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 h (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( h ) denote the counting function. That is,
the number of transaction that have occurred
over the time period 0 to h .
The setup
The ith observed transaction price at time ti
is given by:
pi  ptii
Where pt denotes the full information
price and i denotes deviations of the
transaction price from the full information
price.
Taking logs and differencing yields:
ln  pi   ln  pi 1   ln  pi   ln  pi 1   i  i 1
where i  ln i 
or ri  ri   i
 
 
where ri  ln pti  ln pti1 and  i  i  i 1
Assumption 1: the price process
(1) The log price process is a continuous
semimartingale:
ln  pt   At  M t
Where At is a continuous
finite variation
t
component and M t    s dws
0
(2) The spot volatility  t is a cadlag process.
Assumption 2: the microstructure
noise
(1)  i is a mean zero covariance stationary
process.
 j for j  k
(2) E ii  j   
0 for j>k


(3) E  jl m n   for all j , l , m, n
Lemma 1
Under Assumptions 1 and 2 we obtain:
  E  2 
k 1
 1 k 
2
 
E

   s  1 E  i i k  s 

 2 
s 0
 
Theorem 1
Assume assumptions 1 and 2 are satisfied.
Conditional on a sequence of transaction arrival
times such that


max ti 1  ti , i  1,..., N  h   1  0 as N  h   
we obtain
 N h  2 
 N h 
  ri  k 1
  ri ri  k  s
 1  k   i 1 
 i  k  s 1
ˆ
  

s

1





 N h 
 2  N h 
s 0







 p


 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 Op  dt  .
• The noise returns are Op 1
• In the limit (as the intervals go to zero) the observed
returns are dominated by the Op 1 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
Full Information Price
•
•
•
•
•
•
•
Ask
Effic. Price
•
Bid
t1
t2
t3
t4
t5
t6
t7
t8
Assumption 3
s
i  Qi i  1,..., N  h 
2
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.
s 2 denotes a constant distance that trades occur from
the full information price.
Furthermore, Pr(Qi=1)=Pr(Qi=-1)=.5
Corollary to Theorem 1
Under the assumptions of theorem 1 and
assumption 3 we obtain
p
s
ˆ 
N ( h ) 2
Using this interpretation we refer to ˆ 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 2002.
• Data are filtered to remove outliers.
Distribution of average number of seconds
between trades for S&P100 stocks
40
Frequency
30
20
10
0
0
10
20
30
40
Intertrade Duration
50
60
70
Histograms of t-stats for
1,2,3,5,10,and 15 autocorrelations.
30
60
30
20
30
20
Frequency
40
Frequency
Frequency
50
10
20
10
10
0
0
0
-400
-300
-200
-100
-30
0
-20
-10
-8
-6
-4
-2
0
2
4
6
1.5 2.0
2.5
t3
t2
t1
20
15
10
5
0
Frequency
15
Frequency
Frequency
-10
0
10
5
0
-2
-1
0
1
2
3
t5
4
5
6
7
10
0
-2
-1
0
1
t10
2
3
4
-2.5 -2.0 -1.5 -1.0 -0.5 0.0
t15
0.5 1.0
8
Distribution of FITCs
50
20
Frequency
Frequency
40
30
20
10
10
0
0
0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44
cost %
0.00
0.01
0.02
0.03
0.04
cost $
0.05
0.06
0.07
0.08
Some specific FITCs
Duration
Avg Price
FITC
$ FITC
Eff.
Spread
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%
14.22
40.70
0.09%
$0.0307
0.06%
0.25%
Symbol
Average
100 stocks
Ann SD
Turnover
37.30%
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.
FITC skip 1
Plot of estimates of the FITCs
using all and every other trade
0.005
0.0045
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0
0.001
0.002
0.003
FITC all data
0.004
0.005
FITC skip 30
Estimates based on taking every
30th transaction don’t look so good
(nor should we expect them to).
0.005
0.0045
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0
0.001
0.002
0.003
FITC all data
0.004
0.005
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
ˆ  daily standard deviation (estimated by realized volatility)
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:


ln ˆ    0  1 ln Turn    2 ln ˆ  true price   3 ln  $vol 
 4 ln  # trades   5 Nasdaq   6 Price
Regression Results 1
Coef
StDev
t
Prob
-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
Intercept
Regression Results 2
Intercept
Coef
3.2426
StDev
0.6640
t
-4.882
P
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
ln  FITC   ln  Eff . Spread   0  1lturn
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.