Markowitz’s mean-variance portfolio selection

In this subsection, we present the mean-variance model, which is one of the most commonly used models for the portfolio optimization problem. We begin by considering a stable investment market with N investment outlets, where w_k represents the portfolio (or investment ratio) of asset k(= 1, ⋯, N), and denotes the return rate of asset k in scenario μ(= 1, ⋯, p). However, for simplicity, we do not include short sales, that is, −∞ < w_k < ∞, and we assume that the probability distribution of the return rate is known for each asset. In Markowitz’s mean-variance model, given p scenarios, the investment risk is defined to be the sum of squares of the difference between the gross return for a scenario and its expectation ; determining an investment strategy by minimizing the risk creates a hedge. That is to say, the investment risk of a portfolio with N assets w = (w₁, ⋯, w_N)^T ∈ R^N is defined as (1) where T denotes the transpose of a matrix or vector, and E[f(x)] is the expectation of f(x). Since we have assumed that the probability distribution of the return rate of each asset is known, we represent the return rate as and the return rate matrix as . Also, note that although we introduce the coefficient in Eq (1) for simplicity of the discussion below, since is the summation of N random variables x_kμ w_k (w_k can be interpreted as the coefficient of a random variable x_kμ), even if we do not assume that the return rates of the assets are independent, if w is fixed, the correlation between the returns is small, and the third and higher moments of the return rate are finite, then we expect that as the number of investment outlets N increases, asymptotically approaches a multidimensional Gaussian distribution according to the central limit theorem.

In the mean-variance model, in the absence of constraints (such as budgets), an obvious optimal portfolio is obtained by minimizing the risk function 𝓗(w∣X) with w₁ = ⋯ = w_N = 0. Since this is equivalent to not investing, there is no investment risk; however, in this paper, we use the budget constraint (2) Moreover, although in the actual management of assets, it is necessary to impose expected return restrictions in addition to budget constraints, for simplicity, we will consider only budget constraints. Therefore, the portfolio optimization problem is formulated as determining the portfolio w that minimizes 𝓗(w∣X), that is, the risk function in Eq (1) with the constraint of Eq (2). In the case of p > N, the optimal solution can be analytically determined: (3) where the unit vector e = (1, 1, ⋯, 1)^T ∈ R^N and J⁻¹ is the inverse of the variance-covariance matrix J = {J_ij} = XX^T ∈ 𝓜_{N × N}, where element i, j of matrix J is (4) If p ≤ N, then since matrix J is not a regular matrix, the optimal solution of this portfolio optimization problem cannot be uniquely determined.

Using the definition of 𝓗(w∣X) in Eq (1), for each scenario μ, we can estimate the sum of squares of the difference between the gross earnings and its expectation ; this can be interpreted as the investment potential of portfolio w, and the concentrated investment level q_w is defined as follows [6, 8, 9]: (5) With an equipartition investment strategy w = (1, 1, ⋯, 1)^T ∈ R^N, we obtain q_w = 1; with a concentrated investment strategy, for example, investing only in asset 1, w = (N, 0, ⋯, 0)^T ∈ R^N, so q_w = N is obtained; if one investor invests equally in m of N possible outlets, q_w = N/m. Thus we have (6) and as portfolio w approaches equipartition, q_w decreases to 1, and as it approaches a concentrated investment strategy, q_w increases.

We note that although is widely used as a budget constraint in operations research, we do not use one in this paper. Since the optimal solution to the portfolio optimization problem with a budget constraint that is widely used in operations research is , and the optimal solution defined in Eq (2) is , the relation is proved, that is, in the optimal portfolios of each method, the investment ratios are in agreement. Furthermore, the concentrated investment level q_w can be interpreted as an indicator of diversification when using the budget constraint of Eq (2).

Optimization for Stochastic Phenomena

In this subsection, we consider this optimization problem from a different viewpoint. We analyze the behaviour of minimal investment risk ɛ and the concentrated investment level q_w of the mean-variance model, and we discuss the optimization of stochastic phenomena which have not been addressed in the operations research approach to this problem. Let us consider the following variant of the well-known game of rock-paper-scissors. Rule 1: Two subjects, Alice and Bob, play rock-paper-scissors 300 times. Rule 2: Alice can freely choose to display rock, paper, or scissors. On the other hand, Bob’s choice is randomly assigned by the toss of a fair dice: rock when the dice shows 1 or 2, paper for 3 or 4, and scissors for 5 or 6. Moreover, Alice knows that Bob’s choice is randomly and independently determined by the dice. Rule 3: The winner adds a point, the loser subtracts a point, and if they tie, there is no change to the score. We now consider whether it is expected that Alice will win overall.

Operations research approach for portfolio optimization

From the above argument, we see that optimization of stochastic phenomena is handled differently for an annealed disorder system than it is for a quenched disorder system. Using this core concept, let us reconsider the portfolio optimization problem. In the standard analytical approach of operations research to the portfolio optimization problem, one first averages the risk function 𝓗(w∣X) with the return rate on the assets and then minimizes the expectation of the risk function E[𝓗(w∣X)] with a budget constant. Here, for simplicity, we presume that return rate is independently and identically distributed with a standard normal distribution. Thus, the expectation of the correlation between asset i and asset j is (8) Using this, the expected investment risk function E[𝓗(w∣X)] with a return rate on the assets is (9) where the ratio α = p/N is used. In addition, from the symmetry of this model, the optimal investment strategy of Eq (9), using the budget constraint of Eq (2), describes an equipartition investment strategy. The minimum expected investment risk per asset is evaluated as follows: (10) The concentrated investment level is (11) This analytical approach, which is widely used in operations research, does not provide insight into the optimal investment strategy in an actual market; the reason is not just that the model was simplified by assuming the return rate is independently and identically distributed with a standard normal distribution. Since this analytical approach is equivalent to case (a) in the rock-paper-scissors game with two subjects and the annealed disorder system, it is not clear that this approach could be used to minimize the investment risk 𝓗(w∣X) with respect to a realistic individual return rate matrix X, that is, w = argmin_w𝓗(w∣X). In particular, the equality argmin_w𝓗(w∣X) = argmin_w E[𝓗(w∣X)] is not always satisfied. As we discussed with the rock-paper-scissors game, if we average the investment risk with the return rate, we can avoid the complication of optimizing individual return rates; on the other hand, this approach does not evaluate the optimal strategy based on individual return rates. Even though it is not guaranteed mathematically that the solution to the minimal expected investment risk optimizes the investment risk for each set of return rates 𝓗(w∣X), this approach is widely used in operations research and might provide a misleading investment strategy.

On the other hand, let us consider case (b) in the rock-paper-scissors game with two subjects and the quenched disorder system. In a stable investment market, even if at μ = 0 one had prior information about the probability distribution of the return rate of each asset during the next period (μ = 1 to μ = p), since it is not possible to know the actual return rate, it is difficult to select the optimal investment strategy. However, if we have prior information about the return rates, as discussed in the rock-paper-scissors example, we can minimize the investment risk and obtain an optimal investment strategy. In particular, if p/N ≤ 1, since it is well known that the optimal solution is a linear sum of the eigenvectors of the minimal eigenvalue of the variance-covariance matrix J, the investment risk per asset is 0, since the minimal eigenvalue of the matrix J is 0 since J is nonsingular. We next consider the concentrated investment level q_w. Let 𝓥 be variance of the sample variance ; it is evaluated as follows: (12) When α = p/N is small, 𝓥 is large, and one should invest heavily in blue-chip assets for which the return rates have smaller sample variances than those of the other N investment outlets; in this way, the risk is decreased, and the optimal investment strategy is asymptotically close to a concentrated investment strategy, namely q_w ≫ 1.

When p/N > 1, using the optimal solution for Eq (3), the two indicators can be analytically assessed. That is, the minimal investment risk per asset ɛ(X) and the concentrated investment level q_w(X) can be written as follows: (13) (14) where we use the explicit return rate matrix as the argument since these indicators depend on the return rate matrix. The variance-covariance matrix J = XX^T ∈ 𝓜_{N × N} has already been defined. In actual investments, assuming fair dealing, since we do not have prior knowledge of the actual return rate, we cannot precisely determine the two indicators. However, we can evaluate the previous risk in an investment system and thus support the strategy of an investor. In order to provide useful insight, we need to precisely analyze ɛ(X) and q_w(X). For the reasons noted here, we assume that during the initial period, we have prior knowledge of the return rate; although this assumption is impossible, we note that we will show below that this assumption is not required to evaluate the potential of an investment system. Although we need to assess the optimal solution or the inverse of the variance-covariance matrix in order to assess the potential risk of an investment market, it is difficult to do this since the computational complexity of finding the inverse matrix is proportional to the cube of the matrix size N, which is the number of investment outlets. In addition, we need to find each inverse matrix for each return rate in order to evaluate the minimal investment risk of each set; however, if the minimal investment risk randomly fluctuates with the return rates, we would need to average the minimal investment risk with the return rate set X. We now note that we can use the self-averaging property to simplify evaluation of the potential investment risk.

Statistical mechanics

First, using the Boltzmann distribution of the inverse temperature β(> 0), which is widely used in statistical mechanics [13], the posterior probability of portfolio w given return rate matrix X, P(w∣X), is defined as follows: (15) where the prior probability P₀(w) is 1 if portfolio w satisfies Eq (2), and it is 0 otherwise; e^{−β𝓗(w∣X)} is the likelihood function; and Z(β, X), the partition function, is a normalized constant and is defined as follows: (16) From this, it is found that the posterior probability P(w∣X) satisfies the property of a probability measure, that is, P(w∣X) ≥ 0 and . Furthermore, it is well known that w* = arg max_w P(w∣X), which is obtained using the maximum a posteriori estimation, is consistent with the portfolio obtained by minimizing the investment risk function 𝓗(w∣X). By taking the limit of the inverse temperature β, we obtain (17) where is the optimal investment ratio for asset i. Thus, we can average the portfolio w and the investment risk 𝓗(w∣X) using the a posteriori probability P(w∣X) and allow the inverse temperature β to become sufficiently large: (18) (19) where δ(u) is the Dirac delta function and this holds for any function f(x) such that (see appendix 2). From this reformulation and using the posterior probability defined in Eq (15), the portfolio optimization problem can be solved using the framework of probabilistic reasoning.

Chernoff inequality

Next, we introduce one of the probability inequalities, the Chernoff inequality, as follows. For a random variable Y with known probability measure and a constant number η, the probability that η ≤ Y satisfies the following inequality for any u > 0 is [14, 15]: (20) This can be easily proved; for example, consider the step function Θ(W), which is 1 if W ≥ 0 and 0 otherwise. First, for u > 0, we obtain Θ(W) ≤ e^uW. From this, we derive Pr[η ≤ Y] = E[Θ(Y − η)] ≤ e^−uη E[e^uY]. In addition, for Pr[η ≥ Y], we obtain Pr[η ≥ Y] ≤ e^−uη E[e^uY] for u < 0.

From Eq (20), we could derive a tighter upper bound. Since the right-hand side in Eq (20) is guaranteed for an arbitrary u > 0, there necessarily exists a minimum value for the right-hand side for any u > 0, and we obtain (21) Here R(η) is the rate function and is defined as (22) The cumulative generating function ϕ(u) = logE[e^uY] is a convex function of u, and the rate function R(η), defined by the Legendre transformation of a convex function, is also a convex function. It is also known that R(η) is nonnegative, R(η) = 0 if η ≤ E[Y], and R(η) > 0 if η > E[Y]. These properties of the rate function are proved in appendix 1.

Self-averaging property supported by large deviation theory

When the portfolio w depends on the posterior probability P(w∣X) defined in Eq (15), the probability that the investment risk per asset is less than or equal to a constant number satisfies the Chernoff inequality; that is, satisfies (23) Here, is a positive number and is defined by Eq (16) (with β replaced by ). Thus, the probability inequality of the tighter upper bound of Eq (23) is derived using the following rate function: (24) and we have [16]. In a similar way, for the probability of , , thus is also obtained, where we use the rate function (25) In order to analyze the rate functions in Eqs (24) and (25), it is also necessary to assess and , which depend on the return rate matrix X. Based on the definition in Eq (16), assessing these partition functions analytically is more difficult than assessing the optimal solution analytically. In order to resolve this difficulty, we consider the cumulative distribution of or the Helmholtz free energy f(β, X), as defined in the following equation: (26) The Helmholtz free energy f(β, X) fluctuates randomly with the probability of the return rate matrix X. Thus, it is necessary to evaluate the Chernoff inequality for the Helmholtz free energy and its rate function: (27) where n > 0 has already been defined. In a similar way, we have (28) where n < 0. In conclusion, we obtain the two probability inequalities, and where (29) (30) In both inequalities, it is necessary to analyze .

Similarity to the Hopfield model

In this subsection, in order to determine whether we can use replica analysis to evaluate E[Zⁿ(β, X)], let us consider briefly the problem of recalling a pattern stored in a neural network constructed of N neurons; the mathematical structure of this problem is similar to the portfolio optimization problem [10, 17]. Let S_k be the state of neuron k; then S_k = 1 if neuron k has been fired and S_k = −1 otherwise. Additionally, x_kμ, (k = 1, ⋯, N, μ = 1, ⋯, p) is the memory of neuron k for pattern μ included in p stored patterns, and it is randomly assigned ±1 with equal probability.

Then, for p patterns, the Hebb rule is defined as follows: (31) where J_ij is the correlation between neuron i and neuron j. Thus, it is well known that the neuron state S that minimizes the Hamiltonian 𝓗(S∣X) in Eq (32) is consistent with each stored pattern: (32) If neuron state S is consistent with pattern 1, that is, S_k = x_k1, the Hamiltonian 𝓗(S∣X) can be written as (33) where the overlap between pattern μ and pattern ν in limited by the number of neurons N and satisfies (34) Intuitively, each pattern is orthogonal with each of the others, since the stored patterns are independent and are randomly assigned. This problem of recalling patterns stored in a neural network and of accounting for the number of identifiable patterns is called the associative memory problem, and the model defined in Eq (32) is called the Hopfield model.

In the analysis of the Hopfield model, the upper limit of the number of identifiable patterns is estimated using E[Zⁿ(β, X)], which evaluates the learning potential of the neural network. We also note the mathematical similarity between this model and the mean-variance model, which indicates that we could adapt the analytical approach used for the Hopfield model; that is, we could use replica analysis for the portfolio optimization problem and to assess the potential of the investment system.

Main results obtained in replica analysis

For the detailed calculations of replica analysis, please see appendix 2. We will limit the number of investment outlets N such that α = p/N ∼ O(1). For n ∈ N, we have (35) where , k_a, q_wab, and are order parameters, e = (1, ⋯, 1)^T ∈ Rⁿ is a constant vector, I ∈ 𝓜_{n × n} is the identity matrix, and Extr_A f(A) are the extrema of f(A) with respect to A. From this, the extrema of k, Q_w, and are assessed as follows: (36) (37) (38) where D = e e^T ∈ 𝓜_{n × n} is a square matrix, all of whose components are 1. Based on these results, we do not need to assume replica symmetry ansatz with respect to the order parameters of this model (). Thus, substituting these result into Eq (35), we have (39) Here we should note that in appendix 2, we require that the replica number n in Eq (35) is a natural number, that is, since E[Zⁿ(β, X)] at replica number n ∈ N can be estimated comparatively easily, the replica number n in Eq (39) should also be a natural number. However, in the optimization in Eqs (29) and (30), we need to have n ∈ R to adequately discuss the solution. Thus, we assume here that the replica number n in Eq (39) is a real number and use this to discuss our approach in detail. In below subsection, we will compare this result with the result to justify that this is applicable.

The two rate functions are calculated as follows: (40) (41) where (42) (43) This result satisfies the properties of a rate function, as shown in appendix 1. Moreover, using Gibbs inequality, s − 1 − log s ≥ 0, if α > 1, in the limit as the number of investment outlets N becomes sufficiently large, we obtain (44) (45) From Eqs (44) and (45), since f(β, X) is localized around the constant , (46) is verified for a realistic set of return rates. Namely, the Helmholtz free energy per asset f(β, X), which is a function of the random variable X, becomes a definite value in the limit of sufficiently large N. Thus, we have (47) In addition, because of localizing around the definite value, f^m(β, X) = E[f^m(β, X)] is also satisfied. This property in which a statistic or function of a random variable localizes around a definite value (or its average) is called a self-averaging property. By substituting Eq (47) into Eq (26), we obtain (48) and the two rate functions (49) (50) where . Thus, for a sufficiently large N, since the investment risk per asset is also localized around , the investment risk is self-averaging. Moreover, for a sufficiently large β, from Eq (17) we can derive the minimal investment risk, as follows: (51)

Furthermore, since ɛ(X) is also derived analytically from an identical equation, , we have validated our method in another way (see appendix 2). In addition, from the self-averaging property of the investment risk, since we can ignore the dependency of the investment risk ɛ(X) on the return rate matrix X, we will replace ɛ with ɛ(X). In a similar way, q_w(X) is also self-averaging, and so then (52) is obtained, where q_w(X) has been replaced by q_w.

We also note that since the minimal investment risk per asset and the concentrated investment level are both self-averaging (since their dependency on the return rate matrix X is ignored), we can estimate the potential of this investment system. In a stable investment market, this implies that the minimal investment risk with respect to a realistic return rate averaged over an investment period and the minimal investment risk defined by the return rate are in agreement since the minimal investment risk is self-averaging. Because of this, we do not need the assumption of the quenched disorder system that during the initial period, we have prior knowledge of the return rates that is, we only need to know a priori the previous return rates. This is another advantage of the self-averaging property.

Comparison with results obtained by the operations research approach

In this subsection, we compare the two indicators that were derived in Eqs (10) and (11) using the analytical approach of operations research. For example, we consider and with the two feature indicators derived in above subsection and using the self-averaging property, that is, if α > 1 and ɛ = 0 otherwise, and if α > 1 and q_w ≫ 1 otherwise. Thus, for any α, we have (53) (54) First, the minimal expected investment risk ɛ^OR is not smaller than the expected minimal investment risk ɛ, that is, Eq (53) is consistent with the relationship in Eq (7). Next, from both of the concentrated investment levels and using the analytical procedure of operations research, we find that the risks for each investment outlet are averaged and negated by the returns matrix. We thus conclude that the optimal strategy is equipartition investing. On the other hand, when using our proposed method, since it is possible to find the optimal solution for each investment outlet, the best return rate is found for an investment outlet that has little variation, especially if α is small, and this implies that the optimal strategy is concentrated investing.

Furthermore, we provide another intuitive interpretation using another mathematical argument. By the definition of q_w and the N eigenvalues of matrix J = XX^T ∈ 𝓜_{N × N}, λ_k, (k = 1, ⋯, N, λ₁ ≤ λ₂ ≤ ⋯ ≤ λ_N), then q_w = E[λ⁻²]/(E[λ⁻¹])² and where ; see appendix 3 for the derivation. In addition, since the minimum eigenvalue of the asymptotic distribution is approximately close to +0 when α → +1, if m eigenvalues are regarded as minimum eigenvalues, where m ∼ O(1), then by using L’Hôpital’s rule, we can estimate the asymptotic form of q_w as follows: (55) Since E[λ⁻²] increases faster than (E[λ⁻¹])², q_w increases. This is consistent with our finding that . Moreover, if α ≫ 1, then 1 ≤ q_w ≤ (λ_max/λ_min)², and (56) where the maximum asymptotic eigenvalues are and . This is also consistent with our finding that .

Numerical simulation

Although we presented a theoretical discussion of the potential of an investment system, using the self-averaging property of the investment risk per asset ɛ and the concentrated investment level q_w, we assumed that the replica number n in Eq (39) was a real number in above subsection. In the previous subsection, we presented some mathematical interpretations for our findings. However, it is not guaranteed mathematically that the replica number n ∈ R is applicable; we thus need to verify that we may use this assumption in order to legitimize the findings based on our proposed method. In this subsection, we perform a numerical simulation, and we then compare the results of our proposed method, the numerical results, and the results from the analytical operations research procedure.

In this numerical simulation, the number of investment outlets was N = 10³, and the number of scenarios was p ∈ [1200, 8000]; the scenario ratio was α ∈ [1.2,8.0]. In addition, we assessed J⁻¹ = (XX^T)⁻¹, the inverse of the variance-covariance matrix defined by the randomly assigned return rate matrix X; the return rates on assets were independently and identically distributed with a standard normal distribution. We then solved Eq (3) for the optimal portfolio in order to estimate the minimal investment risk per asset ɛ(X) and the concentrated investment level q_w(X). Finally, we averaged them over 100 sets of the return rate matrix.

In Fig 1, three minimal investment risks per asset and three concentrated investment levels are shown. The horizontal axis indicates the scenario ratio α = p/N, and the vertical axis shows the two indicators. The results of our proposed approach are indicated by solid lines, the numerical results are indicated by markers with error bars, and the results of the operations research approach are indicated by dotted lines. The results of our method (solid lines) and the numerical results (markers with error bars) are in agreement. For this numerical simulation, we considered the case in which we have a priori knowledge of the return rates. Thus, it turns out that our proposed approach can precisely assess the potential of an investment system. On the other hand, the dotted lines are based on a scenario in which the expected utility is maximized, and these results do not coincide with the others. Unfortunately, this indicates that the approach based on maximizing the expected utility is unable to determine the optimal investment strategy and may instead provide a misleading portfolio which is not guaranteed to be optimal with respect to particular set of return rates.

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Fig 1. The investment risk ɛ and the concentrated investment level q_w are shown for the case in which the return rate x_kμ is independently and identically distributed with a standard normal distribution.

The horizontal axis indicates the scenario ratio α = p/N, and the vertical axis shows the investment risk and the concentrated investment level q_w. The two solid lines (results obtained by our proposed approach) and the two dotted lines (results obtained by the operations research approach) are theoretical results. The markers with error bars are the numerical results evaluated using the optimal solution according to a return rate which was randomly assigned. In the simulation, the number of investment outlets N was 10³, and we averaged 100 return rate matrices . This figure shows that the results obtained by our proposed approach (solid lines) and the numerical results (markers with error bars) are in agreement. On the other hand, the results obtained by the operations research approach (dotted lines) do not coincide with the others. Thus, unfortunately, the approach based on maximizing the expected utility cannot propose an optimal investment strategy.

https://doi.org/10.1371/journal.pone.0133846.g001

R(η) ≥ 0 is held

For any η ∈ R, R(η) ≥ 0 holds. Also, for any u > 0, R(η) = max_{u > 0}{uη − ϕ(u)} ≥ uη − ϕ(u) = R(η, u) in the limit as u goes to +0, that is, we obtain lim_{u → +0} R(η, u) = 0 for R(η) ≥ 0. In addition, if R(η) < 0, since Pr[η ≤ Y] ≤ 1 < e^−R(η), R(η) ≥ 0 implies an intuitive upper bound for the cumulative distribution.

When E[Y] ≥ η, R(η) = 0

When η is less than or equal to E[Y], the expectation of Y, that is, E[Y] ≥ η, then R(η) = 0. Since e^uY is a convex function of Y, for any u > 0, ϕ(u) ≥ loge^uE[Y] = uE[Y], and we obtain 0 ≥ uE[Y] − ϕ(u). If η = E[Y], then we obtain R(E[Y]) = 0 from R(E[Y], u) = uE[Y] − ϕ(u) ≤ 0. In addition, if E[Y] ≥ η, then 0 ≥ uE[Y] − ϕ(u) ≥ uη − ϕ(u) = R(η, u), and we obtain R(η) = 0. That is, this property may intuitively imply E[Y] = sup{η∣R(η) = 0}.

R(η) is a convex function

R(η), which is derived from the Legendre transformation of a convex function, is a convex function of η. Thus, for any ∀λ ∈ [0,1), R(η) and R(ξ), (58) where u is nonnegative. Thus, the λR(η) + (1 − λ)R(ξ) ≥ R(λη + (1 − λ)ξ) is obtained by maximizing both sides of Eq (58) for u > 0.