chapter 5. stock portfolio selection using genetic...

Chapter 5 Stock Portfolio Selection using Genetic Algorithm

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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


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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


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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.


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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


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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


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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.


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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 𝑝𝑗𝑛𝑗=1 𝑥𝑗 (5.4)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑤𝑗

𝑛

𝑗=1

𝑥𝑗 ≤ 𝐶

𝑤ℎ𝑒𝑟𝑒 𝑥𝑗 ∈ 0,1 , 𝑗 = 1,2, … , 𝑛

where the binary decision variables xj 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) It can be viewed as the simplest Integer Linear Programming problem

(b) It appears as a sub-problem in many more complex problems

(c) It may represent a great many practical situation.


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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.


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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


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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


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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


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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


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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


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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



Best chromosome = 8 12 3 17 1 3 1 7 4 8



Best chromosome = 4 17 16 3 6 5 1 10 2 11



Best chromosome = 4 17 13 3 8 6 3 2 14 6



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Best chromosome = 6 17 4 9 1 12 1 8 14 10



Best chromosome = 6 17 4 9 1 12 1 8 14 10



Best chromosome = 6 17 4 9 1 12 1 8 14 10


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.


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For,






Best chromosome = 16 16 16 1 1 1 1 1 16 1



Best chromosome = 1 17 17 1 1 1 1 17 17 1



Best chromosome = 15 13 15 2 1 1 2 14 15 2



Best chromosome = 3 13 15 3 3 3 1 11 17 11



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Best chromosome = 9 16 17 2 1 0 1 14 15 11



Best chromosome = 9 16 17 2 1 0 1 14 15 11



Best chromosome = 9 16 17 2 1 0 1 14 15 11


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


management. Finally at generations 3000, 4000 and 5000 the investor can take decision

for investment in various companies for portfolio management.


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For,






Best chromosome = 14 13 16 3 3 3 3 3 16 16



Best chromosome = 2 13 16 2 17 2 2 17 17 2



Best chromosome = 15 15 15 7 4 4 2 15 15 2



Best chromosome = 3 17 17 3 3 3 10 12 17 11



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Best chromosome = 11 16 17 0 1 0 0 17 17 17



Best chromosome = 11 16 17 0 1 0 0 17 17 17



Best chromosome = 11 16 17 0 1 0 0 17 17 17


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


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.

chapter 5. stock portfolio selection using genetic...

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