will deal with transactions costs and turnover more broadly (how they arise), more concretely (how to estimate them), and more strategically (how to reduce these costs while preserving as much of the strategy's value added as possible). We will attack the strategic issue in two ways: reducing transactions costs by reducing turnover while retaining as much of the value added as possible, and reducing transactions costs through optimal trading. Basic insights we will cover in this chapter include the following: • Transactions costs increase with trade size and the desire for quick execution, which help to identify the manager as an informed trader and require increased inventory risk by the liquidity supplier. • Transactions costs are difficult to measure. At the same time, accurate estimates of transactions costs, especially distinctions in transactions costs among different stock trades, can significantly affect realized value added. • Transactions costs lower value added, but you can often achieve at least 75 percent of the value added with only half the turnover (and half the transactions costs). You can do better by distinguishing stocks by their transactions costs. • Trading is itself a portfolio optimization problem, distinct from the portfolio construction problem. Optimal trading can lower transactions costs, though at the expense of additional short-term risk. • There are several options for trade implementation, with rules of thumb on which to use when. Turnover occurs whenever we construct or rebalance a portfolio, motivated by new information (new alphas) or risk control. Transactions costs are the penalty we pay for transacting. Transactions costs have several components: commissions, the bid/ask spread, market impact, and opportunity cost. Commissions are the charge per share paid to the broker for executing the trade. These tend to be the smallest component of the transactions costs, and
Page 447 the easiest to measure. The bid/ask (or bid/offer) spread is the difference between the highest bid and the lowest offer for the stock; it measures the loss from buying a share of stock (at the offer) and then immediately selling it (at the bid). The bid/ask spread is approximately the cost of trading one share of stock.3 Market impact is the cost of trading additional shares of stock.4 To buy one share of stock, you need only pay the offer price. To buy 100,000 shares of stock, you may have to pay much more than the offer price. The 100,000-share price must be discovered through trading. It is not known a priori. Market impact is hard to measure because it is the cost of trading many shares relative to the cost of trading one share, and you cannot run a controlled experiment and trade both many shares and one share under identical conditions. Market impact is the financial analog of the Heisenberg uncertainty principle. Every trade alters the market. Market Microstructure The field of market microstructure studies the details of how markets work and transactions occur, in order to understand transactions costs, especially bid/ask spreads and market impact. This is a field of current active research that as yet lacks a single complete and widely accepted model. There is no CAPM or Black-Scholes model of trading. However, there are at least two ideas from this field that can illuminate the source of transactions costs. When a portfolio manager trades, he or she must go to the marketplace to find someone to trade with. That other person, possibly a specialist on the New York Stock Exchange or a market maker on NASDAQ, will trade ("provide liquidity"), but for a price. Often this liquidity supplier's only business is providing short-term liquidity, i.e., he or she is not a long-term investor in the market. Several considerations determine what price the liquidity supplier will charge (and can charge— after all, providing liquidity is a competitive business). First, the liquidity supplier would like to know why the manager is trading. In particular, does the manager 3You can never be certain of this cost until you actually trade. 4Some authors include the bid/ask spread in market impact, given that it is a market phenomenon (as opposed to commissions and taxes).
Page 448 possess any unique nonpublic information that will soon change the stock price? Is the manager an "informed trader"? If so, the liquidity supplier would want to trade at the price the stock will reach once the information is public. Typically, though, the liquidity supplier can't tell if the manager has valuable information, has worthless information, or is trading only for risk control purposes. He or she can only guess at the value of the manager's information by the volume and urgency of the proposed trade. The larger and more urgent the trade, the more likely it is that the manager is informed, and the higher the price concession the liquidity supplier will demand. Market impact increases with trading volume. A second consideration influencing transactions costs is inventory risk. Even without informed traders, market impact would still exist. Liquidity suppliers have no intention of holding positions in inventory for long periods of time. When the liquidity supplier trades, her or his goal is to hold the position in inventory only until an opposing trade comes along. Every minute before that opposing trade appears adds to the risk. The liquidity supplier has a risk/return trade-off, and will demand a price concession (return) to compensate for this inventory risk. The calculation of this risk involves several factors, but certainly the larger the trade size, the longer the period that the position is expected to remain in inventory, and hence the larger the inventory risk and the larger the market impact. Market impact increases with trading volume. The theory of market microstructure can provide basic insights into the sources of transactions costs. The details of how these influences combine to produce the costs observed or inferred in actual markets is still under investigation and is beyond the scope of this book. However, we will utilize some of these basic insights throughout the chapter. Analyzing and Estimating Transactions Costs Analyzing and estimating transactions costs is both difficult and important. Market impact is especially difficult to estimate because, as we have discussed, it is so difficult to measure. Estimating transactions costs is important because accurate estimates can significantly affect realized value added, by helping
Page 449 the manager choose which stocks to trade when constraining turnover, and by helping him or her decide when to trade when scheduling trades to limit market impact. This endeavor is sufficiently important that analytics vendors like BARRA and several broker/dealers now provide transactions cost estimation services. Ideally, we would like to estimate expected transactions costs for each stock based on the manager's style and possible range of trade volumes. The theory of market microstructure says that transactions costs can depend on manager style, principally because of differences in trading speed. Managers who trade more aggressively (more quickly) should experience higher transactions costs. Wayne Wagner (1993) has documented this effect, and illustrated the connection between information and transactions costs, by plotting transactions costs versus short-term return (gross of transactions costs) for a set of 20 managers. The most aggressive information trader was able to realize very large short-term returns, but they were offset by very large transactions costs. The slowest traders often even experienced negative short-term returns, but with small or even negative transactions costs. (To achieve negative transactions costs, they provided liquidity to others.) Estimation of expected transactions costs requires measurement and analysis of past transactions costs. The best place to start is with the manager's past record of transactions and the powerful ''implementation shortfall" approach to measuring the overall cost of trading.5 The idea is to compare the returns to a paper portfolio with the returns to the actual portfolio. The paper portfolio is the manager's desired portfolio, executed as soon as he or she has devised it, without any transactions costs. Differences in returns to these two portfolios will arise as a result of commissions, the bid/ask spread, and market impact, as well as the opportunity costs of trades that were never executed. For example, some trades never execute because the trader keeps waiting for a good price while the stock keeps moving away. Wayne Wagner has estimated that such opportunity costs often dominate all transactions costs. 5Jack Treynor first suggested the approach, and Andre Perold later embellished it.
Page 450 Many services that currently provide ex post transactions cost analysis do not use the implementation shortfall approach, because it involves considerable record keeping. They use simpler methods, such as comparing execution prices against the volume-weighted average price (VWAP) over the day. Such an approach measures market impact extremely crudely and misses opportunity costs completely. The method simply ignores trade orders that don't execute. And, as a performance benchmark, traders can easily game VWAP: They can arrange to look good by that measure. The most difficult approach to transactions cost analysis is to directly research market tick-by-tick data. Both the previously described methods began with a particular manager's trades. When analyzing tick-by-tick data, we do not even know whether each trade was buyer- or seller-initiated. We must use rules to infer (inexactly) that information. For example, if a trade executes at the offer price or above, we might assume that it is buyer-initiated. The tick-by-tick data are also full of surprising events—very large trades occurring with no price impact. But the record is never complete. Did the price move before the trade, in anticipation of its size? Did the trade of that size occur only because the trader knew beforehand that existing limit orders contained the necessary liquidity? Researchers refer to this as censored, or biased, data. The tick-by-tick data show trades, not orders placed, and certainly not orders not placed because the cost would be too high. Realized costs will underestimate expected costs. There are several other problems with tick-by-tick trade data. The data set is enormous, generating significant data management challenges. But in spite of all these data, we only rarely observe some assets trading. These thinly traded assets are often the most costly to trade. So this record is missing information about the assets whose costs we often care about most. Finally, tick-by-tick data are very noisy, because of discrete prices, nonsynchronous reporting of trades and quotes, and data errors. All of these challenges affect not only the estimation of market impact but also the testing of models forecasting market impact. Clearly, building an accurate, industrial-strength transactions cost model is a very significant undertaking. One approach to transactions costs which has proven fruitful, models costs based on inventory risk. The inventory risk model
Page 451 estimates market impact based on a liquidity supplier's risk of facilitating the trade. Here heuristically is how that works. First, given a proposed trade of size Vtrade, the estimated time before a sufficient number of opposing trades appears in the market to clear out the liquidity supplier's net inventory in the stock is where is the average daily volume (or forecast daily volume) in the stock. Equation (16.1) states that if you want to trade one day's volume, the liquidity supplier's estimated time to clear will be on the order of one day, and so on. This time to clear implies an inventory risk, based on the stock's volatility: where Eq. (16.2) converts the stock's annual volatility σ to a volatility over the appropriate horizon. Equation (16.2) assumes that we measure τclear in days, and that a year contains 250 trading days. The final step in the model assumes that the liquidity supplier demands a return (price concession or market impact) proportional to this inventory risk: where c is the risk/return trade-off, and we measure return relative to the bid price for a sellerinitiated trade and relative to the offer for a buyer-initiated trade. Since there exists some competition between liquidity suppliers, the market will help set the constant c. For a seller-initiated trade, the transactions cost will include not only the price concession from the offer to the bid, but an additional concession below the bid price, depending on the size of the trade. The argument for the buyer-initiated trade is similar.
Page 452 Combining Eqs. (16.1) through (16.3), adding commissions, and converting to units of return, leads to where ctc includes the stock's volatility, a risk/return trade-off, and the conversion from annual to daily units. In general, this approach and Eq. (16.4) are consistent with a trading rule of thumb that it costs roughly one day's volatility to trade one day's volume. This rule of thumb implies that . One consequence of this inventory risk approach is that market impact should increase as the square root of the amount traded. This agrees remarkably well with the empirical work of Loeb (1983)].6 Because the total trading cost depends on the cost per share times the number of shares traded, it increases as the 3/2 power of the amount traded. Loeb, a passive manager at Wells Fargo Investment Advisors, collected bids on different size blocks of stock. Figure 16.1 displays his results against a square root function. His observed dependence of cost on trade size clearly follows the square root pattern (plus fixed costs at low volume). There are several ways to forecast transactions costs, starting with Eq. (16.4). A simple approach is to choose ctc such that trades of typical size experience about 2 percent round-trip transactions costs, or to develop a better estimate based on analysis of the manager's past transactions. If your optimizer requires that transactions costs be expressed as a piecewise linear or quadratic function of trade size, then approximate Eq. (16.4) appropriately, in the region of expected trade sizes. But the above approach leaves much on the table. This inventory risk approach can support more elaborate structural models, 6Researchers at BARRA, especially Nicolo Torre and Mark Ferrari, are responsible for identifying the square root dependence of the Loeb data. They have also used empirical methods to find the best-fitting exponent (square root corresponds to 1/2) in the dependence of market impact on trade volume. The square root provides the best fit. See BARRA (1997) for details.
Page 453 Figure 16.1 which can provide more dynamic and accurate estimates.7 They can also provide more industrialstrength estimates: As we have seen, pure empirical approaches face many problems as a result of poor data quality; poor coverage, especially of illiquid and new assets; and poor timeliness. A structural attack can separate out easier-to-measure elements, facilitate extensions to all assets, benefit from cross-sectional estimation, impose reasonable behavior, and generally limit problems. The inventory risk approach suggests a certain structure. It depends on a forecast of inventory risk and an estimate of the liquidity supplier's charge per unit of risk.