The importance of optimal investment allocations and judicious historical data management are discussed in this article. These are everyday issues that financial operators, financial managers, financial planners and financial regulators have been addressing for decades. A considerable body of work exists on optimal investment allocations (also called portfolio optimisation, portfolio selection, etc.), which focuses on specific hypotheses and assumptions. Examples are provided in the referenced papers (Bielecki et al. 2003; Li & Wan-Lung 2000; Lim 2004; Lim & Zhou 2002; Markowitz 1952; Zhou & Li 1999; Xia 2005; Xun & Zhou 2006). Most of the examples deal with single or short-period cases, which generally require continuous risk management for investment allocations.

Even when financial data describing features of portfolio assets are used to optimise a portfolio over a specified period, such data may impact the future. Thus, this data may be needed in some way to predict future financial data of the same type, as the historical data may contain relevant hidden or implicit information. Autoregressive (AR) models are one of the tools suitable to forecasting and prediction of data.

Autoregressive models are used to study the correlation between various random variables constituting a time series. They are extensively applied in finance, which is the reason why AR models for financial data are critical. The use of these models can facilitate the investigation of financial systems by revealing meaningful and useful information about financial processes. For example, AR models can be used to predict asset prices. This article also proposes the use of AR models to forecast expected profit vectors for financial assets, as well as the covariance matrices related to asset profits.

Certain mathematical, statistical and econometric models for mean forecasting, as well as statistical and econometric models for volatility, covariance, etc., are available. This article notes that every element of the covariance matrix is an expected value and can thus be forecast by using a mean forecasting model. Consequently, this article deals with the computation of optimal investment allocations to minimise risk for a specified minimum level of profit.

Markowitz was one of the first researchers to propose a model for solving single-period portfolio optimisation problems (Markowitz 1952; Zenios 1993). After further refinement and extension, his model contributed to the development of a method for quantifying risk by using variance, which enables investors to maximise profit for a given threshold of the risk.

Merton (1972) uses this model to compute an analytic solution when the covariance matrix is positive, and short-selling is allowed. The task of selecting an optimal investment is not easy, especially when dealing with a large set of assets while aiming to manage risk. One of the critical approaches involves designing reliable mathematical models that describe the problem precisely and accurately, and then computing analytical or numerical solutions to obtain associated computational simulations.

Given a long time horizon [0,T] and N assets i = 1,···, N, such that for every asset i the profit is a random variable with an expected value equal to µ_i, the aim is to compute the optimal investment allocation, ω1*,i=1,…,N, with 0≤ωi*≤1, to drive the portfolio in the period [0, T] and fulfil the following conditions: minimiseω1,…,ωN F(ω1,…,ωN)=12∑i=1N∑j=1Nωiσij ωj[Eqn 1]

Subject to: ∑i=1Nμiωi ≥μp[Eqn 2] ∑i=1Nωi=1[Eqn 3] where ∀i, R_i(t) is the profit of asset i at time t; µ_i(t) is the expected value of profit R_i(t); σ_i(t) is the volatility of R_i(t); µ_p(t) is the specified minimum level for the portfolio expected rate profit; and σ_ij(t)=cov(R_i(t), R_j(t)) is the covariance between the profit of asset i and the profit of asset j.

The above problem can be reformulated as a sequence of the following equality constrained quadratic optimisation problems: minimiseω1,…,ωN F(ω1,…,ωN)=12∑i=1N∑j=1Nωiσij ωj[Eqn 4]

Subject to: ∑i=1Nμiωi=μp,μp ε Q[Eqn 5] ∑i=1Nωi=1[Eqn 6] where µ_p, as stated in [Eqn 5], is a variable that can take many values. For every given value of µ_p, the corresponding optimal investment allocations, the minimum risk σ_p and the point (µ_p, σ_p) must be computed to give a solution curve expressing the trade-off between minimum risk and expected profit. In matrix form, the above problem sequence can be expressed as: min(ω1,…,ωN)( ω1 … ωN )( σ11⋯σ1N⋮⋱⋮σN1⋯σNN )( ω1⋮ωN ) Subject to:( μ1 ⋯μN1 ⋯1 )( ω1⋮ωN )=( μp1 ),μp ε Q where μ = (μ₁ μ₂ … μ_N) is a real-value N-dimensional vector, and Σ = (σ_ij) 1 ≤ i, j ≤ N is a real symmetric matrix. Various methods are available to solve the problem: the interior point method, the active set method, the augmented Lagrangian method, the conjugate gradient method, the gradient projection method, the extension of the simplex algorithm method, the simulated annealing method, the particle swarm optimisation method, the genetic algorithm method, the controlled random search algorithm method, the genetic programming method and the Newton’s method.

For the case in which Σ is positive definite, the above problem becomes a special case of the more general field of convex optimisation. By applying the Lagrangian method, the problem sequence (4)–(6) can be reformulated as the computation of (ω1*,ω2*,…,ωN*) such that the following condition is satisfied: minimiseω1,…,ωN L(ω1,…,ωN,α1,α2)=12∑i=1N∑j=1Nωiσij ωj−α1C1−α2C2[Eqn 9] where: C1=∑i=1Nμiωi−μp[Eqn 10] C2=∑i=1Nωi−1[Eqn 11]

Without loss of generality, consider a fixed integer i, then the derived first-order necessary conditions for optimality are as follows: ∂L∂ωi=∑j=1Nσijωj−α1μi−α2=0[Eqn 12] ∂L∂α1=∑j=1Nμiωi−μp=0[Eqn 13] ∂L∂α2=∑j=1Nωi−1=0[Eqn 14]

As an example, a hypothetical data set (containing 10 time series on financial asset profits for a period of 12 months, generated randomly using Matlab Software) has been processed according to the optimal investment allocations, and the trade-off between risk and profit is given in Figure 1, which, as expected, shows that profit is an increasing function of risk.

A further data set is presented in Table 1, which contains 10 risky financial asset profits for a period of 12 months. For each i, Profit i is the profit of the financial asset i. From Table 1, a vector of 10 expected profits μ = (μ₁ μ₂ … μ₁₀) and a covariance matrix Σ = (σ_ij)_{1 ≤ i, j ≤ N} are computed to serve as inputs for the problem. The associated results are given in Tables 2 and 3, respectively.

The optimal allocations in Table 3 show that the investor does not need to allocate all the money they have because for Asset 1 and Asset 4 the optimal allocation is 0.

As for the first example, additional examples were considered to support the study of the trade-off between the portfolio minimum risk and maximum expected profit. The related computational simulations are illustrated by Figures 2 and 3.

For long-period investment allocations, the first-period investment allocations problem must be solved. Then the historical data (the expected rate-of-profit vector and the covariance matrix) can be forecast to build a new expected rate-of-profit vector and a new covariance matrix for the second period.

At every period other than the second period, the expected rate-of-profit vector and the covariance matrix are obtained by forecasting the expected rate of profit and the covariance matrix for the previous period.

Many techniques exist for forecasting data. Such techniques are described by autoregressive (AR) models, moving average (MA) models and autoregressive moving average (ARMA) models. These are all captured by autoregressive integrated moving average (ARIMA) models. This article uses AR to obtain the asset expected rate-of-profit vector μ = (μ₁ μ₂ … μ_N) and the covariance matrix Σ = (σ_ij)_{1 ≤ i, j ≤ N} for each period.

The following section illustrates the use of forecasting from one period to another.

This section deals with deriving future expected rate--of-profit vectors and future covariance matrices for the portfolio assets of the previous period. The second expected rate-of-profit vector and the covariance matrix are derived from those of the first period. AR models, which are widely used in finance, are applied in this derivation. At every period other than the second period, the expected rate-of-profit vector, as well as the expected covariance matrix, is obtained from the estimated rate-of-profit vector and the estimated covariance matrix of the previous period.

The first period uses the expected rate-of-profit vector and the covariance matrix for the asset profits obtained from given historical data. The second period applies first-order AR modelling to the first period expected rate-of-profit vector and covariance matrix to obtain the second period expected rate-of-profit vector and covariance matrix and then finally to obtain the second-period investment allocations. The third period applies second-order AR modelling simultaneously to the first and second period expected rate-of-profit vectors, and covariance matrices, to obtain the third-period expected rate-of-profit vector and covariance matrix and then finally to obtain the third-period investment allocations.

In general, the kth period simultaneously applies (k-1)-order AR modelling to all the previous (k-1)-period expected rate-of-profit vectors and covariance matrices to obtain the kth period expected rate-of-profit vector and covariance matrix and then to obtain the kth period investment allocations. For every period, the application of the AR model can be performed as follows:

The concern is the following: find the investment allocations ωi[k],i=1,…,N satisfying the following conditions: minimiseω1,…,ωN F[k](ω1[k],…,ωN[k])=12∑i=1N∑j=1Nωi[k]σij[k]ωj[k][Eqn 34]

Subject to: ∑i=1Nμi[k]ωi[k]=μp[k],μp[k]∈Q,[Eqn 35] ∑i=1Nωi[k]=1[Eqn 36] and with the following substitutions: μi[k]=∑s=1p∅sμμi[k−i]+ε[k][Eqn 37] Σi[k]=∑s=1p∅sμΣi[k−i]+ε[k][Eqn 38] where if the following designations apply:

For the first period, the optimal investment allocations can be computed from collected historical data and some specified level of expected profit. Subsequently, for period k>1,μi[k],i=1,…,N and Σ[k]=σij[k]=cov(Ri[k],Rj[k]) can be obtained from the forecasts that are generated for μi[k−1] and Σ[k−1]=σij[k−1]=cov(Ri[k−1],Rj[k−1]) based on an AR model configuration.

The computational simulations are based on ‘Sequential investment allocations’ section to show the effectiveness of the theory. The first period investment allocations are based on historical data of Table 1. From that table, an expected value is computed for each asset and a covariance matrix for the whole table is also computed to serve as input to the investment allocation problem.

By applying a fifth-order autoregressive model to forecast the financial data of Table 1 (considered for the first period investment allocations), the autoregressive coefficients ∅_jπ (Equation 33) for the assets are given in Table 4.

For each of the 10 considered assets, the associated autoregressive coefficients enable one to perform forecasting and obtain four time series of length 12 each (time series for the next four periods where each period has 12 months). For each period, an investment allocation was performed and was summarised by an efficient frontier curve. An investment allocation (portfolio selection) is associated with each point of the efficient frontier (each pair of [risk, profit]).

Figures 4 to 8 are the computational simulations for the sequential investment allocations from period 1 to period 5 respectively.

By browsing and comparing the efficient frontier of Periods 1–5 defining the sequential optimal investment allocation for portfolio optimisation, one can notice that the more we progress in time, the more the portfolio risk (the portfolio variance) decreases and the portfolio profit increases, which shows a good process for the optimisation of the considered investor’s portfolio.

The more time is passing the more risk is getting minimised and profit is getting maximised. One may continue forecasting and analyse the mean-variance performance.

The aim of this article was to compute the optimal investment allocations of a number of portfolio assets over a long-term period. The trade-off between risk and profit for such a long-term formulation was investigated, and computational simulations corroborate the notion that the higher the risk, the higher the profit. For efficient risk management, the given period was subdivided into discrete sub-periods so that every sub-period addressed an optimal investment allocations problem. The forecast data could be computed iteratively for consecutive periods.