Price Impact Costs and the Limit of Arbitrage

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

2. Arbitrageur Price Anomalies

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

6. Estimation of Price-Impact Function • Price Impact where Qt = Quote midpoint at transaction time t Vt = 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. Example: FHT Figure 1 • Overfitting problem except for the Box-Cox model

8. Figure 2 Estimated Price Impacts

9. Estimates for Individual Stocks Table 2

10. Figure 3 Box-Cox vs. Linear PI Functions

11. Table 3 Linear vs. Nonlinear PI Functions

12. Portfolio PI Functions by Size Table 4 • For both buys and sells, • Slope coefficient b decreases with size • Concavity coefficient  has a U-shape

13. 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

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. Investment Strategy Criteria • 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

16. Portfolio Accounting • Initial fund size: 0 • At the beginning of period t1, invest bt= t-1 – PILt– PISt– TCLt– TCSt Volume to compute PIs converted to year 1993 dollars • At the end of period t, t = (1 + rl,t– rs,t+ 0.8 rFF,t) bt • Excess return after cost Rt = t / t-1 – 1 – rFF,t • Break-even fund size • Below, 0 is reported in year 2001 dollars

17. Size Strategy Table 5 • Huge, but is this really attainable?

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

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

20. Book-to-Market Strategy Table 7

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

22. At a glance… Table 6

23. Combined/No-small-stock Strategies 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

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. Actual Hedge Fund Size Table 13

26. Conclusions • 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

27. Future Research • “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)