Gamma Modelling on internal rate of return (properties sector)

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).

 

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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.

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