Wajeeh Mustafa Sarsour¹⁺ --- Shamsul Rijal Muhammad Sabri²

^1,2School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia.

Keywords:Simulation, Markov chain, Investment, Sufficient sample size, Comparison for matrices.

JEL Classification: G11; G17; E22.

ARTICLE HISTORY: Received:18 May 2020, Revised:26 June 2020, Accepted:28 July 2020, Published:12 August 2020

Contribution/Originality: This study is one of very few studies that has investigated how to determine an adequate sample size using a Markov chain simulation model. It presents a selection of companies, in order to study the performance of ROA in Malaysia’s industrial sector, and to assist investors in their decision making.

1. INTRODUCTION

Over the past decades, there has been a considerable amount of debate regarding the extent to which the past can be used to forecast the future. Markov chain models, developed by Russian scientist Andrey Markov in 1906, have been used extensively to forecast the future. A Markov chain, which is a type of Markov process, is a stochastic model that describes a sequence of potential events in which the probability of every event depends entirely on the state accomplished in the previous event (Asmussen, 2008).

In stochastic analysis, the appeal of the Markov chain model is not new. Many stochastic processes used to model biological, physical, financial, engineering systems are Markovian, which means that it is easy to simulate, compared to other models. For example, Anderson and Goodman (1957) estimated the transition probability matrix in a Markov chain model using maximum likelihoods, asymptotic distributions, and test hypotheses on model parameters. Moreover, Fama (1965) discussed the theory of random walks and provided strong evidence to support the stochastic nature of stock prices. Furthermore, Goldfeld (1973) applied a Markov model to switching regressions in order to study growth dynamics that rely on an extended period. Some studies have dealt with small sample sizes, such as Abidin and Jaffar (2014), who used a Geometric Brownian motion to forecast share prices in Bursa Malaysia.  

The majority of Markov models are based on a sufficient data set, either in terms of sample size or frequency of time, as demonstrated by Fama (1965); Zhang and Zhang (2009); Mettle, Quaye, and Laryea (2014); Sarsour and Sabri (2020). However, these models would be difficult to calibrate for data that are characterized by a short frequency of time, which results in an unreliable estimation of the transition probability matrix. Increasing the sample size in a system of transition could help to overcome this shortcoming.

The present study proposes a new Markov chain simulation method to determine the required sample size, in order to obtain a reliable estimate of the transition probability matrix in cases with only a short frequency of time. From an applicative point of view, the main feature of the proposed simulation method is that it allows the estimation of the transition probability matrix in the short-, medium-, and long-term. This provides analysts with all the required information about the system of transitions, in order to implement risk analyses and evaluations.

Investors are interested in gaining profit from their investments, but they face many challenges when making decisions due to price fluctuations and unstable financial situations. Financial analysts study the performance of prices and investments using net present value (NPV), internal rate of return (IRR), and return of asset (ROA). Recently, Sabri and Sarsour (2019) studied a new strategy for modeling stock investment valuations by developing the modified internal rate of return. In this strategy, they divided and shared issuance, through split shares and consolidation, as the financial analysts understood the great importance of reducing investment risks and making better informed decisions on future prices. Raheem and Ezepue (2016) predicted the movements of asset returns of a Nigerian Bank by dividing asset returns into three states—positive, moderate, and negative—using the Markov chain model. In the months of May and October, they revealed the fact that their maximum trading cycle was 18 days and their minimum trading cycle was 7 days in February. Additionally, many studies have examined investment behavior (Helms, Salm, & Wüstenhagen, 2020; Qolbi, Karisma, & Rosyadi, 2020).

Furthermore, a considerable number of studies, such as Vázquez-Quintero et al. (2016), Rimal et al. (2018), Ahmed, Kamruzzaman, Zhu, Rahman, and Choi (2013), have used simulations such as forecasting changes in land cover through a simulation technique based on the Markov chain. Kumar, Trehan, and Joorel (2018) used simulation studies to estimate the population mean using stratified random sampling and two auxiliary variables. It can also be used in studies where it is difficult to collect data, such as the relationship between an earthquake and human activities, as in Albano et al.’s study (2017), as well as medical studies that use the Markov chain Monte Carlo, such as Karami et al. (2019), Hamdy, El-Azab, Al-Saeed, Hassan, and Solouma (2017), and Ricci et al. (2019). It can also be used when multiple data patterns exist, and so a simulation comparison of imputation methods for quantitative data is required (Solaro, Barbiero, Manzi, & Ferrari, 2018). The finance industry can also benefit from such a simulation, as seen in Ye, Zhu, Wu, and Miao’s study (2016), which employed the Markov financial contagion detection and regime switch quantile with a regression model.

This study attempts to determine the number of companies that are required in order to examine the behaviour of ROA in the entire Malaysian industrial sector using a simulation Markov chain model. Previous similar studies have considered ROA for each company separately; however, as ROA values are yearly, their frequency is too low to study. In other words, if we assume that we want to study ROA performance in the industrial sector between 2007 and 2018, the number of ROA values would be twelve for each company, which is too few to study. Therefore, in this study, we established the number of companies required to obtain adequate ROA values, which, based on the time period of our study, is 50. Therefore, the number of ROA values is 600.

The proposed Markov chain simulation method was then applied to determine the number of companies that should be involved in order to perform an accurate forecasting analysis on ROAs in the Malaysian industrial sector. The results revealed that, in order to obtain reliable estimates of the parameters in the transition probability matrix, the number of companies should at least be 69, 37 or 24 companies with more than 3, 6, or 10 years of frequency of time, respectively. MATLAB software was used to implement the proposed simulation.  

2. METHODOLOGY

2.1. Markov Chain Model

The Markov chain model is widely used in many fields. It is a type of stochastic process that was introduced by Andrey Markov in the 1900s and developed by Kolmogorov in 1936. A fundamental part of the stochastic process is the Markov chains model, in which the occurrence of every event depends only on the past event. The state space is the set of values ​​that the Markov process takes, which may be a discrete or continuous value.

2.1.1. Transition Matrix and Transition Probability Matrix

2.2. Comparison for the Transition Probability Matrices

In this section, we will display some of the measures taken when comparing the transition probability matrices, in order to determine the nearest closed matrix. Some researchers have presented measures for the comparison of credit migration matrices based on eigenvalues, eigenvectors, singular values, or Manhattan and Euclidean metrics (Jafry and Schuermann, 2004; Trueck and Rachev, 2009;Israel, Rosenthal [22] Bangia, Diebold, Kronimus, Schagen, and Schuermann, 2002).

a. Eigenvector Distance Metric

The present study determines whether or not the matrices are equivalent, based on their eigenvectors and regardless of their eigenvalues, which computes a ratio of the matrix norm (Arvanitis, Gregory, & Laurent, 1999). The following equation will be used:

b. Metrics Based on Singular Values

We can also compare transition probability matrices using the average of all singular values of the mobility matrix (), as follows:

c. Metrics Based on Eigenvalues

We will illustrate a variety of measures that have been taken to make comparisons between matrices, which depend on eigenvalues (Geweke, Marshall, & Zarkin, 1986). For example:    

d. Manhattan and Euclidean Distance Metrics

These measures are popular approaches to comparing two matrices using the cell by cell distance technique. The Euclidean metric calculates the average root mean square difference, whereas the Manhattan metric calculates the average of the absolute difference between the corresponding elements of the matrices. Specifically,

3. SIMULATION MODEL

The model must contain a sufficient sample size with a high enough frequency of time to ensure the correct estimation of the model’s parameters. The present study, however, has a shorter frequency time. Increasing the sample size would be one solution to this problem. However, to what extent should the sample size be increased in order to obtain reliable estimates of the parameters of the transition matrix? In order to suitably increase the sample size, a novel simulation technique was proposed and implemented, which is summarized in the following steps:

Figure 1. A simulation of the Markov chain model.

4. RESULTS OF SIMULATION AND DISCUSSION

4.1. Eigenvectors

Figure 2 shows the simulation results of the short-term experiment reaching a sufficient sample size to estimate the transition matrices. Typically, the experiment has two states, a frequency time of three years, and is replicated 1000 times. This experiment was performed using different sample sizes. About 63% of the replications are achieved at a  less than 0.08 when the sample size is 15, which indicates that the overall matrices have a high level of differences, meaning that the sample size is insufficient. Using a sample size of 24, 30, 39 and 50 failed to achieve more than 95% of the trials, and therefore, we were unable to acquire a sufficient sample size. However, when the sample size was 69 and above, more than 95% of the trials had performance measures that were less than 0.08, thereby achieving a sufficient sample size. This suggests that studies that involve a three-year frequency time should have a sample size of at least 69. Hence, once the years are merged, the number of movements becomes 207.

Figure 2. Simulation for the medium-term (three years).

The experiments were performed for the medium-term frequency, using two states that have six years of frequency time and were replicated 1000 times. As seen in Figure 3, about 78%, 88% and 91% of the replications were achieved at a  that was less than 0.08 when the sample size was 15, 24 and 30, respectively. This indicates that the overall matrices have a high level of difference, meaning that these experiments failed to reach the required sample size. On the other hand, having a sample size of 37 or more resulted in more than 95% of the trials having performance measures that were less than 0.08. Therefore, one should select a sample size of at least 37 when the studies involved have data that spans six years. Once all the years were merged, the number of movements was 222.

Figure 3. Simulation for the medium-term (six years).

The experiments were also performed to examine long-term frequency, with two states, a frequency time of ten years, and 1000 replications. The simulation results are shown in Figure 4, and it is indicated that about 88% of replications have been achieved with a  that is less than 0.08 when the sample size is 15, which indicates that it failed to reach a sufficient sample size. However, when the sample size is 24 or more, 95% of the trial performances measured less than 0.08, meaning that they have achieving a sufficient sample size. This means that studies that involve three years of frequency time should have a sample size of at least 24. After all the years are merged, the number of movements becomes 240. 

Figure 4. Simulation for a long-term (ten years).

4.2. Result Based on Singular, Manhattan and Euclidean Distance Metrics

Figure 5. Simulation for the short-term (three years).

The simulation results in Figure 6 present an experiment that uses two states, has a frequency of time of six years, and is replicated 1000 times to evaluate the error. This differentiates between matrices using the value of a singular metric in order to reach a sufficient sample size, so that the transition matrices can be estimated. Typically, the experiment is performed using different sample sizes. However, 10 and 20 were considered to be insufficient sample sizes to determine the transition matrix because the error would still be more than On the other hand, using a sample size of 31 or more will result in the trials having errors that are less than. Hence, having 31 samples or more is considered to be a sufficient sample size.

Figure 6. Simulation for the medium-term (six years).

The experiments were performed for the medium-term using two states, a frequency of ten years, and being replicated 1000 times. As seen in Figure 7, the sample size is ten or less, which indicates that the overall matrices have a high level of differences. That means that these experiments have failed to reach the required sample size. On the other hand, having a sample size of 20 or more resulted in less than 10% error. Therefore, one should select a sample size of at least 20 when the study involves data that spans ten years.

Figure 7. Simulation for a long-term (ten years).

We can conclude that a sufficient sample size can be achieved rapidly as we increase the frequency of time and decrease the number of states. The results of the simulation of the adequate sample size can be determined using the above metrics. Typically, simulations with a higher frequency of time result in adequate sample sizes with lower values.

5. ILLUSTRATION OF A TWO-STATE MARKOV MODEL THAT FORECASTS CHANGES TO RETURN ON ASSETS (ROA)

This section provides an example to illustrate how the proposed simulation method achieves a sufficient sample size to obtain reliable estimates of the transition matrix based on the eigenvector’s metric. In this example, the frequency of time is short. Empirical calculations were based on studying the performance of investments through ROAs, from the state  to another state . These two states were studied in each company operating in the Malaysian industrial sector between 2009 and 2018, where data on ROA were collected on a yearly basis. Hence, the main problem in this example is the relatively small amount of data for each company, which requires us to perform the proposed simulation method: State one for a negative ROA and state two for a positive ROA or a ROA equal to zero in the long-term.

Where ROA is an indicator that measures the company’s profitability relative to its total assets, which demonstrates the efficiency of the administration in using its assets to achieve higher profits. The ROA, sometimes referred to as the return on investment, is calculated as a percentage by dividing the company’s annual profits by the total of its assets.

The transition matrices of successive pairs (T-1) for the ROA of 147 companies between 2007 and 2018 have been calculated using Equation 1 and are shown in Table 1.

5.1. Long-term Performance

Based on the simulation results derived from Section 4, we have to select at least  companies to study the long-term performance of  years. Therefore, a random sample of 24 companies was selected from the same sector, in order to study the behavior of ROA over ten years to check the accuracy of the proposed simulation.
The transition probability matrix after carrying out the stationarity test on nine pairs is as follows:

5.2. Middle-Term Performance

5.3. Short-Term Performance

Therefore, we can conclude that all matrices in all three cases were very close to each other, meaning that the result of the simulation was acceptable.

6. CONCLUSION

Sometimes a system of transition may have data that have been obtained over a short frequency of time. The present study aimed to gain reliable estimates of the transition probability matrix. Therefore, a Markov chain simulation method was developed to obtain the required sample size needed to achieve the aims of the study. Three variant models were used to determine the required sample sizes (short-term, medium-term, and long-term). Finally, the application of the proposed simulation method has been demonstrated using yearly ROA data from companies operating in the Malaysian industrial sector. The results in all three cases were close.  

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