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									Yale School of Management




                      Price Impact Costs and
                      the Limit of Arbitrage

                               Zhiwu Chen
                              Werner Stanzl
                            Masahiro Watanabe


                                                1
Arbitrageur    Price
              Anomalies
Yale School of Management




                            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
                                                        3
Yale School of Management




       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)
                                            4
Yale School of Management




                            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
                                                         5
Yale School of Management




      Estimation of Price-Impact Function
  Price Impact                     Qt 1  Qt       Vt   1
                             PI t              ab           t
                                        Qt               
     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                             6
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



         Estimates for Individual Stocks
    Table 2                 GE       KO        BONT        CSII           S        INGR       MIKE
    (a) Buys
    Nobs                23,265      23,157         518      1,212       10,826      2,329      4,704
    aB (×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)
    bB (×10-4)         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)
    B                    0.468       0.410     0.000       0.000        0.302      0.000      0.000
                        (12.51)     (15.05)         (--)       (--)      (7.97)        (--)       (--)
    (b) Sells
    Nobs                25,543      25,029         523         692      16,368      1,710      4,362
    aS (×103)             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)
    bS (×104)          0.000774    0.00406        2.74        3.72     0.00392        1.20       0.47
                         (1.95)       (3.13)    (1.30)      (3.01)        (2.38)    (3.83)     (3.68)
    S                    0.575       0.499     0.000       0.000         0.502     0.000      0.000
                        (13.17)     (18.11)         (--)        (--)    (13.79)        (--)       (--)


                                                                                                 9
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            Box-Cox vs. Linear PI Functions
    Figure 3




                                              10
Yale School of Management


                 Linear vs. Nonlinear PI Functions
       Table 3                                      Ticker Symbol
                             GE        KO       BONT     CSII     S          INGR      MIKE
         Nobs
                            23,265    23,157       518    1,212    10,826     2,329     4,704
         aB (×10-4)           0.614     0.822    3.464    1.288      1.394    1.467     1.200
                            (24.22)   (21.81)    (1.11)   (1.16)   (23.12)    (3.57)    (6.83)
         bB (×10-8)         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       14.57      1.71      2.81      1.40
            Box-Cox         0.83     1.62     17.85       14.23      1.87      3.43      2.37
            Difference      0.01    -0.38     -9.01        0.35     -0.16     -0.62     -0.97

         (b) Price impact from $300,000 trade (bp)
            Linear           1.98    3.31    35.72        80.99      3.29      9.54      2.41
            Box-Cox          2.20    4.05    29.55        23.02      4.24      4.81      4.05
            Difference      -0.22   -0.74      6.17       57.97     -0.96      4.73     -1.65


                                                                                        11
Yale School of Management


            Portfolio PI Functions by Size
   Table 4
              Size aB(x10-3) bB(x10-4)      B aS(x10-3) bS(x10-4)      S
             Small     -1.98     4.56    0.245     -0.16     2.41    0.285
                 2     -1.95     3.15    0.198     -1.16     2.29    0.206
                 3     -1.69     2.48    0.155     -1.13     2.03    0.160
                 4     -1.65     2.53    0.121     -1.13     2.03    0.157
                 5     -1.53     2.44    0.113     -1.10     2.00    0.148
                 6     -1.59     2.33    0.108     -1.41     2.26    0.108
                 7     -1.52     2.10    0.108     -1.24     1.89    0.137
                 8     -1.22     1.49    0.133     -1.19     1.61    0.119
                 9     -1.00     1.11    0.168     -0.99     1.21    0.162
               Big     -0.19     0.22    0.268     -0.25     0.35    0.239

       For both buys and sells,
        Slope coefficient b decreases with size
        Concavity coefficient  has a U-shape                          12
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

       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

                                                          14
Yale School of Management




              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

                                                                  15
Yale School of Management




                        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
                                   1 T
                    sup{ 0  0 | t 1 R t ( 0 )  0}
                                   T
      Below, 0 is reported in year 2001 dollars             16
Yale School of Management


     Table 5
      (a) Equally Weighted, without Costs
                                                 Size Strategy
                         Mean
                        Excess                  Standard                       Sharpe
                        Return      (t-stat)    Deviation   Max / Min           Ratio
           Size          0.067      (1.36)        0.265   0.733 / -0.369        0.252
        Strategy        [0.058]    [(1.46)]      [0.247] [0.733 / -0.369]      [0.236]
         CRSP            0.092      (1.67)        0.295   0.972 / -0.473        0.311
     Equally Weighted   [0.093]    [(2.24)]      [0.259] [0.972 / -0.473]      [0.358]

      (b) Equally Weighted, with Costs

         Arbitrage        Mean                                                                Long Portfolio         Short Portfolio
           Fund         Excess                  Standard                       Sharpe     Mean Price    Mean     Mean Price     Mean
           Size          Return      (t-stat)   Deviation      Max / Min         Ratio     Impact     Turnover    Impact      Turnover
            1M            0.059      (1.21)       0.263      0.722 / -0.373      0.225      0.004       0.661      0.001        0.542
                         [0.051]    [(1.28)]     [0.245]    [0.722 / -0.373]    [0.207]    [0.004]     [0.649]    [0.001]      [0.530]
           10M            0.053      (1.10)       0.261      0.714 / -0.376      0.204      0.008       0.663      0.001        0.544
                         [0.045]    [(1.14)]     [0.244]    [0.714 / -0.376]    [0.185]    [0.008]     [0.651]    [0.001]      [0.533]
          100M            0.043      (0.90)       0.259      0.698 / -0.381      0.167      0.016       0.667      0.002        0.549
                         [0.036]    [(0.91)]     [0.242]    [0.698 / -0.381]    [0.148]    [0.016]     [0.655]    [0.002]      [0.538]
          500M            0.032      (0.68)       0.256      0.680 / -0.386      0.126      0.025       0.671      0.003        0.554
                         [0.025]    [(0.66)]     [0.239]    [0.680 / -0.386]    [0.107]    [0.023]     [0.659]    [0.003]      [0.543]
            1B            0.026      (0.56)       0.254      0.670 / -0.389      0.103      0.029       0.674      0.004        0.557
                         [0.020]    [(0.52)]     [0.237]    [0.670 / -0.389]    [0.084]    [0.028]     [0.661]    [0.003]      [0.546]
            5B            0.008      (0.18)       0.249      0.637 / -0.397      0.034      0.043       0.682      0.005        0.566
                         [0.004]    [(0.10)]     [0.233]    [0.637 / -0.397]    [0.016]    [0.040]     [0.668]    [0.005]      [0.554]
            7B            0.004      (0.08)       0.248      0.628 / -0.398      0.016      0.046       0.684      0.006        0.568
                         [0.000]   [(-0.01)]     [0.232]    [0.628 / -0.398]   [-0.001]    [0.043]     [0.670]    [0.005]      [0.556]
            9B            0.000      (0.01)       0.247      0.621 / -0.400      0.001      0.049       0.685      0.006        0.570
                        [-0.003]   [(-0.09)]     [0.231]    [0.621 / -0.400]   [-0.015]    [0.045]     [0.671]    [0.005]      [0.558]
           10B           -0.001     (-0.03)       0.247      0.618 / -0.401     -0.005      0.050       0.686      0.006        0.570
                        [-0.005]   [(-0.13)]     [0.230]    [0.618 / -0.401]   [-0.021]    [0.047]     [0.672]    [0.006]      [0.559]


      Huge, but is this really attainable?                                                                                              17
Yale School of Management


            Trading and Holding Restrictions
    Figure 5




        Realistically implemented size strategies will not
         accommodate more than several hundred million dollars   18
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                Higher Rebalancing Frequencies
        Figure 6




        The potential benefit of “fine tuning” does not cover higher
         price impact costs                                        19
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                      Book-to-Market Strategy
   Table 7
    (a) Equally Weighted, without Costs

                       Mean
                      Excess                  Standard                       Sharpe
                      Return      (t-stat)    Deviation   Max / Min           Ratio
         B/M           0.093      (3.57)        0.139   0.320 / -0.234        0.664
      Strategy        [0.093]    [(3.76)]      [0.152] [0.320 / -0.272]      [0.610]
       CRSP            0.092      (1.67)        0.295   0.972 / -0.473        0.311
   Equally Weighted   [0.093]    [(2.24)]      [0.259] [0.972 / -0.473]      [0.358]

    (b) Equally Weighted, with Costs

       Arbitrage        Mean                                                                Long Portfolio         Short Portfolio
         Fund         Excess                  Standard                       Sharpe     Mean Price    Mean     Mean Price     Mean
         Size          Return      (t-stat)   Deviation      Max / Min         Ratio     Impact     Turnover    Impact      Turnover
          1M            0.061      (2.30)       0.142      0.296 / -0.254      0.428      0.013       1.030      0.012        0.984
                       [0.061]    [(2.45)]     [0.153]    [0.296 / -0.304]    [0.397]    [0.013]     [0.970]    [0.012]      [0.985]
         10M            0.037      (1.37)       0.145      0.280 / -0.268      0.254      0.024       1.046      0.022        0.999
                       [0.038]    [(1.53)]     [0.155]    [0.280 / -0.320]    [0.248]    [0.022]     [0.985]    [0.021]      [0.999]
         50M            0.013      (0.45)       0.151      0.267 / -0.280      0.084      0.035       1.067      0.032        1.018
                       [0.017]    [(0.64)]     [0.159]    [0.267 / -0.333]    [0.104]    [0.032]     [1.001]    [0.030]      [1.014]
         90M            0.002      (0.07)       0.155      0.262 / -0.305      0.013      0.040       1.077      0.036        1.028
                       [0.007]    [(0.28)]     [0.162]    [0.262 / -0.337]    [0.045]    [0.036]     [1.010]    [0.034]      [1.022]
        100M            0.000      (0.00)       0.156      0.262 / -0.313      0.000      0.040       1.079      0.037        1.030
                       [0.006]    [(0.21)]     [0.163]    [0.262 / -0.338]    [0.035]    [0.037]     [1.012]    [0.035]      [1.023]
        300M           -0.023     (-0.73)       0.170      0.254 / -0.418     -0.135      0.051       1.110      0.047        1.057
                      [-0.014]   [(-0.48)]     [0.174]    [0.254 / -0.418]   [-0.079]    [0.046]     [1.035]    [0.043]      [1.045]


                                                                                                                           20
Yale School of Management



                            Momentum Strategies
     Table 9: Maximum fund sizes, non-overlapping strategies
      (a) 1964-1991
      Equally Weighted                               Value Weighted                (in $ millions)
                                 K                                             K
                    1           3        6      12                 1          3         6      12
           1      <1         <1       <1       <1         1     <1         <1        <1      29.0
           3      <1         <1       <1      32.2        3     <1         1.8     340.6  7,217.8
      J    6      <1         <1       2.4    161.0   J    6     <1       255.2   2,290.9 14,653.0
           9      <1         1.4     32.7     67.8        9     1.8      872.2   8,862.3  6,568.0
          12      <1        15.3     43.2      4.6       12   115.0    4,809.8 10,038.3   2,420.8

      (b) 1964-2001
      Equally Weighted                               Value Weighted
                                 K                                             K
                    1           3        6      12                 1          3            6      12
           1      <1         <1       <1       <1         1      <1        <1           1.3    320.4
           3      <1         <1       <1     102.9        3      <1       13.2      2,433.0 21,445.8
      J    6      <1         <1       4.3    289.4   J    6      <1      666.4      5,928.0 61,647.1
           9      <1         1.9     30.3     31.5        9      <1    3,926.5     17,339.6  7,740.5
          12      <1         6.8     18.2      <1        12      <1    4,171.0      8,926.5    883.7


    Momentum strategies could accommodate billions of dollars
     if no trading restrictions are imposed
                                                                                                  21
Yale School of Management


                                        At a glance…
                                          Rebalancing                           (in $ millions)
            Table 6                       frequency     Equally Weighted      Value Weighted
                        Size Strategy     Annual                 9,173.1                 421.0
                                                                [6841.4]               [129.8]
                                          Semiannual             3,812.7                 173.2
                                                                [4384.0]                 [75.0]
                                          Quarterly                 634.0                  16.6
                                                                  [827.3]                  [8.2]
                        B/M Strategy      Annual                  100.17                     <1
                                                                  [158.5]                   [<1]
                                          Semiannual                25.78                    <1
                                                                    [69.2]                  [<1]
                                          Quarterly                 12.01                    <1
                                                                    [23.0]                  [<1]
                        Size-B/M          Annual                    768.6                758.9
                        Combined                                [1395.9]               [636.5]
                        Strategy          Semiannual                294.6                208.6
                                                                  [946.9]              [325.0]
                                          Quarterly                   44.4                 17.7
                                                                  [175.1]                [37.4]
                        Momentum          Annual                        4.6           2,420.8
                        Strategy                                       [<1]            [883.7]
                                          Semiannual                    2.4           2,290.9
                                                                      [4.3]          [5928.0]
                                          Quarterly                     <1                   1.8
                                                                       [<1]              [13.2]

                                                                                                   22
Yale School of Management



      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
                                                                    23
Yale School of Management




        No-small-stock Momentum Strategy
    Table 12, VW 12/12 non-overlapping strategy
     (f) Value Weighted, with Costs and 1% Market-Capital Per-trade Restriction

        Arbitrage       Mean                                                                  Long Portfolio         Short Portfolio
          Fund        Excess                    Standard                       Sharpe     Mean Price    Mean     Mean Price     Mean
          Size         Return        (t-stat)   Deviation      Max / Min         Ratio     Impact     Turnover    Impact      Turnover
          10M           0.036        (0.94)       0.203      0.486 / -0.387      0.178      0.006       1.798      0.006        1.786
                       [0.037]      [(0.84)]     [0.269]    [0.809 / -0.816]    [0.138]    [0.006]     [1.765]    [0.006]      [1.778]
         100M           0.026        (0.69)       0.201      0.469 / -0.395      0.130      0.011       1.805      0.010        1.794
                       [0.027]      [(0.62)]     [0.266]    [0.795 / -0.818]    [0.103]    [0.010]     [1.773]    [0.010]      [1.786]
           1B           0.004        (0.12)       0.199      0.442 / -0.462      0.022      0.020       1.826      0.020        1.816
                       [0.008]      [(0.19)]     [0.264]    [0.777 / -0.820]    [0.030]    [0.019]     [1.790]    [0.019]      [1.804]
           3B          -0.025       (-0.66)       0.198      0.430 / -0.561     -0.124      0.033       1.857      0.034        1.847
                      [-0.014]     [(-0.33)]     [0.264]    [0.774 / -0.821]   [-0.055]    [0.028]     [1.813]    [0.029]      [1.828]




        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                                           Size (in millions of dollars)
    Style                   #funds   %total   Mean       Minimum       Maximum             Sum
    Top down macro             362   27.4%    241.4        0.0147        4,122.0       87,396.0
    Bottom up approach         694   52.6%    195.0        0.1898       23,474.4      135,306.2
    Short selling              524   39.7%    201.1        0.0147        4,618.1      105,362.6
    Long bias                  443   33.6%    181.1        0.3780       23,474.4       80,217.2
    Market neutral            313     23.7%   152.0        0.0147        4,122.0       47,563.9
    Opportunities             498     37.8%   139.0        0.1100       23,474.4       69,206.0
    Relative value            360     27.3%   183.0        0.0147       10,194.0       65,862.1
    Arbitrage                 408     30.9%   137.3        0.0602       23,474.4       56,018.6
    Discretionary             275     20.8%   101.1        0.0147       23,474.4       27,803.3
    Trend follower            201     15.2%    72.7        0.3384        3,958.9       14,603.4
    Technical                 401     30.4%    74.9        0.0147       23,474.4       30,036.8
    Fundamental               702     53.2%   169.5        0.1898        4,618.1      118,957.1
    Systematic                323     24.5%    83.4        0.0602       10,194.0       26,940.0
    Diverse                   354     26.8%   140.2        0.0147       23,474.4       49,646.3
    Other                     153     11.6%    98.6        0.0147       23,474.4       15,087.2
    Total                    1319    100.0%   139.9        0.0147       23,474.4      184,492.4


                                                                                           25
Yale School of Management




                            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
                                                                    26
Yale School of Management




                            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)

                                                     27

								
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