Monitoring Multidimensional Risk in the Economy ^†

Abstract

In economics, risk analysis is often associated with certain difficulties. These include the presence of several correlated risk factors, non-stationarity of economic processes, and small data samples. A mathematical model of multidimensional risk is described which satisfies the main features of processes in the economy. In the task of risk monitoring, we represent the analyzed factors as a set of correlated non-stationary time series. The method allows us to assess the risk at each moment using small data samples. For this purpose, risk factors are locally described in the form of parabolic or linear trends. An example of monitoring the risk of reducing the level of socio-economic development of Russia in 2000–2023 is considered. The monitoring results showed that the proposed multivariate risk model was generally sensitive to all the most significant economic shocks and adequately responded to them.

1. Introduction

In a broad sense, “risk” is understood as a possible danger of any unfavorable outcome. It is inherent in all spheres of human and social activity. In economics, risk is usually interpreted as the probability of a person or an organization losing part of its resources, income shortfall, or additional costs because of implementation of certain production and financial policies [1].

Changes in the stability of the economic situation and the external and internal environment of any organization’s activity should be assessed in a timely manner to avoid these adverse outcomes. To do this, it is necessary to be able to quantitatively assess the risk of their occurrence and form recommendations to reduce it [2].

The solution to these issues is constantly becoming more complicated due to the increasing efficiency of functioning of the created systems, with a simultaneous tendency to their complication, and the growth of various challenges in the economy and society. Since the occurrence of an unfavorable outcome is explicitly or implicitly reduced to financial losses, the problem of risk analysis is most acute in the economy. It is necessary to build adequate models and effective methods to solve the problems of risk monitoring [3,4,5].

In the framework of monitoring, the system state is assessed, controlled, and managed depending on the impact of certain factors. Without preventive and protective measures, over time the system has a growing conditional probability of an unfavorable event: $0<\dots \le P(A/H_{k-1})\le P(A/H_k)\dots \to 1$, where A is an unfavorable event and $H_k$ is the state of the system at the k-th moment of time.

In economics, risk analysis is often associated with certain difficulties. These include the presence of several risk factors, their interconnectedness (correlation), non-stationarity of processes, and small data samples. Therefore, risk models in the economy should consider these features.

The aim of the research is to propose and test on real data a model of multidimensional risk that satisfies the main features of processes in the economy.

2. Materials and Methods

Let us consider a model of multidimensional risk [6,7]. We have a set of risk factors in the form of a random vector with correlated components $\mathit{X}=(X_1,\dots ,X_m)$, which here we understand as the main socio-economic indicators affecting the safety of the system.

In this case, under the violation of security, we understand the decrease in the level of socio-economic development of the system. We believe that these risk factors can be correlated (dependent) and their simultaneous manifestation is possible. In fact, we represent the state of the system under study as a vector in the Euclidean space ${\mathbb{R}}^m$. Instead of identifying specific hazardous situations, we will define geometric regions of unfavorable outcomes. These areas may look arbitrary depending on the specific problem and are determined based on available a priori information. In economics, usually undesirable events are deviations of risk factor values from some acceptable areas characterizing the safe state of the system under study. Therefore, we will use the concept of undesirable events as large and unlikely deviations of a random variable from the best values in terms of safety.

To apply the multifactor risk model, it is necessary to define the boundaries of acceptable values (threshold levels) of risk factors, according to which a considerable safe state $\mathit{\theta }=({\theta }_1,\dots ,{\theta }_m)$ of the system is calculated. Threshold levels can interpreted as values of risk factors; the output of at least one of the risk factors beyond them leads to the emergence of unfavorable outcomes in the form of violation of economic security. Since risk is in a certain sense a conditional category, its dynamics—improvement or deterioration of the economic situation and the impact of risk factors on it—are important in the first place.

Determination of threshold levels of indicators is difficult due to the large volume of data. In this regard, the values of thresholds are determined based on the analysis of the median and standard deviation for the period under study, which made it possible to consider the dispersion of values of indicators relative to the average and to determine the values, which will signal the manifestation of a crisis.

We define the area of safe values for each risk factor as $G_j=\{x_j:d_j^{-}\le x_j\le d_j^{+}\}$; $d_j^{-}$, $d_j^{+}$ are the left and right boundaries of acceptable values, limiting the area of favorable outcomes, determined on the basis of expert assessments, $d_j^{-}<d_j^{+}$. Accordingly, the areas of unfavorable outcomes $D_j$ are complements of sets $G_j$, i.e., $D_j=\overline{G_j}=\{x_j:x_j<d_j^{-}\vee x_j>d_j^{+}\}$.

If some boundaries are one-sided, then $d_j^{-}=-\infty$, when only the right boundary is given, and $d_j^{+}=+\infty$, when only the left boundary is given. Let us assign the numbers of all $m$ risk factors to three sets of indices: $J^{\left(0\right)}$ (the factors have both limits of permissible values); $J^{(-)}$ (the factors have only the left limit of permissible values); J⁽⁺⁾ (the factors have only the right limit of permissible values).

Then we define the conditional safest state of the system $\mathit{\theta }=({\theta }_1,\dots ,{\theta }_m)$. In the case of bilateral boundaries of acceptable values ${\theta }_j=(d_j^{-}+d_j^{+})/2$; for unilateral boundaries, the values are set based on available historical data of the risk factors and further (a priori) information provided by e.g., experts [7].

The region of unfavorable outcomes D for all risk factors consists of two sets $D=D^{\left(1\right)}\cup D^{\left(2\right)}$. The set $D^{\left(1\right)}$ forms the situations when at least one of the risk factors $X_j$ was in the region of unfavorable outcomes $D_j$, i.e., $D^{\left(1\right)}=\overline{{\prod }_{j=1}^m{G}_j}$. However, situations of large deviations also can occur when all factors X_j are in the admissible area ${\prod }_{j=1}^m{G}_j$. To account for such situations, we introduce a set:

$$D^{\left(2\right)}=\left\{\mathit{x}=({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_{\mathit{m}}):{\sum }_{\mathit{j}=1}^{\mathit{m}}{\displaystyle \frac{{({\mathit{x}}_{\mathit{j}}-{\mathit{\theta }}_{\mathit{j}})}^2}{{\mathit{b}}_{\mathit{j}}^2}}>1,\forall \mathit{j}\in {\mathit{J}}^{(-)}({\mathit{J}}^{(+)})>(<){\mathit{\theta }}_{\mathit{j}}\right\}$$

where $\forall j\in J^{\left(0\right)}\cup J^{(-)}$ $b_j={\theta }_j-d_j^{-}$ and $\forall j\in J^{(+)}$ $b_j=d_j^{+}-{\theta }_j$.

As risk, we will understand the probability of an unfavorable outcome D:

$$P\left(D\right)=P(\mathit{x}\in D).$$

(1)

From (1), we see that the situations $\mathit{x}\in D$ and $\forall j$ $x_j\overline{\in }{D}_j$. We will interpret the value P(D) as the probability of an unfavorable outcome occurring at the next point in time (in our case, in the next year). The estimation of the probability of entering the risk area P(D) is based on the Monte Carlo statistical testing method [8,9].

First, we consider the random vector $\mathit{X}$ to be Gaussian: (1) the use of the normal distribution law relies on the central limit theorem; (2) the analyzed sample is small, which does not allow us to estimate the distribution law of the random vector.

Second, from the available data sample ${\mathit{X}}_{n\times m}$ we define the mean vector $\overline{\mathit{x}}={({\overline{x}}_1,...,{\overline{x}}_m)}^T$ and the covariance matrix ${\mathit{\Sigma }}_{\mathit{x}}={\left\{s_{jl}\right\}}_{m\times m}$. The matrix ${\mathit{X}}_{n\times m}$ contains the results of observations of $m$ risk factors $X_j$ for $n$ time periods. These completely describe the distribution law $p_{\mathit{X}}\left(\mathit{x}\right)$ of the data according to following formula:

$$p_{\mathit{X}}(\mathit{x})={\displaystyle \frac{1}{\sqrt{\left(2\pi {)}^m\cdot \mathit{det}\left({\mathit{\Sigma }}_{\mathit{X}}\right)\right)}}}{e}^{-{\displaystyle \frac{1}{2}}(\mathit{x}-\overline{\mathit{x}}{)}^T\cdot {\Sigma }_{\mathit{X}}^{-1}\cdot (\mathit{x}-\overline{\mathit{x}})},\mathit{x}\in {\mathbb{R}}^m$$

Third, we repeatedly generate new observations ${\mathit{z}}_i=(z_{i1},...,z_{im})$ with the spreading law $p_{\mathit{X}}\left(\mathit{x}\right)$. This procedure is described in [7]. As a result, the probability estimate $P\left(D\right)$ will be equal to the frequency $P\left(D\right)=M/N$, where $N$ is the total number of generated observations ${\mathit{z}}_{\mathit{i}}(i=1,\dots ,n)$, $M$ is the number of outcomes when the generated observation ${\mathit{z}}_i\in D$.

Let us estimate the contribution to the total risk of its components X_j. Absolute and relative change in the probability of unfavorable outcome of the multidimensional system due to the addition of factor X_j we determine by the following formulas:

$$\mathsf{\Delta }P\left(D_j^{-}\right)=P\left(D_j^{-}\right)-P\left(D\right),\delta P\left(D_j^{-}\right)=\mathsf{\Delta }P\left(D_j^{-}\right)/P\left(D_j^{-}\right),$$

where $P\left(D_j^{-}\right)=P(x\backslash x_j\in D_j^{-})$, $D_j^{-}$ is the area of unfavorable outcomes D after exclusion of risk factor X_j.

The contribution of the correlation between the indicators X_j to the overall risk is calculated as

$$\mathsf{\Delta }P\left({\mathit{\Sigma }}_{\mathbf{x}}\right)=P(\tilde{\mathit{x}}\in D)-P(\mathit{x}\in D),$$

where all components ${\tilde{X}}_j$ of the random vector $\tilde{\mathit{x}}$ are mutually independent and have the same distributions as X_j.

For risk analysis, it may also be useful to estimate the univariate risk for each factor as $P\left(D_j\right)=P(x_j\in D_j)$.

The disadvantage of the model described above is that it is static and does not consider the non-stationarity of risk factors.

3. Results

The behavior of risk factors can be represented as a set of mutually dependent time series. Risk analysis in the economy using this model is often complicated by non-stationarity (presence of trends) of indicators.

The trend in the risk factor X_j leads to the fact that it consists of the sum of two independent components $x_i\left(t\right)=\phi \left(t\right)+\epsilon \left(t\right)$, where $\phi \left(t\right)$ is the trend, which is understood as some continuous quasi-deterministic component from time, and $\epsilon \left(t\right)$ is a random component [10,11]. Then the variance of the risk factor will be equal to the sum of two variances ${\sigma }_{x,j}^2={\sigma }_{\phi }^2+{\sigma }_{\epsilon }^2$. As a result, when estimating risk by formula (1) in the procedure of generating observations ${\mathit{z}}_i=(z_{i1},...,z_{im})$, the dispersions of risk factors $X_j$ with trends will be overestimated by the values ${\sigma }_{x,j}^2$. This leads to significant distortions of covariance matrices and, as a result, to overestimation of risk.

We will consider the trends of risk factors as follows. Since for operational risk monitoring the number of observations Δ in each analyzed interval should be small (about 5–15 observations), it is sufficient to consider two models—linear and parabolic trends. At each analyzed interval $(k-∆+1\le i\le k)$ we determine the model parameters and risk as follows.

For each j-th risk factor X_j, we estimate the conditional mathematical expectation as a parabolic regression equation from time $i$

$${\overline{x}}_{i;j}=a_{k;j}+b_{k;j}i+c_{k;j}{i}^2$$

(2)

and check the statistical significance of the coefficient $c_{k;j}$. If it is statistically significant, then further on this interval we consider the risk factor $X_j$ as a trend $x_{i;j}=a_{k;j}+b_{k;j}i+c_{k;j}{i}^2+e_{k;j}$. Otherwise, instead of (2) we consider the paired linear regression

$${\overline{x}}_{i;j}=a_{k;j}+b_{k;j}i$$

(3)

and check the statistical significance of the coefficient b_k;j. If it is significant, then further on this interval, we consider the risk factor $X_j$ as a trend $x_{i;j}=a_{k;j}+b_{k;j}i+e_{k;j}$. Otherwise, we return to the static variant

$${\overline{x}}_{i;j}=a_{k;j}i,$$

(4)

describing the risk factor as $x_{i;j}=a_{k;j}+e_{k;j}$.

Next, based on models (2)–(4), we calculate the estimates of the risk ${\overline{x}}_{i;j}$, errors $e_{i;j}=x_{i;j}-{\overline{x}}_{i;j}$ and error covariance matrix ${\mathit{\Sigma }}_{\mathit{e}}$. After that, we estimate at any i-th moment of the interval $(k-∆+1;k)$ the probability of an unfavorable outcome P(D).

Note that the most accurate regression estimates correspond to the middle of the range of values of the explanatory variable, so it is desirable to take the time moment i for risk assessment in the middle of the interval. If the problem of risk forecasting is solved, the right boundary of the interval can be increased by considering the error covariance matrix unchanged for the forecast periods, and instead of the actual values of risk factors we should use their forecasts according to the regression Equations (2)–(4).

Consideration on model data has shown that the dynamic variant of the model of multidimensional risk considers the presence of trends in risk factors and allows us to estimate risk correctly. For example, Figure 1 shows the results of risk assessment for two correlated risk factors with trends. The dynamic case considered the presence of trends in risk factors, while the static case did not.

The dynamic model of multivariate risk showed almost identical estimation results with the model in the absence of trends (Figure 1a). Some decrease in risk with the increase in the volume Δ of the sample analyzed shows the convergence of sample estimates to the theoretical result at $∆\to \infty$. As can be seen from Figure 1b, in the presence of trends in risk factors, the static variant of the risk analysis model leads to a shift in $P^{\left(3\right)}\left(D\right)$ estimates towards overestimation of the probability of an unfavorable outcome: at $∆\to \infty$ $P^{\left(3\right)}\left(D\right)\to 1$.

4. Discussion

Let us further monitor the risk of deterioration of the socio-economic condition of the Russian Federation according to annual data. The risk factors are given in

Table 1. Risk factors of socio-economic development.

Designation	Indicator and Unit of Measure
X1	Mortality rate of the population of working age (men aged 16–59, women aged 16–54), persons per 10,000 people of the corresponding
X2	Average per capita cash income of the population per month, thousand rubles 1
X3	Volume of crop and livestock production, million RUB per 1000 people 1
X4	Volume of industrial output, million RUB per 100 people 1
X5	Unemployment rate, %
X6	Consolidated budget revenues of Russian regions, million RUB per 100 people 1

¹ Values recalculated considering the consumer price index.

. The information source is the database of the Federal State Statistics Service for the period from 2000 to 2023. Unilateral limits of acceptable values (threshold levels) of indicators are given in

Table 2. Threshold levels of risk factors $X_j$ and safe values ${\theta }_j$.

Indicator	Lower Level dj−	Upper Level dj+	θj
X1	–	85	62.99
X2	25	–	33.81
X3	25	–	39.15
X4	25	–	40.96
X5	–	9.5	6.45
X6	6	–	8.61

. The lower thresholds are set for the factors whose increasing values indicate a positive trend, otherwise the upper thresholds are set. The rationale for the risk factors used and their threshold levels is given in [7].

The risks were calculated on the basis of multivariate static and dynamic risk models described above. The statistical significance of $c_{k;j}$ in (5) and $b_{k;j}$ in (6) was checked using Student’s t-criterion. Due to the small size of the analyzed samples, the significance level λ = 0.1 was used.

The results of monitoring the decline in the level of socio-economic development of Russia according to annual data for static (Δ = 10) and dynamic (Δ = 8) variants of risk assessment are shown in Figure 2 and Figure 3. The dynamic variant allowed the use of a smaller data sample size.

According to the obtained results, the most significant economic shocks of the Russian Federation, noted by many authors, are clearly traceable. Thus, the crisis of 1998, caused by default on foreign debts, led to instability of the economic situation up to 2006. The global financial crisis of 2008 led to a deep recession of the Russian economy in 2009 and to a drop in industrial production ($X_4$) and a sharp increase in unemployment ($X_5$). Unlike dynamic estimation, the static one, due to the specifics of its implementation, is more tied to Δ intervals, so the 2008 crisis overlaps with the next surge of economic instability and represents a single period of fluctuation in the probability of unfavorable outcome from 2006 to 2019.

Figure 3 shows two spikes—in 2013 and 2017. The difficult geopolitical situation in 2013, the fall in oil prices, and the lingering effects of the 2008 crisis led to economic stagnation. Despite the absence of a strong economic downturn this year, the signs of the crisis manifested themselves through falling household incomes ($X_2$) and rising unemployment ($X_5$). The 2014 sanctions hardly led to an increase in risks, which can be explained by the state’s use of reserves to buy socio-economic problems. In 2017, the country was still in a state of prolonged economic crisis caused by the long-term effects of sanctions, falling oil prices and the ruble exchange rate, as well as rising inflation. The COVID-19 pandemic led to a sharp decline in economic activity in 2020, affecting industrial output ($X_4$) and consolidated budget revenues ($X_6$). The impact of increased sanctions pressure in 2022 on the selected risk factors was not revealed up to 2023 (end of dataset).

5. Conclusions

The model of multidimensional risk oriented on the peculiarities of processes in the economy is proposed. They usually include the presence of several correlated risk factors that can occur simultaneously, non-stationarity of processes, and small data samples. The proposed model is based on the representation of the investigated economic system or phenomenon in the form of multidimensional non-stationary processes, which at each moment of time are considered as Gaussian random vectors. In the article, the probability of an unfavorable outcome—a decrease in the level of socio-economic development—is understood as the risk value. This model is supplemented with a dynamic variant, in which risk factors are locally described in the form of parabolic or linear trends. This allowed us to expand the scope of application of this approach.

The monitoring of the risk of decline in the level of socio-economic development of Russia on an annual basis from 2000 to 2023 was carried out. The results of monitoring showed that the proposed multivariate risk model was generally sensitive to all the most significant economic shocks and adequately responded to them. The model was tested on synthetic data simulating the presence of trends in risk factors.

It can be noted that the dynamic variant of risk monitoring is more sensitive to short-term negative changes in indicators. Less dependence on the interval Δ allows the association of the risk with the actual values of a particular period, facilitating the interpretation and perception of the results of the analysis. The static variant, in its turn, is less sensitive, but allows an assessment of the situation under conditions of a large number of economic shocks in a short period.

In the future, we plan to continue the research in the direction of studying the influence of individual risk factors and their correlation on the reduction in the level of socio-economic development.