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Yale School of Management Price Impact Costs and the Limit of Arbitrage Zhiwu Chen Yale School of Management Price Impact Costs and the Limit of Arbitrage Zhiwu Chen Werner Stanzl Masahiro Watanabe 1

Arbitrageur Price Anomalies 2 Arbitrageur Price Anomalies 2

Yale School of Management Market Anomalies l Size effect (Banz, 1981; Fama & French, Yale School of Management Market Anomalies l Size effect (Banz, 1981; Fama & French, 1993) Smaller size, larger returns Long small-size & short big-size stocks l B/M (value) effect (Basu, 1983; FF 1993 Lakonishok et al. , 1994; La. Porta et al. , 1997) Higher B/M, greater returns Long high-B/M & short low-B/M stocks l Momentum (Levy, 1967; Jegadeesh & Titman, 1993 & 2001) Return continuation Long past winners & short past losers 3

Yale School of Management Empirical Price Impact Literature l Linear Price Impact Breen, Hodrick Yale School of Management Empirical Price Impact Literature l Linear Price Impact Breen, Hodrick & Korajczyk (2001) Sadka (2002) l Nonlinear (Concave) Price Impact Hasbrouck (1991) Hausman, Lo & Mac. Kinlay (1992) Keim and Madhavan (1996) Knez and Ready (1996) 4

Yale School of Management Data l TAQ Price impact estimation 1/1993 -6/1993: Oldest available Yale School of Management Data l TAQ Price impact estimation 1/1993 -6/1993: Oldest available l CRSP Return & portfolio formation 7/1963 -12/2001: Covers Fama & French (1993) and Jegadeesh & Titman (1993) l Compustat Accounting information 4 th Quarter, 1962 - 4 th Qtr, 2001 l TASS Estimation of actual hedge fund size Covers 1330 hedge funds as of 5/2000 5

Yale School of Management Estimation of Price-Impact Function l Price Impact where Qt = Yale School of Management Estimation of Price-Impact Function l Price Impact where Qt = Quote midpoint at transaction time t Vt = Dollar trading volume at t l Nonlinearity b/w log ( = 0) & linear ( = 1) functions inclusive l The only method that can be applied to almost all stocks without overfitting to outliers l Nonlinear least squares, purchases and sales separately l Matching & trade direction: Lee & Ready (1991) Method l Discard the top one-percentile trades 6

Yale School of Management Example: FHT Figure 1 • Overfitting problem except for the Yale School of Management Example: FHT Figure 1 • Overfitting problem except for the Box-Cox model 7

Yale School of Management Estimated Price Impacts Figure 2 8 Yale School of Management Estimated Price Impacts Figure 2 8

Yale School of Management Estimates for Individual Stocks Table 2 (a) Buys Nobs a. Yale School of Management Estimates for Individual Stocks Table 2 (a) Buys Nobs a. B (× 10 -3) b. B (× 10 -4) B (b) Sells Nobs a. S (× 103) b. S (× 104) S GE KO BONT CSII S INGR MIKE 23, 157 23, 518 1, 826 10, 329 2, 704 4, 265 212 -0. 020 -0. 060 -5. 28 -3. 89 -0. 13 -0. 49 -0. 78 (-1. 34) (-3. 71) (-2. 75) (-5. 11) (-3. 07) (-1. 99) (-6. 69) 0. 00308 0. 0109 6. 53 4. 91 0. 0379 0. 770 0. 940 (2. 27) (3. 20) (3. 04) (5. 69) (2. 27) (2. 77) (7. 91) 0. 468 0. 410 0. 000 0. 302 0. 000 (12. 51) (15. 05) (--) (7. 97) (--) 25, 029 25, 523 16, 710 1, 362 4, 543 692 368 0. 018 -0. 020 -1. 37 -2. 83 -0. 030 -0. 87 -0. 30 (2. 49) (-1. 50) (-0. 70) (-2. 47) (-1. 65) (-3. 05) (-2. 36) 0. 00077 0. 00406 4 2. 74 3. 72 0. 00392 1. 20 0. 47 (1. 95) (3. 13) (1. 30) (3. 01) (2. 38) (3. 83) (3. 68) 0. 575 0. 499 0. 000 0. 502 0. 000 (13. 17) (18. 11) (--) (13. 79) (--) 9

Yale School of Management Box-Cox vs. Linear PI Functions Figure 3 10 Yale School of Management Box-Cox vs. Linear PI Functions Figure 3 10

Yale School of Management Linear vs. Nonlinear PI Functions Table 3 Nobs a. B Yale School of Management Linear vs. Nonlinear PI Functions Table 3 Nobs a. B (× 10 -4) b. B (× 10 -8) Ticker Symbol GE KO BONT CSII S INGR MIKE 23, 1, 10, 2, 4, 265 157 518 212 826 329 704 0. 614 0. 822 3. 464 1. 288 1. 394 1. 467 1. 200 (24. 22) (21. 81) (1. 16) (23. 12) (3. 57) (6. 83) 0. 0455 0. 0829 1. 075 2. 657 0. 0631 0. 2690 0. 0403 (37. 04) (42. 43) (0. 90) (4. 02) (27. 56) (1. 77) (1. 64) (a) Price impact from $50, 000 trade (bp) Linear 0. 84 1. 24 8. 84 Box-Cox 0. 83 1. 62 17. 85 Difference 0. 01 -0. 38 -9. 01 14. 57 14. 23 0. 35 1. 71 1. 87 -0. 16 2. 81 3. 43 -0. 62 1. 40 2. 37 -0. 97 (b) Price impact from $300, 000 trade (bp) Linear 1. 98 3. 31 35. 72 Box-Cox 2. 20 4. 05 29. 55 Difference -0. 22 -0. 74 6. 17 80. 99 23. 02 57. 97 3. 29 4. 24 -0. 96 9. 54 4. 81 4. 73 2. 41 4. 05 -1. 65 11

Yale School of Management Portfolio PI Functions by Size Table 4 Size a. B(x Yale School of Management Portfolio PI Functions by Size Table 4 Size a. B(x 10 -3) b. B(x 10 -4) Small -1. 98 4. 56 2 -1. 95 3. 15 3 -1. 69 2. 48 4 -1. 65 2. 53 5 -1. 53 2. 44 6 -1. 59 2. 33 7 -1. 52 2. 10 8 -1. 22 1. 49 9 -1. 00 1. 11 Big -0. 19 0. 22 B a. S(x 10 -3) b. S(x 10 -4) 0. 245 0. 198 0. 155 0. 121 0. 113 0. 108 0. 133 0. 168 0. 268 -0. 16 -1. 13 -1. 10 -1. 41 -1. 24 -1. 19 -0. 99 -0. 25 For both buys and sells, • Slope coefficient b decreases with size • Concavity coefficient has a U-shape 2. 41 2. 29 2. 03 2. 00 2. 26 1. 89 1. 61 1. 21 0. 35 S 0. 285 0. 206 0. 160 0. 157 0. 148 0. 108 0. 137 0. 119 0. 162 0. 239 12

Yale School of Management Price Impacts by Size Decile Figure 4 • Buy trades Yale School of Management Price Impacts by Size Decile Figure 4 • Buy trades have positive price impacts, sells negative • Absolute price impact increases with the size of trade • Price impact monotonically decreases with firm size 13

Yale School of Management Implementation of Strategies l Set up a long-short portfolio based Yale School of Management Implementation of Strategies l Set up a long-short portfolio based on each strategy l Measure excess return after cost, where volume to compute PIs converted to year 1993 dollars l Since price impact increases in fund size, there is a maximal fund size at which excess return after cost = 0 l The maximal fund size reported in year 2001 dollars 14

Yale School of Management Investment Strategy Criteria l Portfolios are formed annually, semiannually, or Yale School of Management Investment Strategy Criteria l Portfolios are formed annually, semiannually, or quarterly l Value of long position = Value of short position l Rebalance when stocks are either added to or dropped from a portfolio; also when weights change l Commisions: 15 bp for purchases and sales 25 bp for short-selling l Short-sale rebate: 80% of Fed Fund Rate l Maximum $ volume / trade: Maximum holding: 1% of market cap 5% of market cap 15

Yale School of Management Portfolio Accounting l Initial fund size: 0 l At the Yale School of Management Portfolio Accounting l Initial fund size: 0 l At the beginning of period t 1, invest bt = t-1 – PILt – PISt – TCLt – TCSt Volume to compute PIs converted to year 1993 dollars l At the end of period t, t = (1 + rl, t – rs, t+ 0. 8 r. FF, t) bt l Excess return after cost Rt = t / t-1 – r. FF, t l Break-even fund size l Below, 0 is reported in year 2001 dollars 16

Yale School of Management Table 5 Size Strategy • Huge, but is this really Yale School of Management Table 5 Size Strategy • Huge, but is this really attainable? 17

Yale School of Management Trading and Holding Restrictions Figure 5 • Realistically implemented size Yale School of Management Trading and Holding Restrictions Figure 5 • Realistically implemented size strategies will not accommodate more than several hundred million dollars 18

Yale School of Management Higher Rebalancing Frequencies Figure 6 • The potential benefit of Yale School of Management Higher Rebalancing Frequencies Figure 6 • The potential benefit of “fine tuning” does not cover higher price impact costs 19

Yale School of Management Book-to-Market Strategy Table 7 20 Yale School of Management Book-to-Market Strategy Table 7 20

Yale School of Management Momentum Strategies Table 9: Maximum fund sizes, non-overlapping strategies (in Yale School of Management Momentum Strategies Table 9: Maximum fund sizes, non-overlapping strategies (in $ millions) • Momentum strategies could accommodate billions of dollars if no trading restrictions are imposed 21

Yale School of Management At a glance… Table 6 22 Yale School of Management At a glance… Table 6 22

Yale School of Management Combined/No-small-stock Strategies Size-B/M Combined Strategy (Table 10) l Smaller break-even Yale School of Management Combined/No-small-stock Strategies Size-B/M Combined Strategy (Table 10) l Smaller break-even fund sizes than the size-only strategy because of higher turnover in the long position l Because of this and the smaller # stocks in both the long and short positions, the 1% trade-size and 5% position-size restrictions will make the fund sizes even smaller than those for size-only strategies in Figure 5 No-small-stock B/M Strategy (Table 11) l Restricts the available stocks to only those in the biggest 5 deciles l Mediocre performance, due to much lower returns before cost than with all stocks 23

Yale School of Management No-small-stock Momentum Strategy Table 12, VW 12/12 non-overlapping strategy • Yale School of Management No-small-stock Momentum Strategy Table 12, VW 12/12 non-overlapping strategy • Still works. • Both the EW & VW strategies accommodate b/w $1 and 3 billions with the 1% trade-size restriction. 24

Yale School of Management Actual Hedge Fund Size Table 13 Style Top down macro Yale School of Management Actual Hedge Fund Size Table 13 Style Top down macro Bottom up approach Short selling Long bias Market neutral Opportunities Relative value Arbitrage Discretionary Trend follower Technical Fundamental Systematic Diverse Other Total #funds 362 694 524 443 %total 27. 4% 52. 6% 39. 7% 33. 6% Mean 241. 4 195. 0 201. 1 181. 1 313 498 360 408 275 201 401 702 323 354 153 1319 23. 7% 37. 8% 27. 3% 30. 9% 20. 8% 15. 2% 30. 4% 53. 2% 24. 5% 26. 8% 11. 6% 100. 0% 152. 0 139. 0 183. 0 137. 3 101. 1 72. 7 74. 9 169. 5 83. 4 140. 2 98. 6 139. 9 Size (in millions of dollars) Minimum Maximum 0. 0147 4, 122. 0 0. 1898 23, 474. 4 0. 0147 4, 618. 1 0. 3780 23, 474. 4 0. 0147 0. 1100 0. 0147 0. 0602 0. 0147 0. 3384 0. 0147 0. 1898 0. 0602 0. 0147 4, 122. 0 23, 474. 4 10, 194. 0 23, 474. 4 3, 958. 9 23, 474. 4 4, 618. 1 10, 194. 0 23, 474. 4 Sum 87, 396. 0 135, 306. 2 105, 362. 6 80, 217. 2 47, 563. 9 69, 206. 0 65, 862. 1 56, 018. 6 27, 803. 3 14, 603. 4 30, 036. 8 118, 957. 1 26, 940. 0 49, 646. 3 15, 087. 2 184, 492. 4 25

Yale School of Management Conclusions l Price impact reduces returns substantially l For size Yale School of Management Conclusions l Price impact reduces returns substantially l For size and B/M strategies, only about one hundred million dollars can be accommodated under realistic trading restrictions l This is marginal compared to the actual hedge fund size l However, some momentum strategies may be implemented profitably with about one billion dollars l Market is minimally efficient to allow for size & B/M anomaly; persistence of momentum is still a challenge 26

Yale School of Management Future Research l “Working” the order l VWAP l Time Yale School of Management Future Research l “Working” the order l VWAP l Time variation in liquidity suggests change in price impacts l Change in other costs (bid-ask spread, short sale carry cost, transactions fees) 27