Trading Probability and Turnover as measures of Liquidity Risk: Evidence from the U.K. Stock Market. Ian McManus. Peter Smith.

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1 Trading Probability and Turnover as measures of Liquidity Risk: Evidence from the U.K. Stock Market. Ian McManus (Corresponding author). School of Management, University of Southampton, Highfield, Southampton, UK. SO17 1BJ. Tel: +44 (0) Fax: +44 (0) Peter Smith Professor of Economics and Finance, Department of Economics and Related Studies, University of York, Heslington, York, UK. YO10 5DD. Tel: +44 (0) Fax: +44 (0) Stephen Thomas Professor of Finance, Cass Business School, 106 Bunhill Row, London, UK. EC1Y 8TZ. Abstract This paper utilises monthly data from the U.K. stock market to examine possible regularities in variables (trading probability and fractional turnover) which may be (or may be hypothesised to be) associated with liquidity risk. Modelling is based upon extensions to the CAPM involving variables closely similar to those employed in the Fama-French three factor model (albeit using Dividend Yield in place of the Book to Market ratio). Our findings suggest that the measure of trading probability used here is a close substitute for the market capitalisation measure associated with the Size effect; and that the fractional turnover variable makes a significant contribution as an addendum to the model, albeit conveying information of a different kind. Key Words: Liquidity, Trading Probability, Turnover JEL classification codes: G12, G15 Electronic copy available at:

2 Introduction Following the identification of the Size effect (Banz, 1981) and its eventual incorporation into the Fama-French (1993) three factor model, a body of literature has developed over the course of the last decade which seeks to rationalise the presence of the size effect in terms of the notion of risk associated with (il)liquidity. Investors show themselves to be unprepared to trade in (perceived) illiquid stocks at the level of prices which would normally be associated with similar cash flow patterns in more liquid stocks (Bekaert et al, 2005). Generally, this phenomenon is associated with the stocks of smaller firms; however, these same stocks are generally also characterised by their infrequent trading patterns. Liu (2006) identifies the complex nature of the notion of liquidity; highlighting the four dimensions of trading quantity, trading speed, trading cost and price impact, citing a number of contributions to the literature which shed light on each of these factors (Amihud and Mendelson, 1986; Datar (1998); Amihud, 2002; Pastor and Stambaugh, 2003). Using data from the U.S. stock market, Liu develops a measure of liquidity designed to capture this multidimensional aspect of liquidity. The composite measure seeks to incorporate the infrequent trading aspect of illiquidity by examining the number of zero trading volumes over the prior period; and the turnover (trading volume) aspect by adjusting the measure to impound volume information in order to differentiate between stocks having similar levels of trading frequency. Our measures of these effects are somewhat different (given the alternate dataset employed, and its differing characteristics); however, they set out to capture the same aspects of the liquidity phenomenon in the context of a different dataset. This is outlined in our Methodology section, below. Data Our data is obtained from the London Share Price Database (LSPD) provided by the London Business School. Data is at monthly frequency for individual firms; we utilise Returns data (for Stocks and for 3-month Treasury Bills) to form the basis of our dependent variable (and to form an overall monthly Market Return). Independent variables are formed from data relating to a non-trading indicator, Turnover 1, Dividends, Transaction Prices and Market Capitalisation. In addition, an industry classification vector enables us to screen for firms which we wish to exclude from consideration (e.g. financials, in order to avoid double counting of returns 2 ). We require that only firms which trade in a given month are admitted to the set of portfolios for that month, and that a valid Return, Dividend (this may be a zero dividend), Transaction Price and Market Capitalisation are posted. Survival bias is accounted for by writing down the value of a firm to zero in the month following its last month of valid data 3, and determining its closing return in the final month as 1 Turnover data is available from December 1992 until the end of This excludes, e.g. Investment Trusts, Investment Funds, Currency Funds, Venture Capital Trusts, REITS, etc. 3 Value preservation tests are carried out to differentiate between, e.g. Failure versus Merger. Electronic copy available at:

3 -100%. However, we do not impose a pre-qualification period on stocks being admitted into the portfolio structure 4. In addition to the data directly from the LSPD database, we form a calendar ( inclusive) which enables us to identify precisely those days when exchanges are closed; this enables a refinement to the non-trading indicator provided in the database itself, and facilitates the calculation of our trading probability measure. Methodology Our primary liquidity measure is based essentially upon a measure of the number of non-trading days related to individual stocks. In the LSPD, given its focus on monthly data, this is not provided explicitly on a daily basis, but rather on the basis of an endof-month sample which, for each individual stock, measures the number of calendar days between the last actual trade in the month and the last available trading day of the month. For frequently-traded stocks, this sampling period represents a small fraction of the whole month, but that fraction progressively increases for infrequentlytraded (assumed less liquid) stocks which are the main focus of interest here. The final measure (TradProb) takes the form of the probability, on any particular trading day, of the particular stock being traded on that day 5. This approach provides an unbiased but noisy measure of the desired quantity, which may potentially be smoothed over time (at the risk of exacerbating serial correlation effects); however, with an overall mean number of 141 firms populating each portfolio, the option is taken to perform the averaging over the cross-section of stocks comprising each portfolio. Our results seem to indicate that this approach is satisfactory. The above (probability) measure is used to form the basis of a primary sort of the data (within each month), with individual firms fractional turnover (turnover as a fraction of Market Capitalisation) used as a secondary sort criterion. The ranked observations are then formed into deciles, with decile #1 comprising the (assumed) most liquid stocks (on the basis of the two-level sort) and decile #10 the least 6. In essence, this approach is similar to that of Liu (2006, p. 636) in the sense that the probability measure captures the Speed (or trading continuity) dimension of Liquidity, and the fractional turnover measure reflects the Quantity dimension. 4 This would eliminate a high proportion of the illiquid stocks of primary interest here. 5 After correcting the non-trading indicator featuring in the database for weekends and public holidays the corrected figure (NTI) provides, effectively, the result of a series of binomial trials (working backwards from the last trading day of the month). The probability (as defined above) is simply provided by the relationship Prob = 1/(1+NTI). Thus frequently-traded stocks (NTI = 0) imply Prob =1, etc. Provided that a stock is traded in a given month, the maximum value of NTI is 22; these maxima will occur, however, only in those 31-day months with no public holiday (March, July, October) and having only 8 weekend days. 6 In general, deciles 1 to 7 are populated by frequently-traded stocks (NTI=0, Prob=1) which are differentiated (in the sorting process) by fractional turnover; deciles 8 to 10 exhibit a monotonic decline in average (portfolio) probability. Electronic copy available at:

4 With portfolio formation taking place in a given month (t-1), subsequent portfolio returns for the following month (t), less the risk-free rate Rf(t) pertaining to that same month, form the dependent variable in the following general equation: R pt R ft = β 0 + β 1 (R mt R ft ) + β 2 (FracTurn (t-1) ) + β 3 (DivYld (t-1) ) + β 4 (LogSize (t-1) ) + β 5 (Prob (t-1) ) Portfolio returns (R pt ) for the month (t) are calculated on the basis of equally-weighted portfolios (deciles); returns on the Market (R mt ) are calculated on the basis of an equal weighting of all stocks in the sample in month (t). The variable FracTurn is the ratio (for the portfolio) of the aggregate turnover of all stocks in the portfolio (decile) divided by the aggregate Market Capitalisation of all stocks in the decile. DivYld is the average of the individual stocks dividend yields (which realises the overall yield for the equally-weighted portfolio); LogSize is the natural logarithm of the aggregate Market Capitalisation of all stocks in the decile. Finally, TradProb is the average probability of all stocks in the portfolio, and serves as an aggregate measure of the desired quantity. Portfolios are re-balanced every month, in order to maintain the chosen characteristics of the liquidity-based portfolios through time. In general, analysis of the data takes place as a stacked regression, comprising all deciles in the full sample over 131 months (1,310 observations); various subsets of the complete equation are studied in the process of modelling the returns-generation process, equivalent to imposing zero restrictions on some parameters at various stages in that process. Initial OLS regressions (and associated diagnostic tests for Autocorrelation and Heteroscedasticity) indicate that moderate levels of these phenomena bias the values of the standard errors; accordingly, the estimator of choice here is Least Squares including Newey-West HAC Standard Errors & Covariance 7. Results Discussion of the results of the analysis begins with a review of the (stacked) regression associated with the full-sample of 1310 observations (131 months * 10 deciles). Table 1 displays the Coefficient, T-statistic and associated probability value for each of the variables included in the particular model under scrutiny. Thus model #1 represents the classic CAPM market model ; in this instance, the usual null hypothesis associated with the excess returns form of the equation: H 0 : β 0 =0 is not rejected; β 1 is close to unity (as would be expected by virtue of the portfolio construction employed and the definition of the market portfolio). Before considering the full model specification, an examination of the crosscorrelation matrix (Table 2) shows the LogSize variable to be highly correlated with the TradProb varable (ρ = 0.769); small firms are generally traded infrequently. Whilst the two variables fall short of being collinear, they clearly convey similar 7 This is equivalent to GMM (with HAC) but with the set of regressors also employed as instruments.

5 Table 1 Regression results for the full-sample (stacked) regression. Coefficient (Variable) β 0 (Const) β 1 (MktXS) β 2 (FracTurn) β 3 (DivYld) β 4 (LogSize) β 5 (TradProb) Market Model (#1) Full Model (#2) Model (#3) (Omit T/Prob) Model (#4) (Omit L/Size) Coeff T-stat Prob Coeff T-stat Prob Coeff T-stat Prob Coeff T-stat Prob Coeff T-stat Prob Coeff T-stat Prob Adj. R S.E. of Reg No. of Obs Table 2 Cross-correlation matrix for the variables of interest. PortXS MktXS LogSize FracTurn DivYld TradProb PortXS 1 MktXS LogSize FracTurn DivYld TradProb

6 information into the regression; this reflects in both associated coefficients (β 4, β 5 ) being insignificant in the full model. Elimination of one or the other, as we progress to models #3 or #4, results in the remaining regressor demonstrating a moderate level of significance. In essence, either of these variables could act as a proxy for the other. This leads to the proposition that small firm size implies illiquidity, as investors are unwilling to pay the normal price to capture the cash flow generated by the level of returns associated with such firms. Were we to opt for model #3 as the model of choice, we would be staying close to the spirit of the 3 factor model (albeit using Dividend Yield rather than Book to Market as a proxy for value 8 ); model #3 differs, however, in that the highly significant Fractional Turnover variable features as an additional term in the equation. In so doing, it contributes to the equation as a measure of liquidity. However, it indicates that the more liquid stocks / portfolios are associated with a higher level of return, implying a negative liquidity premium relative to this particular measure. Examination of the literature documents that this is associated with the High-volume Return Premium identified by Gervais et al (2001); it is also a feature of the U.S. stock market data used by Liu (2006), when using portfolio formation on the basis of 1-month prior data and 1-month holding period (effectively the combination used here). These Return Premium effects thus appear to feature strongly in the U.K. data used here. Conclusions The two proposed new liquidity variables (Trading Probability and Fractional Turnover) appear to play very different roles in the modelling of portfolio returns. The Trading Probability measure seems to function as an alternative to the usual logarithm of Size variable. We conclude from the evidence that either the Trading Probability or the Size variable is capable of acting as a proxy for one of the dimensions of liquidity (Trading Speed); the Fractional Turnover variable appears to contribute significantly to the model by conveying separate information into the regression (see, however, the paragraph above in relation to the Return Premium effect). This would appear to be representative of a separate aspect of the complex liquidity picture, and clearly warrants further research, both in terms of the short-term (1-month portfolio formation / holding period) effects and in terms of the likely effect (as the liquidity measure initially proposed) when either or both of the portfolio formation and/or holding periods are adjusted to longer horizons. Postscript This (working) paper has recently been extended to investigate an augmented mimicking-portfolio approach based upon the Fama-French (1993) 3- factor model. The decile portfolios load appropriately on the factors; near future research (forthcoming) will report in detail the findings. 8 The required data to generate Book to Market is not available in the LSPD; Fama and French (1993) acknowledge DY to be a close substitute for B/M in reflecting the Value-Growth dimension.

7 References Amihud, Y., (2002) Illiquidity and stock returns: cross-section and time-series effects. Journal of Financial Markets 5, pp Amihud, Y., Mendelson, H., (1986) Asset pricing and the bid-ask spread. Journal of Financial Economics 17, pp Banz, R., (1981) The relationship between Return and Market Value of Common Stocks. Journal of Financial Economics 9, pp Bekaert, G., Harvey, C.R., Lundblad, C., (2005) Liquidity and Expected Returns: Lessons from Emerging markets NBER Working Paper Datar, V., Naik, N., Radcliffe, R., (1998) Liquidity and Asset Returns: an alternative test. Journal of Financial Markets 1, pp Fama, E., French, K., (1993) Common risk factors in the return on stocks and bonds. Journal of Financial Economics 33, pp Gervais, S., Kaniel, R., Mingelgrin, D., (2001) The high-volume return premium. Journal of Finance 56, pp Liu, W., (2006) A liquidity-augmented capital asset pricing model. Journal of Financial Economics 82, pp Pastor, L., Stambaugh, R., (2003) Liquidity Risk and Expected Stock Returns. Journal of Political Economy 111, pp

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