Some Evidences of Power-Law Distribution in

Indian Capital Market

DEBASIS BAGCHIBENGAL ENGINEERING AND

SCIENCE UNIVERSITY, SHIBPUR

PURPOSE

TO EXAMINE :

1.     Whether a power-law distribution emerges at high market capitalization of the firms and how the distribution behaves for the firms with relatively lower market capitalization.

PURPOSE (contd..)

2.   How exponent of power-law distribution behaves across different levels of firms’ performance in the capital market over time.

MODEL

Pareto Distribution can be expressed as :

xn = C . n - 1/γ

where xn is the wealth of the nth ranked

individual, C is a constant while is

the exponent.

MODEL

ln xn = ln (C ) – (1/γ) ln n • where, xn is the market capitalization of the firm with rank n ln xn = a + b ln n + • can be used as a regression equation to estimate the value of the exponent γ

DATA

ET 500 Database for highest market capitalization of the top 500 companies, listed in The National Stock Exchange of India, are used for the investigation.

REGRESSION RESULTS

Firms’ Market Capitalization Belonging To Top 10% March 2004 August 2003 March 2003 a 11.853

(199.85) * 11.397 (224.591) *

5.00 (138.84) *

b -0.843 (-43.958) *

-0.847 (-51.371) *

-0.973 (-36.47) *

R2 .976 .983 .966 adj R2 .975 .982 .965 1.186 1.181 1.028 ADF Statistic

-3.252* -2.859* -4.153*

t-value in parenthesis * significant at 1% level.

Regression line for top 10% firms in 2004

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

8

9

10

11

12

5 10 15 20 25 30 35 40 45 50

Res idual Ac tual Fitted

Regression line for top 10% firms in 2004

Regression line for top 10% firms of August 2003

-0 .4

-0 .2

0 .0

0 .2

0 .4

7

8

9

10

11

12

5 10 15 20 25 30 35 40 45 50

R es id ua l Ac tua l Fitted

R eg re s s ion l ine fo r to 10 % fi rm s in 2 003

Value of Power –law Exponent at High Wealth Level

March 2004 August 2003 March 2003 Market capitalization of the top (.)% firms

Exponent

Exponent

Exponent

2 1.333 1.408 1.404 3 1.287 1.221 1.199 4 1.328 1.211 1.164 5 1.346 1.239 1.155 6 1.325 1.241 1.155 7 1.304 1.233 1.151 8 1.269 1.230 1.107 9 1.227 1.205 1.104

VALUE OF EXPONENT ( ) FOR FIRMS’ RANKED BETWEEN VARIOUS LEVELS OF MARKET

CAPITALIZATION

Ranks between March 2004 August 2003 51—150 0.659 0.754 151—250 0.631 0.688 251—350 0.535 0.657 351—450 0.505 0.589 451—500 0.468 0.505

VALUE OF FOR POSITIVE & NEGATIVE GROWTH FIRM

(DYNAMIC STATE)

High ( > 200%) growth of Market Negative growth in Market Capitalization of firms Capitalization of firms in 2004 over 2003 in 2004 over 2003 EXPONENT

March 2004

August 2003

March 2004

August 2003

0.797 0.796 0.702 0.805

DISCUSSION

THE VALUES OF EXPONENT AT HIGH WEALTH LEVEL COMPARE FAVOURABLY WITH THE EXPONENT’S VALUE FOR AN EMERGING PARETO DISTRIBUTION OF INDIVIDUAL WEALTH IN THE ECONOMY.

VALUES OF THE EXPONENT GRADUALLY REDUCE

AS WE GO DOWN THE RANKS OF THE FIRMS. THE REDUCTION IN VALUE IS CONSISTENT WITH OTHER RESEARCH FINDINGS.

DISCUSSION ( Contd…)

FASTER GROWTH RATE OF THE FIRMS CHANGES THE VALUE OF EXPONENT OF THE WEALTH DISTRIBUTION VERY MARGINALLY, ALTHOUGH THE PAIRED SAMPLE T-TEST CONFIRMS THAT THERE IS SIGNIFICANT DIFFERENCE IN THE RANKS BETWEEN BEFORE AND AFTER THE GROWTH PERIOD. REASON FOR STATIONARITY ???

DISCUSSION ( Contd…)

THERE HAS BEEN HIGHER DEGREE OF

CHANGE OF THE EXPONENT’S VALUE FOR THE FIRMS WITH DECLINING GROWTH. THE INDEPENDENT SAMPLES T-TEST AFFIRMS THAT THE DIFFERENCE IN RANKS BETWEEN GROWING AND DECLINING MARKET CAPITALIZATION OF THE FIRMS IS STATISTICALLY SIGNIFICANT AND HIGHER RANK DIFFERENCES OF THE DECLINING FIRMS LEAD TO LOWERING OF THE EXPONENT’S VALUE.

CONCLUSION

The results show :

1. Power-law distribution emerges at high wealth level of the firms in Indian capital market. The value of exponent compared well with respect to wealth distribution of the individuals in the economy.

2. Power-law becomes less conspicuous as we

go down the ranks.

CONCLUSION (Contd..)

3. The value of exponent does not change for high growth firms.

Stationarity of the distribution ???

4. The value of the exponents decreases for declining firms due to the fact that differential rank differences are high and statistically significant with respect to rank differences of high growth firms.

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