آیا شکاف ها ایجاد نقدینگی بازار می کنند؟ تئوری و شواهد

We present a market microstructure model of stock splits in the presence of minimum tick size rules. The key feature of the model is that discretionary trading is endogenously determined. There exists a tradeoff between adverse selection costs on the one hand and discreteness related costs and opportunity costs of monitoring the market on the other hand. Under certain parameter values, there exists an optimal price. We document an inverse relation between the coefficient of variation of intraday trading volume and the stock price level. This empirical evidence and other existing evidence are consistent with the model.

U.S. firms split their stocks quite frequently. In spite of inflation, positive real interest rates, and significant risk premiums, the average nominal stock price in the U.S. during the past 50 years has been almost constant. Why would firms keep on splitting their stocks to maintain low prices? This behavior is puzzling since, by doing so, firms actively increase their effective tick size (i.e., tick size/price), potentially exposing their stockholders to larger transaction costs. This paper presents a value maximizing market microstructure model of stock splits. Our model joins practitioners in predicting that firms split their stocks to move the stock price into an optimal trading range in order to improve liquidity. and The driving force of the model stems from the fact that prices on U.S. exchanges are restricted to multiples of 1/8th of a dollar.3 This restriction on prices creates a wedge between the “true” equilibrium price and the observed price.4 Thus a portion of the transaction costs incurred by traders is purely an artifact of discreteness. Anshuman and Kalay (1998) show that discreteness related commissions depend on the location of the “true” equilibrium price on the real line. In other words, whether the discrete pricing restriction is binding or not depends on the location of the “true” equilibrium price relative to a legitimate price (tick) in a discrete price economy. It may so happen that the “true” equilibrium price (plus any transaction cost) is close to a tick. Discreteness related commissions would be low in such a period. As information arrives in the market, the location of the “true” equilibrium price changes, and discreteness related commissions would, therefore, vary over time. They could be as low as 0 or as high as the tick size. Interestingly, liquidity traders can take advantage of the variation in discreteness related commissions by timing their trades. Of course, such strategic behavior is not costless. It involves close monitoring of the market to take advantage of periods with low discreteness related commissions. In general, liquidity traders differ in terms of their opportunity costs of monitoring the market. Some liquidity traders may prefer not to time the market because the benefits from timing trades do not offset their opportunity costs of monitoring. In contrast, other liquidity traders who are endowed with low opportunity costs of monitoring may find it beneficial to time their trades. Such discretionary traders would trade together in a period of low discreteness related commissions. The presence of additional liquidity traders in this period (a period of concentrated trading) forces the competitive market maker to charge a lower adverse selection commission than otherwise. Thus, discretionary liquidity traders save on execution costs – adverse selection as well as discreteness related commissions. Because the tick size is fixed in nominal terms (at 1/8th of a dollar), the economic significance of the savings in discreteness related commissions depends on the stock price level. At low stock price levels, the savings in execution costs due to timing of trades may be significant enough to offset the opportunity costs of monitoring of most liquidity traders. There would be highly concentrated trading at low price levels as most liquidity traders would exercise the flexibility of timing trades. Conversely, at high stock price levels, few liquidity traders would time trades because the potential savings in execution costs are economically insignificant. The key implication of the model is that the stock price level affects the distribution of liquidity trades across time, and consequently, the transaction costs incurred by them. In particular, we show that there exists an optimal stock price level that induces an optimal amount of discretionary trading. This optimal price results in the lowest (total) expected transaction costs incurred by all liquidity traders. Because investors desire liquidity (Amihud and Mendelson, 1986; Brennan and Subrahmanyam, 1995), a value-maximizing firm should choose a stock price level that maximizes liquidity (minimizes the total transaction costs incurred by all liquidity traders). By splitting (or reverse splitting) its stock, a firm can always reset its stock price to the optimal price level. We present numerical solutions of the model to show that, under certain parameter values, an optimal price exists. The numerical solutions show that the optimal price is increasing in the volatility of the underlying asset and decreasing in the fraction of liquidity traders. We also show that the optimal price is (linearly) increasing in the tick size. Finally, using intraday transaction data, we document a cross-sectional inverse relation between the coefficient of variation of time-aggregated trading volume (a measure of the degree of concentrated trading in a stock) and the stock price level. This empirical evidence and other existing evidence are consistent with the model. The paper is organized as follows. Section 2 discusses a numerical example that illustrates the key features of the model. The model is developed in Section 3. Section 4 presents numerical solutions of the model. Section 5 discusses empirical evidence relevant to the model, and Section 6 concludes the paper.

This paper presents a theory that argues that splits improve liquidity. The driving force behind the model is the discrete pricing restriction on organized exchanges. Execution costs arising from discreteness vary over time. Liquidity traders have an incentive to time their trades to lower their execution costs. The asset price level affects their incentives because discreteness related costs are determined by a nominally fixed tick size. By splitting, a firm lowers its stock thereby increasing the incentives of liquidity traders to time their trades. The resulting concentration of trades reduces overall transaction costs incurred by liquidity traders. Our analysis also establishes the existence of an optimal price level that minimizes transaction costs. Stock split factors can be adjusted to reset the stock price level to the optimal level.