A Multivariate Statistical Analysis of Stock Trends. Abstract

Transcription

1 A Multivariate Statistical Analysis of Stock Trends Aril Kerby Alma College Alma, MI James Lawrence Miami University Oxford, OH Abstract Is there a method to redict the stock market? What factors determine if a comany s stock value will rise or fall in a given year? Using the multivariate statistical methods of rincial comonent analysis and discriminant analysis, we aim to determine an accurate method for classifying a comany s stock as a good or a oor investment choice. Additionally, we will exlore the ossibilities for reducing the dimensionality of a comlex financial and economic dataset while maintaining the ability to account for a high ercentage of the overall variation in the data. Introduction The stock market is a financial game of winners and losers. Is your stock tried and true or here today gone tomorrow? How can one ick out a golden nugget like Microsoft from the hundreds of dot-comers that went bust after an all-too-brief moment of glory? Indeed, it may seem like there is really no way to tell a seemingly uncountable number of variables influence our markets and comanies; how can one take all of them into account? Is there a simler way of looking at the market madness? This aer seeks to use statistical methods to survey and analyze financial and economic data to discover such a method of simlification. Using Princial Comonent Analysis, we will combine related factors into a smaller number of key comonents largely resonsible for the variations observed. Then, using Discriminant Analysis, we will develo a model for searating comanies into two categories based on their redicted stock erformance: good and oor investment choices. The data we use in this analysis comes from the Federal Reserve Bank of St Louis, Big Charts Historical Stock Quotes, as well as each comany s annual reorts. We use four comany-secific and six macroeconomic variables that we feel might account for a articular comany, at a secific time, being a good or a oor investment. The ten variables are as follows: net revenue, net income, rice er earnings ratio of stock, diluted earnings er share, consumer sending, consumer investment, unemloyment rate, inflation rate, federal funds rate, and the Dow Jones industrial average.

2 Assessment of Normality In order to run the analysis tests, we first check the normality of the data. Normality testing involves regressing the variables, centered about their mean, against the z-values of each variable. A strong linear relationshi imlies normal data. The samle correlation coefficient, r, is calculated by: cov( x( i), q( i) ) r =, (var( x( i) ) var( q( i) ) where x is the data value and q is the associated normal quartile. The null hyothesis, H 0 : ρ =, imlies normality and is rejected if r is less than the critical value. The original variables must be tested individually for univariate normality. Once the variables are tested (using a Q-Q lot), those that are not normal may be thrown out or ket deending on the discretion of the researcher. The variables which do not dislay a normal distribution may be transformed in order to achieve normality. The function or the square root function may be alied to hel obtain normal variables. For our data, the critical value for r is 0.977, at the α =. 0level of significance. The variables called transformed revenue, transformed earnings er share, transformed consumer sending, inflation and unemloyment rate are found to be normal. Transformed investments and the ercent growth of the DJIAM are rejected with r-values of.969 and.967, resectively. Although five variables failed to ass the normality test, this was to be exected of financial/economic data, and the robust characteristics of the tests allows us to continue with the analysis. Theory of Princial Comonent Analysis Princial Comonent Analysis (PCA) reduces the dimensionality of the data set by linearly combining the original correlated variables into new variables, some of which are ignored. These new variables are linearly indeendent of one another, whereas the original variables may have been deendent. Reducing the dimensionality allows fewer variables to be used to obtain a similar amount of information, strengthening the results of any subsequent statistical tests used. Additionally, since the rincial comonents are comosed of multile variables, they allow for a more accurate sense of how the variables are interacting. After the variables have been tested for normality, the eigenvalues of the variance-covariance matrix Σ corresonding to the original variables are calculated. Each eigenvalue λi, with i=,, corresonds to a articular eigenvector. This vector of coefficients is then used in transforming the linear combination of x-variables into rincial comonents. The ideal goal is to maximize the var( l ' i x ), where l ' i is the vector of eigenvalues and x reresents the original vector. Generally, only eigenvectors with corresonding eigenvalues greater than one are used since smaller eigenvalues contribute little to the total variance. To determine which rincial comonents to kee, the

3 eigenvalue scree lot may be used. There is tyically an elbow in the shae of the scree lot. It is customary to kee those rincial comonents rior to the elbow. The ercent of variation exlained by an individual rincial comonent is equal to the ratio λ i, λ and the total variance exlained by the first k rincial comonents is k i= j= λ i. λ After determining how many rincial comonents to use, the selected comonents, denoted y i for the i th comonent, are calculated. Individually, each y i equals l ij x j, where l ij corresonds to the j th element of the i th eigenvector, and as a whole, j= r r y = Lx. Each yi is orthogonal to the other rincial comonents, ensuring linear indeendence within the rincial comonent variables. Princial Comonent Analysis Alication Before alying the PCA test, we must first check to see that the variables are, in fact, deendent. Therefore, the test for indeendence must be executed. Using the correlation matrix R we test H 0 : P = I against H : P I. The likelihood ratio test is used to test H 0 at α =.0. The null hyothesis is rejected if the test statistic is greater than the critical value. In our case the critical value is χ. 45,.0 = The test statistic is calculated with the formula + 5 ( n ) ln R. Our test statistic is and the P-value is.3467x0-34 which strongly suggests that H 0 be rejected. Corresondingly, we reject the null hyothesis since the test statistic is greater than the critical value. This confirms that our variables are in fact deendent, and rincial comonent analysis may be carried out. Princial comonent analysis, run on the data using Minitab, rovides the results in Table.. The corresonding eigenvalues, as well as the variance accounted for by each of the ten rincial comonents are given. j j= j 3

4 Table.: Princial Comonents PC PC PC 3 PC 4 PC 5 PC 6 Eigenvalue Proortion Cumulative PC 7 PC 8 PC 9 PC 0 Eigenvalue Proortion Cumulative In this case, four rincial comonents have eigenvalues greater than one and are ket, accounting for 78.9% of the total variation. Next, to hel confirm the retention of only four rincial comonents, the eigenvalue scree lot may be useful. However, in this articular case, the scree lot does not clearly determine the number of comonents to be used, and we will base our model on the eigenvalues themselves. The rincial comonents are comuted using the given coefficients. Table.: Princial Comonent Coefficients Variables PC PC PC 3 PC 4 Log Log Diluted Earnings Log Net Income Unemloyment Federal Funds DJIAM Log Consumer Investments Log Consumer Sending Inflation sqrt(price er Earnings) It can be seen that the first comonent is affected largely by the macroeconomic variables and could be renamed Overall Economic Condition. The second rincial comonent cannot be easily named because there is no discernable attern in the variables that affect it. The third rincial comonent is comrised mostly of diluted earnings and the rice er earnings ratio, and could hence be called Strength of Stock. The net income and revenues dominate the fourth rincial comonent, and this comonent could be renamed Profitability. Theory of Discriminant Analysis A tool is needed to classify a multivariate data vector into one of two oulations. Examles of uses for this tool would be to classify students as likely to succeed or fail in college or to classify an organ translant atient as likely to survive or not. Discriminant analysis rovides a rule for classifying observations from a multivariate data set into two or more oulations. 4

5 In the case of two oulations defined by Π N ( µ, Σ) and Π (, ) N µ Σ, we can derive a classification rule that can be used to classify an element x into one of the oulations. Each x is assumed to be -variate normal. When the oulation arameters are unknown, as is often the case, one must obtain training samles for estimation of the mean and covariance of each oulation, as well as derive the classification rule. The estimates of µ, µ, Σ and Σ, are x, x, S and S, resectively. If the covariance matrices for the oulations, Σ and Σ, are equal, then the common ( n ) S + ( n ) S covariance matrix, Σ, is relaced by the ooled estimate, S =. n + n The estimates x, x, S, S, and S are unbiased estimators. Next comes the classification of x into Π or Π. We use an otimal linear discriminant function, a`x, where aˆ = S ( x x ), to assign x into a oulation based on the decision rule: Classify x into Π if aˆ ' x > ( x x )' S ( x x ) otherwise, classify x into Π if aˆ ' x ( x x )' S ( x x ) After we have determined the decision rule, we calculate the aarent error rate (AER) based on the training samle used to derive the decision rule. The AER is the ercentage of observations in the training samle that are misclassified by the decision rule. However, since the AER uses the secific data observations that were used to create the decision rule, there is a chance that the linear discriminant function and its corresonding decision rule will have a higher error rate in ractice. The total robability of misclassification (TPM) denotes the robability that an individual observation will be misclassified using the derived function. This is related to the Mahalanobis distance between the two oulations,, calculated by: = ' Σ a a = ( µ µ )' Σ ( µ µ ) Using this, we calculate: TPM = Φ ( ), where Φ(x) is the cumulative distribution function of the standard normal distribution. For a data samle, TPM is well aroximated by: ˆ α = Φ( ˆ ), where ˆ = ( x x )' S ( x x ). Additionally, the Mahalanobis distance can be used as a classification rule equivalent to that of the linear discriminant function. If the Mahalanobis distance (D ) 5

6 from x to Π is less than the Mahalanobis distance (D ) from x to Π, then we assign x to Π. Otherwise, we assign x to Π. When using a linear discriminant function, it is assumed that Σ = Σ. If this hyothesis is rejected, then we must use a quadratic discriminant function. Discriminant Analysis Alication Before erforming discriminant analysis, we first need a method of classifying a comany as a good or oor investment, for a given year. While there is no definitive method for defining a market investment as good or oor, here is a method that is simle and objective: if the value of a comany s stock over a given year rose, it is classified as a good investment and otherwise it is classified as a oor investment. To obtain the stock rices at the end of each calendar year, we used the Big Charts database online [6]. Our training samle was based on a random selection of 5 comanies, for all years from , where data was rovided in their annual reorts. This made a samle size of 88 distinct comany-year observations. To create a test samle, we removed all eleven of the 00 entries as well as seven randomly selected entries from the data. The remaining 70 entries were used as the training samle. First, it was necessary to test the assumtion for using a linear discriminant function that Σ = Σ. Using the likelihood ratio test, we tested H 0 : Σ = Σ against H : m Σ Σ. Our test statistic, = χ obs, was tested against χ. 45,.0 = We obtained c a value of.6807 for χ obs with S m = { ( ni )}{ln }, and S i= i + 3 = ( + ), c 6( + ) n n n + n with a P-value of aroximately 0. This is strong evidence that we should reject H 0. Therefore, we cannot use a linear discriminant model to classify the data; instead we must use a quadratic discriminant model. Initially, we used the rincial comonents to attemt to create the discriminant model (see Table.3). 6

7 Table.3: Quadratic Model (Princial Comonents) True Grou Poor True Grou Good Classified into Grou Poor 6 4 Good 8 Total N 38 3 N Correct 6 8 Proortion N =70 N correct = 44 Proortion Correct = 0.69 However, this model roduces consistently high error rates. Most notably, it correctly labels the oor grou only 4.% of the time and has an overall error rate of 37.%. Using the ten variables resulting from the first set of transformations (see Aendices) roduces better results, and therefore is ket for the final analysis. The following table summarizes the results of the quadratic model s classification of our training samle. Table.4: Quadratic Model (Original Variables) True Grou Poor True Grou Good Classified into Grou Poor 36 5 Good 7 Total N 38 3 N Correct 36 7 Proortion N =70 N correct = 53 Proortion Correct = The quadratic model has an aarent error rate (AER) of 4.3% ercent. The theoretical TPM for this model was 34.8%. Considering the highly volatile nature of economic data, and the stock market in articular, this is regarded as a low error rate. However, as the urose of the model was to redict how a stock would erform in the ucoming year, when the result is unknown, it is necessary to test our quadratic model on a test samle. For the test samle of 00 data, lus seven random selections from , the discriminant model correctly classified 3 of the 8 stocks as a good or oor investment. The following table shows how the quadratic model classified the 8 stocks in our unknown test samle. 7

8 Table.5: Classification of Test Samle by Quadratic Model Observation Stock Predicted Grou Actual Grou Walt Disney 00 Good Good UPS 00 Good Good 3 Tyson 00 Good Poor 4 Motorola 00 Good Good 5 Microsoft 00 Good Poor 6 Kelg 00 Good Good 7 JCPenney 00 Good Poor 8 Intel 00 Good Good 9 Ford 00 Good Good 0 Comuter Associates 00 Good Good Boeing 00 Good Good Walt Disney 995 Good Good 3 UPS 000 Poor Poor 4 Motorola 000 Poor Poor 5 Kelg 00 Poor Good 6 IBM 999 Poor Poor 7 Ford 00 Good Poor 8 Boeing 998 Good Good The quadratic model laced 3 out of the 8 test stocks from outside the training samle into the correct oulation. The error rate of 7.7% means that this model correctly redicts whether the value of a articular stock will rise or fall over the next year nearly three out of four times in times like these, enough to make a small fortune! Conclusion We are satisfied with our last discriminant model because it redicts a high ercentage of the outcomes for various stocks, and does not lose much accuracy when alied to a samle from outside the training samle. However, desite accounting for a high 8

9 ercentage of the overall variation, the four rincial comonent model is unable to generate an accurate discriminant model for classification of a stock as a good or oor investment. Subsequent tests reveal that the PCA-based models do not achieve high accuracy until most of the comonents are emloyed in the discriminant model, which defeats the urose of using the PCA data. Hence, we are unable to reduce the dimensionality of the data set. We believe that our discriminant model still has a great amount of room for imrovement. Such imrovement can be attained by adding additional variables, articularly those showing asects of the comany not related to rofitability or earnings-share relationshis. We might want to consider comany size, distribution, or even the integration of online sales and marketing as sulementary factors. Additional global economic variables, such as GDP, trends in the Fed s urchase/sale of securities, and variables affecting world trade could rovide additional clarity. As these variables are added, and erhas with the usage of transformations not emloyed in this model, it may still be ossible to reduce the dimensionality of the data set using PCA. We believe that multivariate statistics can be used to analyze a comany s likely erformance in the stock market with resect to global economic conditions as well as its own financial erformance the revious year. Acknowledgements We would like to thank Dr. Vasant Waikar for his tremendous hel in mentoring our research. Without his guidance and teaching, none of this would have been ossible. We would also like to thank Candace Porter, for her suort and dedication. Additionally, we would like to acknowledge Dr. Marc Rubin and Dr. Larry Rankin for their exertise in the area of cororate finance. Finally, we would like to thank all of our rofessors, graduate assistants, and fellow SUMSRI students for their suort, exertise, and camaraderie throughout the duration of the rogram. 9

10 Bibliograhy [] 3M Annual Reorts (995-00). Retrieved June 7, 003 from htt:// [] Boeing Annual Reorts (995-00). Retrieved June 7, 003 from htt:// [3] Comuter Associates Annual Reorts (995-00). Retrieved June 7,003 from htt:// [4] Ford Motor Comany Annual Reorts (998-00). Retrieved June 8,003 from htt:// htt:// [5] Hershey s Annual Reorts (997-00). Retrieved June 8, 003 from htt:// [6] IBM Annual Reorts (6-00). Retrieved June 8, 003 from htt:// [7] Intel Annual Reorts (998-00). Retrieved June 8,003 from htt:// [8] JCPenney Annual Reorts (999-00). Retrieved June 7, 003 from htt:// [9] Kelgs Annual Reorts (998-00). Retrieved June 8, 003 from htt:// [0] Microsoft Annual Reorts (996-00). Retrieved June 7, 003 from htt:// [] Motorola Annual Reorts (998-00). Retrieved June 7, 003 from htt:// [] Tyson Foods, Inc. Annual Reorts (995-00). Retrieved June 8, 003 from htt:// [3] UPS Annual Reorts (000-00). Retrieved June 9, 003 from htt:// [4] Walmart Annual Reorts ( ). Retrieved June 5, 003 from htt:// [5] Walt Disney Annual Reorts (995-00). Retrieved June 5, 003 from htt://disney.go.com/cororate/investors/shareholder/faq_investmentlan.html 0

11 Bibliograhy (continued) [6] Big Charts Historical Quotes. (003). Retrieved June 30, 0003 from htt://bigcharts.marketwatch.com/historical/ [7] Federal Reserve Bank of St. Louis Economic Database. (003). Retrieved June 6, 003 from htt://research.stlouisfed.org/fred/ [8] Johnson, Richard. Wichern, Dean. Alied Multivariate Statistical Analysis: Fourth Edition. Prentice Hall, Inc: New Jersey. 998.

12 Aendix.A Data for 3M Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Poor Good Good Poor Poor Good Aendix.B Data for Boeing Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Poor Poor Good Good Poor Poor

13 Good Aendix.C Data for Comuter Associates Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Poor Good Poor Good Poor Good Good Aendix.D Data for Ford Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Poor Poor Poor Poor 3

14 Aendix.E Data for Hershey Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Poor Good Good Poor Good Aendix.F Data for IBM Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Poor Good Poor Poor Good Poor 4

15 Aendix.G Data for Intel Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Poor Poor Good Good Aendix.H Data for JCPenney Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Poor Poor Good Poor 5

16 Aendix.I Data for Kelg Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Good Good Poor Poor Aendix.J Data for Microsoft Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Poor Poor Good Poor Poor Good Good 6

17 Aendix.K Data for Motorola Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Poor Poor Poor Good Aendix.L Data for Tyson Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Poor Poor Poor Poor Poor Good Poor Good 7

18 Aendix.M Data for UPS Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Good Poor Aendix.N Data for WalMart Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Poor Good Good Good 8

19 Aendix.O Data for Walt Disney Diluted EPS sqrt PPE NI Unemloyment DJIAM Grou Good Poor Poor Good Good Poor Good Good Aendix.A Linear Correlation Values for Normality Test Samle Correlation Variables Coefficient Diluted Earnings 0.99 Log Net Income Unemloyment 0.98 Federal Funds 0.9 DJIAM Consumer Investments Consumer Sending 0.98 Inflation 0.98 sqrt(price er Earnings) Note: Critical Value: r =.977 Significance Level: α =. 0 9