V-Lab

## Motivation

Classical asset pricing models such as the Capital Asset Pricing Model (CAPM) assume frictionless markets where all securities are perfectly liquid and abstract away any potential impacts of liquidity on asset prices. However, the series of liquidity crises throughout the history of finance, among which the 2007-09 Global Financial Crisis is a notable example, has demonstrated that liquidity plays an important role in asset pricing and the overall security market performance. A security is liquid if it can be traded in a large quantity at a low cost. On the contrary, an illiquid security is difficult, costly and/or time-consuming to trade.

Across securities, investors are willing to pay lower prices, or demand higher returns, for securities that are more costly to trade. This gives rise to a positive relation between securities’ trading costs and expected returns, or a negative relation between trading costs and prices (for any given cash flow that the security generates). As the liquidity of securities rises, so does their price.

There are two concepts of liquidity: market liquidity and funding liquidity. The liquidity concept analyzed in here is referred as market liquidity. There is a separate aspect of liquidity referred as funding liquidity - the ease with which traders can obtain funding. See for example Brunnermeier and Pedersen (2009) for an analysis of how these two aspects of liquidity intertwine.

## Definition

Amihud (2002) develops a measure of illiquidity of a security at time $t$ as$ILLIQt=|Rt|VOLDt$Where ${R}_{t}$ is the stock return and ${\mathrm{VOLD}}_{t}$ is volume in 100 million dollars (V- Lab chooses the scaling of the dollar volume to enhance readability).

This measure captures the intuition that a security is less liquid if a given trading volume generates a greater move in its price, thus can be viewed as a coarse measure of the $\lambda$ coefficient of the formal model of Kyle (1985). Hasbrouck (2009) finds that ILLIQ is highly correlated with Kyle's $\lambda$, as well as other illiquidity measures such as the bid-ask spread.

One advantage of ILLIQ over other illiquidity measures is the great data availability of its required inputs. For instance, estimation of Kyle’s $\lambda$ requires intraday data on quotes and trades, whereas Amihud's ILLIQ requires only daily data on prices and trading volumes.

## Estimation

Forecasting ILLIQ requires an econometric model of non-negative processes. The Multiplicative Error Model (MEM) by Engle (2002), which solves the inconveniences associated with conventional approaches such as ignoring non-negativity and taking logs, specifies an error that is multiplied by the mean. Moreover, the MEM is isomorphic to the GARCH model, hence can be easily estimated using GARCH software.

V-Lab's ILLIQ measures include the historical ILLIQ measure which is the 22 trading days (i.e. 1 month) moving average and GARCH-type MEMs that take the above- mentioned daily ILLIQ series as input to generate predictions.

## References

Amihud, Y. 2002. Illiquidity and stock returns: cross-section and time-series effects. Journal of Financial Markets 5:31 – 56. URL http://www.sciencedirect.com/science/article/pii/S1386418101000246.

Brunnermeier, M. K., and L. H. Pedersen. 2009. Market Liquidity and Funding Liquidity. Review of Financial Studies 22:2201–2238. URL http://rfs.oxfordjournals.org/content/22/6/2201.abstract.

Engle, R. F. 2002. New Frontiers for ARCH Models. Journal of Applied Econometrics 17:425–446. URL http://dx.doi.org/10.1002/jae.683.

Hasbrouck, J. 2009. Trading Costs and Returns for U.S. Equities: Estimating Effective Costs from Daily Data. The Journal of Finance 64:1445–1477. URL http://dx.doi.org/10.1111/j.1540-6261.2009.01469.x.

Kyle, A. S. 1985. Continuous Auctions and Insider Trading. Econometrica 53:pp. 1315– 1335. URL http://www.jstor.org/stable/1913210.