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

2. Arbitrageur Price Anomalies

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

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

5. Yale School of Management 5 Data TAQPrice impact estimation 1/1993-6/1993: Oldest available CRSPReturn & portfolio formation 7/1963-12/2001: Covers Fama & French (1993) and Jegadeesh & Titman (1993) CompustatAccounting information 4th Quarter, 1962 - 4th Qtr, 2001 TASSEstimation of actual hedge fund size Covers 1330 hedge funds as of 5/2000

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

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

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

9. Yale School of Management 9 GEKOBONTCSIISINGRMIKE (a) Buys Nobs 23,265 23,157 518 1,212 10,826 2,329 4,704 a B (×10 -3 ) -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) b B (×10 -4 ) 0.003080.01096.534.910.03790.7700.940 (2.27)(3.20)(3.04)(5.69)(2.27)(2.77)(7.91) B 0.4680.4100.000 0.3020.000 (12.51)(15.05)(--) (7.97)(--) (b) Sells Nobs 25,543 25,029 523 692 16,368 1,710 4,362 a S (×10 3 ) 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) b S (×10 4 ) 0.00077 40.004062.743.720.003921.200.47 (1.95)(3.13)(1.30)(3.01)(2.38)(3.83)(3.68) S 0.5750.4990.000 0.5020.000 (13.17)(18.11)(--) (13.79)(--) Estimates for Individual Stocks Table 2

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

11. Yale School of Management 11 Linear vs. Nonlinear PI Functions Ticker Symbol GEKOBONTCSIISINGRMIKE Nobs 23,265 23,157 518 1,212 10,826 2,329 4,704  B (×10 -4 ) 0.6140.8223.4641.2881.3941.4671.200 (24.22)(21.81)(1.11)(1.16)(23.12)(3.57)(6.83)  B (×10 -8 ) 0.04550.08291.0752.6570.06310.26900.0403 (37.04)(42.43)(0.90)(4.02)(27.56)(1.77)(1.64) (a) Price impact from $50,000 trade (bp) Linear0.841.248.8414.571.712.811.40 Box-Cox0.831.6217.8514.231.873.432.37 Difference0.01-0.38-9.010.35-0.16-0.62-0.97 (b) Price impact from $300,000 trade (bp) Linear1.983.3135.7280.993.299.542.41 Box-Cox2.204.0529.5523.024.244.814.05 Difference-0.22-0.746.1757.97-0.964.73-1.65 Table 3

12. Yale School of Management 12 Table 4 For both buys and sells, Slope coefficient b decreases with size Concavity coefficient has a U-shape Portfolio PI Functions by Size Sizea B (x10 -3 )b B (x10 -4 ) B a S (x10 -3 )b S (x10 -4 ) S Small-1.984.560.245-0.162.410.285 2-1.953.150.198-1.162.290.206 3-1.692.480.155-1.132.030.160 4-1.652.530.121-1.132.030.157 5-1.532.440.113-1.102.000.148 6-1.592.330.108-1.412.260.108 7-1.522.100.108-1.241.890.137 8-1.221.490.133-1.191.610.119 91.110.168-0.991.210.162 Big-0.190.220.268-0.250.350.239

13. Yale School of Management 13 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 Price Impacts by Size Decile

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

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

16. Yale School of Management 16 Initial fund size:  0 At the beginning of period t  1, invest b t =  t-1 – PIL t – PIS t – TCL t – TCS t Volume to compute PIs converted to year 1993 dollars At the end of period t,  t = (1 + r l,t – r s,t + 0.8 r FF,t ) b t Excess return after cost R t =  t /  t-1 – 1 – r FF,t Break-even fund size Below,  0 is reported in year 2001 dollars Portfolio Accounting

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

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

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

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

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

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

23. Yale School of Management 23 Size-B/M Combined Strategy (Table 10) Smaller break-even fund sizes than the size-only strategy because of higher turnover in the long position 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) Restricts the available stocks to only those in the biggest 5 deciles Mediocre performance, due to much lower returns before cost than with all stocks Combined/No-small-stock Strategies

24. Yale School of Management 24 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.

25. Yale School of Management 25 Actual Hedge Fund Size Table 13 Size (in millions of dollars) Style#funds%totalMeanMinimumMaximumSum Top down macro36227.4%241.40.01474,122.087,396.0 Bottom up approach69452.6%195.00.189823,474.4135,306.2 Short selling52439.7%201.10.01474,618.1105,362.6 Long bias44333.6%181.10.378023,474.480,217.2 Market neutral31323.7%152.00.01474,122.047,563.9 Opportunities49837.8%139.00.110023,474.469,206.0 Relative value36027.3%183.00.014710,194.065,862.1 Arbitrage40830.9%137.30.060223,474.456,018.6 Discretionary27520.8%101.10.014723,474.427,803.3 Trend follower20115.2%72.70.33843,958.914,603.4 Technical40130.4%74.90.014723,474.430,036.8 Fundamental70253.2%169.50.18984,618.1118,957.1 Systematic32324.5%83.40.060210,194.026,940.0 Diverse35426.8%140.20.014723,474.449,646.3 Other15311.6%98.60.014723,474.415,087.2 Total1319100.0%139.90.014723,474.4184,492.4

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

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