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It's Hard to Distinguish
Skill from Luck

Seven Quantitative
Insights into Active
Management -
A Summary



Forecast Return
Distribution in Aegis
Risk Manager 2.0

The Market
Impact Model™ - Part 3

Automating the
Investment Process



Ron Kahn Looks
Back on 11 Years
at BARRA


The BARRA Brainteaser
for Summer 1998
Solution to the Spring
1998 Brainteaser



A Beautiful Mind

The Market Impact Model™ -
Part 3

by Nicolo Torre

This is the third part of our continuing series on the Market Impact Model.1 In the first two parts we defined the problem that the model addresses and applied economic reasoning to establish the framework of the solution. Now we shall turn to empirical data analysis to provide the details of the solution. To organize the discussion, we begin by stating the conclusion. We then begin building to this conclusion. First we treat data upon which our work rests. Next we treat each model factor in turn, surveying the principal findings of data analysis. Finally, we show how the information is assembled into the final forecast of market impact.

The Market Impact Model

To anticipate our conclusion, we state the final model as:

                          (1)

where:

• is the forecast cost of the trade measured by value
• is the trade volume measured by value
• there are four parameters characterizing the asset being traded:
  is the elasticity; it describes the response of order flow to price signals
  is the volatility of the asset; it describes the variability of the asset’s price
  is the intensity; it describes how often the asset trades
  is the shape; it describes the distribution of trade sizes
• there is a single market parameter:
  is the market tone; it is the price of liquidity
• there is an investor-specific parameter:
  is the skill; it captures the effect of the investor’s trading process on the cost experience
• F() is a complicated function that integrates these different sources of information into the final forecast

The data

There are a number of different data sources that we have used to investigate market impact, among them:

• the market data feed
• trading results achieved by clients
• specialized data sets made available by third parties

  For actually updating the model, however, only the market data feed is available on a timely basis. Thus, our comments will focus on this data source.

Much of this data originates with the specialist’s clerk, who types it in by hand, often under considerable time pressure. Thus, data errors are not uncommon. Some of these errors are corrected, possibly hours after the first report. Other errors remain uncorrected. Adding to the complexity are the cryptic encoding of the data, the reporting of data from multiple markets, and the opportunities for the time order of events to become garbled. A significant, highly technical effort is required to put the market data feed in a form suitable for analysis.

The first stage of analysis aims at detecting data errors and assigning trade initiation. This is essentially a rule-based endeavor, involving such logic as:

• If a trade is reported at a price that is ten times the prior trade price, the decimal point was probably misplaced, so this is a data entry error. Infer that a trade took place at probably one-tenth of the reported price.
• If the trade price is above the latest ask, and the ask was reported recently, then this trade was probably buyer-initiated.

Model factors

In the previous article of this series¹ we identified four characteristics of an asset that would control the risks of a liquidity provider taking a position in that asset. Our purpose now is to capture these characteristics in quantitative terms. We refer to such a quantification of an asset characteristic as a model factor. In general, each factor is itself the product of a submodel. It is convenient to assign a name to each factor which conveys the basic characteristic being quantified. As noted at the commencement of this chapter, the asset factors are elasticity, volatility, intensity, and shape.

Besides the asset-related factors, the model also has a market factor—the market tone—and an investor-related factor—the skill. To discuss these latter two factors, it is first necessary to partially construct the market impact cost function. Accordingly, discussion of these factors will be deferred to a later section.

Elasticity

As we saw in the previous article, it is necessary to model how the direction of order flow responds to price changes.

We define the equilibrium price to be the price at which an order arriving in the market is equally likely to be a buy order or a sell order. While the equilibrium price would be difficult to estimate in real time, in retrospective analysis it can be determined with fair accuracy. We define the price deviation to be the difference between the logarithms of the midquote and the equilibrium price :

                          (2)

The elasticity function () gives the probability that an incoming market order will be a sell order, conditional on the current price deviation. By definition, we have:

We might expect that, as becomes large, () should move toward 1, and that, as becomes a large negative number, then () should go to 0. Thus, we might expect the shape of the function to resemble FIGURE 1.

()

Figure 1 - Expected form of the elasticity function

Actually, however, when we measure the market’s response to price deviation, we find that the elastic function is compressed within narrower vertical limits. This compression is probably due to the presence of price-takers in the market—investors who are content to trade at whatever price is quoted in the market, and who, therefore, do not respond to price signals.

We find it convenient to parameterize the elasticity function as:

                          (3)

where:

f is the elastic fraction; it describes the maximum strength of the market’s response to price signals

c is the elastic coefficient; it describes the sensitivity of the market to price deviations

tanh() is the hyperbolic tangent function

We also define the elasticity S as the slope of the elasticity function at the point = 0. For small deviations :

                          (4)

The slope is related to the elastic fraction and coefficient by S = fc. S is the elasticity reported in the Market Impact Model.

Referring to FIGURE 2, we see that the vertical range of the function is 2f, and S is the steepness of the function at the origin. The elastic coefficient c determines how quickly the function flexes in response to price deviations.

Figure 2 - Actual form of the elasticity function

When we measure these various parameters for different assets (see FIGURE 3), we find some interesting tendencies. For the larger companies the elastic fraction tends to be lower, while the elastic coefficient tends to be higher. In other words, more of the orders for these assets are generated by price-takers, but the non-price-takers seem to be fairly sensitive to price deviations. By contrast, for smaller companies it seems that fewer orders are generated by price-takers, but larger price deviations are required to generate a response. Possibly investors are less certain of what the "right" price is for these assets, and so they are slower to perceive a bargain.

Figure 3 - Elasticity parameters as a function of capitalization

Although it requires fairly sophisticated econometric technique to back the elasticity function out of the available data, we may note that the specialist has a considerable informational advantage in this regard. Simply by looking at the limit order book, he can gauge the strength of near market demand. Furthermore, by moving his quotes around he can test the market's response to price changes. Thus, he can assess the likely reaction of order flow to price change.

Volatility

Elasticity describes the reaction of order flow to deviation of the market price from the equilibrium price. However, the equilibrium price itself varies due to the changing level of demand for an asset. The volatility factor provides a description of this variability.

The standard model for asset prices is that they follow a lognormal diffusion process. This process is characterized by two parameters and , such that if is the price of the asset at time t, then for

r is normally distributed with mean and variance . Several improvements can be made on this basic model:

• In general is not a fixed constant, but will vary through time.
• For short holding periods , the size of r will typically be small, and so the distribution of r will deviate from normality due to the discreteness of price changes.
• Empirically, return distributions deviate from normality in that the probability of large returns, albeit rare, is somewhat greater than a normal distribution would allow.

Intensity

There is considerable variability in how quickly orders for a given asset flow to the market. FIGURE 4 illustrates this variation for GE on a daily time scale, and FIGURE 5 shows the variation on an intraday time scale.

Figure 4 - Number of trades per day for General Electric

Figure 5 - Number of trades per half-hour for General Electric

This variability may at first seem random, but careful data analysis reveals a rich degree of structure. For instance:

• Trading activity exhibits calendar effects, tending to be light near certain holidays and heavy on days when exchange-traded options and futures contracts expire (i.e., "witching hours").
• Trading activity has a characteristic daily and weekly tempo.
• Increasing price volatility correlates with increasing trading activity.
• On a daily time scale, trading activity exhibits long-term trends punctuated by spurts of activity which decay over a few days.
• On an hourly time scale, we observe a tendency for bursts of activity to cluster together, possibly as a result of a single order generating multiple trades.

The intensity factor is actually an entire submodel of trading activity. It provides a forecast of the level of trading activity and of the typical variation around that level.

Shape

For a given asset, there is considerable variation in trade sizes. Again we turn to GE to illustrate this effect (see FIGURE 6).

Figure 6 - Trade size distribution for General Electric, showing percent of trades made

Obviously only the largest portfolios can place the largest orders and hence give rise to the largest transactions. We are led, therefore, to hypothesize a relationship between the distributions of portfolio sizes, order sizes, and transaction sizes. FIGURE 7 makes the comparison. In fact, we see that there is a rough similarity between these distributions. We note also, however, that there is a progressive shift leftward. Basically, as positions are moved through the market, they get broken up into multiple orders and an order may give rise to multiple transactions. Thus, a process of fragmentation shifts the distributions leftward. Here we are most probably observing the efforts of traders to minimize realization of market impact costs.

Figure 7 - Distribution of order sizes for NYSE system orders, fill sizes for those orders, and portfolio sizes for U.S. equity portfolios of $500 million or more

Another interesting feature of the data is the preference for round numbers, as indicated by the peaks at 100, 1000, and 5000 shares. We can also measure sizes in value rather than share terms, and then we find some preference for positions of $1 million. Probably these preferences arise from the experience that it is easier to find counterparties at these sizes.

A third feature of the data we may remark on is its tendency to carry through time. As we have noted, ultimately there is a connection between the distribution of portfolio sizes and of trade sizes. The mean of the distribution of portfolio sizes changes, of course, with the rise and fall of the overall market. Aside from this effect, however, the shape of the portfolio size distribution is quite stable. We expect the order and transaction size distributions also to be stable through time. Indeed, this is basically the case. However, two factors lead to greater variability in these measures:

• We have remarked on the fragmentation process, which involves a leftward shift in the distributions. The degree of fragmentation can vary with time, most likely in response to the overall level of liquidity in the market, and thus some variability in the distributions can arise.
• There can be some variation in the relative participation of institutions and individuals in the marketplace. Since the largest trades are placed by institutions, the right-hand tail of the distribution responds to the level of institutional activity in the market. If, for instance, institutions have a tendency to trade significantly more at the end of quarter than at the beginning, then the greater weight will fall into the tail of the trade size distribution.

Figure 8 - Trade size distribution for General Electric, showing values traded

The shape factor models the distribution of possible trade sizes. We have chosen to model this distribution as a mixture of continuous and discrete distributions. The continuous part captures the overall shape of the distribution, and in particular the tail. The discrete part captures the extra weight that occurs at round numbers.

The liquidity provider's risk

Collectively elasticity , volatility , intensity , and shape constitute a careful model of the uncertain trading environment faced by the liquidity provider. Suppose the liquidity provider assumes a position of size at a price that represents a deviation from the equilibrium. As trading occurs, there are many possible paths by which the liquidity provider can clear this position out of inventory. By applying our model we can calculate the probability of each individual path, much as was done in exhibit 1 of the previous excerpt in this series.¹ In this way, we can find the distribution of possible returns that might be earned by the liquidity provider. This distribution is characterized by a certain mean and standard deviation . The term is what we have meant by the liquidity provider's risk.

Recall that the liquidity provider seeks to maximize the ratio of return () to risk () subject to competitive pressures. For = 0 we have = 0, and as increases, increases. Furthermore, as increases, r(V,d,e,s,f,z) decreases. This combination causes the return/risk ratio to increase with , which at a unique for each stock will yield a target ratio () characteristic of the market as a whole:

                          (5)

We denote this unique as . Since , , , and are all characteristics of the particular asset, we abbreviate as for asset i. As we shall now see, the ratio is the market tone.

Market tone

The market tone () is a market-wide number. To estimate it, we form a sample of trades in a cross-section of assets on a given day. Since we require separate market tones for buyer- and seller-initiated transactions and for different market mechanisms, we restrict the sample accordingly. For , a trade in the sample, we suppose that K() is the estimated impact of the trade and i() is the asset traded.

Then we set by the requirement that it minimize:

                          (6)

where the sum is taken over all in the sample . The estimate of is based on data from a single day. From day to day, however, tends to change by only a few percent (at least during reasonably steady market conditions). The implication is that today's estimate of the market tone provides an adequate forecast of tomorrow's market tone.

Preliminary calculation of impact

Given the market tone and the asset parameters , , , , we may calculate the market impact cost of a trade of size as:

                          (7)

These estimates are made under the assumption that the market is at equilibrium at the time of the trade and that the liquidity provider holds a neutral inventory position. If the liquidity provider was looking to unwind a certain position and this trade provided such an opportunity, then the impact would probably be overestimated. Similarly, in non-equilibrium conditions, this estimate could be an underestimate.

We make the assumption of trading at equilibrium for practical reasons. Our goal is to provide forecasts valid for one to a few days ahead. Over this time frame there is little ability to predict in which direction (buy or sell) market disequilibrium will lie. Furthermore, to the extent there is any ability to predict the market's momentum, our research shows that it is already being priced by the market. This finding is entirely consistent with market efficiency. Thus, an equilibrium assumption is appropriate for the application we have in mind. For a randomly constructed sample of trades, we would expect the discrepancy between the hypothesis and reality to balance out over time. Thus, the impact estimates should retain their value, provided they are interpreted as expectations rather than on a trade-by-trade basis.

Skill

Investors' actual processes do not generate random samples of orders, however. For instance:

• "Contrarian" investors may tend to systematically trade in the opposite direction from the market's overall momentum.
• A trade list that is being worked creates liquidity that can be provided to the market opportunistically. The investor's trading operation may have the ability to efficiently manage this opportunity so that trading costs are reduced.
  The investor may hold non-consensus preferences, with the result that the opportunity costs and risks of working orders or using search market mechanisms are judged lower than the direct cost of market orders.

Accordingly, we introduce a skill factor that is used to adjust the forecast impact as:

                          (8)

and the impact cost is adjusted similarly. Thus, a skill of 0 corresponds to the preliminary model forecast, while a skill of 1 implies the ability to eliminate market impact costs through skillful exploitation of one's trading opportunities.

Naturally, the skill factor needs to be estimated for each investor separately by comparing average trade costs to the forecast for the generic investor.

The market impact function

We have completed the description of the model. In summary, we see that the market impact cost function F() is:

=                           (9)

Although the functional form of F() is quite complex, we can easily determine the effect of each variable on the forecast cost.

• As the volume increases, the liquidity provider has to carry inventory longer, so impact rises. Cost is the product of impact and volume, so cost also rises and at a faster than linear rate.
• As the elastic coefficient increases, order flow responds more quickly to price signals, so time-to-clear decreases and thus cost decreases. As the elastic fraction increases, the maximum response of order flow increases, also leading to lower cost. Thus, in general, as elasticity increases, cost decreases.
• As volatility increases, a liquidity provider's risk increases and cost increases.
• As the intensity of trading increases, time-to-clear shortens, and so cost decreases.
• As the mean of the shape distribution increases, time-to-clear shortens and so cost decreases. As the dispersion of the shape distribution increases, the uncertainty in the time-to-clear increases and so cost rises.
• As the market tone increases, liquidity becomes more costly and cost increases.
• As the skill increases, the investor either purchases less liquidity or purchases it more economically, so costs decrease.

The piecewise-linear approximation

The full market impact function F() is very time-consuming to compute. Accordingly, we often replace it by an approximation that can be computed quickly. There are many different ways of constructing an approximating function. For some applications, however, it is convenient for the approximating function to be piecewise-linear, and so we use a function of this class. Constructing an approximation always means accepting some discrepancy between the base function and its approximation. We have constructed the approximation by requiring that:

• the approximation is always equal to or greater than the base value
• the approximation is within a fixed tolerance of the base value

The realized spread

For the smallest trades, we find that the predicted impact is generally less than the minimum tick size. Accordingly, we need to increase the cost function to reflect the reality of minimum impacts. We rely on a measure of the spread for this adjustment. Many trades actually occur inside the quoted spread. Accordingly, we deflate the quoted spread by a factor that measures the volume of trading inside the quoted spread. We term this deflated spread the realized spread, since it reflects the spread investors actually pay. For the smallest trades, the forecast impact is rounded up to the realized spread, and the piecewise-linear cost function is adjusted accordingly.

Summary

There is considerable fine detail to market conditions for an asset. We model this detail with attention to the features that are of particular significance to a liquidity provider. Thus, we produce four submodels of elasticity, volatility, intensity, and shape that characterize market conditions for an asset. From this information, we can evaluate the liquidity provider's risk. Combining the quantity of risk with the market tone leads to a forecast cost for the generic investor. A process-level adjustment, the skill, then adapts the forecast to a specific investor. Finally, a piecewise-linear approximation is introduced to achieve a computationally efficient implementation. The final result is a forecast of market impact costs.