In this paper, we study on the Malaysian properties sector of public listed companies from years 2008 to 2018. The lists of Malaysian public listed companies (Shariah compliances) are yearly updated and can be downloaded from the Security Commission Malaysia (www.sc.com.my). The updated lists consist of the name of companies with their respective stock codes and are divided into several sectors such as consumer products, industrial and properties to name of few. From this information, we can then download the annual reports that can be obtained from the Bursa Malaysia website (www.bursamalaysia.com) and the stock prices are accessible from The Wall Street journal website (www.wsj.com).

Table 1: The Malaysian public listed companies stock codes – Properties sector

Stock Code (Number of yearly dividends distributed from 2008 to 2018) |
||||

1007(6) | 8206(3) | 7323(11) | 8893(5) | 4375(0) |

5959(3) | 3557(0) | 5038(7) | 6114(10) | 3743(0) |

7007(0) | 8613(1) | 8494(11) | 3913(3) | 1538(11) |

4057(2) | 6815(2) | 5789(7) | 9539(0) | 4022(1) |

2305(11) | 5020(11) | 3573(0) | 5073(8) | 2259(0) |

6173(10) | 7010(0) | 7617(10) | 5827(11) | 2429(0) |

7187(0) | 9962(11) | 8583(11) | 1724(11) | 7889(1) |

5738(0) | 7077(0) | 8141(1) | 6912(3) | 7079(0) |

5049(6) | 5062(11) | 1651(11) | 8664(11) | 5401(11) |

6718(11) | 4251(11) | 6181(9) | 4596(5) | 5148(5) |

3484(0) | 5084(8) | 7189(5) | 5207(10) | 7066(0) |

7198(0) | 9687(0) | 5040(2) | 4286(0) | |

3417(10) | 1589(0) | 1694(0) | 6017(11) |

Notes: For more information, the companies’ names can be referred from the lists of Shariah compliances (www.sc.com.my). The bracket indicates the number of yearly dividends that the companies have declared over 11 years.

In determining the companies’ internal rate returns (IRR), it is very important for us extract the companies’ information such as dividend rate declared (in RM) as well as the stock issuances that are updated in yearly basis. Table 1 indicates the number of dividends declared for each company in yearly basis. However, it is too difficult to present the respective yearly dividend rate per stock unit for each company as it is too tedious to assemble it in a spreadsheet. We will be considering to let them out upon requests. However, we present the events of stock issuance over 11 years study informatively that are presented in Table 2.

Table 2: The Malaysian public listed companies share issuances – Properties sector

Stock Code |
Year |
Stock Issuance |

7187 | 2007 | Bonus Issue (1 bonus share by 8 ordinary shares) |

3484 | 2009 | Share consolidation (500 ordinary shares to 160 shares) |

3417 | 2015 | Bonus Issue (1 bonus share by 10 ordinary shares) |

8206 | 2015 | Share Split (1 ordinary share to 2 shares) |

5020 | 2012 | Share Split (1 ordinary share to 2 shares) |

5062 | 2011 | Bonus Issue (1 bonus share by 5 ordinary shares) |

2012 | Bonus Issue (1 bonus share by 3 ordinary shares) | |

2013 | Bonus Issue (1 bonus share by 4 ordinary shares) | |

2017 | Bonus Issue (1 bonus share by 3 ordinary shares) | |

4251 | 2014 | Share Split (1 ordinary share to 2 shares)
Bonus Issue (1 bonus share by 5 ordinary shares) |

2015 | Bonus Issue (1 bonus share by 5 ordinary shares) | |

5084 | 2015 | Share Split (1 ordinary share to 2 shares)
Bonus Issue (2 bonus shares by 5 ordinary shares) |

7323 | 2014 | Share Split (1 ordinary share to 2 shares) |

5038 | 2014 | Bonus Issue (1 bonus share by 1 ordinary share) |

8494 | 2018 | Treasury Share (4 Treasury shares by 100 ordinary shares) paid as dividend at RM1 per unit |

5789 | 2018 | Bonus Issue (1 bonus of 1 existing share) |

8583 | 2007 | Share Split (1 ordinary share to 2 shares)
Bonus Issue (1 bonus of 5 existing share) |

2009 | Bonus Issue (1 bonus of 5 existing share) | |

2013 | Bonus Issue (1 bonus of 5 existing share) | |

2015 | Bonus Issue (1 bonus of 4 existing share) | |

6181 | 2012 | Bonus Issue (1 bonus of 5 existing share) |

1694 | 2018 | Bonus Issue (4 bonus of 5 existing share) |

6114 | 2011 | Bonus Issue (1 bonus of 10 existing share) |

2012 | Bonus Issue (1 bonus of 10 existing share) | |

5827 | 2016 | Bonus Issue (3 bonus of 5 existing share) |

1724 | 2011 | Share Split (1 ordinary share to 2 shares)
Bonus Issue (2 bonus of 5 existing share) |

8664 | 2011 | Bonus Issue (1 bonus of 2 existing share) |

2014-2018 | Dividend reinvested to share unit | |

5207 | 2015 | Bonus Issue (1 bonus of 2 existing share) |

1538 | 2013 | Treasury Share (1 Treasury share by 20 ordinary shares) paid as dividend at RM1 per unit |

2259 | 2010 | Share Split (1 ordinary share to 3 shares) |

7889 | 2013 | Bonus Issue (2 bonus of 3 existing share) |

2015 | Bonus Issue (1 bonus of 2 existing share) | |

5401 | 2015 | Treasury Share (1.3 Treasury share by 100 ordinary shares) paid as dividend at RM1 per unit |

2017 | Treasury Share (1.2 Treasury share by 100 ordinary shares) paid as dividend at RM1 per unit |

Table 2 reflects the accumulation of the share unit. For instance, if there is no share issuance announced for a particular year, the share unit (i.e. ) equals one. The share split from one (1) ordinary share to two (2) means the current share units earned are multiplied by two units (i.e. ). The stock price here is reduced to half to enhance liquidity of the share capital traded in the market. Hence, the share capital remains unchanged. On the other hand, bonus issue of one (1) by two (2) existing ordinary shares indicates the company awards half of the share units to accumulate 1.5 share units at the end of the year. If these two events are practiced simultaneously, then the shareholder will be awarded by times as previous (i.e. ). Some companies practice treasury shares which buys back the shareholders’ shares at random for a certain rate of share price. Here, as our main objective is to accumulate our wealth by growing up our share units and capital, we ignore this practice by denying the offer to sell share back to the company. In practice, most dividends are being distributed in cash that are credited into the shareholders’ bank accounts. Some of the companies practice a mandatory treasury share by distributing the shareholders dividend instead, that can be called as treasury share dividend. Here, the shareholder gets dividend in cash, but the share units reduce to a certain portion that is reported in the company’s annual reports. For further example, please refer to the annual reports of stock code 8494 (year 2018), 1538 (2013) and 5401 (2015 and 2017).

Sabri and Sarsour (2019) demonstrated the computation of mathematically which consist of bonus issue and share split (or share consolidation) which can be referred in equation (4) of their article. Table 3 indicates the computation of share accumulation by factors of share split (A), bonus issue (B) and treasury share (C). Anyone of stocks for a particular year that does not experience these events, then .

Table 3: The Malaysian public listed companies share accumulation – Properties sector

Stock Code | Year | Split (A) | Bonus (B) | Treasury (C) | |

7187 | 2007 | 1 | 0.125 | 0 | 1.125 |

3484 | 2009 | 0.32 | 0 | 0 | 0.32 |

3417 | 2015 | 1 | 0.1 | 0 | 1.1 |

8206 | 2015 | 2 | 0 | 0 | 2 |

5020 | 2012 | 2 | 0 | 0 | 2 |

5062 | 2011 | 1 | 0.2 | 0 | 1.2 |

2012 | 1 | 0.333 | 0 | 1.333 | |

2013 | 1 | 0.25 | 0 | 1.25 | |

2017 | 1 | 0.333 | 0 | 1.333 | |

4251 | 2014 | 2 | 0.4 | 0 | 2.4 |

2015 | 1 | 0.4 | 0 | 1.4 | |

5084 | 2015 | 2 | 0.8 | 0 | 2.8 |

7323 | 2014 | 2 | 0 | 0 | 2 |

5038 | 2014 | 1 | 1 | 0 | 2 |

8494 | 2018 | 1 | 0 | 0.04 | 0.96 |

5789 | 2018 | 1 | 1 | 0 | 2 |

8583 | 2007 | 2 | 0.4 | 0 | 2.4 |

2009 | 1 | 0.2 | 0 | 1.2 | |

2013 | 1 | 0.2 | 0 | 1.2 | |

2015 | 1 | 0.25 | 0 | 1.25 | |

6181 | 2012 | 1 | 0.2 | 0 | 1.2 |

1694 | 2018 | 1 | 0.8 | 0 | 1.8 |

6114 | 2011 | 1 | 0.1 | 0 | 1.1 |

2012 | 1 | 0.1 | 0 | 1.1 | |

5827 | 2016 | 1 | 0.6 | 0 | 1.6 |

1724 | 2011 | 2 | 0.8 | 0 | 2.8 |

8664 | 2011 | 1 | 0.5 | 0 | 1.5 |

2014-2018 | Not Stated exactly | ||||

5207 | 2015 | 1 | 0.5 | 0 | 1.5 |

1538 | 2013 | 1 | 0 | 0.05 | 0.95 |

2259 | 2010 | 3 | 0 | 0 | 3 |

7889 | 2013 | 1 | 0.667 | 0 | 1.667 |

2015 | 1 | 0.5 | 0 | 1.5 | |

5401 | 2015 | 1 | 0 | 0.013 | 0.987 |

2017 | 1 | 0 | 0.012 | 0.988 |

If we wish to hold the stock for the company chosen in long term period, the stock valuation can also be seen by computing the (modified) internal rate of return ((M)IRR). The investment strategy for holding the stock is by allocating a level amount of contribution at the beginning of the year for years. At the same time, if the company declares dividend yearly, the cash dividends are reinvested and together deposited with the level contribution to enlarge the share units. At the end of years, we let all the our share units to earn the share capital which indicates the profit of our investment for years. If our share capital is less than our total contribution, we may expect our MIRR to be in negative form. The detail procedure of the investment return was documented by Sabri and Sarsour (2019), and is presented as follows,

(1) |

For year ,

net present value of stock investment which is computed at time 0.

accumulated share unit after share issuance at the end of year . Here, where is the function of share issuance and is the share units at the beginning of year

date of share purchased and sold.

date of dividend and share issued based on the stock reported on year.

stock price at date .

Cash Balance of at year .

Cash dividend at year .

modified internal rate of return.

yearly fixed contribution, .

terminal value invested fund to be let at the end of year . Here,

For further example, we compute the mirr of company Tropicana Corporation Berhad (Trop) that was formerly known as Dijaya Corporation Berhad (Dijacor) with its stock code 5401. From years 2011 to 2018, the company was committed in distributing the share dividends. Furthermore, it also implements treasury share dividend in the year 2015 and 2017 that reflects to the decrement of 0.013 and 0.012 share portions. Table 4 presents the share prices, dividend rates, and share issuance functions for Trop company.

Table 4: Dividend, share price and share issuance indicator extracted from the Trop’s annual reports and daily share prices obtained from www.wsj.com

Date | Share Price | ||||||||

1 | 3/1/11 | 31/12/11 | 1.01448 | 1.34614 | 0.0375 | 1 | 0 | 0 | 1 |

2 | 3/1/12 | 31/12/12 | 1.34614 | 1.00473 | 0.0225 | 1 | 0 | 0 | 1 |

3 | 3/1/13 | 31/12/13 | 0.99497 | 1.21933 | 0.08175 | 1 | 0 | 0 | 1 |

4 | 3/1/14 | 31/12/14 | 1.19982 | 1.03399 | 0.0400 | 1 | 0 | 0 | 1 |

5 | 3/1/15 | 31/12/15 | 1.04374 | 0.98814 | 0.0630 | 1 | 0 | 0.013 | 0.987 |

6 | 3/1/16 | 31/12/16 | 0.98814 | 0.98814 | 0.0450 | 1 | 0 | 0 | 1 |

7 | 3/1/17 | 31/12/17 | 0.98814 | 0.91500 | 0.0320 | 1 | 0 | 0.012 | 0.988 |

8 | 3/1/18 | 31/12/18 | 0.92000 | 0.91500 | 0.0160 | 1 | 0 | 0 | 1 |

By this investment strategy, we allocate RM10,000 every beginning of the year for five-year period from 3/1/2011 to 3/1/2018, and withdraw our investment fund on 31/12/2018. Table 5 illustrates how investment information in Table 4 can be utilized in constructing the investment cash inflows

and outflows.

Table 5: The cash flow of investing to Trop company from the years 2011 to 2018.

1 | 9,941.90 | 9,800 | 58.10 | 9,800 | 9,800 | 364.48 | 10,000 |

2 | 10,365.28 | 7,700 | 57.30 | 17,500 | 17,500 | 392.80 | 10,000 |

3 | 10,447.19 | 10,500 | 2.91 | 28,000 | 28,000 | 2,281.94 | 10,000 |

4 | 12,238.16 | 10,200 | 46.69 | 38,200 | 38,200 | 1,524.65 | 10,000 |

5 | 11,481.14 | 11,000 | 90.20 | 49,200 | 48,560 | 3,093.90 | 10,000 |

6 | 13,142.26 | 13,300 | 41.84 | 61,860 | 61,860 | 2,780.44 | 10,000 |

7 | 12,747.01 | 12,900 | 75.28 | 74,760 | 73,863 | 2,388.94 | 10,000 |

8 | 12,420.00 | 13,500 | 44.22 | 87,363 | 87,363 | 1,396.04 | 10,000 |

From the information in table 5, we may then compute the respective terminal investment, and MIRR, by setting our investment period, . For example, if , then . From equation (1), by equating zero valued of NPV, we may search the MIRR, , that is,

This results in . For further information, table 6 presents terminal investment and MIRR respectively based on investment periods from one to eight years, by contributing RM10000 every year starting from year 2011.

Table 6: Terminal investment and MIRR of Trop company based on yearly contribution, RM10000 starting from year 2011

1 | 13,614.75 | 0.3650 | 5 | 51,168.58 | 0.0077 |

2 | 18,032.87 | -0.0673 | 6 | 63,949.02 | 0.0183 |

3 | 36,426.10 | 0.1006 | 7 | 70,049.11 | 0.0002 |

4 | 41,069.76 | 0.0106 | 8 | 81,377.65 | 0.0038 |

For stocks, and the years of study starts at , the random variable of MIRR for -th stock at investment year for -years investment period, can be denoted as where .

It is very important to choose the best potential stocks to hold in a long term. Furthermore, holding a stock for -years period of investment may vary in terms of MIRR. Some might choose the best point of times to start investing, but it is very difficult to identify it as the MIRR measure can only be observed by yearly basis. Therefore, by assuming the MIRR for all companies and starting time to invest are common, we may define the MIRR, denoted as , as random variable having the mean and variance and respectively.

In investment, we may obtain a positive value of profit (or even be greater than our capital investment) as well as poorly earn nothing. This indicates our capital of investment, , could be infinite or even zero valued. For a period time of , our terminal investment, is in between 0 to infinite. Hence,

Since , we define a non-negative transformed rate of return, such that,

In this paper, we illustrate the transformed MIRR, by assuming Gamma distribution. Let , then

(2) |

Equation (2) gives us and . If we wish to write the MIRR distribution, we may have . This will also indicates that, and . For a period of investment, , the methods of moments estimation of s and s can be defined as and respectively. Thus, it gives us and .

Log-likelihood function…

(3) |

About 63 companies from properties sector are observed in our study. Here, we only study the behaviour of MIRR from one to eight investment year periods (i.e., ). As the period of our study starts from year 2008 and lasts to 2018, we may set and and hence, . Furthermore, all companies’ MIRR are counted in our study. For example, for one year investment period (i.e. ) we manage to obtain a maximum of multiplied with 63 companies to obtain the sample size of 693, and for we manage to get 252 sample size of MIRR. Table 7 summarizes the parameters estimates of transformed by using methods of moments towards Gamma distributions as in equation (2) over eight period of investment of properties sector in Malaysia.

Table 7: Parameters estimates of transformed MIRR by assumption of Gamma distributions

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

693 | 630 | 567 | 504 | 441 | 378 | 315 | 252 | |

1.159 | 1.156 | 1.145 | 1.136 | 1.134 | 1.137 | 1.126 | 1.114 | |

1.871 | 1.488 | 1.399 | 1.351 | 1.333 | 1.330 | 1.297 | 1.265 | |

2.545 | 8.811 | 14.902 | 21.329 | 27.335 | 34.723 | 43.534 | 51.700 | |

0.455 | 0.131 | 0.077 | 0.053 | 0.041 | 0.033 | 0.026 | 0.022 | |

1.158 | 1.154 | 1.147 | 1.130 | 1.121 | 1.146 | 1.132 | 1.137 | |

0.158 | 0.154 | 0.147 | 0.130 | 0.121 | 0.146 | 0.132 | 0.137 | |

0.527 | 0.151 | 0.088 | 0.060 | 0.046 | 0.038 | 0.029 | 0.025 |

From table 7, both means and variances of transformed MIRR as displayed in figure 1, decline over the investment period. This means purchasing the companies’ shares of this sector is very attractive for a short term period, but may have high risk to hold them. Meanwhile, we may be trust to hold longer as seen the declining of the variance of the MIRRs. Thus, we are interested in observing how much they change over periods of investment for this sector. Furthermore, the log likelihood for this model is -380.993.

Figure 1: mean and variance of by assumption of Gamma distributions

As of this motivation, we let as Gamma distribution with mean and variance . Then, for , we impose a new growth rate, by letting This results in the development of three-parameters Gamma distribution as follows,

(4) |

having the mean and variance . For instance, if , then the transformed MIRR of follows Gamma distribution with the pdf of with the mean and variance . If we consider the period of investment to be studied until at years, for the sample size of and the maximum of the years that the data has been collected is , the likelihood function of transformed MIRR is

and, the log-likelihood function is then be simplified as,

(5) |

It is very hard to search the Gamma parameters simultaneously from equation (5) by deriving , and and equating to zero valued. As mentioned by Abbasi et al (2006 and 2011), it is also quite exhaustive to derive the gradient of the Gamma model above to attain the complicated objective function. Based on Abbasi et al (2006) in estimating three-parameters Weibull distribution, we also incorporate simulated annealing (SA) which was developed by Bertsimas and Tsitsiklis (1993) in finding our Gamma based on Metropolis et al (1958). The algorithm of SA procedure in estimating the three-parameters Gamma distribution is as follows:-

An initial value of Gamma parameters can be searched by generating up to sets of . For , the best parameters, ** **initially is searched by,

where, is the log-likelihood function as in equation (5). In table 7, is uniformly integer generated from ; whereas and are uniformly continuous generated from and respectively. The larger the value of , the more accurate the initial value determined. In this paper, we use which is about time consuming in determining .

Set an initial value of control parameters, temperature, , cooling, and .

Set and thus, compute

While the stopping criterion is not reached, do:

Set

While do:

Generate solution in the neighbourhood of

Calculate

If

Else

Generate a random number,

If :

End

End

If

End

End

End

Stopping criterion can be set by taking the norm of to be less than 0.001, with condition, **.**

Sometimes, the new solution does not meet the term that is its objective function less than the initial objective function. As long as it is closer, the final solution is adopted.

, and

and

and

Log-likelihood for saturated model is -380.99 whereas the log-likelihood for the dynamic model is -987.91. The deviance is 1214, which significantly reject the dynamic model at chi-squared distribution of 13 degrees of freedom (i.e. ). Thus, the model should be improved. It might be due to the difference in mean and variance of the Gamma distribution for 1-year and 2-years investment periods. However, it is very important to incorporate the dynamic Gamma distribution as it indicates the increasing of period of investment may lead to the reduction of rate of return by 0.0081 as can be seen not only the reducing of the return, but the risk of investment, that can be observed by the variance can also be improved. Thus, we may argue that the longer the investment period, the lower the risk of investment for this sector.