq E[f(w^{*}(X), X)]. g(X). .. Since. Moreover. probability of. \displaystyle \min_{w\in W}f(w, X). \displaystyle \arg\min_{w\in W}f(w, X). \displaystyle \min_{w\in W}f(w, X). =. f(w, X) \geq f(w^{*}(X), X). both sides with random variable means. \in. and random variable X , the real‐valued. (not always convex). W. \in. the. as. w. w. W is the feasible solution subset.. this,. is held. we. can. \displaystyle \geq\min_{w\in W}f(w, X) we can. ,. ran‐. Or use. and. average the. where the notation. E[f(w, X)] \geq E[f(w^{*}(X), X)]. is held when any. w\in W,. \displaystyle \min_{w\in W}E[f(w, X)] \geq E[f(w^{*}(X), X)] is obtained.. Further, from. f(w^{*}(X), X)=\displaystyle \min_{w\in W}f(w, X). ,. \displaystyle \min_{w\in W}E[f(w, X)] \geq E[\min_{w\in W}f(w, X)] is also obtained. That. is, The. From the argument in the. (1). ,. ,. (2). minimum of average is not smaller than the average of minimum.. previous work [5], for. any. X,. \displaystyle \min_{w\in W}f(w, X)=E[\min_{w\in W}f(w, X)]. ,. (3).

(3) 26. is. guaranteed mathematically (we call self‐averaging).. Thus, from \displaystyle \min_{w\in W}E[f(w, X)] \geq. E[\displaystyle \min_{w\in W}f(w, X. \displaystyle \min_{w\in W}E[f(w, X)]\geq\min_{w\in W}f(w, X) is rewritten. The solution of ordinary research. From. \displaystyle \min_{w\in W}E[f(w, X)]\geq\min_{w\in W}f(w, X). (OR). is determined. side is the cost of the. always. not. can. (5). optimal solution which the. in the left hand side is the. \displaystyle \min_{w\in W}f(w, X). minimize the cost.. in the. investor wants to know.. right. In the. hand. previous. [5]). work. opportunity. Portfolio. 3. no. restrictions. and the full. optimally diversified. on. short. portfolio of N. transpose of. \cdots , p). a. vector. For. .. given. (6). ,. or. [5].. selling.. matrix.. simplicity of X=. our. \vec{x}_{1},. \cdots. ,. and. to in the. investment market. \cdots , N). \vec{w}=\{w_{1}, \cdots , w_{N}\}^{\mathrm{T} \in \mathrm{R}^{N}. discussion, similar. \displayst le\{ frac{x_l}$\mu$}{\sqrt{N}\ \in \mathrm{R}^{N\times p}. model,. an. ,. in the. period (or scenario). previous work,. identically normally. portfolio. where \mathrm{T} indicates the. x_{i $\mu$} is the return rate of asset i under. independently. the p return rate vectors. In the mean‐variance. investment in N assets in. w_{i} is the amount of asset i (=1,. assets is denoted. that each return rate x_{i $\mu$} is. and variance 1. los =\displaystyle \frac{E[f(w^{\mathrm{O}\mathrm{R} ,X)]}{\min_{w\in W}f(w,X)}=\frac{ $\alpha$}{ $\alpha$-1}. optimization problem. This talk considers. with. $\mu$ (=1,. w^{\mathrm{O}\mathrm{R}}=\displaystyle \arg\min_{w\in W}E[f(w, X. ,. E[f(w^{\mathrm{O}\mathrm{R}}, X)]. of cost management,. case. cost of the solution which. as. w^{\mathrm{O}\mathrm{R} =\displaystyle \arg\min_{w\in W}E[f(w, X)],. and. E[f(w^{\mathrm{O}\mathrm{R} , X)]\displaystyle \geq\min_{w\in W}f(w, X) is also obtained. In the. (4). ,. it is assumed. distributed with. is the return rate matrix where under this. mean. assumption,. \vec{x}_{p}\in \mathrm{R}^{N}, \vec{x}_{ $\mu$}=\{x_{1 $\mu$}, \cdot\cdot , x_{N $\mu$}\}^{\mathrm{T} \in \mathrm{R}^{N}.. the investment risk. \mathcal{H}(\vec{w}|X). of. portfolio \vec{w}. is defined. as. follows:. \displaystyle\mathcal{H}(\vec{w}|X)=\frac{1}{2N}\sum_{$\mu$=1}^{p}(\sum_{i=1}^{N}w_{i}(x_{i$\mu$}-0) ^{2} =\displayst le\frac{1}2N}\sum_{$\mu$=1}^{p}(\sum_{\dot{x}=1}^{N}w_{i}x_{i$\mu$})^{2} where. E[x_{x $\mu$}]. =. 0 and. E[x_{i $\mu$}^{2}]. =. 1. .. =. XX^{\mathrm{T}}. \in. \mathrm{R}^{N\times N} be. a. since J does not. unique optimal portfolio. for. a. to be. non‐singular matrix,. simply p>N However, .. (7). Note that the necessary and sufficient condition for the. optimal portfolio for portfolio optimization problem that J. 0. that. always need. uniquely determined given is,. to be. in. [5]. the rank of matrix J be N a. portfolio optimization problem. regular. matrix to. with several. guarantee. constraints,. as. is or a. in.

(4) 27. the present. \mathcal{H}(\vec{w}|X). work,. do not. we. is rewritten. as. adopt. the. regular. matrix. assumption here. The. investment risk. follows;. \displaystyle\mathcal{H}(\vec{w}|X)=\frac{1}{2N}\sum_{$\mu$=1}^{p}(\sum_{i=1}^{N}w_{i}x_{i$\mu$})^{2}. =\displayst le\frac{1}2\sum_{\dot{$\iota$}=1}^{N}\sum_{j=1}^{N}w_{i}w_{j}(\frac{1}N}\sum_{$\mu$=1}^{p}x_{i$\mu$}x_{j$\mu$}). =\displaystyle\frac{1}{2}\vec{w}^{\mathrm{T}J\vec{w} where i ,. J=\{J_{ij}\}\in \mathrm{R}^{N\times N}. jth component of. (8). ,. is defined. J_{ij}=\displaystyle\frac{1}{N}\sum_{$\mu$=1}^{p}x_{i$\mu$^{X}j$\mu$} In the ment. previous work [5],. we. consider the. follows,. as. (9). .. optimal portfolio which. can. minimize the invest‐. risk,. \displaystyle \mathcal{H}(\vec{w}|X)=\frac{1}{2}\vec{w}^{\mathrm{T} J\vec{w} with the. budget constraint,. \displaystyle\sum_{i=1}^{N}w_{i}=N When. p>N this optimal solution ,. where \vec{e}=. \{1, 1, \cdots , 1\}^{\mathrm{T}. is calculated. as. \in. \mathrm{R}^{N}. (11). .. is. \displayst le\vec{w}^{*=\frac{NJ^{-1}\vec{ }{e\rightarow\mathrm{r}J^{-1}\vec{ } \mathcal{H}(\vec{w}^{*}|X). (10). ,. is used.. Using \vec{w}^{*}. (12). ,. =. \displaystle\frac{NJ^{-1}\vec{} \mathrm{e}\rightarow\mathrm{}J^{-1}\vec{}. ,. the minimal investment risk. follows;. \displaystyle \mathcal{H}(\vec{w}^{*}|X)=\frac{1}{2}(\vec{w}^{*})^{\mathrm{T} J(\vec{w}^{*}) =\underline{N^{2}}. (13). 2訝 J^{-1}\vec{e}'. where. J=XX^{\mathrm{T}} \in \mathrm{R}^{N\times N}. matrix of J ,. this In. however,. .. In order to. since the. assess. computation. it. accurately,. amount. O(N^{3}). is. we. need to calculate the inverse. required,. it is hard to. implement. approach. a. similar way,. we. also consider the. optimal portfolio which. can. minimize the investment. risk,. \displaystyle \mathcal{H}(\vec{w}|X)=\frac{1}{2}\vec{w}^{\mathrm{T} J\vec{w}. ,. (14).

(5) 28. with the constraints of. budget. and investment concentration. \displaystyle\sum_{i=1}^{N}w_{i}=N \displayst le\sum_{\dot{$\iota$}=1^{N}w_{\dot{l}^{2}=N$\tau$. as. follows;. (15). ,. where. Eq. (15). means. straint and $\tau$\geq 1. budget. constraint and. (16). ,. Eq. (16). means. The minimal investment risk is assessed. .. as. investment concentration. follows;. \displaystyle\mathcal{H}(\vec{w}^{*}|X)=\frac{N^{2_{e}^{\lrcorner}$\Gam a$}(J-$\theta$I_{N})^{-1}J( -$\theta$I_{N})^{-1}\vec{e}{2(e\rightar ow\mathrm{r}(J-$\theta$I_{N})^{-1}\vec{e})^{2} where the. optimal portfolio \vec{w}^{*}. is determined. 面. as. con‐. (17). ,. follows,. *=\displayst le\frac{N(J-$\theta$I_{N})^{-1}\vec{e}{e\rightarow\mathrm{r}(J-$\theta$I_{N})^{-1}\vec{e}. (18). ,. and parameter $\theta$ satisfies. N$\tau$=\displaystle\frac{N^2_{e}\ovalbox{\t smal REJ CT}(J-$\thea$I_{N})^{-2}\vec{ } (e^{$\Gam a$}\rightarow(J-$\thea$I_{N})^{-1}\vec{ })^{2} where. I_{N}\in \mathrm{R}^{N\times N}. is the. identity. matrix.. Boltzmann distribution. 4. We here reconsider this lem is formulated. as. follows;. regarded. Let. us. s.t.. Bayesian inference,. as. the. constraint W is. as. as. regarded. (20). the feasible. as a. portfolio. subset. In. prior P_{0}(\vec{w}) and \mathcal{H}(\vec{w}|X). is. loglikelihood.. denote the Boltzmann distribution. probability). \left{bginary}{l \sum_dot{$\ia}=1^{Nw_i}=\ sum_{i=1}^Nw_{i2}=N$\tau end{ary}\ight.. W=\displaystyle \{\vec{w}\in \mathrm{R}^{N}|\sum_{i=1}^{N}w_{i}=N, \sum_{i=1}^{N}w_{l}^{2}=N $\tau$\}. the context of also. approach. optimal problem using Bayesian inference. The optimization prob‐. \mathcal{H}(\vec{w}|X). We prepare. (19). ,. (or. the. posterior probability. or. the conditional. follows;. P(\displaystyle\vec{w}|X)=\frac{P_{0}(\vec{w})e^{-$\beta$\mathcal{H}(\vec{w}|X)} {Z(X,$\beta$)}. ,. (21).

(6) 29. where. P_{0}(\vec{w})=\left\{ begin{ar y}{l 1\vec{w}\inW\ 0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e} \end{ar y}\right. Z(X, $\beta$)=\displaystyle \int_{-\infty}^{\infty}d\vec{w}P_{0}(\vec{w})e^{- $\beta$ \mathcal{H}(\vec{w}|X)} are. used. We call. the property of. Z(X, $\beta$). the. (22) (23). ,. partition function and $\beta$(>0) the. inverse temperature. From. exponential,. \mathcal{H}(\vec{w}_{a}|X)<\mathcal{H}(\vec{w}_{b}|X) \Leftrightarrow P(\vec{w}_{a}|X)>P(\vec{w}_{b}|X) From. \mathcal{H}(\vec{w}_{a}|X) <\mathcal{H}(\vec{w}_{b}|X)\Leftrightarrow P(\vec{w}_{a}|X) >P(\vec{w}_{b}|X). \displaystyle \lim_{ $\beta$\rightar ow\infty}\int_{-\infty}^{\infty}d\vec{w}P(\vec{w}|X)\vec{w} is is also held. In. held.. addition, using. the. Furthermore,. ,. in the. of. case. ,. sufficiently large $\beta$, \vec{w}^{*}. =. \displaystyle \mathcal{H}(\vec{w}^{*}|X)=\lim_{ $\beta$\rightar ow\infty}\int_{-\infty}^{\infty}d\vec{w}P(\vec{w}|X)\mathcal{H}(\vec{w}|X). partition function Z(X, $\beta$). ,. -\displaystyle \frac{\partial}{\partial $\beta$}\log Z(X, $\beta$)=\int_{-\infty}^{\infty}d\vec{w}P(\vec{w}|X)\mathcal{H}(\vec{w}|X). (24). ,. is obtained. From. \displaystyle \mathcal{H}(\vec{w}^{*}|X)=\lim_{ $\beta$\rightar ow\infty}\int_{-\infty}^{\infty}d\vec{w}P(\vec{w}|X)\mathcal{H}(\vec{w}|X) -\displaystyle \frac{\partial}{\partial $\beta$}\log Z(X, $\beta$)=\int_{-\infty}^{\infty}d\vec{w}P(\vec{w}|X)\mathcal{H}(\vec{w}|X). (25). ,. (26). ,. then. \displaystyle \mathcal{H}(\vec{w}^{*}|X)=\lim_{ $\beta$\rightar ow\infty}\{-\frac{\partial}{\partial $\beta$}\log Z(X, $\beta$)\} is obtained.. property,. Further,. we. since the minimal investment risk. will examine the. \mathcal{H}(\vec{w}^{*}|X). (27). ,. is satisfied with. self‐averaging. following;. \mathcal{H}(\vec{w}^{*}|X)=E[\mathcal{H}(\vec{w}^{*}|X)]. =\displaystyle \lim_{ $\beta$\rightar ow\infty}\{-\frac{\partial}{\partial $\beta$}E[\log Z(X, $\beta$)]\}. (28). .. Replica analys |\mathrm{s}. 5 So. as. to. assess. E[\log Z(X, $\beta$ we. the minimal investment risk. since it is not easy to average the. apply replica. trick. \displaystyle \log Z=\lim_{n\rightarrow 0\frac{Z^{n}-1}{n}. ,. \mathcal{H}(\vec{w}^{*}|X). ,. logarithm. we. need to evaluate. analytically. function of the partition. function,. then,. E[\displaystyle \log Z(X, $\beta$)]=\lim_{n\rightar ow 0}\frac{E[Z^{n}(X, $\beta$)]-1}{n}. ,. (29).

(7) 30. is obtained. Or. we. allow to. use. \left{begin{ar y}l \im_{n\rightarow0}\frac{logE[Z^{n}(X,$\beta$)]}{n\ lim_{n\rightarow0}\frac{prtial}{\prtialn}\ogE[Z^{n}(X,$\beta$)] \end{ar y}\ight.. E[\log Z(X, $\beta$)]= In any case,. however,. we. need to. assess. when n\in \mathrm{Z} , it is. E[Z^{n}(X, $\beta$. For any n\in \mathrm{R} , it is difficult to. comparatively. E[Z^{n}(X, $\beta$)]=E. (30). easy to calculate it. as. assess. E[Z^{n}(X, $\beta$. follows;. [ (\displaystyle \int_{-\infty}^{\infty}d\vec{w}P_{0}(\vec{w})e^{- $\beta$ \mathcal{H}(\vec{w}|X)} れ]. =E[\displaystyle\prod_{a=1}^{n}(\int_{-\infty}^{\infty}d\vec{w}_{a}P_{0}(\vec{w}_{a})e^{-$\beta$\mathcal{H}(\vec{w}_{a}|X)} ]. =\displaystyle\int_{-\infty}^{\infty}\prod_{a=1}^{n}d\vec{w}_{a}P_{0}(\vec{w}_{a})E[e^{-$\beta\Sigma$_{a=1}^{n}\mathcal{H}(\vec{w}_{a}|X)}] From. this, the configuration. implemented first.. average is. Thus,. we. (31). .. use. the saddle. point. method,. \log E[Z^{n}(X, $\beta$)]. { -N^{d}e^{$\Gam a$}\displaystyle\vec{k}-\frac{N$\tau$}{2}\mathrm{T}\mathrm{r}$\Theta$+\frac{N}{2}\mathrm{T}\mathrm{r}Q_{w}\tilde{Q}_{w}-\frac{p}{2} \displaytle\frac{N}2 |\displaystyle \tilde{Q}_{w}- $\Theta$|+\frac{N}{2}\vec{k}^{\mathrm{T} (\tilde{Q}_{w}- $\Theta$)^{-1}\vec{k} }, Extr. \simeq\vec{k} $\theta$^{\rightar ow},Q_{w},Q_{w}^{-} ). ‐. is. \vec{k}=\{k_{1}, k_{2}, \cdots , k_{n}\}^{\mathrm{T}}, \vec{e}=\{1, 1, \cdots , 1\}^{\mathrm{T} \in \mathrm{R}^{n} and. Q_{w}=\{q_{wab}\}, \tilde{Q}_{w}=\{\tilde{q}_{wab}\}\in \mathrm{R}^{n\times n} auxiliary. \mathrm{E}\mathrm{x}\mathrm{t}\mathrm{r}_{m9(m)}. variables which. constraint of ath case. (32). logdet. assessed, where. the notation. log det |I+ $\beta$ Q_{w}|. means are. are. used and I\in \mathrm{R}^{n\times n} is the. 9(m). the extremum of. related to the. replica, respectively.. that the number of assets N is. In. budget. addition,. diag \{$\theta$_{1}, $\theta$_{2}, \cdots , $\theta$_{n}\},. $\Theta$=. identity. with respect to. matrix. m. Moreover,. and k_{a} and. $\theta$_{a}. constraint and investment concentration. q_{wab}=. \displaystyle \frac{1}{N}\sum_{i=1}^{N}w_{ia}w_{ib}. is. prepared.. In the. sufficiently large,. $\Phi$(n)= \displN\rightarrow\infty aystyle \lim \underl ine{1}\log E[Z^{n}(X, $\beta$)] N Extr. =\vec{k}, $\theta$,Q_{w},Q_{w}^{-}\rightar ow ‐. is obtained where. \displayte\frac{1}2 log. { -e^{$\Gam a$}\displaystyle\vec{k}\lrcorner-\frac{$\tau$}{2 $\Theta$+\displaystyle\frac{1}{2}\mathrm{T}xQ_{w}\tilde{Q}_{w}-\frac{$\alpha$}{2} |\displaystyle \tilde{Q}_{w}- $\Theta$|+\frac{1}{2}\vec{k}^{\mathrm{T} (\tilde{Q}_{w}- $\Theta$)^{-1}\vec{k} },. log det |I+ $\beta$ Q_{w}|. Tr. (33). det. $\alpha$=p/N\sim O(1). .. When a,. b(=1,2, \cdots , n). ,. we assume. (q_{wab},\tilde{q}_{wab})=\left\{ begin{ar y}{l ($\chi$_{w}+q_{w},\tilde{$\chi$}_{w}-\tilde{q}_{w})&a=b\ (q_{w},-\tilde{q}_{w})&\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e} \end{ar y}\right. (k_{a}, $\theta$_{a})=(k, $\theta$). ,. the. following;. (34). (35).

(8) 31. where it is called the ansatz of. replica symmetry solution. We substitute the replica symmetry. Eq. (33), then. solution into. $\Phi$(n)=k $\theta,\chi$_{Extr w},q_{w},\overline{ $\chi$}_{w} ). ). \displaystyle \overline{q}_{w}\{-nk-\frac{n $\tau \theta$}{2}+\frac{n($\chi$_{w}+q_{w})(\tilde{ $\chi$}_{w}-\tilde{q}_{w}) {2}. -\displaystyle \frac{n(n-1)q_{w}\tilde{q}_{w} {2}-\frac{n-1}{2}\log(\tilde{ $\chi$}_{w}- $\theta$)-\frac{1}{2}\log(\tilde{ $\chi$}_{w}- $\theta$-n\tilde{q}_{w}). +\displaystyle \frac{nk^{2} {2(\tilde{ $\chi$}_{w}- $\theta$-n\tilde{q}_{w}) -\frac{(n-1) $\alpha$}{2}\log(1+ $\beta \chi$_{w})-\frac{ $\alpha$}{2}\log(1+ $\beta \chi$_{w}+n $\beta$ q_{w})\}. is obtained. Note that. form is real. From. n. in. Eq. (36). is. integer. at. first,. but it. might. assume. that. n. at. ,. (36). present. this,. $\phi$=\displaystyle \lim_{N\rightar ow\infty}\frac{1}{N}E[\log Z(X, $\beta$)] =\displaystyle \lim_{N\rightar ow\infty}\frac{1}{N}\lim_{n\rightar ow 0}\frac{\partial}{\partial n}\log E[Z^{n}(X, $\beta$)]. { -k-\displaystyle \frac{ $\tau \theta$}{2}+\frac{($\chi$_{w}+q_{w})(\tilde{ $\chi$}_{w}-\tilde{q}_{w}) {2}+\frac{q_{w}\tilde{q}_{w} {2} (37) -\displaystyle \frac{1}{2}\log(\tilde{ $\chi$}_{w}- $\theta$)+\frac{\tilde{q}_{w}+k^{2} {2(\tilde{ $\chi$}_{w}- $\theta$)}-\frac{ $\alpha$}{2}\log(1+ $\beta \chi$_{w})-\frac{ $\alpha \beta$ q_{w} {2(1+ $\beta \chi$_{w}) \} $\phi$=\displaystyle \lim_{N\rightar ow\infty}\frac{1}{N}E[\log Z(X, $\beta$)] \displaystyle \mathcal{H}(\vec{w}^{*}|X)=\lim_{ $\beta$\rightar ow\infty}\{-\frac{\partial}{\partial $\beta$}E[\log Z(X, Extr. =k, $\theta,\chi$_{w},q_{w},\overline{ $\chi$}_{w},\tilde{q}_{w}. ,. is. analyzed.. $\beta$. From. and. $\epsilon$=\displaystyle \lim_{N\rightar ow\infty}\frac{1}{N}\mathcal{H}(\vec{w}^{*}|X) =. \left{\begin{ar y}{l \frac{$\alph \tau$+ \tau$-12\sqrt{$\alph \tau$( \tau$-1)}{2&1-\frac{1}$\tau$}\leq$\alph$\ 0&\mathrm{o}\mathrm{t}\ athrm{}\mathrm{e}\mathrm{}\mathrm{w}\mathrm{i}\ athrm{s}\mathrm{e} \nd{ar y}\right.. (38). is assessed.. 6. Discussion When. 1-\displaystyle \frac{1}{ $\tau$}. \leq $\alpha$. ,. we. calculate. \displaystyle \frac{ $\alpha \tau$+ $\tau$-1-2\sqrt{ $\alpha \tau$( $\tau$-1)} {2}=\frac{ $\alpha$-1}{2}+\frac{(\sqrt{ $\alpha$( $\tau$-1)}-\sqrt{ $\tau$})^{2} {2} From this. budget. finding. constraint. ,. then. $\tau$=. (39). and the minimal investment risk per asset of the mean‐variace model with. only,. \displaystyle\frac{$\alpha$-1}{2}. ,. discussed in. [5],. it is. interpreted that. term induced from the investment concentration constraint.. \sqrt{ $\tau$}=0. .. \displayst le\frac{$\alpha$}{$\alpha$-1}. \displaystyle \frac{(\sqrt{ $\alpha$( $\tau$-1)}-\sqrt{ $\tau$})^{2} {2} is the risk. Moreover, when. \sqrt{ $\alpha$( $\tau$-1)}-. is obtained.. $\tau$=$\chi$_{w}+q_{w}=\displaystyle\frac{1}{N}\sum_{\dot{$\iota$}=1}^{N}w_{i}^{2}. ,. (40).

(9) 32. diagonal of Q_{w}. where $\chi$_{w} +q_{w} is the mal. investment risk per asset. expected. ordinary. reseach. (OR),. ,. that. is,. q_{w $\alpha$ a}. In similar way,. .. assess. we. $\epsilon$^{\mathrm{O}\mathrm{R} =N\displaystyle\rightar ow\infty\mathrm{h}\mathrm{m}\frac{1}{N}\min_{\vec{w}\inW}E[\mathcal{H}(\vec{w}|X)] using. the. the mini‐. approach of. \left{\begin{ar y}{l \frac{$\alph \tau$+ \tau$-12\sqrt{$\alph \tau$( \tau$-1)}{2&1-\frac{1}$\tau$}\leq$\alph$\ 0&\mathrm{o}\mathrm{t}\ athrm{}\mathrm{e}\mathrm{}\mathrm{w}\mathrm{i}\ athrm{s}\mathrm{e} \nd{ar y}\right.. $\epsilon$=. (41). $\alpha \tau$. 0R. $\epsilon$ =\overline{2}. (42). . \displayte\frac{$\epsilon$^{\mathrm{O}\mathrm{R} $\epsilon$},. Thus, the opportunity loss. \displayt e\frac{$\epsilon$^{\mathrm{O}\mathrm{R} $\epsilon$}= \left {\begin{ar y}{l \frac{$\alph \tau$}{ \alph \tau$+ \tau$-12\sqrt{$\alph \tau$( \tau$-1)} &1-\frac{1} $\tau }\leq$\alph$\ +\infty&\mathrm{o}\mathrm{}\ athrm{}\mathrm{e}\mathrm{}\ athrm{w}\mathrm{i}\ athrm{s}\ athrm{e} \nd{ar y}\right. where. $\tau$-. 1. -2\sqrt{ $\alpha \tau$( $\tau$-1)}. proposed approach. our. replica analysis),. we. \leq 0 is held when 1. based. the. on. assumption. compare the results derived. numerical simulation and those obtained. (OR).. From. Eq. (17). to. investment risk per asset. Eq. (19), given. of. \displayte\frac{1}$\tau} a. \leq. $\alpha$. and. replica. $\tau$. \geq 1. X. (or simply. in. ordinary research. \mathrm{R}^{N\times \mathrm{p} and J=XX^{\mathrm{T}} the minimal. \in. ,. $\epsilon$=\displaystyle\frac{1}{N}\mathcal{H}(\vec{w}^{*}|X)=\frac{Ne\rightar ow\mathrm{r}(J-$\theta$I_{N})^{-1}J( -$\theta$I_{N})^{-1\rightar ow}e{2(e\rightar ow\mathrm{r}(J-$\theta$I_{N})^{-1}\vec{e})^{2} where. verify. using the proposed method with those from. \displayst le\{ frac{x_l}$\mu$}{\sqrt{N}\. =. In order to. .. symmetry solution. using the standard approach. is,. $\epsilon$. -. (43). (44). ,. $\tau$=\displaystyle\frac{N^{\ve $\Gam a$}e(J-$\theta$I_{N})^{-2}\vec{e}-{(\overline{ }^{\mathrm{T}(J-$\theta$I_{N})1\vec{e}).. As the numerical setting; The number of assets N=500 and the number of periods p=1000, that. is, $\alpha$=p/N=2 ; As the sample sets, C= 100 where X^{c}=. are. prepared. the. probability. respect. with. \displaystle\{ frac{x_\mathrm{t}$\mu$}^{c}\sqrt{N}\ \in \mathrm{R}^{N\times p};x_{i $\mu$}^{c}. mean. 0 and variace. to each return rate. matrix;. is. matrices, X^{1}, X^{2}. return rate. and. independently. 1, respectively; $\epsilon$^{\mathrm{C} =. ,. ,. X^{100},. identically assigned. \displaystyle \frac{1}{N}\mathcal{H}(\vec{w}^{*}|X^{c}). with. is evaluated with. The minimal investment risk per asset is estimated. by. $\epsilon$=\displaystyle \frac{1}{c}\sum_{c=1}^{c}$\epsilon$^{c}. Our. analytical procedure. 1. Stochastic 2. Boltzmann. is:. E[f(w^{\mathrm{O}\mathrm{R}}, X)] \displaystyle \geq\min_{w\in W}f(w, X). optimization:. P(\vec{w}|X). distribution:. \displaystyle \int_{-\infty}^{\infty}d\vec{w}P(\vec{w}|X)\mathcal{H}(\vec{w}|X). ). =. \displaystyle\frac{P_{\mathrm{O}(\vec{w})e^{-$\beta$?\{(\vec{w}|X}{Z(X,$\beta$)}. .. and. −. \displaystyle \frac{\partial}{\partial $\beta$}\log Z(X, $\beta$). .. \displaystyle \mathcal{H}(\vec{w}^{*}|X)=E[\mathcal{H}(\vec{w}^{*}|X)]=\lim_{ $\beta$\rightar ow\infty}\{-\frac{\partial}{\partial $\beta$}E[\log Z(X, $\beta$)]\}.. 3.. Self‐averaging:. 4.. Replica. 5.. Replica symmetry. trick: We estimate ansatz:. E[Z^{n}(X, $\beta$)]. of n\in \mathrm{R}. using E[Z^{n}(X, $\beta$)] of n\in \mathrm{Z}.. q_{wab}=$\chi$_{w}+q_{w} if a=b otherwise q_{wab}=q_{w}.. 6. Numerical simulations: Our. ,. proposed approach. is. supported.. =.

(10) 33. 1.4. \underli{=wgtx（pm$\oea^{}hrLcisubt\e^{)}. I.2. \chek{$omga \tex{（}mahrU$\oeg}. 1. 8 董. \overline{\mathrm{c} 0.8 \in$\omega$. \overline{$\omega$}. \subset q\suc \mathrm{t}\mathrm{D}0.6 0.4 investment concentration. Figure. Minimal investment risk per asset. 1. $\epsilon$. at. $\alpha$. (tau). p/N. =. analysis (orange line), numerical simulation (sky‐blue asterisks erations research. of. approach (green. replica analysis. with 100. are. dashed. line). versus. with. bars),. error. investment concentration. consistent with the averages obtained from. samples and N=500. 2 results from. =. a. numerical. $\tau$. replica and op‐ Results. .. experiment. assets.. Conclusion and the future works. 7. In this mization. talk,. we. problem. have discussed the minimal investment risk per asset for with. which. replica analysis,. a. budget. was. constraint and. an. constraint which has been discussed in. previous work,. approach. failed to. identify accurately. we. assessed. budget. investment concentration constraint.. an. portfolio opti‐. constraint, using. portfolio optimization problem. of the minimal investment risk per asset from the inclusion of. a. developed for and improved during interdisciplinary research. Un‐. like the minimal investment risk per asset and. research. investment concentration. with. quantitatively. constraint. only. case. budget. a. the deviation. caused. by. the standard operations. In contrast,. the minimal investment risk of the. portfolio. optimization problem, since the obtained optimal portfolio only minimizes the expected vestment. risk,. not the investment. investors information about the. As the future treat. the. a more. realistic. we. need to. depiction of. portfolio optimization problem. assets of. varying. concentration. the. work,. case. risk, making. optimal. it clear that this. investment. an. we. need to. in‐. provide. in order to be able to. the investment market. For instance, investment market. risk levels. As alternative constraints to. constraint,. approach. cannot. strategy.. improve and develop the model in. the. a. budget. consider, for instance,. that the return rate is not normalized and linear. comprising. an. we. a. inequality. analyze. risk‐free asset and. constraint. expected. need to. or an. investment. return constraint for. constraints..

(11) 34. Acknowledge The author. appreciates the fruitful. Kobayashi.. This paper is. [8].. This work is. author. arranged. an. at Akita Prefectural. Research. project. 2 of the. No.. Scholarship. Xiao, a. H. Yamamoto and K.. previous. paper. 15\mathrm{K}20999 :. No.. 5 of the. No.. by. 1414 of the. No.. project No. 2068 of the. Zengin. the. same. The President No. 50 of the. Japan. Institute of. project of the Institute of Economic Research Foundation. project. and Finance: Research. X.. University: Research project. Informatics, Japan: Research project. Life Insurance: Research. University:. Tada,. partly supported by Grant‐in‐Aid. Project for Young Scientists National Institute of. comments of D.. and introduced version of. Kyoto. at. Foundation for Studies in Economics. Institute of Statistical Mathematics: Research. Kampo Foundation: Research project of the Mitsubishi UFJ. Trust. Foundation.. Reference [1]. Phys.. [2]. S.. [3]. I.. J. \mathrm{B} ,. Ciliberti,. under. I. Kondor and M. Mézard. measures. ,. S. Pafka and I. Kondor \mathrm{B} ,. T. Shinzato. through portfolio replication,. Eur.. 27, 175. Kondor, S. Pafka and G.. Phys. J.. [5]. Risk minimization. (2007). expected shortfall, Quant. Fin., 7,. various risk. [4]. (2007). S. Ciliberti and M. Mézard. J. Bank.. easibility of portfolio optimization. 389. (2007). Nagy. On the. Noise. sensitivity of portfolio selection under. Fin., 31, 1545. (2002) Noisy. covariance matrices and. portfolio optimization,. Eur.. 27, 277. (2015) Self‐averaging property. of minimal investment risk of mean‐variance. model, PLoS One, 10, e0133846. [6]. T. Shinzato and M. Yasuda. (2015). Belief. propagation algorithm for portfolio optimization. problems, PLoS One, 10, e0134968. [7]. T. Shinzato returns. [8]. (2016). Portfolio optimization. problem with. nonidentical variances of asset. using statistical mechanical informatics, Phys. Rev.. T. Shinzato. (2016). Minimal investment risk of. and investment concentration. \mathrm{E} ,. 94, 062102. portfolio optimization problem. constraints, https: // arxiv. \mathrm{o}\mathrm{r}\mathrm{g}/\mathrm{a}\mathrm{b}\mathrm{s}/1605.06845. with. budget.

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