UNIVERSAL PORTFOLIO GENERATED BY IDEMPOTENT MATRIX AND SOME PROBABILITY DISTRIBUTION LIM KIAN HENG MASTER OF MATHEMATICAL SCIENCES

Transcription

1 UNIVERSAL PORTFOLIO GENERATED BY IDEMPOTENT MATRIX AND SOME PROBABILITY DISTRIBUTION LIM KIAN HENG MASTER OF MATHEMATICAL SCIENCES FACULTY OF ENGINEERING AND SCIENCE UNIVERSITI TUNKU ABDUL RAHMAN APRIL 2015

2 ii UNIVERSAL PORTFOLIO GENERATED BY IDEMPOTENT MATRIX AND SOME PROBABILITY DISTRIBUTION By LIM KIAN HENG A project report submitted to the Department of Mathematical and Actuarial Sciences, Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, in partial fulfillment of the requirements for the degree of Master in Mathematical Sciences in April 2015

3 iii DECLARATION I Lim Kian Heng hereby declare that this project report is based on my original work except for quotations and citations which have been duly acknowledge. I also declare that it has not been previously or concurrently submitted for any other degree at UTAR or other institutions. (LIM KIAN HENG) Date

4 iv APPROVAL FOR SUBMISSION I certify that this project report UNIVERSAL PORTFOLIO GENERATED BY IDEMPOTENT MATRIX AND SOME PROBABILITY DISTRIBUTION was prepared by LIM KIAN HENG has met the requirements for the award of Master of Mathematical Sciences at Universiti Tunku Abdul Rahman. Approved by, Signature : Supervisor : Date :

5 v ABSTRACT UNIVERSAL PORTFOLIO GENERATED BY IDEMPOTENT MATRIX AND SOME PROBABILITY DISTRIBUTION Lim Kian Heng Universal portfolio is a robust trading strategy. Two methods were used to generate the universal portfolios in order to obtain higher return in this thesis. First method universal portfolio which generated by the symmetric idempotent matrix. The second method universal portfolio generated by probability distribution such as Gamma, Beta, Lognormal, Gaussian and Inverse Gaussian distributions. By empirical study, the best value of u in symmetric idempotent matrix to generate highest return from the universal portfolio after a certain period is studied. Both results are used to compare with the CSD universal portfolio which studied by Tan and Lim (2011) to find out a better method to generate the universal portfolio.

6 vi ACKNOWLEDGEMENTS I would like to convey my appreciation to those who have supported me for this project. Firstly thanks to my supervisor Dr Tan Choon Peng who provided guidance to me since the very beginning till end. He teach and advise me patiently so that I can understand further about this topic, I couldn t finish this project without his invaluable guidance and advice. Secondly I would like to thank you University Tunku Abdul Rahman to provide me a delightful study environment and group of dedicated lecturers during my study. Furthermore, thank you very much indeed to families and friends with their assistant to me during this research.

7 vii TABLE OF CONTENT Page DECLARATION ABSTRACT ACKNOWLEDGEMENTS LIST OF TABLES iii iv v viii CHAPTER 1 INTRODUCTION Universal portfolio Objective Problem definition 3 2 LITERATURE REVIEW 4 3 AN ADDITIVE-UPDATE UNIVERSAL PORTFOLIO Chi-square Divergence (CSD) Universal Portfolios Research by Tan and Lim(2011) The MAHALANOBIS Additive-Update Universal Portfolios 12 4 LOW ORDER UNIVERSAL PORTFOLIOS Introduction to Low Order Universal Portfolios The Finite Order Universal Portfolios Low Order Universal Portfolios Order 1 Universal Portfolio Order 2 Universal Portfolio Order 3 Universal Portfolio The Order Universal Portfolio Wealth Function 21 5 RESEARCH METHODOLOGY Introduction Universal Portfolios generated by idempotent Matrix Idempotent matrix Idempotent matrix construction 27

8 viii 5.3 The Universal Portfolio Generated by Five Special Distribution Order 1 Universal Portfolio Order 2 Universal Portfolio Order 3 Universal Portfolio The Five Special Distributions 44 6 COMPUTATIONAL RESULT ANALYSIS Result of Universal Portfolio generated by idempotent matrix Performance of Finite Order Universal Portfolio 56 7 CONCLUSIONS 73 APPENDICES 75 REFERENCES 76

9 ix LIST OF TABLES TABLE TITLE PAGE Value of (max), and, where 11 (0.3333, , ) for the CSD universal portfolios The 5 probability distributions and its corresponding probability 44 density functions The First four moments of the selected probability distributions Values of the principal and the generated by 47 idempotent matrix for Data Set A 6.2 Values of the principal and the generated by 48 idempotent matrix for Data Set B 6.3 Values of the principal and the generated by 49 idempotent matrix for Data Set C 6.4 Values of the principal and the generated by 50 idempotent matrix for Data Set D 6.5 Values of the principal and the generated by 51 idempotent matrix for Data Set E

10 x TABLE TITLE PAGE 6.6 Values of the principal and the generated by 52 idempotent matrix for Data Set F 6.7 Values of the principal and the generated by 53 idempotent matrix for Data Set G 6.8 Values of the principal and the generated by 54 idempotent matrix for Data Set H 6.9 The return after 500 days for the finite order universal portfolio 58 generated by Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when µ = The return after 500 days for the finite order universal portfolio 58 generated by Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when µ = The return after 500 days for the finite order universal portfolio 59 generated by Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when σ = 1

11 xi TABLE TITLE PAGE 6.12 The return after 500 days for the finite order universal portfolio 59 generated by the Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when σ = The return after 500 days for the finite order universal portfolio 60 generated by the Lognormal distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when µ = The return after 500 days for the finite order universal portfolio 60 generated by the Lognormal distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when µ = The return after 500 days for the finite order universal portfolio 61 generated by the Lognormal distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when σ = 1

12 xii TABLE TITLE PAGE 6.16 The return after 500 days for the finite order universal portfolio 61 generated by the Lognormal distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when σ = The return after 500 days for the finite order universal portfolio 62 generated by the Inverse Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when = The return after 500 days for the finite order universal portfolio 62 generated by the Inverse Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when = The return after 500 days for the finite order universal portfolio 63 generated by the Inverse Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when = 1

13 xiii TABLE TITLE PAGE 6.20 The return after 500 days for the finite order universal portfolio 63 generated by the Inverse Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when = The return after 500 days for the finite order universal portfolio 64 generated by the Beta distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when The return after 500 days for the finite order universal portfolio 64 generated by the Beta distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when The return after 500 days for the finite order universal portfolio 65 generated by the Beta distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when The return after 500 days for the finite order universal portfolio 65 generated by the Beta distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when 10

14 xiv TABLE TITLE PAGE 6.25 The return after 500 days for the finite order universal portfolio 66 generated by the Gamma distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when The return after 500 days for the finite order universal portfolio 66 generated by the Gamma distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when The return after 500 days for the finite order universal portfolio 67 generated by the Gamma distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when The return after 500 days for the finite order universal portfolio 67 generated by the Gamma distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when 10

15 1 CHAPTER 1 INTRODUCTION 1.1 Universal portfolio Universal portfolio is a robust investment strategy. This portfolio suggested that investor to distribute investment principal into m numbers of shares with portfolio vector and price relative vector accordingly on the n-th trading day. Where, and is the ratio of the closing price to the opening price for the m shares respectively on n-th trading day At the end of first trading day, the portfolio vector price relative vector, the total principal of investor and the S 1 = =. At the end of second trading day, the portfolio vector the price relative vector, the total principal of investor and S 2 = = =. By applying the same method, at the end of n trading day, the portfolio vector and the price relative vector, the total principal of investor S n = =. This project, we carried out a research on the universal portfolio which portfolio vector is generated by the symmetric Idempotent matrix and some probability

16 2 distributions. We used these methods to generate the universal portfolio which maximise the profit by a best parameter u value selected. First method, universal portfolio which portfolio vector generate by using the symmetric idempotent matrix via the Mahalanobis formula The symmetric idempotent matrix is a special matrix where the matrix multiplied by itself, the result is itself, that is x =. In this research, we consider only the symmetric idempotent matrix. We will study the best value of parameter u in symmetric idempotent matrix M(u) for the universal portfolio which generate maximum profit after certain trade period. The second method we used 5 common probability distributions to generate the universal portfolio where the portfolio vector is given by In this research, we study only universal portfolios of order v = 1, 2, 3 due to the limitation of computational time and memory. Gamma distribution, Beta distribution, Lognormal distribution, Gaussian distribution and Inverse Gaussian distribution were used in this research to generate the universal portfolio, where each distribution has 2 parameters. We will study how the parameter affected the return of investment. 1.2 Objective (i) To establish the Mahalanobis universal portfolio which generated by symmetric Idempotent matrix ( non-negative definite matrix). (ii) To establish universal portfolio which generated by Gamma distribution, Beta distribution, Lognormal distribution, Gaussian distribution and Inverse Gaussian distribution

17 3 (iii) To achieved higher wealth of investor by selected best parameter value through the above (i) and (ii) method. (iv) To compare the both result in (iii) with CSD universal portfolio which studied by Tan and Lim (2011) 1.3 Problem definition (i) How to develop a symmetric idempotent matrix? (ii) Can the Mahalanobis universal portfolio generated by symmetric idempotent matrix (non negative definite matrix)? (iii) How to generate universal portfolio by using Gamma distribution, Beta distribution, Lognormal distribution, Gaussian distribution and Inverse Gaussian distribution? (iv) How good/bad is the wealth of investor by using these methods?

18 4 CHAPTER 2 LITERATURE REVIEW Many past researchers in the world carried out research on universal portfolios. Thomas M. Cover (1991) was the earliest researcher who study this topic. He uses sequential portfolio selection procedure for stock market investment. Throughout his research he didn t made any statistical assumption on the behaviour of the stock market. This is the initial idea of universal portfolio. In the universal portfolios research of Thomas M. Cover and Erik Ordentlich (1996), they used a -weighted side information in the sequential investment algorithm. As a result, they establish a close connection between universal data compression and universal investment. Avrim Blum and Adam Kalai (1999) study the universal portfolios with and without transaction cost. They found that the proportion of total wealth generate by a stock is same at the beginning of each trade period. They also discovered the constant rebalanced portfolio (CRP) investment strategy.

19 5 Alexei A. Gaivoronski and Fabio Stella (2000) used the BCRP (best constant rebalanced portfolio) and SCRP (successive constant rebalanced portfolio) to generate the universal portfolio. The advantages of these portfolios are less demanding in computationally compare to other previously known portfolios. The stock data they used in NYSE (New York Stock Exchange) show that their portfolio included the consideration of possible nonstationary market behaviour. Suleyman S. Kozat and Andrew C. Singer (2006) carried out research on universal portfolios with side information and switching. He study further in the best constant rebalanced portfolios, he manage to establish a less complexity portfolio algorithm and show that significant gains on historical stock pairs can be achieves with this portfolio algorithm Patrick O Sullivan and David Edelman (2011) study and introduced the adaptive universal portfolios. Their result can achieve greater early growth compare to Cover s universal portfolio which relatively poor in the early growth. Tan and Lim (2011) carried out research on additive-update universal portfolio.they use the chi-square divergence (CSD) distance measure to minimizing and maximizing the same objective functions in order to generate the universal portfolios. Later on, Tan and Lim (2012) focusing on the Mahalanobis squared divergence (or quadratic divergence) universal portfolio, by using the Mahalanobis squared divergence method they derived many group of additive-update universal portfolio. They used symmetric,

20 6 positive definite matrices to generate these groups of Mahalanobis squared divergence additive-update universal portfolio. They manage to obtain sufficient bound for valid parametric values. They also show that the Mahalanobis additive-update universal portfolio can achieve a wealth higher than that of the best constant rebalanced portfolio (BCRP) with a sufficient condition. In year 2013, in order to solve the long implementation time and huge memory problem, Tan (2013) introduced universal portfolio with performance bounds which generated by distribution. His finite order universal portfolio advantage in much lesser time and memory for implementation due to less past data require for storage and computation.

21 7 CHAPTER 3 ADDITIVE UPDATE UNIVERSAL PORTFOLIO 3.1 Chi-square Divergence (CSD) Universal Portfolios The sequence portfolio vectors of chi-square divergence (CSD) universal portfolio is given by where is a real number so that for all and The xi-parametric family chi-square divergence (CSD) universal portfolios can be define by Equation (3.1) for real number so that for and Please take note that xi-parametric family can only defined with some bounded values and not all (real number). By maximizing and minimize the objective functions using the chi-square divergence distance measure we can generate the xi-parametric family of chi-square divergence (CSD) universal portfolios. The logarithms which applied below are all with base e. Proposition Let the objective functions and

22 8 where is the chi-square divergence distance measured (Kullback-Leibler distance measure) and. In order to simplify both the objective functions in (3.2) and (3.3), we let approximate to then, the maximum of the objective function is achieved at given by (3.2) and similarly, the minimum of is also achieved at given by (3.3) where is replaced by. Proof. We introduce the Lagrange multiplier to maximize the objective functions for portfolio vector is where and When we applied the m partial derivatives and let it to be zero for the maximum objective function,

23 9 Solving (3.8), we get summing over, we obtain which results in (3.2). Also with the same approach it can be shown that the minimum of achieved at in (3.2). Proposition A sufficient condition for for all and all positive integers n where defined in (3.2) is that where is the relative stock price of stock i on the trading day and is given. Proof. For in (3.2), for and we have, thus

24 10 Please note that is true and valid for all and any satisfying (3.11) will imply that (3.12) is satisfied as well. Remark Any real number satisfy the equation (3.11) will generate a xiparametric family of chi-square divergence (CSD) universal portfolios. Please note that if the minimum and the maximum price relatives are 0.92 and 1.00 respectively, then the condition in (3.12) says that = That mean a parametric family of chi-square divergence (CSD) universal portfolios can be generated for range Research by Tan and Lim Tan and Lim (2011) had carried out the research of chi-square divergence (CSD) universal portfolios for different data sets. These data sets are Set A, Set B and Set C where Set A consist of listed company in Kuala Lumpur Stock Exchange(KLSE) which is Malayan Banking, Genting and Amway (M) Holdings. Set B consist of Public Bank, Sunrise and YTL Corporation. Set C consist of Hong Leong Bank, RHB Capital and YTL Corporation. The trading period is chosen from January 1, 2003 to November 30, 2004, total 500 trading days. The starting principal is assumed to be 1. The final principal after n trading days is as shown in the table below [ ]. The initial portfolio verctor (0.3333, , ) for Tan and Lim research on Set A, Set B and Set C. The maximum returns, vector portfolio, the value of where is maximum and the range are shown in Table

25 11 Table Value of (max), and, where (0.3333, , ) for the CSD universal portfolios. Data Set A B C Normal range of determined by (3.12) (max) = at = = (0.2106, , ) (max) = at = = (0.0000, , ) (max) = at = = (0.0000, , ) Extended range of (max) = at = = (0.2106, , ) (max) = at = = (0.0000, , ) (max) = at = = (0.0000, , )

26 12 MAHALANOBIS UNIVERSAL PORTFOLIO 3.3 The MAHALANOBIS Additive-Update Universal Portfolios The Mahalanobis Additive-Update Universal Portfolio has three parameters, that is positive definite symmetric matrix A, the initial portfolio vector and a scalar parameter. The sequence portfolio vector of Mahalanobis Additive-Update Universal Portfolio is given by formula as follow:, where the initial portfolio vector is given, and real number so that all value in 0 for n=0,1,2,3.. Matrix A is a positive definite symmetric matrix and =. Proposition The sequence portfolio vector of Mahalanobis Additive-Update Universal Portfolio can be replace by formula as shown below:, where is given by

27 13 Matrix E = Matrix = ), matrix is symmetric and the sum of row of matrix is zero. Proof Given by, where E= let Then by comparing both formula we should get. Identify the i-th row in the above equation then we get. Comparing the coefficients of we get,.

28 14 Proposition Given a portfolio vector define under equation, the sequence portfolio vector to be a portfolio, it is necessary and sufficient that: or, for i=1,2..,m, where represent the i-th element of vector. Proof Given that portfolio vector and, since, then, we get. (since ) if > 0, then. if < 0, then. The wealth of investor At the end of n trading day, the portfolio vector price relative vector, the total principal of investor and the S n = =.

29 15 CHAPTER 4 LOW ORDER UNIVERSAL PORTFOLIOS 4.1 Introduction to Low Order Universal Portfolios Thomas M. Cover and Erik Ordentlich (1996) had presented a sequential investment algorithm, the -weighted universal portfolio with side information, which achieves, to first order in the exponent, the same wealth as the best side-information dependent investment strategy determined in hindsight from observed market and side-information outcomes. Based on this concept, Tan (2013) adopted this approach and introduced the theory of finite and moving order universal portfolios generated by some special probability distributions. In this chapter, it is our main objective to study the empirical performance of low order universal portfolios. The advantage of low order universal portfolio is that it can help in saving a lot of computational time and computer memory when performing calculation compared to the method which is used in Cover-Ordentlich universal portfolio. For this reason, low order universal portfolio is a better approach than the method in Cover- Ordentlich universal portfolio to generate the total wealth achieved in investment..

30 The Finite Order Universal Portfolios Suppose Y1, Y2,..., Y m are mutually independent random variables with probability density functions f ( y ), f ( y ),..., f ( y ) respectively. Then the joint probability density Y1 1 Y2 2 Ym m function of Y1, Y2,..., Y m is: f ( y, y,, y ) f ( y ), f ( y ),..., f ( y ) (4.1) 1 2 m Y1 1 Y2 2 Ym m for, where D is defined by: 1 2 D y, y,, y : f (y ) 0, for all i 1, 2,..., m (4.2) m Y i i Let denote the price-relative vector on day n. Then the inner product for is well defined for an m-stock market. Let v be a fixed positive integer. Then the order v universal portfolio generated by the m independent random variables is sequence given by: for ;. Notice that we have to assume that the moments are positive for ;.

31 Low Order Universal Portfolios In universal portfolio (4.3), the order of v is the number of days of the past stock-price information to be taken into account in calculating the next-day portfolio. We introduced a low order universal portfolio because it can save a lot of computational time and computer memory if the order v is small, say By doing so, we are able to generate results effectively and efficiently. Therefore, we will investigate the universal portfolios with orders 1, 2, 3 which is generated by some special probability distributions. The specific formula of (4.3) for each will be obtained in the following sub-sections. Also, we need the moments of each probability distribution to help in our calculation as shown in the formula (4.3) Order 1 Universal Portfolio From (4.3), the portfolio proportion for stock k on day n+1 for the order 1 universal portfolio is given by:

32 18 where the normalizing constant, We note that by the independence of, Order 2 Universal Portfolio From (4.3), the order 2 portfolio proportion for stock k on day n+1 is given by:

33 19 where the normalizing constant, We note that by the independence of,, where the values of are distinct.

34 Order 3 Universal Portfolio From (4.3), the order 3 portfolio proportion for stock k on day n+1 is given by: for k = 1, 2,, m, where the normalizing constant, We note that by the independence of,

35 21 where the values of are distinct. 4.4 The Order Universal Portfolio Wealth Function The wealth function can be calculated recursively as follows: where. From (4.3), the wealth increase on day n+1, namely can be evaluated as follows for the order v universal portfolio:

36 22 From (4.4), for the special case of the order 1 universal portfolio for 3 stocks, the wealth increase on day n+1 is given by: where from equation (4.5),

37 23 CHAPTER 5 RESEARCH METHODOLOGY 5.1 Introduction For the first part we used the symmetric idempotent matrix (non-negative definite matrix) to generate the universal portfolio. The second part universal portfolio is generate by some probability distributions. Each of the data set contain of 3 or 5 stocks. For the first part, by using the Mahalanobis universal portfolio formula, the symmetric idempotent matrix is used to generate the universal portfolio. The Mahalanobis universal portfolio formula as follow: where all elements in the matrix is given by where is the i-th row, j-th column element in the idempotent matrix, is the sum of i-th row of idempotent matrix,

38 24 is the sum of j-th column of idempotent matrix, is the sum of all elements in the idempotent matrix. The second part we use some probability distributions to generate the portfolio. Let v be a positive integer, function be m random variables having a joint probability density defined over B, where Let the measure have a p.d.f.. The universal portfolio which generated by is defined as follow: for. The case we consider here, the random variables are mutually independent from the probability density function, namely has the probability density function for where is a parameter. So the joint probability density function is given as follow: where is defined over.

39 25 The numerator of : now where is the number of in the product, for and is the moment of.

40 26 The denominator of : Therefore universal portfolio of order v which generated by as follow: where where, for is the number of in the product

41 Universal Portfolios generated by idempotent Matrix Idempotent matrix Idempotent matrix is a very special matrix where the matrix multiplied by itself, the result is itself. That is idempotent matrix x idempotent matrix = idempotent matrix. Some characteristics of idempotent matrix as follow: 1. Idempotent matrix is a non-negative definite matrix and its eigenvalues are either 0 or Sum of the matrix diagonal is equal to the rank of the matrix. 3. A full rank idempotent matrix must be identity matrix. Example: 4. A zero rank idempotent matrix must be zero matrix. Example: 5. Idempotent matrix must be a square matrix Idempotent matrix construction Base on the idempotent matrix s characteristic 2, 3, 4 and 5, we assume a 2x2 rank 1 symmetric idempotent matrix as follow: M(u) = Since M(u) x M(u) = M(u), then. By matrix multiplication, we can obtain simultaneous equations as follow:,.

42 28 Solved these simultaneous equation we get, hence a 2x2 rank 1 symmetric idempotent matrix as follow: M(u) =, where 0 Assume a 3x3 rank 2 symmetric idempotent matrix as follow: M(u) =, since M(u) x M(u)=M(u), then By matrix multiplication, we can obtain simultaneous equations as follow:,,, By solving these equations we get. Hence 3x3 rank 2 symmetric idempotent matrix as follow: M(u) =, where 0.

43 29 Let assume 4x4 rank 3 symmetry idempotent matrix M(u) as follow: M(u) =, since M(u) x M(u)=M(u), then, by matrix multiplication, we can obtain simultaneous equations as follow:,,, by solving these equations we get, hence 4x4 rank 3 symmetric idempotent matrix as follow: M(u) =, where 0 by repeating the same process we can obtain a 5x5 rank 4 symmetry idempotent matrix as follow:

44 30, where 0 In this research, we consider special idempotent Matrix such that it is symmetric which as shown below: where 0 for all real numbers. We consider real number for u as a parameter for the universal portfolio. matrix: The group of 3-stock data Set A, B and C is generated by symmetry idempotent Apply the formula in (5.2 ), we can generate a 3x3 matrix.

45 31 where all the elements in matrix shown as follow: After obtained the matrix, let vector = (0.3333, , ), by using the 3 stocks data set for 500 days to generate the, select a valid value by using equation so that all values in vector satisfying condition we can generate all the vector, where n=1, 2, 3,4 500 and m= 3. Also the final principal S 500 can be obtain via equation

46 32 S n = =. We will use this result to compare with other later. The group of 5-stock data Set D, E and F, G, H is generated by idempotent matrix: Apply the formula in (5.2), we can generate a 5x5 matrix.

47 33 After obtained the matrix, let vector = (0.2, 0.2, 0.2, 0.2, 0.2), use the 5 stocks data set D, E, F, G and H for 1500 days to generate the, select a valid value by using equation

48 34 so that all values in vector satisfying condition we can generate all the vector, where n=1, 2, 3, and m= 5. Also the final principal S 1500 can be obtain via equation S n = =. We will use this result to compare with other later. 5.3 The Universal Portfolio Generated by Five Special Distributions Order 1 Universal Portfolio Let s derive the formula for order 1 universal portfolio.

49 Order 2 Universal Portfolio The formula derivation for order 2 universal portfolio.

50 36

51 where 37

52 Order 3 Universal Portfolio The formula derivation for order 3 universal portfolio.

53 39

54 40

55 where 41

56 42

57 43

58 The Five Special Distributions In our research, we choose five probability distributions to generate the moments required in. The distributions are lognormal distribution, Gamma distribution, Beta distribution, Gaussian distribution and the inverse Gaussian distribution. Table contain the corresponding probability density functions which we selected. Table The 5 probability distributions and its corresponding probability density functions Distribution Probability Density Function f( x ) Gaussian, Lognormal, Inverse Gaussian, Beta, Gamma, 1 e 2 1 e x 2 2 x 3 e x x ln x x 2 2 2x x. is a gamma function 1 x 1 e x. is a gamma function

59 45 Table The First four moments of the selected probability distributions Distribution Gaussian , Lognormal, Inverse Gaussian, Beta, 1 E(X) E(X) E(X ) E(X) E(X ) E(X ) 3 Gamma,

60 46 CHAPTER 6 COMPUTATIONAL RESULTS ANALYSIS 6.1 Result of Universal Portfolio generated by idempotent matrix We prepare 8 sets of share price data to run the universal portfolio, that is data set A, B, C, D, E, F, G and H. Where the data set A, B and C consists of 3 different listed company stocks from Kuala Lumpur Stock Exchange (KLSE). Portfolio set A contain Maybank (1155), Genting(3182) and Amway (6351). Portfolio set B contain PBBank(1295), Sunrise(6165) and YTL(4677). Portfolio set C contain HLBank(5819), RHBCAP(1066) and YTL(4677). Recorded period(500 trade days) of these stocks is taken from 1 st January year 2003 until 30 th November year Portfolio set D contain IOIcorp (1961), Carlsbg(2836), BAT(4162), Nestle(4707) and DIGI(6947). Portfolio set E consist of PBBank(1295), Kulim(2003), KLCC(5235SS), AEON (6599) and KLK(2445). Portfolio set F contain AMBank(1015), BJTOTO(1562), AIRASIA(5099), GAMUDA(5398) and Genting(3182). Portfolio set G consist of listed company AEON(6599), BAT(4162), Kulim(2003), Nestle(4707) and DIGI(6947) while portfolio set H consist of listed company DIGI(6947), PBBank(1295), KLCC(5235SS), Carlsbg(2836) and KLK(2445). Each of these data sets consists of 5 different listed company from KLSE. The recorded period of these stocks is 1500 trading days. Assuming the starting principal is 1, that is and let the portfolio vector. We use the data set A, B, C and symmetric idempotent

61 47 matrix to generate the universal portfolio for 500 trading days. Use data set D, E, F, G, H and symmetric idempotent matrix to generate the universal portfolio for 1500 trading days. All the computation results are as shown below: Table 6.1: Values of the principal and the generated by idempotent matrix for Data Set A Set u Min Max Best A

62 48 Table 6.2: Values of the principal and the generated by idempotent matrix for Data Set B Set u Min Max Best B

63 49 Table 6.3: Values of the principal and the generated by idempotent matrix for Data Set C Set u Min Max Best C

64 50 Table 6.4: Values of the principal and the generated by idempotent matrix for Data Set D Set u Min Max Best D

65 51 Table 6.5: Values of the principal and the generated by idempotent matrix for Data Set E Set u Min Max Best E

66 52 Table 6.6: Values of the principal and the generated by idempotent matrix for Data Set F Set u Min Max Best F

67 53 Table 6.7: Values of the principal and the generated by idempotent matrix for Data Set G Set u Min Max Best G

68 54 Table 6.8: Values of the principal and the generated by idempotent matrix for Data Set H Set u Min Max Best H

69 55 Lets analyze the result of 3 stock portfolios for Set A,B and C. Based on data set A Table 6.1, the maximum principal for 500 trading days can be achieved when parameter u is 0 with. The correspondent best value is which is within the range ( , ). Base on the observation a decreasing trend in when the parameter u is increasing from 0 to 0.7. After the maximum is achieved, there is an increasing trend in for parameter u range from 0.70 to 1.0. The same nature goes to data set B table 6.2 and data set C table 6.3. For data Set B, the maximum principal for 500 trading days is achieved when the parameter u is 0 with. The correspondent best value is which is within the range ( , ). The decreasing trend occur in range u from 0 to 0.9, the increasing trend occur in range u from 0.9 to 1.0. For data Set C, the maximum principal for 500 trading days can be achieved when the parameter u is 0 with. The correspondent best value is which is within the range ( , ). The decreasing trend occur in range u from 0 to 0.55 and from 0.9 to 1.0, the increasing trend occur in range u from 0.55 to 0.9. Therefore, we conclude that Universal Portfolio generated by symmetric idempotent Matrix with highest wealth after 500 trading days achieved when the parameter u is 0 for the same 3 stocks portfolio data sets A,B and C. On the other hand, we analyse the performance of 5 stock portfolios data set D,E,F,G and H. Based on data set D Table 6.4, the maximum wealth after 1500 trading days = achieved when the parameter u is 1. The correspondent best value is which is within the range ( , ). There is an increasing trend in for parameter u range from 0.05 to 0.50 and parameter u range from 0.8 to 1.0. There is a decreasing trend for parameter u range from 0.00 to 0.05 and parameter u range from 0.50 to 0.8. Based on data Set E table 6.5, the maximum = is achieved when the parameter u is 0. The

70 56 correspondent best value is which is within the range ( , ). There is an increasing trend in for parameter u range from 0.90 to 1.0. The decreasing trend in in for parameter u range from 0.0 to 0.9. Based on data set F table 6.6, the maximum = is achieved when parameter u is The correspondent best value is which is within the range ( , ). There is an increasing trend in for parameter u range from 0.00 to The decreasing trend in for parameter u range from 0.85 to 1.0. Based on data set G table 6.7, the maximum = is achieved when the parameter u is The correspondent best value is which is within the range ( , ). There is an increasing trend in when the parameter u range from 0.00 to The decreasing trend in for parameter u range from 0.85 to Based on data set H table 6.8, the maximum = is achieved when the parameter u is 1.0. The correspondent best value is which is within the range ( , ). There is an increasing trend in when the parameter u range from 0.40 to The decreasing trend in for parameter u range from 0.00 to Performance of Finite Order Universal Portfolio There are 8 sets of data were used to generate the universal portfolio, that is data set A, B, C, D, E, F, G and H. Each set of data consists of 3 different stocks from Kuala Lumpur Stock Exchange(KLSE). Set A is the portfolio contain stocks Maybank, Genting and Amway (M) Holdings. Set B is the portfolio contain stocks Public Bank, Sunrise and YTL

71 57 Corporation. Set C is the portfolio contain stocks Hong Leong Bank, RHB Capital and YTL Corporation. The trading period recorded for these stocks is 500 days, which taken from 1 st January 2003 to 30 th November On the other hand, data set D is the portfolio contain Industrial Oxygen Industries (IOI), Carlsberg, British American Tobacco (BAT), Nestle and DIGI. Data set E is the portfolio contain Public Bank, Kulim, Kuala Lumpur City Centre (KLCC), AEON and Kuala Lumpur Kepong Berhad (KLK). Data set F is the portfolio contain AmBank, BJTOTO, AIRASIA, GAMUDA and Genting. Data set G is the portfolio contain AEON, BAT, Kulim, Nestle and DIGI. Data set H is the portfolio contain DIGI, Public Bank, KLCC, Carlsberg and KLK. Each set of data contain 5 different stocks from Kuala Lumpur Stock Exchange(KLSE). The trading period recorded for these sets is 1500 days. Assuming the initial capital is 1 unit, that is and the initial portfolio vector,. The universal portfolios generated by Gaussian, Inverse Gaussian, Lognormal, Beta and Gamma distribution for 500 days for data Set A, B and C. All the computation results are shown as tables below.

72 58 Table 6.9: The return after 500 days for the finite order universal portfolio generated by Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when µ = 1 µ = 1 Set A Set B Set C Order 1 Order 2 Order 3 Order 1 Order 2 Order 3 Order 1 Order 2 Order Table 6.10: The return after 500 days for the finite order universal portfolio generated by Gaussian distribution with different parameter values for a portfolio of 3 stocks which is Set A, Set B and Set C when µ = 10 µ = 10 Set A Set B Set C Order 1 Order 2 Order 3 Order 1 Order 2 Order 3 Order 1 Order 2 Order