Stock Portfolio Selection using Genetic Algorithm

Chapter 5. Stock Portfolio Selection using Genetic Algorithm In this study, a genetic algorithm is used for Stock Portfolio Selection. The shares of the companies are considered as stock in this work. In the first stage good quality of stocks are identified by stock ranking. In the second stage investment allocation in the selected good quality stocks is optimized using genetic algorithm. Hence by using genetic algorithm an optimal portfolio can be determined. This application provides a very feasible and useful tool to assist the investors in planning their investment strategy and constructing their portfolio. 5.1 Markowitz Portfolio Theory Modern portfolio theory (MPT) is a theory of finance which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. Although MPT is widely used in practice in the financial industry and several of its creators won a Nobel memorial prize for the theory, in recent years the basic assumptions of MPT have been widely challenged by fields such as behavioral economics. MPT is a mathematical formulation of the concept of diversification in investing, with the aim of selecting a collection of investment assets that has collectively lower risk than any individual asset. That this is possible can be seen intuitively because different types of assets often change in value in opposite ways. For example, to the extent prices in the stock market move differently from prices in the bond market, a collection of both types of assets can in theory face lower overall risk than either individually. But diversification lowers risk even if assets returns are not negatively correlated indeed, even if they are positively correlated. More technically, MPT models assets return as a normally distributed function (or more generally as an elliptically distributed random variable), define risk as the standard 159

deviation of return, and model a portfolio as a weighted combination of assets, so that the return of a portfolio is the weighted combination of the assets' returns. By combining different assets whose returns are not perfectly positively correlated, MPT seeks to reduce the total variance of the portfolio return. MPT also assumes that investors are rational and markets are efficient. MPT was developed in the 1950s through the early 1970s and was considered an important advance in the mathematical modeling of finance. Since then, many theoretical and practical criticisms have been leveled against it. These include the fact that financial returns do not follow a Gaussian distribution or indeed any symmetric distribution, and that correlations between asset classes are not fixed but can vary depending on external events (especially in crises). Further, there is growing evidence that investors are not rational and markets are not efficient. 5.2 Design and Implementation Genetic algorithms are probabilistic, robost and heuristic search algorithms premised on the evolutionary ideas of natural selection and genetic. The basic concept of genetic algorithms is designed to simulate the processes in natural system necessary in for evolution, specifically for those that follow the principle of survival of the fittest. They represent the intelligent exploitation of a random search within a defined search space to solve a problem. Genetic Algorithm is developed by John Holland and his students at Michigan University during 1965-1975. Implementation of Problem The Complete Genetic Algorithm design has been prepared for the problem in C language and now it is applied to some real data. Total 32 listed Companies are considered as a data set. Financial indicators namely ROCE, LR and P/E Ratio are given as a input to stock ranking model. Here share is considered as stock for our problem. By applying Stock Ranking Model with the financial indicators ROCE, LR and P/E Ratio for 160

consecutively 5 years (2007 to 2011) as an input to genetic algorithm, rank of a particular company is obtained. After getting the rank and applying sorting to ranks we select top 10 companies as a input to genetic algorithm. Input Size: Top 10 companies as objects are taken as input to genetic algorithm. Hence the chromosome size will be 10. Encoding Scheme: Value Encoding has been applied to the problem. Fitness Function: The total value of the permutation if its weight is max capacity. Parent Selection: After finding the fitness value of each member of the population first Elitism is applied and few best chromosomes are selected and copied to the new generation and then Roulette wheel selection is applied to copy rest of the population. Crossover/ Mutation point: Whether to do the crossover or mutation is determined by generating a random number and comparing it with the user entered probability. Once it is decided to do crossover/mutation, the crossover/mutation points are also determined randomly, by generating a random number. Here one point crossover and exchange a position is used for crossover and mutation. 161

5.3 Stock Ranking Model The aim of this stage is to identify the quality of each stock so that investors can choose some good ones for investment by using stock ranking. In this study, total 32 companies are considered as shown in table 5.1. Some financial indicators of these listed companies are employed to determine and identify the quality of each stock. That is, the financial indicators of the companies are used as input variables while a score is given to rank the stocks. The output variable is stock rank. Through the study of Markowitz Portfolio Theory, three important financial indicators, Return On Capital Employed (ROCE), Price/Earnings ratio (P/E Ratio), and Liquidity Ratio are utilized in this study. The real data for the year 2007 to 2011 are utilized in this study. The real data is described in Annexure 1. The Stock Ranking Model is developed in C language. The definition of financial indicators are given as follows Financial Indicators ROCE = (Profit) / (Shareholder s Equity) * 100 % (5.1) P/E Ratio = (Stock Price) / ( Earning Per Share) * 100 % (5.2) Liquidity Ratio = (Current Assets) / (Current Liabilities) * 100 % (5.3) Table 5.1 Companies with its Sr.No. and Company Name Sr. No. Company Name 1 3I Infotech 2 Aarti Drugs 3 Ashok Leyland 4 Cipla 162

5 Bajaj Telefims 6 Crisil 7 Blue Star 8 Fulford (India) 9 Gujarat Gas 10 JSW Steel 11 Sun Pharmaceutical Industries 12 Sonata Software 13 UltraTech Cement 14 Aptech 15 Amtek India 16 Apollo Tyres 17 Bharti Airtel 18 Cholamandalam I & FC 19 Crompton Greaves 20 Finolex Industries 21 Gabriel India 22 Kansai Nerolac Paints 23 Granules India 24 Graphite India 25 India Oil Corporation 26 Infomedia 18 27 Kajaria Ceramics 28 Surya Roshni 29 Tata Steel 30 Uttam Galva Steel 31 UTV Software Communications 32 ACC 163

Status Here 8 statuses are designed representing different qualities in terms of different interval varying from 0 ( Extremely Poor ) to 7 ( Very Good ). The statuses for financial indicators are as shown in the status table 5.2 below. Table 5.2 Status Table VALUE STATUS - to -30 % 0-30 % to -10 % 1-10 % to +10 % 2 10 % to 30 % 3 30 % to 50 % 4 50 % to 70 % 5 70 % to 90 % 6 90 % to + 7 The output of the three financial indicators for each year is compared with the status table to get individual ranking of each financial indicator. After obtaining individual rank of each financial indicator for each year all are added to get rank. After getting the rank of each year, all the individual rank of all years are again added. After adding all the individual rank of all years an average is taken to get the final rank. Here the lowest final rank is 0 and highest final rank is 21. Hence in this way final stock rank of each and every company is obtained. After getting final stock rank of each and every company sorting of final ranks id done to get top 10 companies. 164

5.4 Stock Portfolio Selection using Knapsack Problem with Genetic Algorithm 5.4.1 Knapsack Problem The knapsack problem is defined as follows: We are given a set of n items, each item j having an integer profit pj and an integer weight wj. The problem is to choose a subset of the items such that their overall profit is maximized, while the overall weight does not exceed a given capacity c. We may formulate the model as the following integer programming model: maximize n j =1 p j x j (5.4) subject to n w j j =1 x j C where x j 0,1, j = 1,2,, n where the binary decision variables x j are used to indicate whether item j is included in the knapsack or not. Without loss of generality it may be assumed that all profits and weights are positive, that all weights are smaller than the capacity c, and that the overall weight of the items exceeds c. Knapsack problem is one of the most intensively studied discrete programming problems. The reason for such interest basically derives from three facets. (a) (b) (c) It can be viewed as the simplest Integer Linear Programming problem It appears as a sub-problem in many more complex problems It may represent a great many practical situation. 165

In the previous stage, some good quality stocks can be revealed in terms of stock return ranking. These good qualities of stocks are used as an input to genetic algorithm. The three basic questions regarding investment decision occurs as follows:- In which company should I invest? How much money in which company should I invest? 5.4.2 Steps of Genetic Algorithm for Stock Portfolio Selection:- 1) Generate initial random population. 2) Calculate the fitness value of each chromosome. 3) Fitness value = (Profit)*(Units) where Profit = ((Current share value Previous share value) / previous share value) 1 Unit = Rs. 1000 Maximum Units = 100 Total Investment = Rs. 1 Lac 4) Apply Roulette wheel selection method to select good chromosomes. 5) Apply crossover and mutation to good chromosomes. 6) Again calculate fitness value of chromosomes. 7) Repeat from step 2 to 6 until a best chromosome is found. 166

After completing genetic algorithm the answers of all the three questions specified above are obtained. 5.4.3 Selection Criteria Encoding Technique : Value Encoding Number of Chromosomes : 30 Number of Companies : 10 Number of Units = 100 Maximum Investment = Rs. 1 Lac Minimum Crossover Probability :0.60 Maximum Crossover Probability : 0.95 Minimum Mutation Probability : 0.001 Maximum Mutation Probability : 1.00 Selection Method: - Roulette Wheel Selection Crossover Method: - 1-point crossover Mutation Method:- Exchange a position 167

5.5 Computational Results 5.5.1 Results of Stock Ranking Model Here the stock ranking model gives the output as top 10 companies with its Sr.No. Rank and Company Name in table 5.3. Now from the 32 companies, 18 companies are listed below as a top 10 companies. Below listed companies are considered for the investment for the investor. The investor can select any 10 companies from the below given 18 companies for the investment. Table 5.3 Top 10 companies with its Sr.No. Rank and Company Name Sr. No. Rank Company Name 2 21 Aarti Drugs 4 21 Cipla 5 21 Balaji Telefims 6 21 Crisil 7 21 Blue Star 8 21 Fulford (India) 10 21 JSW Steel 11 21 Sun Pharmaceutical Industries 13 21 Ultratech Cement 16 21 Apollo Tyres 19 21 Crompton Greaves 21 21 Gabriel India 22 21 Kansai Nerolac Paints 24 21 Graphite India 25 21 India Oil Corporation 27 21 Kajaria Ceramics 29 21 Tata Steel 32 21 ACC 168

5.5.2 Stock Portfolio Selection From the table 5.3 top 10 companies can be selected. Here A to J represents the top 10 companies from as shown in table 5.4. The description of these top 10 companies is as shown in table 5.4. Table 5.4 Top 10 companies with its company name Sr. No. Company Company Name 1 A Aarti Drugs 2 B Cipla 3 C Balaji Telefims 4 D Crisil 5 E Fulford (India) 6 F JSW Steel 7 G Sun Pharmaceutical Industries 8 H Crompton Greaves 9 I India Oil Corporation 10 J Tata Steel 169

Executing Genetic Algorithm By executing the Genetic Algorithm the Initial Random Population is generated which is shown below in table 5.5. Here companies are A to J. The values of A to J are represented in Units. 1 Unit is Rs. 1000. Total represents the total investment in rupees (thousands) in the various companies A to J. Fitness represents the fitness value of the chromosome. The initial random population gives the output Best Chromosome, Maximum Fitness and Total investment as shown in table 5.5. Table 5.5 Initial Random Population generated by Genetic Algorithm A B C D E F G H I J Total Fitness 2 12 8 16 13 17 5 4 5 2 84-0.7835 6 16 7 12 13 8 0 2 10 13 87 3.1825 14 7 4 1 11 6 17 0 2 1 63-4.8308 15 2 12 12 2 13 10 7 15 8 96-2.3039 9 7 16 12 1 16 4 4 8 7 84 1.3347 5 14 17 7 16 8 2 6 7 12 94 3.4474 14 11 13 0 14 6 9 3 3 14 87 0.3830 8 3 5 10 10 1 9 13 16 13 88 0.3370 5 6 1 10 7 3 14 12 11 15 84-2.6729 7 5 5 16 0 11 6 10 15 13 88 0.4752 12 17 11 15 5 3 2 10 2 17 94 2.4413 9 3 4 14 8 10 5 11 12 8 84 0.2516 2 13 11 4 8 13 4 2 16 11 84 2.9864 11 3 8 0 16 8 1 0 1 12 60 1.4736 1 11 1 10 3 7 12 1 10 14 70-2.4598 9 10 5 6 3 12 9 4 2 13 73-1.6438 10 7 5 2 16 15 9 6 10 3 83-0.7359 170

2 17 2 1 8 11 9 2 5 13 70-0.8785 8 2 13 15 12 13 10 5 1 11 90-2.7115 3 3 1 8 5 2 14 16 15 12 79-2.1404 5 0 16 9 10 9 13 13 5 11 91-2.2182 17 6 9 16 1 16 3 5 2 15 90 0.2386 0 16 7 17 14 14 13 7 1 8 97-3.8805 9 11 11 9 17 6 3 10 10 9 95-0.2316 9 8 5 8 5 13 7 11 9 7 82 0.9231 2 10 7 3 6 15 10 11 8 13 85-0.7347 7 3 1 6 12 0 7 9 9 2 56-0.6742 7 0 11 9 10 1 14 7 9 16 84-2.2697 16 2 15 5 14 8 16 9 13 0 98-1.9110 10 12 0 9 9 10 7 12 5 1 75 0.6507 Best Chromosome = 5 14 17 7 16 8 2 6 7 12 Maximum Fitness = 3.4474 at Total = 94 171

5.5.3 Results at different Crossover Probability and Mutation Probability For, Minimum Crossover Probability = 0.00 Maximum Crossover Probability = 1.00 Minimum Mutation Probability = 0.001 Maximum Mutation Probability = 0.08 Results for 800 generations Best chromosome = 3 15 12 0 4 10 2 1 10 5 Maximum Fitness = 3.4308 at Total = 62 Results for 1000 generations Best chromosome = 8 12 3 17 1 3 1 7 4 8 Maximum Fitness = 3.5253 at Total = 64 Results for 1500 generations Best chromosome = 4 17 16 3 6 5 1 10 2 11 Maximum Fitness = 3.7315 at Total = 75 Results for 2000 generations Best chromosome = 4 17 13 3 8 6 3 2 14 6 Maximum Fitness = 3.8699 at Total = 76 172

Results for 3000 generations Best chromosome = 6 17 4 9 1 12 1 8 14 10 Maximum Fitness = 3.9388 at Total = 82 Results for 4000 generations Best chromosome = 6 17 4 9 1 12 1 8 14 10 Maximum Fitness = 3.9388 at Total = 82 Results for 5000 generations Best chromosome = 6 17 4 9 1 12 1 8 14 10 Maximum Fitness = 3.9388 at Total = 82 From the above results it is found that at generations 800, 1000, 1500 and 2000 we are getting variations in Best chromosome, Maximum Fitness and Total investment. But at generations 3000, 4000 and 5000 we are getting the same Best chromosome, Maximum Fitness and Total investment. This concludes that at generations 800, 1000, 1500 and 2000 investor may not take decision for investment in various companies for portfolio management. Finally at generations 3000, 4000 and 5000 the investor can take decision for investment in various companies for portfolio management. 173

For, Minimum Crossover Probability = 0.40 Maximum Crossover Probability = 1.00 Minimum Mutation Probability = 0.001 Maximum Mutation Probability = 0.50 Results for 800 generations Best chromosome = 16 16 16 1 1 1 1 1 16 1 Maximum Fitness = 5.7575 at Total = 70 Results for 1000 generations Best chromosome = 1 17 17 1 1 1 1 17 17 1 Maximum Fitness = 6.2364 at Total = 74 Results for 1500 generations Best chromosome = 15 13 15 2 1 1 2 14 15 2 Maximum Fitness = 5.5143 at Total = 80 Results for 2000 generations Best chromosome = 3 13 15 3 3 3 1 11 17 11 Maximum Fitness = 6.8366 at Total = 80 174

Results for 3000 generations Best chromosome = 9 16 17 2 1 0 1 14 15 11 Maximum Fitness = 6.2364 at Total = 86 Results for 4000 generations Best chromosome = 9 16 17 2 1 0 1 14 15 11 Maximum Fitness = 6.2364 at Total = 86 Results for 5000 generations Best chromosome = 9 16 17 2 1 0 1 14 15 11 Maximum Fitness = 6.2364 at Total = 86 From the above results it is found that at generations 800, 1000, 1500 and 2000 we are getting variations in Best chromosome, Maximum Fitness and Total investment. But at generations 3000, 4000 and 5000 we are getting the same Best chromosome, Maximum Fitness and Total investment. This concludes that at generations 800, 1000, 1500 and 2000 investor may not take decision for investment in various companies for portfolio management. Finally at generations 3000, 4000 and 5000 the investor can take decision for investment in various companies for portfolio management. 175

For, Minimum Crossover Probability = 0.60 Maximum Crossover Probability = 0.95 Minimum Mutation Probability = 0.001 Maximum Mutation Probability = 1.00 Results for 800 generations Best chromosome = 14 13 16 3 3 3 3 3 16 16 Maximum Fitness = 5.3926 at Total = 90 Results for 1000 generations Best chromosome = 2 13 16 2 17 2 2 17 17 2 Maximum Fitness = 5.8160 at Total = 90 Results for 1500 generations Best chromosome = 15 15 15 7 4 4 2 15 15 2 Maximum Fitness = 5.9143 at Total = 94 Results for 2000 generations Best chromosome = 3 17 17 3 3 3 10 12 17 11 Maximum Fitness = 6.6003 at Total = 96 176

Results for 3000 generations Best chromosome = 11 16 17 0 1 0 0 17 17 17 Maximum Fitness = 7.5880 at Total = 96 Results for 4000 generations Best chromosome = 11 16 17 0 1 0 0 17 17 17 Maximum Fitness = 7.5880 at Total = 96 Results for 5000 generations Best chromosome = 11 16 17 0 1 0 0 17 17 17 Maximum Fitness = 7.5880 at Total = 96 From the above results it is found that at generations 800, 1000 and 1500 we are getting variations in Best chromosome, Maximum Fitness and Total investment. But at generations 2000, 3000, 4000 and 5000 we are getting the same Best chromosome, Maximum Fitness and Total investment. This concludes that at generations 800, 1000 and 1500 investor may not take decision for investment in various companies for portfolio management. Finally at generations 2000, 3000, 4000 and 5000 the investor can take decision for investment in various companies for portfolio management. From the above all results it has been concluded that for Minimum Crossover Probability = 0.60, Maximum Crossover Probability = 0.95, Minimum Mutation Probability = 0.001 and Maximum Mutation Probability = 1.00 the genetic algorithm gives far better results for stock portfolio selection. 177