Joint Distribution of Stock Market Returns and Trading Volume

Transcription

1 Rev. Integr. Bus. Econ. Res. Vol 5(3) 0 Joint Distribution of Stock Market Returns and Trading Volume Muhammad Idrees Ahmad * Department of Mathematics and Statistics, Sultan Qaboos Universit, Muscat, Oman Amadou Sarr. Department of Mathematics and Statistics, Sultan Qaboos Universit, Muscat, Oman ABSTRACT Bivariate normal distribution is fitted to monthl stock market returns and trading volume data from Muscat Securities Market (MSM). Marginal and conditional distributions are derived. The graphs of the fitted distributions of both the joint and marginal distributions appear to be normal. The empirical joint probabilities are worked out which show a significant joint movement of the volume and returns. The results indicate that the joint probabilities are higher when the trading volume is close to monthl average and the returns are positive. The marginal probabilities show that the returns are positive with probabilit of 0.43 and it will occur on average with a return period of two months. In addition the conditional epectation of the returns indicates a significant correlation of volume with the returns. Kewords: Bivariate normal distribution; Trading Volume; Stock market returns; Marginal Distributions; Conditional distributions; conditional epectation.. INTRODUCTION The relationship between trading volume and stock returns has been the main issue of theoretical and empirical research for a long time. The markets have to be aware about trading volume which reflects how the informed traders and uniformed traders interact with each other in the marketplace (Fauzia Mubarik & Attia Y. Javid, 009). This stud attempts to eplain the effect of trading volume on the probabilit of the stock returns of Muscat Securit Market. The nature of the stock return distribution has been ver vigorousl investigated in a univariate framework (Richard et. Al.,00; Kevin. S. 03). Trading volume which is the total number/quantit of stock contracts sold during a trading da is a ver strong indicator of the market activit. This might significantl affect the shape of the distribution of returns and consequentl the probabilities of the return. Therefore relationship between the stock return and trading volume as determined b their joint distribution needs to be investigated. This stud focus on the empirical joint distribution of trading volume and stock returns assuming that the bivariate normal distribution would adequatel represents this joint process. Copright 06 GMP Press and Printing ( ISSN: (Online); (CDROM); (Print)

2 Rev. Integr. Bus. Econ. Res. Vol 5(3). DATA AND DESCRIPTIVE STATISTICS The Muscat Securities Market (MSM) was established b the Roal Decree (53/88) issued onst June 988 to regulate and control the Omani securities market and to participate, effectivel, with other organizations for setting up the infrastructure of the Sultanate's financial sector. The main inde of Muscat Securities Market has been established in 99. A number of companies included in the inde sample have changed overtime to reach currentl 30 companies, the most liquid in the market. Five ears monthl data on the MSM inde in the form of log differenced returns () and the trading volume () as measured b log of the number of shares traded was taken from the official website of the MSM. The trading volume could be measured either b number of stocks traded or b the value of the stocks traded. We have taken the number of shares as the trading volume, because the number of shares better shows the market activit. The summar statistics and the histograms of the univariate distributions of and are presented in figures and. For the logarithm of the number of shares data, the mean is.737 with standard deviations 0.58.The distribution of these data is normal because the Anderson-Darling test of normalit gives p-value >.05. For the returns data, the mean is with standard deviations The univariate distribution of these data is also normal because the Anderson- Darling normalit test gives p-value >.05. The Pearson correlation of volume and return was calculated to be which was highl significant at % level. This makes it necessar to investigate the joint distribution because the probabilities of returns based on marginal distribution of stock returns would be misleading in the presence of significant correlation. Fig: Summar statistics for Volume() Anderson-Darling Normalit Test A-Squared 0.46 P-Value Mean.737 StDev 0.58 Variance Skewness Kurtosis N 58 Mean 95% Confidence Intervals Minimum.453 st Quartile.307 Median.683 3rd Quartile 3.44 Maimum % Confidence Interval for Mean Median Copright 06 GMP Press and Printing ( ISSN: (Online); (CDROM); (Print)

3 Rev. Integr. Bus. Econ. Res. Vol 5(3) Fig: Summar statistics of returns() Anderson-Darling Normalit Test A-Squared 0.4 P-Value Mean StDev Variance Skewness Kurtosis N 58 Mean 95% Confidence Intervals Minimum st Quartile -.65 Median rd Quartile.7634 Maimum Median 95% Confidence Interval for Mean BIVARIATE NORMAL DISTRIBUTION The bivariate normal distribution is used in this section to find the joint probabilit distribution of correlated returns of the share price () and trading volumes(). The joint pdf, the marginal and conditional distributions are as below (Wilks, D. S. 006). ( ) ( )( ) ( ) f µ ρ µ µ µ, ) = ep + π σ ( ) ( ) σ ρ ρ σ σ σ <, < () Where: µ = E( ) σ = V ( ) µ = E( ) ρ = E [( µ )( µ )] σ σ σ = V ( ) 3. MARGINAL DISTRIBUTIONS The marginal distributions are derived b integrating the joint distribution over other variable and are given as below: ( ) ( µ ) f = ep < < () πσ σ f ( ) ( µ ) = ep < < πσ σ ( µ, σ ) Y ~ N( µ, σ ) ~ N (3) 3. CONDITIONAL DISTRIBUTIONS The conditional probabilit distribution of Y given X can be derived b dividing the joint distribution of and with marginal distribution of and is epressed as follows: Copright 06 GMP Press and Printing ( ISSN: (Online); (CDROM); (Print)

4 Rev. Integr. Bus. Econ. Res. Vol 5(3) 3 ( µ ) f ( ) = ρσ ep ( ) ( ) µ + πσ ρ ρ σ σ < < Where ( µ ) ρσ Y = ~ N µ +, σ ( ρ ) σ (4) Using these conditional distributions we can work out the conditional epectation as below: E( ] = f( )d = µ + ρσ σ ( μ ) var[ ] = σ ( ρ ) (5) Then the conditional epectation is linear with β 0 = µ ρσ μ σ and β = ρσ σ This implies that the regression model would be: = β 0 + β + ε and E( ]=β 0 + β where E(ε) =0, var(ε) = σ This justifies the use of regression analsis when the volume as a regressor is also a random variable. (Koed, A. 008) 4. JOINT PROBABILITIES The empirical joint probabilities are computed on the same principle of single variable. A bivariate frequenc classification table was first constructed b taking suitable class limits of returns and volume. Then the joint class frequencies were counted and the frequencies in the limits of row i and column j of the table is defined as the joint empirical probabilit function of the two random variables and is estimated b (Yue, et al.,990): f i, j = n ij (6) N+0. Where N is the total number of observations, and n ij is the number of occurrences of the combinations of i and j. The mid points of class limits are denoted b i and j. Theoretical joint probabilities are estimated using Eq. () b replacing parameters b their respective moment estimates and mid points b i and j. The empirical joint distributions estimated from the bivariate table are presented in table (3) along with the bivariate frequencies. This table also contains the theoretical joint probabilit worked out using the bivariate normal distribution in the brackets. The Copright 06 GMP Press and Printing ( ISSN: (Online); (CDROM); (Print)

5 Rev. Integr. Bus. Econ. Res. Vol 5(3) 4 marginal distribution of and for the observed data and the theoretical marginal distribution are also presented in this table. This table shows that the theoretical probabilit is higher than the observe one for the marginal probabilit of the trading volume. In contrast, the theoretical probabilit is smaller than the observe one for the marginal probabilit of the return. We ma need to Table#: Bivariate frequenc, empirical joint probabilities along with theoretical bivariate normal probabilities. The numbers inside the brackets show the theoretical bivariate normal probabilit. Mid Mid Marginal probabili t (0.0033) 0.07 ( ) ( ) (0.035) (0.033) (0.8773).3 ( ) (0.059) ( ) ( ).8 (0.005) 0.07 ( ) (0.0700) ( ) ( ) ( ) (0.0354) ( ) 0.07 (0.060) 0.07 ( ) ( ) ( ) ( ) 0.07 ( ) (0.9068) Marginal probabilit (0.0039) 0.03 ( ) 0.43 ( ) ( ) (0.0353) 0.07 ( ) choose an improved method of estimation of empirical distribution since the method we are using is biased. 5. CONDITIONAL PROBABILITIES The conditional probabilit of returns given trading volume is estimated based on equation(4) and is shown in Table (4 ).These probabilities are estimated for selected values of volume. Table# The conditional probabilit of return of share price inde given trading volume f( =3.3) f( =.3) f(,) Copright 06 GMP Press and Printing ( ISSN: (Online); (CDROM); (Print)

6 Rev. Integr. Bus. Econ. Res. Vol 5(3) 5 We then eamined the return(r)-volume(v) relationship in regression framework. Since the volume is also a random variable there we used the conditional epectation and the fitted regression results are are below: The estimated regression equation is E(r t v)= V t (7) Predictor Coef SE Coef T P Constant V S = R-Sq = 3.4% R-Sq(adj) =.8% It is clear from this equation that the epected returns would be negative when the volume is low (Timoth J. Brailsford, 994). 6. CONCLUSION This stud investigates the relationship between stock returns and trading volume based on the monthl data of Muscat Securit Market from 009 to 03. The empirical results verif that there is a significant interaction between trading volume and returns. The results indicate that the joint probabilities are higher when the return is positive and trading volume is close to the average. That is the return is higher when the trading volume is between about 0 to 60 million. The marginal probabilities show that the returns are positive with probabilit of 0.43 and it will occur on average after two months. In addition, the cumulative probabilities indicate the returns are positive with higher probabilities when the trading volume is higher than 60 million, and it would occur on average after 3 months. Various regression models were fitted to investigate the relationship between returns and the trading volume. All these models showed that the trading volume affects the return significantl. However the coefficient of determinations was not ver high. This ma be due to some other problems of regression such as stationarit and autocorrelation which needs to be further investigated. REFERENCES [] Ainde, Kaode. (008). Performances of Some Estimators of Linear Model when Stochastic Regressors are Correlated with Autocorrelated Error Terms. European Journal of Scientific Research;Ma008, Vol. 0 Issue 3, p558. [] Fauzia Mubarik & Attia Y. Javid.(009). Relationship between Stock Returns, Trading Volume and Volatilit Evidence From Pakistani Stock Market. Asia Pacific Journal of Finance and Banking Research. Vol. 3. No. 3. pp-7. [3] Kevin. S. (03) A Stud of the Relation Between Market Inde, Inde Futures and Inde ETFs: A Case Stud of India. Rev. Integr. Bus. Econ. Res. Vol (). [4] Richard D. F. Harris and Coskun C. K. (00) The Empirical Distribution of UK and US Stock Returns. Journal of Business Finance & Accounting, 8(5) & (6), June/Jul 00, X. Copright 06 GMP Press and Printing ( ISSN: (Online); (CDROM); (Print)

7 Rev. Integr. Bus. Econ. Res. Vol 5(3) 6 [5] Timoth J. Brailsford.(994). The empirical relationship between the trading volume, return and volatilit. [6] Wilks D. S. (006). Statistical Methods in Atmospheric Sciences. Elsevier Inc. UK. [7] Yue, S. Ouarda, T. Bobe,B. Legendre,P Bruneau, P. (990). The Gubmel Mied Model for flood frequenc analsis. Journal of Hdrolog.6, Copright 06 GMP Press and Printing ( ISSN: (Online); (CDROM); (Print)