the nyse opening mechanism and portfolio trading

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The NYSE Opening Mechanism And Portfolio Trading

Peter Bossaerts0

August 1999

0California Institute of Technology and CEPR. Financial support from State Street Bank is gratefully acknowledged.

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The NYSE Opening Mechanism And Portfolio Trading

Abstract

In principle, implementation of portfolio investment strategies through market orders at the NYSE open would

be problematic because of execution price uncertainty. This paper measures the impact, by comparing the

actual value at the end of the trading day against the value one would have obtained if it were possible to

observe opening prices when submitting orders. For positively weighted portfolios of twenty-five securities, for

instance, the one-year cumulative risk of daily portfolio trading at the NYSE open is found to be 7 cents per

dollar invested. This is only one-third of the risk of holding a typical security overnight during the year. Incontrast to the latter, however, execution price risk appears not to be compensated.

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1 Introduction

Since Markowitz first formalization, portfolio theory has focused on optimal allocation in a multi-security

environment, emphasizing the trade-off between risk and return. By now, portfolio theory is well-understood

and widely applied.

Less attention has been paid, however, to the implementation of the portfolio allocations that portfolio

theory prescribes. This is not a simple matter of sending out market orders, because virtually all established

exchanges do not operate as marketplaces for portfolios. Instead, they are organized as a set of parallel markets

in single securities.

Problems rise when the execution of orders in the market for one security can generally not be made

dependent on trades or prices in other markets. Therefore, the implementation of desired portfolio allocations

requires a substantial amount of coordination of orders across the markets of each of the component securities.

This paper studies the ability of one market mechanism to facilitate portfolio trading, namely, the NYSE

opening. This mechanism works as a Walrasian auction, where the specialist operates as the auctioneer. There

is one important difference with the Walrasian auction, however: the specialists is allowed to (and generally

will) take a position. Among other things, this gives the specialist the opportunity to adjust the clearing price

(the NYSE open in a particular stock) to developments in other markets, and, hence, to facilitate portfolio

trading.

The NYSE open has been the subject of academic research before, as in [9].1 The latter, however, focuses

on the NYSE opening procedure as a single-security price setting mechanism. As a matter of fact, both the

theoretical and empirical market microstructure literature has been dealing almost exclusively with single-

security trading mechanisms. Exceptions include [2, 4].

The focus on the NYSE open is further justified by its importance in terms of daily volume. [9] reports that

the volume at the open is a negligible fraction of total daily volume only for the largest and most liquid stock.

To understand theoretically to what extent a trading mechanism with parallel markets facilitates portfolio

trading, one has to analyze the covariance matrix of the clearing prices in cross-section.

If, as is the case with prices during the continuous-time trading mechanism after the NYSE open, a single

eigenvalue dominates the covariance matrix (we will document this fact later on), then clearing prices will be

1[5] studies the NASDAQ opening, and [1] studies the Paris Bourse opening.

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highly correlated across issues, as if a single common factor drives all the uncertainty. If, in addition, the

correlation is positive, then market orders (to trade a given quantity at any price) that aim to establish a

positively-weighted portfolio will not execute well. While the eventual composition of the portfolio (weights

assigned to each component stock) will be highly correlated with the desired composition, the amount of

dollars invested will be wrong (overinvestment if prices went up, underinvestment if prices went down). In

contrast, if there is little cross-sectional correlation in overnight price changes, the total dollar investment will

be approximately as desired, even if the eventual weights of the portfolio may be slightly off.

The outcome would be different for swaps, where a number of securities are to be exchanged for a number

of other securities. Swaps would be necessary, among others, when rebalancing an existing positively-weighted

portfolio, or to establish an zero-net-weight (arbitrage) portfolio. When clearing prices are highly positively

correlated, implementation of swaps by means of market orders would generate small errors. On days when prices

open generally higher, both the long and short end would become overweighted, leading to a net cancelation.

In the absence of cross-sectional correlation, large errors would result.

The purpose of this study is to establish empirically how the NYSE open performs from the point of view of

portfolio allocation. We will compare the end-of-day valuesof (i) the actual portfolio established with market

orders at the open, against (ii) the hypothetical portfolio that would have been obtained if orders at the open

could have been submitted based on the actual opening prices. We could also have chosen to compare actual

against desired weights. But it is not clear what metric should be used in such a multi-dimensional comparison.

Moreover, sometimes issues do not trade at all, and, hence, no open is available, in which case a comparison of

actual (zero) versus desired (positive or negative) weights would become problematic.

In determining the market orders, we will compute quantities (size of the orders) based on the previous-day

close, adjusted for stock splits and dividends. In terms of our criterion (end-of-day portfolio values), one could

do better, because there appears to be a highly significant, albeit small, negative correlation between overnight

price changes and price changes during the trading day. This correlation could be taken into account. In future

research, one could also attempt to determine what type of limit orders would improve upon market orders.

The analysis is based on a recent history of 95 days of NYSE open activity in S&P 1500 stock during the

first half of 1999. In total, 97,973 complete records (containing at least the previous-day close, the open and

the days close) are retained in the dataset. We will study randomly chosen positively weighted portfolios as

well as swaps, with 5, respectively 25, securities each.

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An analysis of the capability of trading mechanisms in implementing desired portfolio compositions is not

only of importance for practical investments analysis. It appears to be relevant for market equilibration as

well. Recent experiments ([3]) provide direct evidence of the difficulties of thin trading in securities markets

organized as parallel multiple-unit double auctions. Execution uncertainty made it hard for subjects to assess

the likelihood of implementing a desired portfolio re-allocation, to the point that they often stopped submitting

orders. This, of course, made it impossible for markets to reach equilibrium. The phenomenon was very apparent

in the experimental financial markets of [3], where distance from equilibrium could readily be computed (unlike

for the NYSE).

It is no surprise that we will find it to be risky to construct portfolios by trading at the NYSE open. We

will address the question whether the risk is significant. We will also investigate whether the market manages

to compensate traders who choose to incur the risk.

The remainder of this paper is organized as follows. The next section provides a few statistical facts about

the risk embedded in NYSE overnight price change (close-to-open), in comparison to that of price changes

during the trading day (open-to-close). Section 3 explains the methodology used to evaluate the capability of

the NYSE opening to accomodate portfolio trades. Section 4 presents the empirical results. Section 5 concludes

with evidence on risk compensation.

2 The Data

Ninety-five days of NYSE activity in S&P 1500 stock were collected, covering the period January 8 1999 to

May 28 1999. Each record contains at least the close of the previous trading day (adjusted for stock splits

and dividends), the opening that day, and the days close. The data were obtained from Interquote. Spot

checks with other data providers confirmed their reliability. Not all of the 1,045 issues that were retained in

the dataset have a complete record, because, (i) some days no trade was recorded, (ii) unavailability because of

computer network problems. In total, 97,973 records were retained, somewhat less than the theoretical 99,275

(=1045

95).Table 1 provides a few descriptive statistics for the close-to-open return, and compares them to those for the

open-to-close return. Of importance are the differences in the volatility (standard deviation) and the kurtosis.

The volatility of the open-to-close return is about double that of the close-to-open return. Of course, this is a

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well-known phenomenon, explored more extensively in [7]. But the kurtosis of the close-to-open return is an

order of magnitude higher than that of the open-to-close. In fact, there are a large number of cases where the

close-to-open return is zero, meaning that the stock opened at the previous days close. That would be good

for portfolio trading: the previous days close provides an excellent basis with which to determine the size of

ones market orders. On the other hand, there are also quite a few outliers, which may affect the outcome.

The contrast in volatility and kurtosis between the close-to-open and open-to-close is illustrated graphically

in Figure 1, which plots the respective histograms and fitted normal curves. The scales of the two subplots were

made to match, in order to facilitate comparison. The fitted normal curve for the open-to-close is more spread

out, indicating higher volatility. The histogram of the close-to-open, however, is far more leptokurtotic when

compared to the fitted normal curve, indicating higher kurtosis.

There is a small, but highly significant amount of predictability in the data. Table 2 displays the results

from a least squares projection of the open-to-close return onto the previous close-to-open return. The slope

coefficient has a t-statistic of about 10, indicating that the relationship is highly significant. The R2 of the

projection is only 0.10%, however, so that the economic relevance of the predictability is minor.

Still, the negative slope coefficient implies that the specialist over-reacts when setting opening prices relative

to closing prices. If the specialist systematically buys when the open is above the close, and sells when the close

is above the open, then (s)he could profit on average by reverting the position at the close.

Because the economic significance of the price reversals are minor, we will ignore it when determining the

size of the market orders to be submitted at the open. We could marginally improve the performance of the

order submission strategy if we did take the negative correlation of the open-to-close and previous close-to-open

into account.

In order to form an idea about the extent to which price changes are correlated, Table 3 displays descriptive

statistics of the ratio of the largest eigenvalue to the second largest one in the covariance matrix of 25 securities.

Two hundred sets of 25 randomly selected securities were used to compute the statistics in Table 3. We compare

the ratio of eigenvalues for the close-to-open return against that of the open-to-close return.

It is well known that the covariance matrix of stock returns is dominated by a single large eigenvalue. The

largest eigenvalue is often one or two orders of magnitude higher than the next one. See, e.g., [8]. This can

be interpreted as meaning that stock price changes are driven by one pervasive factor, and, hence, that stock

prices are highly correlated. The correlation is positive.

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When investigating the close-to-open and open-to-close returns, there seems at first not to be much evidence

that the largest eigenvalue is much bigger than the second one. See Table 3. Still, the minimum value of the

ratio is very small in the case of the open-to-close.

In fact, the statistics do not tell the entire story. Figure 2 compares the histogram of the ratio of the largest

eigenvalue over the next one for the close-to-open against that for the open-to-close. There is a pronounced

difference. While the histogram of the ratio is concentrated around relatively high values (0.7-0.8) for the close-

to-open, 20% of the randomly selected sets of 25 securities generated a ratio below 0.10 for the open-to-close.

It appears that the presence of a single, pervasive factor that drives price changes is only characteristic of

the open-to-close, and not of the close-to-open. The behavior of the close-to-open price change cannot simply

be explained in terms of one dominant force.

As mentioned in the previous section, this implies that implementation of positively-weighted portfolios by

market orders at the open is likely to generate diversification effects, because opening prices do not seem to be

determined in unison (even if the specialists could possibly do so). That is, the larger the portfolio, the lower

the impact of execution price uncertainty. In contrast, no diversification effects can be expected for swaps,

because erroneous allocations at the long end will not be offset by mistakes at the short end.

Let us turn to measuring how bad (if at all) execution of portfolio trades is at the NYSE open.

3 Methodology

We will evaluate the ability of the NYSE opening mechanism to accomodate portfolio trades by comparing the

end-of-day value of (i) the actual portfolio that could be formed by submitting market orders at the open, based

on quantities determined from the previous close, (ii) the hypothetical portfolio that would be obtained if the

investor could condition on the actual opening prices.

By comparing end-of-period values, we not only compare actual and desired compositions, but, more im-

portantly, how differences in the compositions lead to different performance as measured by a one-day return.

To be explicit, let p

c denote the vector of previous-day closing prices for I

securities. Let p

o denote thevector of opening clearing prices. And let pc be the vector of the days closing prices.

An investor wishes to buy a portfolio characterized by a vector of desired weights x. We will consider two

cases:

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1. The weights inx are all positive and add up to 1. The resulting portfolio is to be referred to as a positively

weighted portfolio. We will look at randomly constructed portfolios, where the weights are based on I

independent draws from a uniform distribution on [0, 1], normalized to add up to 1. Let xi and ui denote

the weight on security i and uniform draw i, respectively. Then:

xi= uiIi=1 ui

.

2. The weights in x add up to zero. The resulting portfolio is to be referred to as a swap. We will consider

randomly constructed swaps, where I 1 weights are based on independent draws from a uniform distri-

bution on [1, 1], and weight Iis given by minus the sum of the weights of the other I 1 weights. That

is, for i= 1,...,I 1,

xi= ui,

and

xI= I1i=1

xi.

The actualmarket orders used to establish the portfolio are based on the previous-day closing prices. Let

ni denote the size of the market order submitted for execution at the opening of security i. ni solves

xi = nipc,i

Ni=1 nipc,i. (1)

Contrast this with the hypotheticalmarket orders one would want to submit knowing the opening prices. The

size of hypothetical order i, denoted ni, solves

xi = nipo,iNi=1 nipo,i

. (2)

When an issue does not trade on a particular day (or when we did not have a record), we set

ni= 0.

(This will not have an effect on the performance measurement; it is included only for the sake of completeness.)

The quantities in (1) define what we have been calling the actual portfolio. Those in (1) define thehypothetical

portfolio.

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The end-of-day values of the two portfolios are measured simply as the sums of closing prices times quantities.

Let Jbe the number of issues that traded during the day or for which we had a trading record (J I). the

end-of-day value of the actual portfolio, V, equals

V =J

i=1

nipc,i. (3)

That of the hypothetical portfolio, V, would have been:

V =J

i=1

nipc,i. (4)

We will report in the next section on the distribution of the absolute value of the difference between V and

V for 200 different portfolios of randomly selected securities.

Here is how these statistics were calculated. Let t index time (trading days; t = 1,...,T = 95), and let

l index the randomly drawn portfolios (l = 1,...,L = 200). Vl,t and Vl,t denote the value of the actual and

hypothetical versions of portfolio l at the end of day t. The difference between the actual and hypothetical

value, to be referred to as the error, is defined as follows:

l,t= Vl,t Vl,t . (5)

For ease of reference, we will normalize the error l,t. For positively weighted portfolios, l,t is already

normalized, by the dollar amount invested. Interpreting the weights of the desired portfolios as dollar values,

the size of our positively weighted portfolios is one dollar. Therefore, when expressed as a percentage, l,t

measures the end-of-day valuation error in cents per dollar invested.

Normalization is more problematic for swaps. We define the size of a swap as the expected sum of the positive

weights. The expected sum of positive weights was estimated to be the following (based on simulations):

E[I

i=1

xi1{xi>0}]

1.5 ifI= 5;

7.1 ifI= 25.(6)

We will use this simple estimate of the size of the swap to scale l,t.

So, the error will be expressed as percentages, or, equivalently, as US cents per dollar investment. That

provides an easily interpreteable measure of the effect on portfolios of execution price uncertainty at the NYSE

open.

Let us turn to the results.

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4 Empirical Results

We will first discuss the results for randomly selected portfolios of 5 securities. Later on, we increase the number

of securities, to 25, in order to assess whether the increase leads to better diversification of execution price risk.

To start, let us look at the errors for one randomly chosen portfolio. Figure 3 plots the error l,t against

time for a randomly chosen portfolio with weights that add to zero (a swap). The error is measured in cents per

dollar expected investment. The swap portfolio contains five securities, so, from (6), the expected investment

is $1.5.

According to Figure 3, the errors may seem fairly small, between +/ one cent. Still, this is significant

if one considers that the overnight volatility of a typical stock is 1.38% (see Table 1). Hence, the effect of

misallocation because of execution price uncertainty at the open is not negligible, when expressed as a fraction

of the overnight risk of a single security.

Moreover, there are outliers. In the plot, the error reaches almost 5 cents at one point. This suggests

leptokurtosis: a larger number of extreme outcomes than predicted by the normal distribution.

The following tables and figures will provide a more comprehensive picture than Figure 3. They will lead

us to confirm the impression that the effect of execution price uncertainty on the end-of-day portfolio values

is a significant fraction of the overnight risk of holding one security, and that the distribution of the error is

leptokurtotic.

Table 4 displays several statistics for the error l,t. Results are given for both positively weighted portfolios

and swaps.

For positively weighted portfolios, Table 4 shows that the mean error is 0.071 cent per dollar invested. The

reason for the positive error for positively weighted portfolios emerges from inspection of the average overnight

price change of S&P 500 issues at the NYSE. Table 1 reports that a typical issue gains 0.085% (cents per dollar).

We did not take this gain into account in formulating the order submission strategy at the open. Since order

size was determined on the basis of closing prices, and opening prices are on average higher, we generally end

up with a higher total dollar investment than desired (about 0.085% more). This also shows up in the value of

the actual portfolio at the end of the day: for portfolios of five securities, it is on average 0.071% higher than

what the portfolio would have been worth had we been able to submit orders at the open conditional on all the

opening prices.

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The standard deviation of the error is high, at 0.732 cents, and there is substantial leptokurtosis, so that

the error can be as high as 12 cents. One should put this in perspective, however. The standard deviation

of the overnight price change of a single security is 1.38 cents (see Table 1). So, the execution price risk is

only one-half the volatility of overnight price changes. In other words, diversification is at play. Of course, this

should be attributed to the lack of a dominant eigenvalue in the covariance matrix of overnight price changes

(see Figure 2).

Likewise, the kurtosis of the error l,t is far less than that of the typical overnight price change: 26 against

185. Diversification explains this.

As for swaps, Table 4 reports that the average error is tiny. The average overnight price increase for a typical

stock of 0.085 cents (see Table 1) does not show up in the error for swaps, because swaps are essentially the

combination of a long and a short positively weighted portfolio. The average overinvestment on the long end is

generally offset by an overinvestment on the short end. The net effect is zero.

Contrary to what we expected from our evidence of the absence of a dominant eigenvalue in the covariance

matrix of overnight price changes, the standard deviation of the error for swaps is small. At 0.371, it is only

about 1/4 the risk of the overnight price change. And it is only 1/2 of the standard deviation of the error for

positively weighted portfolios.

But the beneficial effect of portfolio trading at the open is far less pronounced for the kurtosis. The kurtosis

drops from 185 at the individual stock level (see Table 1) to only 86 in the case of swaps. Contrast this with the

kurtosis for positively weighted portfolios, which drops to 26. Hence, the absence of a dominant eigenvalue in

the covariance matrix of overnight price changes leads to a bigger decrease in the kurtosis for positively weighted

portfolios than for swaps.

The effect of the kurtosis can also be seen from the extrema: whereas the standard deviation of the error

for swaps is only 1/2 that for positively weighted portfolios, the extrema (minimum, maximum) are about the

same.

A better picture may emerge from the histograms. Figure 4 displays the histograms for both positively

weighted portfolios (top panel) and swap portfolios (bottom panel). The histograms are put on the same

scale. The X-axis is truncated, however, at +/ 3 (cents per dollar). The leptokurtosis is obvious, despite the

truncation of the X-axis.

To further assess the impact of diversification, Table 5 displays the results when we increase the number of

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securities from 5 to 25. Of course, the average error does not change much, because it is caused by the average

overnight price increase of a typical stock. Diversification does reduce the standard deviation and kurtosis of

the error further.

For positively weighted portfolios, this is as expected. For swaps, the standard deviation of the error

(expressed as cents per dollar expected investment) is reduced further, but the effect on the kurtosis is only

marginal: it drops from 86 for five securities to 57 for twenty-five securities. With five securities, the kurtosis

was only four times higher than for positively weighted portfolios; with twenty-five securities, it is five times

higher. The extrema are much reduced, however.

Figure 5 plots the histograms of the errors. For ease of reference, the scales are again the same for the two

histograms. Figure 5 is not able to aptly depict the higher kurtosis of the distribution of the error for swaps.

Tables 4 and 5 allow us to investigate the cumulative effect over one year of daily execution of trading at

the NYSE open. Doing so would provide an alternative measure of the size of the errors.

Let us focus on positively weighted portfolios. Summing the average errors in Tables 4 and 5 over one

year (approximately 250 trading days) reveals that the cumulative effect of daily trading at the NYSE open is

18% for portfolios of 5 securities, and 21% for portfolios of 25 securities. Of course, this average reflects the

overnight price gain of a typical issue, which, at 0.085% per day, cumulates to 21% over one year. The standard

deviations of the one-day errors are 0.732% and 0.446% per day for portfolios of five and twenty-five securities,

respectively. Over one year, this cumulates to 12% and 7%, respectively.

In other words, daily trading of positively weighted portfolios at the NYSE open would have generated a

cumulative overinvestment of 1820% on a yearly basis, and between 7% and 12% undesired volatility. The

former could have been avoided by reformulating the order submission strategy to take into account the average

overnight price gain. The latter cannot be avoided, because it is generated by the execution price uncertainty

inherent to the NYSE open.

5 To Conclude: Is Execution Price Risk Compensated?

From a practical point of view, the ultimate question is whether execution price risk at the NYSE open is

significant, and whether it is compensated for. The evidence presented in this paper seems to suggest that it

is significant. However, from the perspective of portfolio analysis, it is much less than what one would infer

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from a study of the overnight price risk of a typical security. That is, diversification definitely plays a role.

Diversification is possible because of the absence of a single dominant eigenvalue in the covariance matrix of

overnight price changes. Diversification even works for swaps. But the effect on kurtosis is less pronounced.

This means that outliers are more likely than what the (small) standard deviation would suggest.

Over one year, the risk of daily execution of trading at the NYSE open is approximately 12% for positively

weighted portfolios of five securities, or 7% for portfolios of twenty-five securities. From this perspective, the

risk from execution price uncertainty at the NYSE open does not appear to be marginal.

There is one puzzling aspect of the results that deserves emphasis, however. While the execution price

uncertainty is nontrivial, investors do not seem to be compensated for it. In particular, the mean difference

in end-of-day value between the actual and hypothetical portfolios can be explained entirely as an effect of

the average overnight price increase. If the mean error had been larger, then the difference could have been

interpreted as compensation for the size (standard deviation; kurtosis) of the error. Such is not the case.

This contrasts with the risk of holding a typical security overnight. Table 1 documents that it amounts to

22% per year, and is compensated at 21% per year.

Of course, compensation of execution price risk would require an inordinate level of sophistication for financial

markets. Still, there is evidence that financial markets manage to compensate complex risks in other contexts.

For instance, [6] reports that holders of small firm stock are compensated only over earnings announcement

periods, when most risk is realized (for that category of stock).

Besides more elaborate order submission strategies than the one considered in this paper (such as limit

orders), an alternative would be to trade in the continuous, hybrid system during the day. It is not clear how

costly this is from a portfolio-analytic point of view. We should leave the analysis to future work.

One wonders whether there are quick fixes to the the opening mechanism, so that investors escape the 712%

annual, uncompensated risk that the present system implies. For instance, if investors were able to directly

trade portfolios instead of individual securities, the execution risk that we focused on in this paper would be

eliminated.

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References

[1] BIAIS, Bruno, Pierre HILLION and Chester SPATT [1999] Price Discovery And Learning During The

Pre-Opening Period In The Paris Bourse, Journal of Political Economy, forthcoming.

[2] BOSSAERTS, Peter [1993]: Transaction Prices When Insiders Trade Portfolios,Finance14, 43-60.

[3] BOSSAERTS, Peter, Dan KLEIMAN and Charles PLOTT [1998]: Price Discovery In Financial Markets:

The Case Of The CAPM, Caltech working paper.

[4] CABALLE, Jordi and Murugappa KRISHNAN [1994]: Imperfect Competition In A Multisecurity Market

With Risk Neutrality, Econometrica62, 695-704.

[5] CAO, Charles, Eric GHYSELS and Frank HATHEWAY [1998]: Price Discovery Without Trading: Evi-

dence From The Nasdaq Pre-Opening, Penn State University working paper.

[6] CHARI, V.V., Ravi JAGANNATHAN and Aharon R. OFER [1988]: Seasonalities in Security Returns:

The Case of Earnings Announcements, Journal of Financial Economics21, 101-122.

[7] FRENCH, Kenneth R. and Richard ROLL [1986]: Stock Return Variances: The Arrival Of Information

And The Reaction Of Traders, Journal of Financial Economics17, 5-26.

[8] GREEN, Richard C. and Burton HOLLIFIELD [1992]: When Will Mean-Variance Efficient Portfolios Be

Well-Diversified? Journal of Finance47, 1785-1810.

[9] MADHAVAN, Ananth and Venkatesh PANCHAPAGESAN [1998]: Price Discovery In Auction Markets:

A Look Inside The Black Box, USC Marshall Business School working paper.

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Table 1: Descriptive Statistics, NYSE, S&P 1500 Stock, January 8 1999 to May 28 1999

Return

Close-To-Open Open-To-Close

Number 97973 97973

Mean Return (%) 0.085 -0.026

(St. Error) (0.004) (0.009)

St. Dev. (%) 1.38 2.69

Skewness 0.01 0.61

Kurtosis 185 15.8

Maximum (%) 58.9 70.5

Minimum (%) -51.2 -44.7

Table 2: Least Squares Projection Of Open-To-Close Return Onto Previous Close-To-Open Return, NYSE,

S&P 1500 Stock, January 8 1999 to May 28 1999, 97973 Observations

Coefficients

Intercept Slope

Estimate (

10

2

) -0.021 -6.328(St. Error, 102) (0.009) (0.623)

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Table 3: Ratio Of Largest To Second Eigenvalue Of The Covariance Matrix Of 25 Randomly Selected Securities,

200 Replications, NYSE, S&P 1500 Stock, January 8 1999 to May 28 1999, 97973 Observations

Return

Close-To-Open Open-To-Close

Number 200 200

Mean Ratio 0.572 0.543

(St. Error) (0.015) (0.023)

St. Dev. 0.218 0.318

Skewness -0.27 -0.79

Kurtosis 2.1 2.1

Maximum 0.978 0.954

Minimum 0.080 0.025 103

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Table 4: Differences In End-of-day Value Between Actual And Hypothetical Portfolios (l,t), For 200 Randomly

Chosen Portfolios Of 5 Securities, Selected From NYSE S&P 1500 Stock, January 8 1999 to May 28 1999.

Portfolio Type

Positively Weighted Swaps

Number 19,000 19,000

Mean 0.071 (cent)a -0.002 (cent)b

(St. Error) (0.005) (0.003)

St. Dev. 0.732 0.371

Skewness -0.67 0.43

Kurtosis 26.2 85.6

Maximum 9.46 10.30

Minimum -12.21 -8.40

aExpressed as number of cents per dollar investment.bExpressed as number of cents per dollar expected investment.

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Table 5: Differences In End-of-day Value Between Actual And Hypothetical Portfolios (l,t), For 200 Randomly

Chosen Portfolios Of 25 Securities, Selected From NYSE S&P 1500 Stock, January 8 1999 to May 28 1999.

Portfolio Type

Positively Weighted Swaps

Number 19,000 19,000

Mean 0.087 (cent)a 3 104 (cent)b

(St. Error) (0.003) (2 104)

St. Dev. 0.446 0.030

Skewness -0.99 -1.35

Kurtosis 11.1 56.9

Maximum 3.70 0.60

Minimum -3.57 -0.79

aExpressed as number of cents per dollar investment.bExpressed as number of cents per dollar expected investment.

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-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3x 10

4

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3x 10

4

Figure 1: Histogram and Fitted Normal Curve, Close-To-Open Return (Top Panel) and Open-To-Close Return

(Bottom Panel), NYSE, S&P 1500 Stock, January 8 1999 to May 28 1999.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

Figure 2: Histogram Of Ratio Of Largest To Second Eigenvalue Of Covariance Matrix Of 25 Randomly Selected

Securities, 200 Replications, Close-To-Open Return (Top Panel) and Open-To-Close Return (Bottom Panel),

NYSE, S&P 1500 Stock, January 8 1999 to May 28 1999.

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10 20 30 40 50 60 70 80 90-2

-1

0

1

2

3

4

5

Time (Days)

Error(l,t)

Figure 3: Errors (Differences Between End-of-day Values Of Actual Portfolio And Hypothetical Portfolio) For

A Randomly Chosen Swap Portfolio Of 5 Issues, Selected From NYSE S&P 1500 Stock, January 8 1999 to May

28 1999. Errors Are Expressed In Cents Per Dollar Expected Investment.

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-3 -2 -1 0 1 2 30

500

1000

1500

2000

2500

3000

-3 -2 -1 0 1 2 30

500

1000

1500

2000

2500

3000

Figure 4: Histogram Of Differences In End-of-day Values Between Actual And Hypothetical Portfolio, In Cents

Per Dollar Invested, For 200 Randomly Chosen Portfolios Of 5 Securities, Selected From NYSE S&P 1500 Stock,

January 8 1999 to May 28 1999. Top Panel: Positively Weighted Portfolios; Bottom Panel: Swaps.

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-1.5 -1 -0.5 0 0.5 1 1.50

500

1000

1500

2000

-1.5 -1 -0.5 0 0.5 1 1.50

500

1000

1500

2000

Figure 5: Histogram Of Differences In End-of-day Values Between Actual And Hypothetical Portfolio, In Cents

Per Dollar Invested, For 200 Randomly Chosen Portfolios Of 25 Securities, Selected From NYSE S&P 1500

Stock, January 8 1999 to May 28 1999. Top Panel: Positively Weighted Portfolios; Bottom Panel: Swaps.

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the nyse opening mechanism and portfolio trading

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