estimate of asset risk and also estimates of the time to clear. To estimate both these quantities, the 7See BARRA (1997).
Page 454 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). Turnover, Transactions Costs, and Value Added 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
Page 455 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 trade-off 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 VAI. 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 8We are allowing for constraints that limit our choices, such as full investment in risky assets, portfolio beta equal to 1, no short sales, etc. The choice set CS is restricted to be closed and convex. We will consider two cases explicitly: CS defined by equality constraints and CS defined by inequality constraints.
Page 456 Figure 16.2 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 9The concavity of VA(TO) follows from the value added function's being concave in the holdings hP, the turnover function's being convex in hP, and the choice set CS's being convex. The fact that VA(TO) is nondecreasing follows from common sense; i.e., a larger amount of turnover will let you do at least as well. The frontier will be made up of quadratic segments (piecewise quadratic) when the choice set is described by linear inequalities.
Page 457 indicates a decreasing marginal return for each additional amount of turnover that we allow. A Lower Bound 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 The Value of Scheduling Trades 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 10The implication is loose for two reasons. First, we derived the relationship between value added and information ratio at optimality, assuming no constraints (e.g., on turnover). Second, in this chapter we have derived a relationship between turnover and incremental value added.
Page 458 Figure 16.3 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. Transactions Costs 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 valueadded/turnover frontier when the
Page 459 Figure 16.4 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. Implied Transactions Costs The slope of the value-added/turnover frontier can be interpreted as a transactions cost. We can reverse the logic and link any level
Page 460 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 valueadded/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. An Example 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. 11We produced 100 samples from the standard normal distribution.
Page 461 TABLE 16.1 Value Added Percentage of TOQ Lower Bound Excess Total Percentage of VAQ Implied Transactions Cost 0.0 –2.73 0.00 –2.73 0.0% 10.0 –1.91 0.87 –1.03 39.2% 8.66% 20.0 –1.17 1.00 –0.18 59.0% 5.12% 30.0 –0.52 0.88 0.36 71.4% 3.40% 40.0 0.04 0.70 0.74 80.2% 2.50% 50.0 0.52 0.51 1.03 86.8% 1.90% 60.0 0.91 0.34 1.24 91.8% 1.43% 70.0 1.21 0.19 1.40 95.5% 1.02% 80.0 1.43 0.09 1.51 98.0% 0.67% 90.0 1.56 0.02 1.58 99.5% 0.33% 100.0 1.60 0.00 1.60 100.0% 0.00% 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 roundtrip 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 12It is possible to make up a two-stock example where the lower bound on the frontier would be exact. Common sense indicates that with more stocks and a reasonable distribution of alphas, there is considerable room to add value by plucking the best opportunities.
Page 462 Generalizing the Result 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 13In a formal sense we are defining if P ∈ CS and VAp = –∞ if P ∈ CS. This means that VA (TO) = –∞ if TO < TOJ. 14See the appendix for justification.
Page 463 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 as a Portfolio Optimization Problem 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
Page 464 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 shortterm 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. An Example 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
Page 465 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 andh(T) = 1. Initially the portfolio is entirely cash, and at the end ofT 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 over each 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
Page 466 interesting transactions costs in terms of influence on trading strategy.15 We will model the cumulative active market impact as 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. 15We can include other transactions costs (commissions and taxes) in our forecasts of stock returns, but they will not influence our trade schedule, since they are the same, no matter what the schedule. 16Risk scales with the square root of time, while expected return scales linearly with time. As the period shrinks, risk will increasingly dominate return.
Page 467 Figure 16.5 Trade Implementation 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). 17The choices for trade implementation are actually much more numerous. They include crossing networks, alternative stock exchanges, and principal bid transactions. Treatment of these alternatives lies beyond the scope of this book.
Page 468 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. Summary 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
Page 469 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 threequarters 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. Problems 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?
Page 470 References 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).
Page 471 ———. "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 Counter-point." Journal of Portfolio Management, vol. 20, no. 3, 1994, pp. 25–29. Technical Appendix 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. Inequality Case, CS = {h|A · h ≤ b} Portfolio Q is optimal for the problem Max{VAP|hP ∈ CS}. This means that we can find nonnegative Lagrange multipliers π ≥ 0 such that Since hI ∈ CS, we have b – A · hI ≥ 0 and π ≥ 0, so If we premultiply Eq. (16A.1) by (hI – hQ) and use Eqs. (16A.2) and (16A.3), we find that
Page 472 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 The slope of VA(δ), Eq. (16A.9), is 2 · ΔVAQ · (a – b · δ), which is positive for 0 ≤ δ < 1 and decreases to κ as δ approaches 1. Equality Case, CS = {h|A · h = b} 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
Page 473 Trade Optimization 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
Page 474 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
Page 475 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: Exercise 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). Applications Exercises 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).
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