Low-dimensional Characterisation of Liquidity of Individual Stocks in the Indian Market

Abstract

There is no single, definitive and universally accepted measure of a stock’s ‘liquidity’. The literature proposes a large number of proxy measures for liquidity. Suitable low dimensional characterisation of liquidity may improve understanding and enable better market regulation. We present factor analyses of stock-specific liquidity measures, using cross-sectional data from the National Stock Exchange of India, for two time periods reflecting different market conditions. Day-wise analyses of eleven liquidity proxies suggest five factors, interpretable as spread, depth, volume, price elasticity and relative activity. Consistency in the results obtained from the two periods suggests that the factor structure is stable.

JEL Classification: G11, C38

Keywords

Liquidity stock market factor analysis National Stock Exchange of India

Introduction

Liquidity is a multi-faceted property of an asset that is easy to intuitively understand but difficult to quantify. Many proxy measures of liquidity have been proposed in the literature in the context of stock markets: Aitken and Comerton-Forde (2003) report 68 extant measures. These include traded volume, volatility, bid-ask spreads, and so on.

How many important underlying dimensions or facets does liquidity really have? How can these be described? Do these facets remain consistent under varying market conditions? In addressing these or related questions, however, a distinction must be made.

Liquidity can be studied both as a market-wide averaged quantity, as well as a security-specific quantity. In the first line of investigation one might ask, is the market today more or less liquid than it was yesterday and in what ways? The answer has implications for policy makers who are looking at macroeconomic trends in the market. In the second line of investigation, one might ask, is stock A more or less liquid than stock B and in what ways? The answer has implications, e.g., for portfolio choices by individual traders and might aid stock exchanges in fine-tuning stock indices or determining margin rates for specific securities.

Several empirical studies of the first issue above (market-wide liquidity) have appeared in the literature, starting with Chordia et al. (2000). In the present work, we consider the second issue. While several proxy measures of liquidity have been proposed and used in numerous studies, the question of their underlying dimensionality or reduction thereof has not been addressed, particularly in the context of specific securities.

The primary aim of this work is to examine underlying dimensions or facets of security-specific liquidity proxies. Specifically, we conduct a ‘factor analysis’ of several proxy variables using intra-day and daily data from the National Stock Exchange (NSE) of India. Our aim is to empirically identify a few underlying factors and to check their consistency across varying market conditions. To this end, we use data for more than a hundred stocks from two different three-month periods: one near the start of a long bull run, in 2004 and the other just preceding the bear market of 2008 (see Figure 1).

We conduct factor analysis for 11 liquidity proxies computed from daily data collated from the trade information and snapshots of the limit order book. Separate factor models are built for each trading day. The methodology followed has been described in detail in the fourth section. We find that five factors emerge consistently with the same liquidity proxies loading on the same factors on most days. Moreover, there is a strong correspondence between the factors, as well as their relative importance, across these two periods.

One fundamental aspect of our cross sectional study is that we study all stocks simultaneously on a given day, then independently for the next day, and so on. We do not follow sequential changes in liquidity measures over time, either within the day or from day to day. Consequently, temporal aspects of liquidity, like resiliency and immediacy, are not directly incorporated in our study. However, proxies for these are included, in the form of volume and frequency of trading. We will discuss this again at the end of the article.

The rest of this article is organised as follows. The second section presents a brief review of the relevant literature; the third section gives a brief outline of the theoretical background; the fourth section describes the data used and the methodology adopted; and the fifth section presents empirical findings and related discussion. Finally, the sixth section concludes the article.

Proxy Measures of Liquidity: A Review of the Literature

Black (1971) defined a liquid market as one that is continuous (small trades can be made instantly) and efficient (large trades can be made slowly, without much loss). Kyle (1985) proposed three transactional properties implicit in liquidity: tightness, depth and resiliency. More recently, many different proxy measures of liquidity have been proposed and used. A review of commonly used proxy measures may be found in Aitken and Comerton-Forde (2003).

In this light, our study is motivated by the following question. If a stock has Kyle’s tightness, does it automatically have depth in appropriate proportion? Or is there meaningful independent variation among the various liquidity proxies corresponding to each stock, when we view a collection of stocks at a given time? And finally, can this variation be captured from among several liquidity proxies in a suitable low dimensional way?

In the context of dimensional reduction, some authors, viz., Chordia et al. (2001), Hasbrouck and Seppi (2001) and Huberman and Halka (2001), used principal component analysis in studies of commonality in liquidity. Lesmond (2005) performed factor analysis on five liquidity measures across 31 emerging markets and found that a single factor represents the common variation in all of the liquidity measures examined. Chollette et al. (2007, 2008) conducted factor analyses on market-wide averaged quantities, using data from the Oslo Stock Exchange (2007) and US stocks (2008), with a view to understanding risk premiums and components thereof. These are all studies of market wide liquidity measures.

There seems to be no study devoted to stock-specific factor analysis of liquidity proxies using cross sectional data and this is the main focus of the present article.

In what follows we list the sources of some of the liquidity proxies used or referred to in our study. Trading volume refers to the number of shares traded daily (Brennan et al. 1998; Chordia et al. 2000, 2001). Turnover, the monetary value of shares traded daily, was used by Rouwenhorst (1999), Domowitz et al. (1997), Bekaert et al. (2003) and Levine and Schmukler (2003) among others. Aitken and Comerton-Forde (2003) mention turnover ratio, defined as the total number of traded shares divided by the number of outstanding shares, used by the International Federation of Stock Exchanges (FIBV). This is also used by Jun et al. (2003) and Beck and Levine (2004). Frequency of trading refers to the number of trades for a stock in a day of trading and has been used by Lehmann and Modest (1994) and Fleming (2001).

Kyle (1985) introduced the concept of ‘price impact’ or ‘market impact cost’ of a single trade calculated from the formula: P(1)y where P is the price paid, μ is the mean price, y is the trade size and λ is the market impact cost. A similar but more aggregated measure for price impact is ‘liquidity ratio’ defined as the ratio of monetary volume of trading to the price change over an interval (Cooper et al. 1985; Grossman and Miller 1988; Kluger and Stephan 1997; Muscarella and Piwowar 2001).

Bid-ask spread is a popular proxy for liquidity and has been discussed by numerous authors. In this study we use relative bid-ask spread, defined as the difference between bid and ask prices, divided by the midpoint of the same. This proxy was discussed by Amihud and Mendelson (1986, 1989), Brockman and Chung (2002) and Acharya and Pedersen (2005).

Immediacy is approximately represented within our study as age of outstanding orders, computed from the limit order book, following Dalvi et al. (2005). Order depth is defined by Aitken and Comerton-Forde (2003) as the total volume of all orders in the order book divided by the total number of shares on issue. Market depth, defined as the sum of the monetary value of ‘buy’ and ‘sell’ orders submitted at the best bid and ask prices, is used here following Goldstein and Kavajecz (1999), Engle and Lange (2001) and Chordia et al. (2001). Price elasticity is represented as the slope of the order book which reflects the depth available at each price level in the order book: see Biais et al. (1995), Danielsson and Payne (2001) and Næs (2005).

The Factor Analysis Model

Factor analysis aims to approximate the covariance relationships among many variables in terms of a few underlying, but unobservable, random quantities called factors (Johnson and Wichern 2002). The factor analysis probability model is described below.

We consider a p × 1 random vector X with mean vector μ and covariance matrix Σ. The factor analysis model is given by: X(2)F where Λ = [λ_ij] is a p × m matrix of factor loadings (‘L’ in Johnson and Wichern 2002)

F = (F_1,, F_2, .........., F_m)' is a vector of m unobservable common factors and

ε = (ε_1, ε_2, .........., ε_p)' is a vector of p specific factors.

The coefficient λ_ij of F_j for X_i is called the loading of the jth factor on the ith variable.

Here we use the orthogonal factor model where F and ε are assumed to satisfy the following: E(3)FE F(4) (5) Cov(ε) = Ψ

where in turn Ψ is diagonal, with elements Ψ_i for i = 1,…, p. Finally, F(6)F

Under these assumptions the covariance matrix of X can be expressed as follows: X(7)^T

Var (X_{i}) = \sum_{j = 1}^{m} λ_{i j}^{2} + Ψ_{i}

(8)

The first term on the right-hand side describes the total variance of X_i explained by the m common factors and is called communality, while Ψ_i is called uniqueness.

Implicit in the above discussion is the fact that the elements of X are individually scaled to unit variance. Moreover, the matrix Λ is determined uniquely only up to multiplication by an arbitrary orthogonal matrix, referred to as a factor rotation; we have used the ‘varimax’ rotation (Johnson and Wichern 2002). Finally, we mention that the parameters in the model are numerically sought using maximum likelihood estimation and the iterative solution algorithm does not always converge, that is, a solution is not always obtainable.

We now consider the number m of factors to extract. In our exploratory analysis, we do not know the number in advance. The choice of m is subjective and is guided by the level of simplification desired, interpretation of the underlying factors and finally a statistical test of the adequacy of the m factor model as outlined in the following.

In testing the goodness of fit of the FA model, the null hypothesis is that the covariance matrix has a certain structure (Equation [7]). The alternative hypothesis is that it has no special structure and for large samples follows a multivariate normal distribution. A likelihood ratio statistic is computed as:

L i k e l i h o o d R a t i o = \frac{m a x i m i z e d l i k e l i h o o d u n d e r H_{0}}{m a x i m i z e d l i k e l i h o o d u n d e r H_{a}}

(9)

A chi-square test is used to accept or reject the null hypothesis (Johnson and Wichern 2002).

In the present work, the number of factors used was increased sequentially until rejection of the null hypothesis was no longer indicated (p-value sufficiently small). The ‘Factor Analysis of Day-wise Data’ section gives further details. All computations in this article were done using the freely available statistical analysis software package ‘R’ (http://www.r-project.org/).

Methodology

Data

The data used for this analysis was obtained from the National Stock Exchange of India (NSE), India’s largest equity market. The data include the following:

summarised daily information for each security

four daily snapshots of the Limit Order Book: at 11 am, 12 noon, 1 and 2 pm

detailed information on every single trade of each security on each day

The summary data has daily information on the opening price, closing price, high price, low price, last traded price, total number of shares traded, total value of shares traded and the total number of trades for each security listed on the NSE. The snapshots of the LOB have information on the kind of order (buy or sell), quantity, price, time of order placement and several flags indicating the day, price and quantity preferences of the trader.¹ The trade database has information on the time, price and quantity traded for each transaction.

Time Periods of the Study

The movement of the NIFTY² index for six years is given in Figure 1. The two time periods chosen for the study are shown using vertical lines. The first period, 1 September to 30 November 2004, was at the beginning of a long bull run. The second period, 4 December 2007 to 29 February 2008, was at the beginning of a bear run that continued well into 2009. The analyses presented below are conducted separately and independently for the two time periods (as well as independently for each day within each time period) to examine the consistency of common liquidity factors in bull and bear phases.

Figure 1

Source: Developed by the authors.

Sample Selection

For the first period in the study, data is collected for a sample of 124 stocks traded on the exchange. In the second period, a sample of 127 stocks is selected. These samples include all the 50 stocks in the NIFTY index in each period and several other stocks of smaller companies. Thus, we cover a wide range of stocks from large cap to small cap stocks.

There is imperfect overlap between the two sets of stocks (106 stocks were included in both sets). Some of the stocks from the first period were no longer traded in the second period of our study; some others, which were NIFTY constituents during the first period, were no longer NIFTY constituents during the second period.³ The companies are further divided into two groups of roughly equal numbers based on market capitalisation, referred to as ‘large cap’ and ‘small cap’ companies.⁴ Our division is based on the market capitalisation of the stocks in our sample at the time of interest and is motivated by the common perception that liquidity characteristics of large and small companies differ significantly.

Proxy Measures Used

Eleven proxy measures are used in this study, based on popularity among other authors as well as ease of computation given available data. These are given below in Table 1. The trade-based measures (measures 1 to 6 in Table 1) are all extracted or computed directly from the summarised daily data. Measures derived from the limit order book (measures 7 to 11) are computed for buy and sell side orders separately. Measure 11 or price elasticity is taken here as slope of the order book, as mentioned in the ‘Proxy Measures of Liquidity: A Review of the Literature’ section. Computation of the slope is outlined in Appendix 1. The measures derived from four snapshots (at four times of the trading day) are averaged to arrive at a daily value (removing any time-of-day effect).

Summary Statistics

We now present some summary statistics for the proxy measures for large and small cap companies, separately for the two time periods. The summarised measures differ across company groups and across time periods as well, as we discuss below. It may be seen that the standard deviation of some of the measures such as the firm size, turnover, daily traded volume and number of trades are larger than the mean values. This indicates that there is a non-symmetric distribution for these measures, with a long right tail.

Table 1

Proxy Measures Selected for the Study

No.	Measure	Definition	Variable Name
Trade-based Measures
1	Trading Volume	Number of shares traded daily	trdvolume
2	Turnover	Monetary value of shares traded daily	turnover
3	Turnover ratio	Ratio of number of traded shares to number of issued shares	turnoverrate
4	Price range	(High price – low price)/Average Price	pricerange
5	Liquidity ratio	Absolute value of price change/traded volume for a day	liqratio
6	Frequency of trading	Number of trades for a stock	notrades
Order-based Measures
7	Relative bid-ask spread	Bid-ask spread divided by midpoint price	relbidask
8	Immediacy	Age of non-executed orders at the time of the snapshot	davgagebuy/davgagesell
9	Order depth	Volume of all orders/Total number of issued shares	dborderdepth/dsorderdepth
10	Market depth	Monetary value of buy and sell orders at best bid and ask price	dbmktdepth/dsmktdepth
11	Price elasticity of orders	Slope of the order book at the snapshot	dbelast/dselast

Source: Developed by the authors.

Table 2 shows that firm size and trading volumes (measured by turnover, number of trades, volume of shares traded, etc.) grew significantly from the first period to the second. In contrast, some other liquidity proxies such as liquidity ratio, order depth, market depth and price elasticity seem to suggest that small cap companies are less liquid in the second period compared to the first. For instance, order depth has decreased unequivocally. Price elasticity has increased for both large and small cap stocks and for both buy and sell orders. Similarly, the average age of orders in the snapshot has increased. Still other proxies, such as bid-ask spread and market depth, show no clear trend across periods.

Table 2

Summary Statistics—Large and Small Cap—Period 1 and 2

	Period 1—1 September 2004 to 30 November 2004				Period 2—4 December 2007 to 29 February 2008				Growth from Period 1 to Period 2
	Large Cap Companies		Small Cap Companies		Large Cap Companies		Small Cap Companies
		Mean	Standard Deviation	Mean	Standard Deviation	Mean	Standard Deviation	Mean	Standard Deviation	Large Cap	Small Cap
Firm size (in ₹ million)	69,163	84,964	2,814	2,886	242,853	298,772	17,631	13,156	251%	527%
Turnover (in ₹ million)	460	632	24	41	1,353	1,593	125	146	194%	417%
Daily trade volume (number of shares)	1,710,684	3,934,177	340,922	711,393	2,823,695	5,464,524	758,945	1,656,129	65%	123%
Number of trades	12,225	12,719	1,724	2,504	26,342	27,483	5,966	7,699	115%	246%
Price range (in ₹)	3.06	0.67	4.87	2.05	6.04	1.17	7.08	1.10	98%	45%
Turnover ratio	0.00888	0.01057	0.00667	0.00690	0.00713	0.00786	0.00700	0.00811	–20%	5%
Liquidity ratio	0.00015	0.00056	0.00814	0.04579	0.00007	0.00018	0.02888	0.16987	–50%	255%
Relative bid ask spread (ratio)	0.00139	0.00067	0.01192	0.01538	0.00204	0.00039	0.00644	0.00690	47%	–46%
Average age of buy orders (secs)	4,222	790	2,230	1,536	4,653	329	3,611	1,235	10%	62%
Average age of sell orders (secs)	4,681	849	2,646	1,794	5,093	310	3,866	1,340	9%	46%
Order depth (buy orders)	0.00060	0.00061	0.00155	0.00126	0.00046	0.00038	0.00085	0.00054	–24%	–45%
Order depth (sell orders)	0.00098	0.00106	0.00240	0.00179	0.00061	0.00069	0.00113	0.00103	–38%	–53%
Price elasticity of buy orders	–0.00092	0.00261	–0.01282	0.08363	–0.00137	0.00303	–0.03213	0.16744	49%	151%
Price elasticity of sell orders	0.00058	0.00181	0.01002	0.06425	0.00118	0.00282	0.06474	0.39298	103%	546%
Market depth (No. of buy orders)	720	737	952	1411	961	966	685	1,460	33%	–28%
Market depth (No. of sell orders)	697	612	1,132	2,075	853	811	686	1,398	22%	–39%

Source: Developed by the authors.

The two periods considered in this study therefore seem to indeed represent quite different market conditions. In this light, we will be interested in the degree to which factor analysis results are (or are not) consistent across the two time periods.

Factor Analysis of Day-wise Data

We compute the values for different liquidity proxies for each stock for each trading day in both periods. Data on X_i is cross-sectional, that is, each X_i is a vector of values for the ith liquidity proxy computed for n stocks for a particular day. Analyses are conducted separately for large cap and small cap stocks and within each group, separately for the buy and sell side proxy measures. For each day for each of these four groups we build separate FA models.

Since the underlying structure of correlations in the data is not known a priori, the factor analysis is exploratory as opposed to confirmatory. The number of latent factors is called ‘m’. We use different values of m to arrive at an optimum and consistent factor structure. Three criteria built into ‘R’, namely, the Cattell Scree plots, the Kaiser Criterion and the Parallel Analysis Criterion, are used for an initial guide on the optimal number of factors. Figure 2 shows a sample scree plot generated by R indicating the optimum number of factors to be four.

Figure 2

Source: Developed by authors.

Table 3

Typical Factor Loading Matrix

	Factor 1	Factor 2	Factor 3	Factor 4	Factor 5
01 trdvolume	0.50	0.53	0.04	0.01	0.41
02 turnover	0.91	0.08	0.08	–0.09	–0.02
03 turnoverrate	0.43	0.85	0.10	–0.04	–0.02
04 pricerange	0.04	0.06	–0.02	0.70	0.02
05 liqratio	–0.08	–0.16	–0.54	0.58	–0.02
06 notrades	0.93	0.32	0.12	–0.11	0.00
07 drelbidask	–0.35	0.03	–0.46	0.49	0.20
08 davgagebuy	0.22	0.08	0.35	–0.61	0.16
09 dborderdepth	0.00	0.64	0.07	0.00	0.35
10 dbmktdepth	–0.02	0.23	0.01	–0.04	0.97
11 dbelast	0.06	0.10	0.97	–0.19	0.05

Source: Developed by the authors.

‘R’ uses maximum likelihood factoring which generates a chi-square goodness-of-fit test for testing the number of factors needed to describe the correlations in the model. We increase the number of factors one at a time (2, 3, 4, 5) until a satisfactory goodness of fit (p value ≤ 0.05) is obtained. We then test the consistency in the factor structure across several days. It is found that a five-factor model meets the goodness of fit criteria on more days and also provides a more consistent structure. A typical factor loading matrix generated is shown in Table 3. In this study, a factor loading cut-off of 0.3 (in magnitude) is used to retain important variables.⁵ The factor loadings that are greater than 0.3 are marked in boldface in Table 3. A visual examination of Table 3 shows us that the important variables loading on each factor are as follows:

Factor 1: Variable numbers 2, 6, 1, 3 and 7

Factor 2: Variable numbers 3, 9, 1 and 6

Factor 3: Variable numbers 11, 5, 7 and 8

Factor 4: Variable numbers 4, 8, 7 and 5

Factor 5: Variable numbers 10, 1 and 9

Identification of Common Factors across Several Days

When the factor loading matrices for several days were visually compared, it was found that the factors emerging were similar to each other in terms of the variables loading on them. Similarity in the factor structure noted across days indicates that there are a few consistent underlying facets of liquidity. We present below in Table 4 four factor loading matrices (reporting only factor loadings > 0.3 in magnitude) to illustrate this.

Table 4

Four Factor Loading Matrices for NIFTY Companies—Buy Orders

28 February 2008	Factor 1	Factor 2	Factor 3	Factor 4	Factor 5
28 February 2008	A	D	B	C	E
01 trdvolume				0.91
02 turnover	0.90
03 turnoverrate		0.95
04 pricerange					–0.43
05 liqratio			–0.64
06 notrades	0.79			0.40
07 drelbidask	–0.37
08 davgagebuy					0.31
09 dborderdepth		0.83
10 dbmktdepth				0.64	0.42
11 dbelast			0.97

31 January 2008	Factor 1	Factor 2	Factor 3	Factor 4	Factor 5
31 January 2008	C	D	E	A	B
01 trdvolume	0.89			0.30
02 turnover				0.77
03 turnoverrate	0.41	0.54
04 pricerange			0.35
05 liqratio					–0.99
06 notrades	0.55	0.34		0.74
07 drelbidask			0.81
08 davgagebuy
09 dborderdepth		0.97
10 dbmktdepth	0.79
11 dbelast					0.46

1 February 2008	Factor 1	Factor 2	Factor 3	Factor 4	Factor 5
1 February 2008	A	C	D	B	E
01 trdvolume	0.31	0.92
02 turnover	0.98
03 turnoverrate	0.33	0.42	0.70
04 pricerange					0.61
05 liqratio				–0.76
06 notrades	0.72	0.57
07 drelbidask					0.46
08 davgagebuy					–0.67
09 dborderdepth			0.97
10 dbmktdepth		0.48
11 dbelast				0.74

10 January 2008	Factor 1	Factor 2	Factor 3	Factor 4	Factor 5
10 January 2008	A	B	D	E	C
01 trdvolume	0.78		0.34		0.51
02 turnover	0.80
03 turnoverrate	0.59		0.77
04 pricerange				0.63
05 liqratio		–0.99
06 notrades	0.95
07 drelbidask				0.64
08 davgagebuy				–0.37
09 dborderdepth			0.55
10 dbmktdepth					0.33
11 dbelast		0.96

Source: Developed by the authors.

To understand the commonality between factors emerging on different days, we do the following. In the first factor loading matrix, successive factors are arbitrarily labelled A, B, C, D and E. Subsequently obtained factor loading matrices are visually examined for matching with the original factors. We group the factors from different factor models that load on the same variables and give them a common name. For instance, we see that the first factor in the first factor loading matrix (arbitrarily named Factor A) is similar to the first factor in the 2nd factor loading matrix, the fourth factor in the 3rd factor loading matrix and the first factor in the 4th factor loading matrix. Similar visual matching was done for the other factors and the corresponding factors were named accordingly in each of the factor loading matrices.

Computation of an Averaged (RMS) Factor Loading Metric

While a visual examination allows us to identify certain commonalities in the factor structure across days, we need some way of quantifying this commonality. Towards this end, we compute a summarised metric that indicates how a certain variable loads on a certain factor ‘on average’. The computed metric should take into account the relative importance of each variable in each factor (given by the magnitude of the individual factor loading coefficients) as well as the consistency with which it appeared when FA was conducted over several days. This metric is a frequency weighted RMS factor loading⁶ and is computed as follows:

RMS Factor Loading = \sqrt{\frac{1}{K_{i j}} (\sum_{k = 1}^{{K^{'}}_{i j}} (λ_{i j k}^{2}))}

(10)

where λ_ijk = loading of the jth factor on the ith variable on the kth day.

K’ = number of days on which |λ_ij| > 0.3 (the cut-off used⁷)

K_ij = number of days for which a 5 factor model could be built with p value > 0.05⁸

${K^{'}}_{i j} \leq K_{i j}$

Results and Discussion

Numerical values computed for the average factor loading of each variable loading on different factors are given in Tables 5, 6, 7 and 8. There are eight groups based on three criteria—large cap and small cap, buy and sell, Period 1 and Period 2. We see from these tables that for each factor there are at least a couple of variables that load on them with a high factor loading. The variables that were found to have consistently high factor loading on the five factors identified are as follows:

Table 5

RMS Factor Loadings—Large Cap, Buy

Large Cap–Buy
V No	Variable	Sign	RMS Factor Loading (Period 1)	RMS Factor Loading (Period 2)
Factor A
2	turnover	+	0.91	0.85
6	notrades	+	0.88	0.70
1	trdvolume	+	0.46	0.47
3	turnoverrate	+	0.41	0.43
7	drelbidask	–	0.25	0.31
Factor B
11	dbelast	+	0.85	0.87
5	liqratio	–	0.75	0.87
7	drelbidask	–	0.50
8	davgagebuy	+	0.40
Factor C
10	dbmktdepth	+	0.93	0.41
1	trdvolume	+	0.63	0.59
9	dborderdepth	+	0.35
6	notrades	+		0.40
Factor D
3	turnoverrate	+	0.85	0.76
9	dborderdepth	+	0.77	0.86
1	trdvolume	+	0.40	0.59
6	notrades	+	0.36	0.47
10	dbmktdepth	+		0.30
Factor E
4	pricerange	+	0.62	0.67
7	drelbidask	+	0.37	0.42
8	davgagebuy	–	0.27	0.50

Source: Developed by the authors.

Table 6

RMS Factor Loadings—Large Cap, Sell

Large Cap–Sell
V No	Variable	Sign	RMS Factor Loading (Period 1)	RMS Factor Loading (Period 2)
Factor A
2	turnover	+	0.91	0.85
6	notrades	+	0.90	0.69
1	trdvolume	+	0.47	0.43
3	turnoverrate	+	0.44	0.34
7	drelbidask	–	0.27	0.37
Factor B
11	dselast	+	0.85	0.90
5	liqratio	+	0.83	0.87
7	drelbidask	+	0.48
8	davgagesell	–	0.38
Factor C
10	dsmktdepth	+	0.87	0.35
1	trdvolume	+	0.64	0.56
9	dsorderdepth	+	0.40
6	notrades	+		0.42
Factor D
9	dsorderdepth	+	0.82	0.89
3	turnoverrate	+	0.80	0.80
1	trdvolume	+	0.40	0.64
6	notrades	+	0.35	0.49
10	dsmktdepth	+		0.38
Factor E
4	pricerange	+	0.58	0.65
7	drelbidask	+	0.59	0.42
8	davgagesell	–	0.48	0.60

Source: Developed by the authors.

Table 7

RMS Factor Loadings—Small Cap–Buy

Small Cap–Buy
V No	Variables	Sign	RMS Factor Loading (Period 1)	RMS Factor Loading (Period 2)
Factor A
2	turnover	+	0.94	0.64
6	notrades	+	0.94	0.61
1	trdvolume	+	0.72	0.42
3	turnoverrate	+	0.67	0.50
8	davgagebuy	+	0.45
Factor B
11	dbelast	+	0.89	0.67
5	liqratio	–	0.92	0.62
Factor C
10	dbmktdepth	+	0.76	0.58
1	trdvolume	+	0.58	0.64
6	notrades	+		0.37
Factor D
3	turnoverrate	+	0.64	0.50
9	dborderdepth	+	0.60	0.64
4	pricerange	+	0.48	0.26
Factor E
7	drelbidask	+	0.74	0.61
8	davgagebuy	–	0.58	0.43
4	Pricerange	+	0.32	0.34

Source: Developed by the authors.

Table 8

RMS Factor Loadings—Small Cap, Sell

Small Cap–Sell
V No	Variables	Sign	RMS Factor Loading (Period 1)	RMS Factor Loading (Period 2)
Factor A
2	turnover	+	0.94	0.70
6	notrades	+	0.94	0.68
1	trdvolume	+	0.76	0.47
3	turnoverrate	+	0.63	0.55
8	davgagesell	+	0.35	0.18
Factor B
11	dselast	+	0.91	0.70
5	liqratio	+	0.94	0.68
Factor C
10	dsmktdepth	+	0.75	0.62
1	trdvolume	+	0.54	0.63
6	notrades	+		0.32
Factor D
3	turnoverrate	+	0.69	0.46
9	dsorderdepth	+	0.66	0.60
4	pricerange	+	0.49	0.13
Factor E
7	drelbidask	+	0.73	0.66
8	davgagesell	–	0.72	0.58
4	pricerange	+	0.33	0.33

Source: Developed by the authors.

Factor A: Variables 2, 6, 1 and 3 (turnover, number of trades, traded volume and turnover ratio)

Factor B: Variables 11 and 5 (buy/sell elasticity, liquidity ratio)

Factor C: Variables 10 and 1 (market depth for buy/sell orders, traded volume)

Factor D: Variables 3 and 9 (turnover ratio and order depth)

Factor E: Variables 4, 8 and 7 (price range, relative bid ask and average age of order)

Next we would like to see whether these variables load on the corresponding factors with consistently high factor loadings. The distribution of the factor loadings across different factor models built for different days is presented using box plots in Figures 3 and 4. For conciseness we have shown here only a few box plots for a couple of variables for each factor, drawn from the FA done for large cap buy orders in period 2.

Figure 3

Source: Developed by the authors.

Figure 3

Source: Developed by the authors.

Figure 3

Source: Developed by the authors.

Figure 3

Source: Developed by the authors.

Figure 4

Source: Developed by the authors.

Figure 4

Source: Developed by the authors.

Figure 4

Source: Developed by the authors.

Figure 4

Source: Developed by the authors.

The box plots indicate the range within which the factor loadings for a particular variable on a given factor vary across different factor models built for 63 days. We can see that the factor loadings of the variables loading on Factors A, B, C and D lie in a narrow range of values. For instance from Figure 3A, we find that the maximum loading for ‘traded volume’ on Factor A is 0.65, while the minimum is 0.99 but 50 per cent of the values lie between 0.79 and 0.92 (interquartile range). The interquartile ranges depicted in Figures 3 and 4 indicate that most of the factor loadings vary within a narrow band (the only exception being the factor loadings for traded volume on Factor C). This shows the overall consistency of the magnitude of the factor loadings across days. Similar consistency is also seen in the other three cases, namely large cap sell, small cap buy and small cap sell orders.

Description of Factors

The factors identified in both periods are described below. The factors have been named based on the variables that had a consistently high loading on them in both periods.

Factor A—Volume

As can be seen from Figure 5, factor A loads primarily on the variables 2, 6, 1, 3 and 7 for the large cap stocks and on variables 2, 6, 1, 3 and 8 for small cap stocks. The corresponding variables are turnover (2), number of trades (6), trading volume (1), turnover ratio (3) and relative bid-ask spread (7) or average age of order (8). The most important variables 2, 6, 1 and 3 are related to daily trading activity for that scrip. Hence this factor is named ‘Volume’.

Figure 5

Source: Developed by the authors.

Factor B—Price Elasticity

As can be seen from Figure 6, Factor B loads primarily on variables 5 and 11. These variables are respectively buy (sell) elasticity and liquidity ratio (ratio of daily price change over the daily volume traded). These are measures of change in price over a unit change in volume and hence this factor is named the ‘Price Elasticity’ factor (ΔP/ΔX, computed as the slope of the order book as given in Appendix 1). Relative bid ask spread (variable 7) is positively correlated; however, this correlation is not observed consistently in both periods. The average age of orders on the LOB (variable 8) is negatively correlated to this factor but again this correlation is not consistent as seen in Figure 6. We can see that the factor loadings for both variables 5 and 11 decrease to some extent in the second period especially for small caps.

Figure 6

Source: Developed by the authors.

Factor C—Depth

This factor predominantly loads on Variables 10 and 1. The corresponding variables are ‘market depth’ (number of shares available at the best bid and ask prices) and trading volume. Both of these variables are representative of the depth available in the market—hence this factor is named ‘Depth’ (Figure 7). In the first period, variable 9, order depth (the number of orders as a ratio of total outstanding shares) is positively correlated to the market depth for large cap companies. But in the second period, this variable is not found to be important—instead variable 6, the number of trades, is positively correlated with this factor. An interesting thing to note here is that the factor loadings for variable number 10 (market depth) among large cap companies drop significantly in the second period.

Figure 7

Source: Developed by the authors.

Factor D—Relative Activity

This factor loads primarily on variables 9 and 3. The corresponding variables are ‘turnover ratio’ which is the ratio of number of traded shares to the total number of outstanding shares and ‘order depth’ which is the ratio of order volume to total number of outstanding shares. Both of these variables are a measure of the proportion of the company that is being traded—hence it is called Relative Activity (Figure 8). Other than the primary variables 3 and 9, variable 1 (trading volume), variable 6 (number of trades) and variable 10 (market depth) also seem to be correlated to this factor in large cap companies. Among small cap companies only variable 4 (relative bid ask spread) is correlated to this factor. The factor loadings are seen to be smaller for the small cap stocks than for large cap stocks.

Figure 8

Source: Developed by the authors.

Factor E—Spread

The two main variables in this factor are ‘price range’ (difference between the highest and lowest prices in a day), variable 4; ‘relative bid ask spread’, variable 7; and ‘average age of orders’, variable 8. The first two variables are both representative of spread in the market and so this factor is termed ‘Spread’ (Figure 9). It is seen that the average age of unexecuted orders at the snapshot (variable 4) is negatively correlated to this factor. This factor has smaller factor loadings within the large cap stocks (especially for the variables bid ask spread and price range) indicating its greater importance among small cap stocks.

Figure 9

Source: Developed by the authors.

Discussion of Results

We have noted above the emergence of a stable five factor structure across several days, on both the buy and sell sides, for stocks of widely varying market capitalisation and in two different periods with different market conditions. We now consider the implications of these results in some greater detail. For example, why are the factor loadings for ‘Spread’ lower than the factor loadings for ‘Volume’ or ‘Price Elasticity’? Why have the factor loadings for ‘Depth’ decreased from the first time period to the second? What are the intuitive, economic, or financial implications of these observations?

To address these issues, we discuss the intuitive meaning behind the structure of the factor analysis model (Equation [2]), reproduced below for ease of reference: XF

In the equation, μ simply accounts for the mean value, while the other quantities address variation and correlation. The magnitude of μ is of little importance in the interpretation of the factor model. Of the remaining terms, ε accounts for uncorrelated variations in the individual components of X also known as specific factors. If X contained completely uncorrelated components, then ε would capture all the variation and Λ would be zero. In factor analysis, we assume that the components of X are, in fact, correlated. Usually the quantity of greatest interest to the analyst is the matrix of factor loadings, Λ. The ‘correlated part’ of the variation in X is reflected in the magnitudes of the elements of Λ.

In light of the above, the relatively small factor loadings observed in the factor ‘Spread’ suggest that bid ask spread and price range are somewhat weakly correlated, while being roughly uncorrelated from the other liquidity proxies in the study. Similarly, the high factor loadings observed in other factors such as ‘Volume’ or ‘Price Elasticity’ indicate that the variables loading on these factors are consistently well correlated even in different market conditions. More interestingly, we consistently find lack of strong correlation between the variables loading on the factors ‘Volume’ and ‘Price Elasticity’, or factors ‘Volume’ and ‘Spread’ indicating that these consistently reflect different aspects of liquidity.

Our study is the first attempt to quantitatively examine correlations between liquidity proxies across stocks, in two different time periods under different market conditions. We did not anticipate obtaining consistent factor models across time periods. Hence it is one of the principal findings of this study that there is in fact a fair degree of consistency in the results across the two time periods. However, we do see some small systematic differences that are discussed below.

As shown in Figure 1, the second period included the beginning of the bear run in 2008 when presumably the investor confidence was faltering. Second, as noted from our summary statistics in Table 2, there were significantly greater volumes traded in the second period. Third, as discussed in the ‘Summary Statistics’ section, small cap companies were relatively less liquid in the second period as compared to the first. It may be a consequence of the above conditions that the factor loadings observed in the second period are by and large smaller than in the first period. Perhaps the larger volumes as well as the uncertainty in the market led to relatively greater uncorrelated individual fluctuations and less correlated variations among the liquidity proxies.

In particular we find that the loadings for the variable ‘market depth’ on Factor C (‘Depth’) have decreased significantly in the second period. Market depth is defined as the number of shares available at the best bid or ask price. In a time of greater uncertainty, fluctuating investor confidence may lead to very rapid changes in the best bid and ask prices and also in the number of traders willing to trade at these prices. The variation in this quantity may thus be quite uncorrelated to the variations in all other proxies. This might account for the much smaller factor loadings in this factor in the second period.

We also note some small differences in the factor structure for large cap and small cap companies. The factor ‘Relative Activity’ has larger loadings in large cap companies, while the factor ‘Spread’ has larger loadings in small cap companies. Thus, the importance of these two factors is reversed among large and small cap companies. We suggest that these are both reflections of the underlying market interest in the companies in question. Low market interest can significantly increase spreads in small cap companies resulting in larger variations. Large cap companies generally have very small spreads (mean values of 0.14 per cent and 0.20 per cent, respectively, in periods 1 and 2 and also very small variances—see Table 2) and so spread may not be a good indicator of market interest among large cap companies. Instead relative activity would be a better indicator of market interest in a particular large cap stock.

We also conducted factor analysis separately for buy and sell side liquidity proxies anticipating that there may be some differences. But interestingly we did not come across any significant differences in the factor models between these two sides.

Our factor analysis has, thus, uncovered several possible interesting relationships between different liquidity proxies. A more detailed investigation of these relationships and the underlying causes for such relationships may be taken up in future research.

Conclusions

We have examined the common factors underlying various stockspecific proxy measures of liquidity in the Indian equity market. Eleven proxy measures were computed for four subsets of data—buy and sell side liquidity measures for large cap and small cap companies. A five factor model was found to capture the correlations between the proxy variables on most days, over two different periods of time and under different market conditions. A key distinguishing feature of our study is that, unlike previous studies concerned with market-wide averaged liquidity measures, we concentrated on stock-specific proxies in a cross-sectional analysis.

Comparison of the daily factor loading matrices across days and across the two periods yielded largely consistent results in terms of the variables loading on the factors, as well as the relative importance of the variables on those factors. This consistency across different market conditions suggests a multidimensional, yet meaningfully identifiable, characterisation of liquidity of stocks. The five factors identified for both the buy and sell side measures, in order of importance, are ‘Volume, Price Elasticity, Depth, Relative Activity and Spread’. Despite an overall consistent structure, we do notice some differences across periods and across large and small cap companies that have been discussed in some detail. The economic causes for such differences have not been explored in this article and could possibly be topics of future research.

Of the three popularly recognised attributes of liquidity, namely depth, spread and resiliency, our results clearly identify the first two. Resiliency, defined as the rapidity with which the stock price recovers after a random and uninformative shock, is a dynamic factor and can be measured using time series data (Degryse et al. 2005; Dong et al. 2007; Kan 2004; Large 2007). In this study we have not explicitly looked at temporal variations in liquidity. However, the three remaining factors, viz., price elasticity, volume traded and relative activity on a particular day, are all related to the resiliency of the stock on a particular day.

We believe our results shed new light and improve our understanding of the multidimensional nature of liquidity. We also observe some interesting ways in which the certain aspects of stock-specific liquidity change under different market conditions and for large cap and small cap stocks. These issues can be further investigated in future research.

Appendix 1. Computation of the Slope of the Order Book

We outline here the method for computing price elasticity, one of the proxy measures of liquidity included in the factor analysis (see the ‘Proxy Measures Used’ section).

The snapshot of the order book can be treated as an ‘instantaneous’ supply and demand curve for the stock in question at a given point of time. The corresponding slope is computed as follows.

Let the order book contain a sell order for n₁ shares at a price P₁, then an order for n₂ shares at a price P₂, etc., with P₁, P₂, and so on, all arranged in increasing order, i.e., P₁< P₂< P₃. Now suppose a market buy order arrives for N shares, where n₁ + n₂< N< n₁ + n₂ + n₃. In such cases, to find the average price of the sale, some sort of interpolation is needed. The calculation is simpler if N exactly equals the sum of, say, the first k orders, for some integer, k. In such cases, we say $N_{k} = \sum_{i = 1}^{k} n_{i}$ and write

P_{a v} (k) = \frac{\sum_{i = 1}^{k} n_{i} P_{i}}{\sum_{i = 1}^{k} n_{i}} = \frac{\sum_{i = 1}^{k} n_{i} P_{i}}{N_{k}}

(11)

Subsequently, a graph of P_av(k) against N_k can be plotted. Here, with no real consequence, we have subtracted the mean price (mean of best bid and best ask). The result showing both buy and sell sides on the same plot, looks, for example, like Figure 1A.1. Note that trading volume (hypothetical, of course) is taken as positive for the sell side orders and negative for the buy side orders, to get a single, approximately continuous, underlying curve. Note also that, leaving out some relatively small proportion of total orders, a linear approximation seems reasonable. For this reason, we have taken the best 80 per cent of order volumes and computed the slope of the order book there from. In this study, the slope was actually computed separately for buy and sell sides (for each stock, for each snapshot and on each day), although it may be justifiable to combine the buy and sell sides into a single slope.

Figure 1A.1

Source: Developed by the authors.

Finally, we note that there are more market sell orders than buy orders for the data in the figure. This asymmetry was commonly observed for most stocks on most days and was included in this study through the variable ‘order imbalance’.

Footnotes

Notes

References

Acharya

Pedersen

L. H.

(2005), ‘Asset Pricing with Liquidity Risk’, Journal of Financial Economics, 77(2): 375–410.

Aitken

Comerton-Forde

(2003), ‘How Should Liquidity be Measured?’, Pacific-Basin Finance Journal, 11(1): 45–59.

Amihud

Mendelson

(1986), ‘Asset Pricing and the Bid–ask Spread’, Journal of Financial Economics, 17(2): 223–49.

Amihud

Mendelson

(1989), ‘The Effects of Beta, Bid-ask Spread, Residual Risk, and Size on Stock Returns’, The Journal of Finance, 44(2): 479–86.

Beck

Levine

(2004), ‘Stock Markets, Banks, and Growth: Panel Evidence’, Journal of Banking & Finance, 28(3): 423–42.

Bekaert

Harvey

C. R.

Lundblad

(2003), ‘Liquidity and Expected Returns: Lessons from Emerging Markets’, Review of Financial Studies, 20(6): 1783–1831.

Biais

Hillion

Spatt

(1995), ‘An Empirical Analysis of the Limit Order Book and the Order Flow in the Paris Bourse’, The Journal of Finance, 50(5): 1655–89.

Black

(1971), ‘Towards a Fully Automated Exchange, Part I’, Financial Analysts Journal, 27(4): 28–35.

Brennan

M.J.

Chordia

Subrahmanyam

(1998), ‘Alternative Factor Specifications, Security Characteristics, and the Cross-section of Expected Stock Returns’, Journal of Financial Economics, 49: 345–73.

10.

Brockman

Chung

D. Y.

(2002), ‘Commonality in Liquidity: Evidence from an Order-driven Market Structure’, The Journal of Financial Research, 25(4): 521–39.

11.

Chatterjee

(2010), ‘Studies on Some Aspects of Liquidity of Stocks: Limit Order Executions in the Indian Stock Market’, unpublished PhD thesis, Indian Institute of Science, Bangalore.

12.

Chollete

Næs

Skjeltorp

J. A.

(2007), ‘What Captures Liquidity Risk? A Comparison of Trade and Order Based Factors’, Working Paper, Norwegian Central Bank.

13.

Chollete

Næs

Skjeltorp

J. A.

(2008), ‘The Risk Components of Liquidity’, EFA 2008 Athens Meetings Paper. Available at SSRN: http://ssrn.com/abstract=1102205

14.

Chordia

Roll

Subrahmanyam

(2000), ‘Commonality in Liquidity’, Journal of Financial Economics, 56(1): 3–28.

15.

Chordia

Roll

Subrahmanyam

(2001), ‘Market Liquidity and Trading Activity’, The Journal of Finance, 56(2): 501–30.

16.

Cooper

S.K.

Groth

J. C.

Avera

W. E.

(1985), ‘Liquidity, Exchange Listing, and Common Stock Performance’, Journal of Economics and Business, 37(1): 19–33.

17.

Dalvi

Krishnan

Nath

G. C.

(2005), ‘Liquidity in an Emerging Market: Evidence from India’s National Stock Exchange’, Working paper. Available at http://www.yu.edu/uploadedFiles/Faculty/Murgie_Krishnan/nse_liquidity.pdf

18.

Danielsson

Payne

R. G.

(2001), ‘Measuring and Explaining Liquidity on an Electronic Limit Order Book: Evidence from Reuters D2000-2’, EFA 2001 Barcelona Meetings. Available at SSRN: http://ssrn.com/abstract=276541 or DOI: 10.2139/ssrn.276541.

19.

Degryse

Jong

F.D.

Ravenswaaij

M.V.

Wuyts

(2005), ‘Aggressive Orders and the Resiliency of a Limit Order Market’, Review of Finance, 9: 201–42.

20.

Domowitz

Glen

Madhavan

(1997), ‘Market Segmentation and Stock Prices: Evidence from an Emerging Market’, The Journal of Finance, 52(3): 1059–85.

21.

Dong

Kempf

Yadav

P. K.

(2007), ‘Resiliency, the Neglected Dimension of Market Liquidity: Empirical Evidence from the New York Stock Exchange’, SSRN eLibrary. Available at SSRN: http://ssrn.com/paper=96726

22.

Engle

R.F.

Lange

(2001), ‘Predicting VNET: A Model of the Dynamics of Market Depth’, Journal of Financial Markets, 4(2): 113–142.

23.

Fleming

M.J.

(2001), ‘Measuring Treasury Market Liquidity’, Economic Policy Review, 9(3): 9–32.

24.

Goldstein

M.A.

Kavajecz

K. A.

(1999), ‘Eighths, Sixteenths and Market Depth: Changes in Tick Size and Liquidity Provision on the NYSE’, White

Rodney L.

Center Working Paper No. 14–98. Available at SSRN: http://ssrn.com/abstract=97748

25.

Grossman

S.J.

Miller

(1988), ‘Liquidity and Market Structure’, The Journal of Finance, 43(3): 617–37.

26.

Hasbrouck

Seppi

(2001), ‘Common Factors in Prices, Order Flows and Liquidity’, Journal of Financial Economics, 59(3): 383–411.

27.

Huberman

Halka

(2001), ‘Systemic Liquidity’, Journal of Financial Research, 14(2): 161–78.

28.

Johnson

R.A.

Wichern

D. W.

(2002), Applied Multivariate Statistical Analysis, 5th ed. Singapore: Pearson Education, Inc.

29.

Jun

S-G.

Marathe

Shawky

H.A.

(2003), ‘Liquidity and Stock Returns in Emerging Equity Markets’, Emerging Markets Review, 4(1): 1–24.

30.

Kan

A.C.N

(2004), ‘Resiliency Ability of the Underlying Spot Market after the Introduction of Index Futures Contracts: Evidence from Hong Kong’, Journal of Emerging Market Finance, 3(3): 269–83.

31.

Kluger

B.D.

Stephan

(1997), ‘Alternative Liquidity Measures and Stock Returns’, Review of Quantitative Finance and Accounting, 8(1): 19–36.

32.

Kyle

A.S.

(1985), ‘Continuous Auctions and Insider Trading’, Econometrica, 53(6): 1315–35.

33.

Large

(2007), ‘Measuring the Resiliency of an Electronic Limit Order Book’, Journal of Financial Markets, 10(1): 1–25.

34.

Lehmann

B.N.

Modest

D. M.

(1994), ‘Trading and Liquidity on the Tokyo Stock Exchange: A Bird’s Eye View’, The Journal of Finance, 49(3): 951–84.

35.

Lesmond

D.A.

(2005), ‘Liquidity of Emerging Markets’, Journal of Financial Economics, 77(2): 411–52.

36.

Levine

Schmukler

(2003), ‘Migration, Spillovers, and Trade Diversion: The Impact of Internationalization on Stock Market Liquidity’, Working Paper No. 9614, National Bureau of Economic Research, Cambridge, MA.

37.

Muilenburg

L.Y.

Berge

Z. L.

(2001), ‘Barriers to Distance Education: A Factor-analytic Study’, The American Journal of Distance Education, 15(2): 7–22.

38.

Muscarella

C.J.

Piwowar

M.S.

(2001), ‘Market Microstructure and Securities Values: Evidence from the Paris Bourse’, Journal of Financial Markets, 4(3): 209–29.

39.

Næs

(2005), ‘Essays on the Microstructure of Stock Markets: Empirical Evidence from Trading Arrangements Without Dealer Intermediation’, unpublished doctoral dissertation, Norwegian School of Economics and Business Administration.

40.

Rouwenhorst

(1999), ‘Local Factors and Turnover in Emerging Markets’, The Journal of Finance, 54(4): 1439–64.