Federico M. Bandi andJeffrey 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 t1, t2, …ti denote a sequence of arrival times where ti denotes the arrival time of the ith trade.

• Let N( ) denote the counting function. That is, the number of transaction that have occurred over the time period 0 to .

h

h

h

The ith observed transaction price at time ti is given by:

Where denotes the full information price and denotes deviations of the transaction price from the full information price.

ii t ip p

The setup

itp

Taking logs and differencing yields:

1

1 1 1

1

ln ln ln ln

where ln

or

where ln ln and i i

i i i i i i

i i

i i i

i t t i i i

p p p p

r r

r p p

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.

0

t

t s sM dw

ln t t tp A M

t

tA

Assumption 2: the microstructure noise

(1) is a mean zero covariance stationary process.

(2)

(3)

i

for j k

0 for j>k

j

i i jE

for all , , ,j l m nE j l m n

Lemma 1

Under Assumptions 1 and 2 we obtain:

1

2 2

0

11

2

k

i i k ss

kE E s E

Theorem 1Assume assumptions 1 and 2 are satisfied.

Conditional on a sequence of transaction arrival times such that

we obtain 1max , 1,..., 1 0 as i it t i N h N h

21

1 1

0

1ˆ 1

2

N h N h

i i i k s pki i k s

s N h

r r rk

sN h N h

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.

pO dt

1pO

1pO

Economic intuition

Ask

Bid

Effic. Price

Full Information Price

• • •

• •

• • •

t1 t5t4t3t2 t7t6 t8

Assumption 3

Where Qi denotes signed order flow. Specifically, Qi=1 denotes trades that occur above the full information price and Qi=-1 denotes trades that occur below the full information price.

denotes a constant distance that trades occur from the full information price.

Furthermore, Pr(Qi=1)=Pr(Qi=-1)=.5

1,...,2i i

sQ i N h

2s

Corollary to Theorem 1

Under the assumptions of theorem 1 and assumption 3 we obtain

( )

ˆ2

p

N h

s

Using this interpretation we refer to as anestimate of what we affectionately call the Full Information Transaction Cost (FITC).

ˆ

• 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 2002.

• Data are filtered to remove outliers.

Distribution of average number of seconds

between trades for S&P100 stocks

706050403020100

40

30

20

10

0

Intertrade Duration

Fre

quen

cy

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

0-100-200-300-400

60

50

40

30

20

10

0

t1

Fre

quen

cy

0-10-20-30

30

20

10

0

t2

Fre

quen

cy

86420-2-4-6-8-10

30

20

10

0

t3

Fre

quen

cy76543210-1-2

15

10

5

0

t5

Fre

que

ncy

43210-1-2

15

10

5

0

t10

Fre

quen

cy

2.52.01.51.00.50.0-0.5-1.0-1.5-2.0-2.5

20

10

0

t15

Fre

quen

cy

Distribution of FITCs

0.440.400.360.320.280.240.200.160.120.080.04

50

40

30

20

10

0

cost %

Fre

quen

cy

0.080.070.060.050.040.030.020.010.00

20

10

0

cost $

Fre

quen

cy

Some specific FITCs

Symbol Duration Avg Price FITC $ FITCEff.

Spread Ann SD Turnover

GE 4.68 37.50 0.06% $0.0215 0.05% 32.52% 0.03%

IBM 6.51 102.82 0.06% $0.0583 0.04% 26.65% 0.10%

NXTL* 1.10 5.07 0.18% $0.0093 0.56% 218.34% 0.16%

Average100 stocks 14.22 40.70 0.09% $0.0307 0.06% 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

00.0005

0.0010.0015

0.0020.0025

0.0030.0035

0.004

0.00450.005

0 0.001 0.002 0.003 0.004 0.005

FITC all data

FIT

C s

kip

1

Estimates based on taking every 30th transaction don’t look so good

(nor should we expect them to).

00.0005

0.0010.0015

0.0020.0025

0.0030.0035

0.004

0.00450.005

0 0.001 0.002 0.003 0.004 0.005

FITC all data

FIT

C s

kip

30

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 tradetrades=average number of trades per day

Asymmetric Information measureturn=(Shares transacted)/(shares outstanding)

Volatility measure

Other variables includedNasdaq=dummy variable for Nasdaq stocksSpread=average log(ask)-log(bid)Price=sample average price level (to capture price discreteness effects).

ˆ daily standard deviation (estimated by realized volatility)

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

0 1 2 3

4 5 6

ˆ ˆln ln ln ln $

ln # Price

true priceTurn vol

trades Nasdaq

Regression Results 1

  Coef StDev t Prob

Intercept -3.4684 0.6968 4.977 0.000

lturn 0.2186 0.0544 4.017 0.000

lsize -0.1846 0.0834 2.212 0.029

lsdprice 0.2177 0.0756 2.878 0.005

ltrades 0.1184 0.1116 1.061 0.291

price -0.0067 0.0016 4.182 0.000

nasdaq -0.7859 0.3123 2.516 0.0136

Regression Results 2

  Coef StDev t P

Intercept 3.2426 0.6640 -4.882 0.000

lturn 0.1812 0.0414 4.369 0.000

lsize 0.1296 0.0654 -1.980 0.050

lsdprice 0.2599 0.0643 4.041 0.000

price 0.0071 0.0015 -4.520 0.000

nasdaq 0.5126 0.1768 -2.898 0.0047

Regression Results 3

  Coef StDev t Prob

Intercept 1.1169 0.6609 -1.689 0.094

lturn 0.0917 0.0380 2.410 0.017

lsize 0.040 0.0573 -0.708 0.480

lsdprice 0.1144 0.0594 1.924 0.057

price 0.0005 0.0018 0.282 0.778

nasdaq 0.1746 0.1597 -1.093 0.277

lspread 0.6528 0.1062 6.143 0.000

Difference regression

0 1ln ln . FITC Eff Spread lturn

  Coef StDev t Prob

Intercept 1.5166 0.2197 6.901 0.000

lturn 0.1834 0.0342 5.354 0.000

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.