Arbitrageurs Rent Seekers & Liquidity Providers

Gaetan Lion

May 2014

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Trading ahead of index fund rebalancing “… estimated [costs] to be at least 21 to 28 bp annually for S&P 500 index funds, and at least 38 to 77 bp per year for Russell 2000 funds.” Wikipedia article on High Frequency Trading

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Annualized net returns over past 1 yr, 5 yr, and 10 yr

1 yr 5 yr 10 yr

Vanguard S&P 500 21.80% 21.15% 7.41%

S&P 500 Index 21.86% 21.16% 7.42%

Vanguard Small Cap Index25.29% 26.80% 9.91%

Small Cap Index* 25.33% 26.74% 9.84%

Yearly operating expenses

Vanguard S&P 500 0.05%

Vanguard Small Cap Index 0.09%

Vanguard Index Funds track the Indices nearly perfectly!

The Small Cap Index consists of: Russell 2000 through May 16, 2003 MSCI US Small Cap 1750 through January 30, 2013 CRSP US Small Cap Index thereafter

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This paper is Wikipedia’s source on costs to Index funds.

“Because the cost is suffered by the benchmark index itself, it will remain hidden from index investors who simply focus on the explicit fees of index funds and their performance relative to the index.” pg. 272

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S&P 500. - 10 to + 10 trading days around Effective Trading Day

S&P 500. Extending horizon to 120 trading days after Effective Trading Day

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Russell 2000. - 30 to + 10 trading days

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The above are for stocks that cross only the lower cut-off of the Russell 2000. The ones crossing the higher cut-off (in-&-out of Russell 1000) are not be affected.

Russell 2000 extending horizon to – 50 + 120 trading days

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Two competing theories and solutions

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Theory/Author(s) Problem Solution

Conventional Chen (2006) Gastineau (2002)

The cost impact on Indexers is solely due to Arbitrageurs.

No pre-announcement to eliminate Arbitrageurs opportunity to extract profits at index’s costs.

Counterintuitive Petajisto (2010)

Indexers run into liquidity constraints buying or selling large positions (i.e. 10%) of stocks.

Make index changes visible with long pre-announcement periods to reduce liquidity constraints and cost to the index.

Petajisto’s Argument

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“If index funds suddenly and unexpectedly rushed out to buy 10% [representative level] of all the shares of a firm [S&P 500 addition], who would sell those shares to them?” For a $10 billion market cap firm, that would be a $1 billion demand shock that would have a dramatic price impact. This is why S&P preannounces the index changes five trading days before the effective day of the change, as they want to give active traders enough time to get ready to meet such large demand. Thus, arbitrageurs provide the liquidity to meet indexers demand.

Supporting Petajisto’s position

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S&P 500, 5 trading day lead time Russell 2000, 21 trading day lead time

The stock cost impact is a lot lower for the Russell 2000 where traders have a 21 trading day lead time vs. the S&P 500 where they have only a 5 trading day lead time.

Measuring impact on S&P 500

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S&P 500 Addition S&P 500 Deletion Add. & Del.

A B C = A x B A B C = A x B Total

Market Cap* Cost per stock Impact M. Cap* Cost per stock Impact Impact

# out of 500 Impact on S&P 500 # out of 500 Impact on S&P 500 on S&P 500

1990 11 0.1% 3.6% 0.00% 7 0.1% -22.3% -0.02% 0.02%

1991 10 0.1% 9.1% 0.01% 5 0.1% -43.1% -0.03% 0.04%

1992 6 0.1% 3.3% 0.00% 5 0.1% -39.4% -0.02% 0.03%

1993 7 0.1% 5.1% 0.00% 6 0.1% -6.0% 0.00% 0.01%

1994 13 0.2% 6.4% 0.01% 13 0.2% -5.0% -0.01% 0.02%

1995 23 0.3% 8.1% 0.03% 14 0.2% -7.8% -0.01% 0.04%

1996 20 0.3% 7.4% 0.02% 14 0.2% -7.4% -0.01% 0.03%

1997 24 0.3% 11.8% 0.04% 7 0.1% -5.3% 0.00% 0.05%

1998 37 0.6% 12.7% 0.07% 10 0.1% -13.1% -0.02% 0.09%

1999 38 0.6% 10.4% 0.06% 11 0.1% -10.9% -0.01% 0.08%

2000 48 0.8% 14.4% 0.12% 23 0.3% -17.8% -0.06% 0.18%

2001 27 0.4% 4.2% 0.02% 14 0.2% -20.2% -0.03% 0.05%

2002 21 0.3% 7.6% 0.02% 16 0.2% -22.1% -0.05% 0.07%

2003 8 0.1% -1.1% 0.00% 2 0.0% -32.1% -0.01% 0.01%

2004 14 0.2% 4.3% 0.01% 7 0.1% -5.0% 0.00% 0.01%

2005 14 0.2% 4.6% 0.01% 2 0.0% -22.2% -0.01% 0.01%

Weighted Avg. 9.2% 0.048% 10 -15.2% -0.026% 0.074%

Petajisto Avg. 8.8% -15.1%

* GL's estimate

Measuring impact on Russell 2000

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Russell Addition from below Russell Deletion from below Add. & Del.

A B C = A x B A B C = A x B Total

Market Cap Cost per stock Impact M. Cap* Cost per stock Impact Impact

# out of 2000 Impact on R 2000 # out of 2000 Impact on R 2000 on R 2000

1990 284 7.3% 3.5% 0.26% 174 2.3% -3.7% -0.09% 0.34%

1991 403 18.3% 2.2% 0.40% 344 2.3% -6.0% -0.14% 0.54%

1992 439 17.4% -1.6% -0.28% 388 3.8% -7.4% -0.28% 0.00%

1993 344 13.0% 1.0% 0.13% 314 3.9% -3.4% -0.13% 0.26%

1994 462 18.1% -0.4% -0.07% 415 6.4% -3.0% -0.19% 0.12%

1995 363 11.6% 7.0% 0.81% 251 3.9% -3.7% -0.14% 0.96%

1996 413 14.1% -3.5% -0.49% 312 4.8% -10.1% -0.48% -0.01%

1997 436 14.3% 8.0% 1.14% 320 4.7% -7.1% -0.33% 1.48%

1998 427 14.0% -0.1% -0.01% 243 4.2% -5.3% -0.22% 0.21%

1999 412 13.7% 9.6% 1.32% 250 3.9% -7.9% -0.31% 1.62%

2000 546 20.0% 31.0% 6.20% 334 4.8% -5.1% -0.24% 6.44%

2001 501 12.3% 5.6% 0.69% 279 1.8% -5.6% -0.10% 0.79%

2002 382 9.0% 8.5% 0.77% 233 1.9% -10.0% -0.19% 0.96%

2003 283 6.3% 1.1% 0.07% 187 1.6% 5.6% 0.09% -0.02%

2004 310 6.8% -2.0% -0.14% 214 2.5% -1.6% -0.04% -0.10%

2005 206 4.1% 6.0% 0.25% 215 2.1% 1.4% 0.03% 0.22%

Weighted Avg. 5.5% 0.905% 280 -5.1% -0.210% 1.115%

Petajisto Avg. 4.7% -4.6%

Different impact estimates over 1990-2005 period

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GL replication is close to Petajisto’s empirical 2 mth lag strategy (instead of his calculated estimates). In the 2 month lag strategy, he uses an Index fund that would rebalance two months later then the Index, thus avoiding much of the earlier impact on the specific stocks that are added or deleted.

S&P 500 R. 2000

Petajisto

Empirical a) 2 mth lag strategy 0.09% 1.20%

Calculated b) Abstract conclusion-Low 0.21% 0.38%

Calculated c) Abstract conclusion-High 0.28% 0.77%

Calculated GL replication 0.07% 1.12%

The 2 month lag strategy

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With the S&P 500 you can see that the directional gap in cost impact between Additions and Deletions has dramatically narrowed between the rebalancing trading day vs. 60 days (42 trading days) later. This shows that waiting out 2 months to rebalance would work. However, when looking at the Russell 2000, you can’t say the same thing. That spread has barely moved.

Why is impact so much greater on R 2000?

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It is because of the much higher stock turnover. Avg. stock turnover = Avg. market cap stock additions/Total market cap + Avg. market cap of stock deletions/Total market cap

S&P 500 R. 2000

A Average stock turnover 0.4% 16.0%

B Median impact per stock addition 6.9% 2.9%

C Median impact per stock deletion -15.5% -5.2%

D = A(B-C) Overall impact on index 0.09% 1.28%

How to reduce the cost impact on Index Funds?

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1) Preannouncements with longer lead time to reduce the stock specific cost (as demonstrated when comparing the 21 trading days of Russell 2000 vs. 5 trading days of S&P 500);

2) Hold broad-based indices to reduce stock turnover. The Wilshire 5000 has a stock turnover a lot lower than the S&P 500.

3) Use the Russell 1000. This is pretty much an arbitrage free index. That’s because the stocks that get in or out of the Russell 1000 are traded with the Russell 2000 indexers. There is no arbitrage opportunity or liquidity constraints here. Petajisto has documented that for the Russell 1000 the stock specific cost is essentially 0% (Russell 2000 upper cutoff).