Transactions costs, turnover, and trading are the details involved in moving your current portfolio to your target portfolio.1 These details are important. Studies show that, on average, active U.S. equity managers underperform the S&P 500 by 1 to 2 percent per year,2 and, as Jack Treynor has argued, that average underperformance can only be due to transactions costs. Given that typical institutional managers seek to add only 2 to 3 percent active return (and charge ∽0.5 percent for the effort), this is a significant obstacle.
Transactions costs often appear to be unimportant. Who cares about a 1 or 2 or even 5 percent cost if you expect the stock to double? Unfortunately, expectations are often wrong. At the end of the year, your performance is the net result of your winners and losers. But winner or loser, you still pay transactions costs. They can be the investment management version of death by a thousand cuts. A top-quartile manager with an information ratio of 0.5 may lose roughly half her returns because of transactions costs. They are important.
Chapter 14 dealt with transactions costs as an input to the portfolio construction exercise. Now we 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 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.
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 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 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 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.
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 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 seller-initiated 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
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, which can provide more dynamic and accurate estimates.7 They can also provide more industrial-strength 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.
Chapter 3 presented structural models of risk. A structural model of inventory risk involves an estimate of asset risk and also estimates of the time to clear. To estimate both these quantities, the market impact model developed by BARRA relies on submodels to estimate asset volatility, trading volume and intensity (trade size and trade frequency), and elasticity.
By separating out each component, the BARRA model can apply appropriate insight and technology. We may have difficulty discerning patterns within the tick-by-tick market impact data. But we can apply our previous understanding of structural risk models to estimate asset risk. We can apply insights into trading volume and intensity which often hold across most stocks. For example, all stock trading exhibits higher volume at the open and close, and low volume in the vicinity of certain holidays.
Elasticity captures the dependence of buy versus sell orders on price. Imagine that a liquidity supplier fills a large sell order and demands a price concession. The trade price moves below the bid, and the supplier’s inventory now has a positive position. But the low price will attract other buyers. Elasticity measures how the number of buyers versus sellers changes as we move away from an equilibrium price.
The BARRA market impact model uses the distribution of trade frequency, trade size, and elasticity to estimate time to clear, given any particular desired trade size. This time to clear, combined with estimated risk, leads to the inventory risk forecast. The final step uses a liquidity supplier’s price of risk to convert inventory risk into an expected price concession. The BARRA model estimates this price of risk separately for buy orders and sell orders, and for exchange-traded and over-the-counter stocks.
As well as developing their model, BARRA researchers have also developed latent-variables methods for testing the accuracy of such models, given the problems with tick-by-tick data. For details, see BARRA (1997).
We now wish to go beyond the simple observation that transactions costs drag down performance, to see how much value added we can retain if we limit a strategy’s turnover. We’ve all heard about the extremely promising strategy that unfortunately requires 80 percent turnover per month. Don’t dismiss that policy out of hand. We may be able to add considerable value with the strategy if we restrict turnover to 40 percent, 20 percent, or even 10 percent per month.
We shall build a simple framework for analyzing the effects of transactions costs and turnover to help us understand the tradeoff between value added and turnover. This framework will provide a lower bound on the amount of value added achievable with limited turnover, and also clarify the link between transactions costs and turnover. It will also provide a powerful argument for the importance of accurately distinguishing stocks based on their transactions costs.
For any portfolio P, consider value added
where ψP is the portfolio’s active risk relative to the benchmark B. The manager starts with an initial portfolio I that has value added VA1. We will limit the portfolios that we can choose to a choice set8 CS. Portfolio Q is the portfolio in CS with the highest possible value added. We will assume for now that portfolio I is also in CS, but later relax that assumption. The increase in value added as we move from portfolio I to portfolio Q is
Now let TOP represent the amount of turnover needed to move from portfolio I to portfolio P. As a preliminary, let us define turnover, since there are several possible choices. If hP is the initial portfolio and 
is the revised portfolio, then the purchase turnover is
and the sales turnover is
These purchase and sales turnover statistics do not include changes in cash position. One reasonable definition of turnover, which we will adopt, is
Turnover is the minimum of purchase and sales turnover. With no change in cash position, purchase turnover will equal sales turnover. This turnover definition accommodates contributions and withdrawals by not including them in the turnover formula.
The turnover required to move from portfolio I to portfolio Q is TOQ. If we restrict turnover to be less than TOQ, we will be giving up some value added in order to reduce cost. Let VA(TO) be the maximum amount of value added if turnover is less than or equal to TO. Figure 16.2 shows a typical situation. The frontier VA(TO) increases from VAI to VAQ. The concave9 shape of the curve indicates a decreasing marginal return for each additional amount of turnover that we allow.
As the technical appendix will show in detail, when we assume that CS is defined by linear equality constraints (e.g., constraining the level of cash or the portfolio beta to equal specific targets) and includes portfolio I, we can obtain a quadratic lower bound on potential value added:
Underlying this result is a very simple strategy, prorating a fraction TO/TOQ of each trade needed to move from the initial portfolio I to the optimal portfolio Q. This strategy leads to a portfolio that is in CS, has turnover equal to TO, and meets the lower bound in Eq. (16.10).
We can express the sentiment of Eq. (16.10) in the value added/turnover rule of thumb:
You can achieve at least 75 percent of the (incremental) value added with 50 percent of the turnover.
This result sounds even better in terms of an effective information ratio, since the value added is proportional to the square of the information ratio (at optimality). It implies that a strategy can retain at least 87 percent of its information ratio with half the turnover.10
We can exceed the lower bound in Eq. (16.10) by judicious scheduling of trades, executing the most attractive opportunities first. For example, suppose there are only four assets. As we move from portfolio I to portfolio Q, we purchase 10 percent in assets 1 and 2 and sell 10 percent of our holdings in assets 3 and 4. Turnover is 20 percent. Suppose the alphas for the four assets are 5 percent, 3 percent, −3 percent, and −5 percent, respectively. Then buying 1 and selling 4 has a bigger impact on alpha than swapping 2 for 3. If we have restricted turnover to 10 percent, we could make an 8 percent trade of stock 1 for stock 4 and a 2 percent trade of 2 for 3 rather than doing 5 percent in each trade.
Figure 16.3 illustrates the situation. The solid line shows the frontier; the dotted line shows the lower bound. The maximum opportunity for exceeding the bound occurs for turnover somewhere between 0 and 100 percent of TOQ.
The simplest assumption we can make about transactions costs is that round-trip costs are the same for all assets. Let TC be that level of costs. We wish to choose a portfolio P in CS that will maximize
Figure 16.4 illustrates the solution to this problem. Let SLOPE(TO) represent the slope of the value-added/turnover frontier when the level of turnover is TO. Since the frontier is increasing and concave, SLOPE(TO) is positive and decreasing. The incremental gain from each additional amount of turnover is decreasing, so the slope of the frontier SLOPE(TO) will decrease to zero as TO increases to TOQ. SLOPE(TO) represents the marginal gain in value added from additional transactions, and TC represents the marginal cost of additional transactions. The optimal level of turnover will occur where marginal cost equals marginal value added, i.e., where SLOPE(TO*) = TC.
As long as the transactions cost is positive and less than SLOPE(0), we can find a level of turnover TO* such that SLOPE(TO*) = TC. If TC > SLOPE(0), it is not worthwhile to transact at all, and the best solution is to stick with portfolio I.
The slope of the value-added/turnover frontier can be interpreted as a transactions cost. We can reverse the logic and link any level of turnover to a transactions cost; e.g., a turnover level of 20 percent corresponds to a round-trip transactions cost of 2.46 percent.
Transactions costs contain observable components, such as commissions and spreads, as well as the unobservable market impact. Because managers cannot be sure that they have a precise measure of transactions costs, they will often seek to control those costs by establishing an ad hoc policy such as “no more than 20 percent turnover per quarter.” The insight that relates the slope of the value-added/turnover frontier to the level of transactions costs provides an opportunity to analyze that cost control policy. One can fix the level of turnover at the required level TOR and then find the slope, SLOPE(TOR), of the frontier at TOR. Our ad hoc policy is consistent with an assumption that the general level of round-trip transactions costs is equal to SLOPE(TOR). If we have a notion that round-trip costs are around 2 percent, and we find that SLOPE(TOR) is about 4.5 percent, then something is awry. We can make three possible adjustments to get things back into harmony. One, we can increase our estimate of the round-trip costs from 2 percent. Two, we can increase the allowed level of turnover TOR, since we are giving up more marginal value added (4.5 percent) than it is costing us to trade (2.0 percent). Three, we can reduce our estimates of our ability to add value by scaling our alphas back toward zero. A combination of these adjustments—a little give from all sides—would be fine as well. This type of analysis serves as a reality check on our policy and the overall investment process.
Consider the following example, using the S&P 100 stocks as a starting universe and the S&P 100 as the benchmark. We generated alphas11 for the 100 stocks, centering and scaling so that they were benchmark-neutral, and also so that portfolio Q would have an alpha of 3.2 percent and an active risk of 4 percent when we used λA = 0.1. The initial portfolio contained 20 randomly chosen stocks, equal-weighted, with an alpha of 0.07 percent and an active risk of 5.29 percent, typical of a situation when a manager takes over an existing account.
Table 16.1 displays the results, including the value added separated into two parts: the lower bound and the excess above the bound. The excess is the benefit we get from scheduling the best trades first. Table 16.1 also displays the implied transactions costs. We see that reasonable levels of round-trip costs (about 2 percent) do not call for large amounts of turnover, and that very low or high restrictions on turnover correspond to unrealistic levels of transactions costs.
Note that the value of being able to pick off the best trades (the difference between the lower bound and the actual fraction of value added) is largest when turnover is 20 percent of the level required to move to portfolio Q. In this case, the rule of thumb is conservative: We achieve 87 percent of the value-added for 50 percent of the turnover.12
We made three assumptions in deriving these results: (1) the initial portfolio was in CS, (2) CS was described by linear equalities, and (3) all round-trip transactions costs are the same. We will reconsider these in turn.
If portfolio I is not in the choice set, then we can think of the portfolio construction problem as being a two-step process. In step 1, we find the portfolio J in the choice set such that the turnover in moving from portfolio I to portfolio J is minimal. The value added in moving13 from portfolio I to portfolio J is not a consideration, so we may have VAI ≥ VAJ or VAI < VAJ. The turnover required to move from I to J is TOJ. The lower bound in Eq. (16.10) will still apply, although we start from portfolio J rather than portfolio I.
This situation can demonstrate the costs of adding constraints. What if portfolio I were not in CS, and we were limiting turnover to 10 percent per month? If the first 4 percent of the turnover is required to move the portfolio back into the choice set, then we have only 6 percent to take advantage of our new alphas.
If the choice set CS is described by inequality constraints, such as a restriction on short selling and upper limits on the individual asset holdings, then the analysis becomes more complicated. However, the value-added/turnover frontier VA(TO) will have the same increasing and concave slope that we see in Fig. 16.2. There will be a quadratic lower bound on VA(TO); however, that lower bound14 is not as strong as the lower bound we obtain in the case with only equality constraints. You are not guaranteed three-quarters of the value added for one-half the turnover. Nevertheless, in our experience, 75 percent is still a reasonable lower bound.
So far, we have made the assumption that all round-trip transactions costs are the same. It is good news for the portfolio manager if the transactions costs differ (and she or he can forecast the difference). Recall that the difference between the lower bound and the value-added/turnover frontier stemmed from our ability to adroitly schedule the most value-enhancing trades first. Our ability to discriminate adds value. Differences in transactions costs further enhance our ability to discriminate.
We have analyzed this effect in our example. We began with the implied transactions costs at 1.90 percent for 50 percent of TOQ. We then set the transactions costs to 75 percent of that, or 1.42 percent, for half of the stocks, and raised the transactions costs for the other stocks to 2.26 percent, so that transactions costs remained constant at 50 percent of TOQ. Taking into account these differing costs when optimizing barely affected portfolio alphas or risk, but did reduce transactions costs by about 30 percent.
Accurate forecasts of the cost of trading can generate significant savings when used as part of the portfolio rebalancing process.
The better our model of transactions costs, the better our ability to discriminate among stocks. In the example above, we distinguished stocks by their linear costs. More elaborate models can distinguish them on the basis of more accurate, dynamic, and nonlinear costs.
Clearly we have seen promise in the approach of lowering transactions costs by lowering turnover while retaining much of the value added. Naïve versions of this can preserve 75 percent of the value added with 50 percent of the turnover, and clever versions, utilizing differences in asset alphas and asset transactions costs, can well exceed the naïve result. We now move on to a second approach to reducing transactions costs: optimal trading.
Trading is a portfolio optimization problem, but not the portfolio construction problem we have discussed at length. Imagine that you have already completed the portfolio construction (or rebalancing) problem. You own a current portfolio, and you desire the output from the portfolio construction problem. You trade to move from your current portfolio to your desired portfolio. Scheduling these trades—what stock to trade first, second, etc.—over an allowed trading period is a portfolio optimization problem. The goal is to maximize trading utility:
defined as short-term alpha minus a short-term risk adjustment, minus market impact. This trading utility function swaps reduced market impact for increased short-term risk. We distinguish short-term alphas and risk from investment-horizon alphas and risk (discussed elsewhere in this book), because stock returns over hourly or daily horizons often behave quite differently from stock returns over monthly or quarterly horizons.
In portfolio construction, the goal is a target portfolio. In trading, the goal is a set of intermediate portfolios held at different times, starting with the current portfolio now, and ending with the target portfolio a short time later.
The benchmark for trading is immediate execution, and we measure return and risk relative to that benchmark in Eq. (16.12). The problem with quick execution, as discussed earlier, is that it increases market impact. Market impact costs increase with trade volume and speed.
What’s the intuition about how risk, market impact, and alphas will affect trade scheduling? Risk considerations should keep the trade schedule close to the benchmark, i.e., push for quick execution. Market-impact considerations will tend to lead to even spacing of trades. Alphas will push for early or late execution.
The details of how to implement this optimal trading process—for example, how to model market impact as a function of how fast you trade—are beyond both the scope of this book and the state of the art of the investment management industry. However, it’s useful to present a very simple example of the idea. Even this simple example, though, involves sophisticated mathematics that is relegated to the technical appendix.
Consider the trading process at its most basic: You have cash, and you want to buy one stock. You think the stock will go up. You want to buy soon, before the stock rises. But to avoid market impact, you are willing to be patient and assume some risk of missing the stock rise. What is your optimal trading strategy?
Here are the details. Start with cash amount M. After T days, you want to be fully invested in stock S, with expected return f and risk σ. The benchmark is immediate execution.
We need to quantify the return, risk, and transactions costs for the implemented portfolio relative to the benchmark. For this simple example, we can completely characterize the implemented portfolio at time t by the fractional holding of the stock h(t). Assume that you can trade at any time and at any speed, so long as you are fully invested by day T. We then seek the optimal h(t) at each time over the next T days. The cash position of the implemented portfolio is simply 1 − h(t). The active portfolio stock position relative to the benchmark is h(t) − 1, since the benchmark is fully invested. The implemented portfolio will have h(0) = 0 and h(T) = 1. Initially the portfolio is entirely cash, and at the end of T days the portfolio is fully invested in stock S.
Over the next T days, the cumulative portfolio active return will be
This integrates (or sums up) the active return over each small period dt to calculate the total active return over the period of T days.
Similarly, the cumulative active risk of the implemented portfolio will be
Once again, this cumulative active risk integrates (or sums up) active risk contributions overeach period dt of the full T-day trading period. This active risk involves the active portfolio position and the stock return risk.
Finally, we must treat cumulative transactions costs. For the example, we will focus specifically on market impact, the only interesting transactions costs in terms of influence on trading strategy.15 We will model the cumulative active market impact as
with
Equation (16.15) models market impact as simply proportional to the square of the stock accumulation rate. The faster the portfolio holding changes, the larger the market impact.
This simple model ignores any memory effects—a big assumption, but only if trades are a significant fraction of daily volume. According to this, the market doesn’t remember what you traded yesterday, it just sees what you are trading this instant. Still, the total market impact over the T-day trading period is the integral (or sum) of the market impact over each subperiod dt.
The technical appendix describes how to analytically solve for the h(t) which maximizes Eq. (16.12). Here we will illustrate a graphical solution for different parameter choices. Three different elements influence the solution: return, risk, and market impact. We are looking at short horizons, and typically the risk and market impact components will dominate the expected returns components.16 Assuming that the expected return is small, we still have two distinct cases: risk aversion dominates market impact, and market impact dominates risk. Figure 16.5 illustrates these two cases, using T = 5 days. When market impact dominates risk aversion, the optimal schedule is to evenly space trades, even though the benchmark is immediate execution. When risk dominates market impact, the optimal schedule will closely track the immediate execution benchmark. Within two days (40 percent of the period), the portfolio’s stock position reaches 75 percent of its target.
After devising a trading strategy, either through the optimization described above or through some more ad hoc approach, the next step is actual trading. You can implement trades as market orders or as limit orders.17
Market orders are orders to trade a certain volume at the current market’s best price. Limit orders are orders to trade a certain volume at a certain price. Limit orders trade off price impact against certainty of execution. Market orders can significantly move prices, but they will execute. Limit orders will execute at the limit price, if they execute (they may not).
There is a current debate over the value of using limit orders in trading. Many argue that placing limit orders provides free options to the marketplace. For example, placing a limit order to buy at a price P constitutes offering to buy the stock at P, even if the stock is rapidly moving to 80 percent of P. Your only protection is the ability to cancel the order as the price begins to move.
Trading portfolios of limit orders has additional problems. For example, in a large market move, all the sell limit orders may execute while none of the buy orders execute. This will add market risk to the portfolio.
Given these concerns about limit orders, plus the typical portfolio manager’s eagerness to complete the trades (implement the new alphas), the general rule is to use limit orders sparingly, mainly for the stocks with the highest anticipated market impact, and with limit prices set very close to the existing market prices.
The opposite side of the debate is that limit orders let value managers sell liquidity and earn the liquidity provider’s profit. The appropriate order type may depend on the manager’s style.
We have discussed transactions costs, turnover, and trading, with a strategic focus on reducing the impact of transactions costs on performance. After discussing the origins of transactions costs and how they rise with trade volume and urgency, we focused on the question of analyzing and estimating transactions costs. This is difficult because measuring transactions costs is difficult, but it can significantly affect realized value added, as the rest of the chapter discussed. The most accurate analysis of transactions costs uses the implementation shortfall: comparing the actual portfolio with a paper portfolio implemented with no transactions costs.
We discussed the inventory risk approach to modeling market impact. This leads to behavior matching market observations, especially the dependence of price impact on the square root of volume traded. It also leads to practical forecasts of market impact, ranging from the fairly simple to the more complex structural market impact models.
One approach to reducing transactions costs is to reduce turnover while retaining value added. We developed a lower bound on value added as a function of turnover, and verified the rule of thumb that restricting turnover to one-half the level of turnover if there is no restriction on turnover will result in at least three-quarters of the value added. We can exceed that bound through our ability to skim off the most valuable trades first, and by accounting for differences in transactions costs between stocks. We also saw that the slope of the value-added/turnover frontier implies a level of round-trip transactions costs.
We then looked at the trading process directly, to see that trading itself is a portfolio optimization problem, distinct from portfolio construction. Optimal trading can reduce transactions costs, trading reduced market impact for additional short-term risk. Choices for trade implementation include market orders and limit orders. The disadvantages of limit orders make them most appropriate for the highest market impact stocks, with limit prices close to market prices.
1. Imagine that you are a stock trader, and that a portfolio manager plans to measure your trading prowess by comparing your execution prices with volume-weighted average prices. How would you attempt to look as good as possible by this measure? Would this always coincide with the best interests of the manager?
2. Why is it more difficult to beat the bound in Eq. (16.10) with a portfolio of only 2 stocks than with a portfolio of 100 stocks?
3. A strategy can achieve 200 basis points of value added with 200 percent annual turnover. How much value-added should it achieve with 100 percent annual turnover? How much turnover is required in order to achieve 100 basis points of value added?
4. How would the presence of memory effects in market impact change the trade optimization results displayed in Fig. 16.5?
5. In Fig. 16.5, why does high risk aversion lead to quick trading?
Angel, James J., Gary L. Gastineau, and Clifford J. Webber. “Reducing the Market Impact of Large Stock Trades.” Journal of Portfolio Management, vol. 24, no. 1, 1997, pp. 69-76.
Atkins, Allen B., and Edward A. Dyl. “Transactions Costs and Holding Periods for Common Stocks.” Journal of Finance, vol. 52, no. 1, 1997, pp. 309-325.
BARRA, Market Impact Model Handbook (Berkeley, Calif.: BARRA, 1997).
Chan, Louis K. C., and Josef Lakonishok. “The Behavior of Stock Prices around Institutional Trades.” Journal of Finance, vol. 50, no. 4, 1995, pp. 1147-1174.
_________. “Institutional Equity Trading Costs: NYSE versus NASDAQ.” Journal of Finance, vol. 52, no. 2, 1997, pp. 713-735.
Ellis, Charles D. “The Loser’s Game.” Financial Analysts Journal, vol. 31, no. 4, 1975, pp. 19-26.
Grinold, Richard C., and Mark Stuckelman. “The Value-Added/Turnover Frontier.” Journal of Portfolio Management, vol. 19, no. 4, 1993, pp. 8-17.
Handa, Puneet, and Robert A. Schwartz. “Limit Order Trading.” Journal of Finance, vol. 51, no. 5, 1996, pp. 1835-1861.
Kahn, Ronald N. “How the Execution of Trades Is Best Operationalized.” In Execution Techniques, True Trading Costs, and the Microstructure of Markets, edited by Katrina F. Sherrerd (Charlottesville, Va.: AIMR 1993).
Keim, Donald B., and Ananth Madhavan. “The Cost of Institutional Equity Trades.” Financial Analysts Journal, vol. 54, no. 4, 1998, pp. 50-69.
Lakonishok, Josef, Andre Shleifer, and Robert W. Vishny. “Study of U.S. Equity Money Manager Performance.” Brookings Institute Study, 1992.
Loeb, Thomas F. “Trading Costs: The Critical Link between Investment Information and Results.” Financial Analysts Journal, vol. 39, no. 3, 1983, pp. 39-44.
Malkiel, Burton. “Returns from Investing in Equity Mutual Funds 1971 to 1991.” Journal of Finance, vol. 50, no. 2, 1995, pp. 549-572.
Modest, David. “What Have We Learned about Trading Costs? An Empirical Retrospective.” Berkeley Program in Finance Seminar, March 1993.
Perold, Andre. “The Implementation Shortfall: Paper versus Reality.” Journal of Portfolio Management, vol 14, no. 3, 1988, pp. 4-9.
Pogue, G. A. “An Extension of the Markowitz Portfolio Selection Model to Include Variable Transactions Costs, Short Sales, Leverage Policies and Taxes.” Journal of Finance, vol. 45, no. 5, 1970, pp. 1005-1027.
Rudd, Andrew, and Barr Rosenberg.. “Realistic Portfolio Optimization.” In Portfolio Theory—Studies in Management Science, vol. 11, edited by E. J. Elton and M. J. Gruber (Amsterdam: North Holland Press, 1979).
Schreiner, J. “Portfolio Revision: A Turnover-Constrained Approach.” Financial Management, vol. 9, no. 1, 1980, pp. 67-75.
Treynor, Jack L. “The Only Game in Town.” Financial Analysts Journal, vol. 27, no. 2, 1971, pp. 12-22.
________. “Types and Motivations of Market Participants.” In Execution Techniques, True Trading Costs, and the Microstructure of Markets, edited by Katrina F. Sherrerd (Charlottesville, Va.: AIMR, 1993).
_________. “The Invisible Costs of Trading.” Journal of Portfolio Management, vol. 21, no. 1, 1994, pp. 71-78.
Wagner, Wayne H. (Ed.). A Complete Guide to Securities Transactions: Controlling Costs and Enhancing Performance (New York: Wiley, 1988).
________. “Defining and Measuring Trading Costs.” In Execution Techniques, True Trading Costs, and the Microstructure of Markets, edited by Katrina F. Sherrerd (Charlottesville, Va.: AIMR, 1993).
Wagner, Wayne H., and Michael Banks. “Increasing Portfolio Effectiveness via Transaction Cost Management.” Journal of Portfolio Management, vol. 19, no. 1, 1992, pp. 6-11.
Wagner, Wayne H., and Evan Schulman. “Passive Trading: Point and Counterpoint.” Journal of Portfolio Management, vol. 20, no. 3, 1994, pp. 25-29.
This technical appendix will cover two topics: the bound on value added versus turnover, and the solution to the example trading optimization problem.
We begin by proving the bound on value added versus turnover for the cases of linear inequality and equality constraints. Eq. (16.10) corresponds exactly to the case of linear equality constraints.
Portfolio Q is optimal for the problem Max{VAP|hP ∈ CS}. This means that we can find nonnegative Lagrange multipliers π ≥ 0 such that
and
Since hI ∈ CS, we have 
and π · ≥ 0, so
If we premultiply Eq. (16A.1) by (hI - hQ) and use Eqs. (16A.2) and (16A.3), we find that
Now consider the family of solutions as we move directly from portfolio I to portfolio Q:
where we have introduced the trade portfolio hT. These solutions are all in CS as long as 0 ≤ δ ≤ 1. The solutions have value added
where σT is the risk of hT. If we use Eq. (16A.4), then Eq. (16A.6) simplifies to
Since VA(0) =VAI, VA(1) =VAQ, and ΔVAQ = VAQ − VAI, we have
Thus Eq. (16A.7) simplifies further to
where
and
The slope of VA(δ), Eq. (16A.9), is 
, which is positive for 0 ≤ δ < 1 and decreases to к as δ approaches 1.
The analysis is as before, except that π in Eq. (16A.1) is unrestricted in sign. Therefore κ = 0 and, from Eq. (16A.8), 
. Thus Eq. (16A.9) simplifies to
We will now describe how to solve analytically for the trading strategy h(t) which maximizes utility in the simple example
Schematically, we can represent the utility as
At the optimal solution, the variation of this utility will be zero:
Integrating the second term by parts (and remembering that the variation is zero at the fixed endpoints of the integral),
Thus we can maximize U by choosing h(t) which satisfies
Applying this to the particular utility function [Eq. (16A.13)], the portfolio holding must satisfy the second-order ordinary differential equation
plus the boundary conditions h(0) = 0 and h(T) = 1. Defining relative parameters
and rearranging terms, Eq. (16A.18) becomes
Applying standard mathematical techniques to Eq. (16A.21), we find that the optimal solution h(t) is
We can characterize various regimes for the solution h(t) by looking at some dimensionless quantities which enter into the optimal h(t):
R1. (g · T)2 >> 1. Risk aversion dominates over market impact.
R2. (g · T)2 << 1. Market impact dominates over risk aversion.
R3. s · T2 >> 1. Alpha is positive and dominant over market impact.
R4. s · T2 << - 1. Alpha is negative and dominant over market impact.
R5. | s/g2| >> 1. Alpha (positive or negative) dominates risk aversion.
R6. |s/g2| << 1. Risk aversion dominates over alpha.
If we assume that the alpha is zero, so that the coefficient s = 0 above, then it’s interesting to see the limiting behavior of Eq. (16A.22) in regimes R1 and R2. When market impact dominates over risk, h(t) follows a straight-line path from 0 to 1. The result is uniform trading. If, on the other hand, risk dominates over market impact, h(t) exponentially approaches 1:
1. Assuming zero alpha, derive the limit of Eq. (16A.22) when market impact dominates over risk. Also show that in the limit that risk dominates over market impact, the optimal trade schedule reduces to an exponential, as in Eq. (16A.26).
Use alphas from a residual reversal model to build an optimal portfolio of MMI stocks. The initial portfolio is the MMI, and the benchmark is the CAPMMI. Use a risk aversion of 0.075 and the typical institutional constraints: full investment, no short sales.
1. What is the value added of the MMI? What is the value added of the optimal portfolio? What is the incremental value added?
2. What is the turnover in moving from the MMI to the optimal portfolio?
3. Now build a portfolio exactly halfway between the MMI and the optimal portfolio:
What is the turnover in moving from the MMI to this intermediate portfolio? What is its value added? Compare the incremental value added of this portfolio over the MMI to that of the optimal portfolio over the MMI. Verify Eq. (16.10).
