Federico M. Bandi and Jeffrey R. Russell University of Chicago, Graduate School of Business

Introduction In a rational expectations setting with asymmetric information two prices can be defined (i.e. Glosten and Milgrom (1985), Easley and O’Hara (1987,1992)). –The equilibrium price that would exist if all agents possessed all public and private information, henceforth, the full information price. –The equilibrium price that exists in equilibrium given all public information, henceforth, the efficient price. Of course, market frictions also induce departures of transaction prices from these equilibrium values.

Measuring transaction costs Ideally, estimates of transaction cost should measure deviations of transaction prices from the full information price. Most estimates of transaction cost, however, measure deviations from the efficient price. –Bid/Ask spread –Effective spread –Realized spread –Roll’s measure This is due to the fact that while both the full information and the efficient price are unobservable the efficient price can, under certain assumptions, be approximated by the midpoint of the bid and ask price.

Our contribution We propose a simple and robust non-parametric estimate of the cost of trade defined as deviations from the full information price. The estimator is simple because it relies only on second moments of the observed transaction price returns. We are robust because we relax all the assumptions imposed by the above mentioned estimators.

Notation We consider a fixed time period (a trading day, for instance) Let t 1, t 2, …t i denote a sequence of arrival times where t i denotes the arrival time of the i th trade. Let N( ) denote the counting function. That is, the number of transaction that have occurred over the time period 0 to.

The i th observed transaction price at time t i is given by: Where denotes the full information price and denotes deviations of the transaction price from the full information price. The setup

Taking logs and differencing yields:

Assumption 1: the price process (1)The log price process is a continuous semimartingale: Where is a continuous finite variation component and (2)The spot volatility is a cadlag process.

Assumption 2: the microstructure noise (1) is a mean zero covariance stationary process. (2) (3)

Lemma 1 Under Assumptions 1 and 2 we obtain:

Theorem 1 Assume assumptions 1 and 2 are satisfied. Conditional on a sequence of transaction arrival times such that we obtain

Technical intuition The estimator relies on the different orders of magnitude of the two components of the observed returns. The full information returns are. The noise returns are In the limit (as the intervals go to zero) the observed returns are dominated by the noise returns. Therefore, as the trading rate increases, sample second moments of observed returns provide consistent estimates of the corresponding moments of the noise returns.

Economic intuition Ask Bid Effic. Price Full Information Price t1t1 t5t5 t4t4 t3t3 t2t2 t7t7 t6t6 t8t8

Assumption 3 Where Q i denotes signed order flow. Specifically, Q i =1 denotes trades that occur above the full information price and Q i =-1 denotes trades that occur below the full information price. denotes a constant distance that trades occur from the full information price. Furthermore, Pr(Q i =1)=Pr(Q i =-1)=.5

Corollary to Theorem 1 Under the assumptions of theorem 1 and assumption 3 we obtain Using this interpretation we refer to as an estimate of what we affectionately call the Full Information Transaction Cost (FITC).

Lemma 3 is the standard approach implemented by Roll. Our measure is similar to Roll’s. We emphasize that our method greatly relaxes the assumptions necessary to derive Roll’s estimator. Specifically, we allow for: –Correlation between full information and noise returns. –Arbitrary temporal dependence in the noise. –Predictability of the full information return process.

Data We obtain transactions data from TAQ for all S&P100 stocks over the month of February Data are filtered to remove outliers.

Distribution of average number of seconds between trades for S&P100 stocks

Histograms of t-stats for 1,2,3,5,10,and 15 autocorrelations.

Distribution of FITCs

Some specific FITCs SymbolDurationAvg PriceFITC$ FITC Eff. SpreadAnn SDTurnover GE %$ %32.52%0.03% IBM %$ %26.65%0.10% NXTL * %$ %218.34%0.16% Average 100 stocks %$ %37.30%0.25%

How good is the asymptotic approximation? Our asymptotic theory suggests that we should sample as frequently as possible. If trade-to-trade sampling is sufficient, then an estimate based on sampling every other transaction price should yield similar results. We next compare two estimates that use all and every other trade.

Plot of estimates of the FITCs using all and every other trade

Estimates based on taking every 30 th transaction don’t look so good (nor should we expect them to).

Cross section regressions Proposition 1: “Operating Costs” theory suggests stocks with higher liquidity should display smaller transaction costs. Proposition 2: “Asymmetric information” theory says that stocks with more private information should have larger transaction costs. Proposition 3: Both “operating cost” and “asymmetric information” theory suggest stocks with higher volatility should have larger transaction costs.

Liquidity measures $vol=average dollar volume per trade trades=average number of trades per day Asymmetric Information measure turn=(Shares transacted)/(shares outstanding) Volatility measure Other variables included Nasdaq=dummy variable for Nasdaq stocks Spread=average log(ask)-log(bid) Price=sample average price level (to capture price discreteness effects).

Taking logs of all variables we run the following cross sectional regression:

Regression Results 1 CoefStDevtProb Intercept lturn lsize lsdprice ltrades price nasdaq

Regression Results 2 CoefStDevtP Intercept lturn lsize lsdprice price nasdaq

Regression Results 3 CoefStDevtProb Intercept lturn lsize lsdprice price nasdaq lspread

Difference regression CoefStDevtProb Intercept lturn

Conclusions This paper proposes a new estimator for the cost of trade as measured by the expected deviation of transaction prices from the full information price. The estimate is consistent under weak assumptions. The proposed estimator is trivial to implement involving nothing more than calculating the second moments of the observed transaction prices.

Our empirical work demonstrates that we obtain sensible estimates for the S&P100 stocks. Skip sampling demonstrates the accuracy of our asymptotic approximations. We find support for the operating cost and asymmetric information theory of transaction cost. We find that the difference between FITCs and effective spreads can be attributed to asymmetric information measures thereby providing support for our economic interpretation of the proposed measure.

We examine market microstruture hypothesis about the determinants of the cost of trade. More liquid stocks have smaller effective spreads. Stocks with a higher proportion of informed traders have larger effective spreads. Stocks with higher volatility have larger effective spreads. We are currently constructing finite sample MSEs for our estimator.