1. The Capacity of Trading Strategies Landier (TSE), Simon (CFM), Thesmar (HEC) AFA 2016, San Francisco

2. Practical and academic debate on capacity –How much can be traded without deteriorating performance too much? –Optimal Trading speed in presence of transaction costs? –How much money should a fund accept? “How abnormal are anomalies?” –If feasible profits are small, not very abnormal (e.g. small caps) –Defining big anomaly as “high capacity anomaly” Many classic signals have lower Sharpe than before –Does it imply less capacity for asset managers? Motivation

3. Example : sorting on Cash-Flows / Assets (“quality” strategy; ) (1990-2013) Most Usual Strategies Have Higher Sharpe Within Mid-Caps LARGE STOCKS MID-CAP STOCKS

4. Trading costs: Korajczyk & Sadka (2004), Frazzini&al (2012), Novy-Marx & Velikhov (2014), Dreschler&Dreschler (2014) … Dynamic trading: Litterman (1990s), Garleanu&Pedersen (2014), Collin- Dufresne&al (2014), Sefton and Champonnois (2015) … “Disappearing Anomalies”: Lean and Pontiff (2014), Harvey & al. (2014) Literature

5. 1. A tractable model to analyze capacity –Optimal portfolio –Solution and approximation –Calibrations: signal persistence illiquidity across size pools 2. Implementation and back-testing of the trading rule –Four “classic” strategies 3. Robustness of the framework –When is model fit for practical application? Outline

6. MODEL

7. Static portfolio optimization (Markowitz) where: x t+s = asset holdings at time t+s r t+s = expected returns at time t+s Σ = variance matrix  = risk aversion (= portfolio size)

8. Dynamic Portfolio Optimization (Garleanu-Pedersen, 2014) Usual Markovitz Some signal s t predicts returns at date t+1: Transaction cost (quadratic) This signal is transient, it mean-reverts with speed φ:

9. Assume  <<  (large investor) Optimal trade takes into account : 1. How costly it is to trade the stock 2. Whether we have time to exploit the signal (in and out) Optimal Trading “target portfolio” “Markowitz portfolio” “trading speed”

10. Markovitz (“pure”) Sharpe: Realized Sharpe: Result 1: Sharpe Deterioration with portfolio scale Scale of the Portfolio = Dollar Volatility

11. Result 2: Defining Capacity Start from a strategy: compute SR*, Sharpe Ratio when $0 are traded (usual academic exercise) Ask: « what Dollar volatility can be traded under constraint that SR > target ? » (For example, target =.3 ) Answer (for small φ) : -Low λ (=high liquidity) is good for capacity -Low φ (=slow signal) is good for capacity

12. Sharpe frontier: Spanning the volatility scale Sharpe when trading 0$ Lower gamma

13. Role of persistence SR / SR* BM in the large pool Low vol in the mid pool Repurchasers in the mid Persistence

14. Role of Liquidity SR / SR* 1991 1996 2001 2006 liquidity

15. Expected Transaction Costs given targeted Sharpe loss Interpretation: if fund is ready to accept a reduction of Sharpe by 50%, it means it “optimally” burns 1-(1/2) 2/3 ≈ 40% of its realized PNL. Also: Gross vs Net Performance

16. EMPIRICAL IMPLEMENTATION

17. 1.Book-to-Market 2.Share Repurchasers 3.Low volatility 4.Cash-Flows / Assets (« profitability», « quality ») Use 1990-2013; CRSP + Compustat (7 months lag) First, trade optimally on past data, and see what capacity we reach Four Classic Strategies

18. 1.59% daily volume  0.10% price impact. (Engle&al. (2008)) Dollar volume smaller in mid-caps  higher λ 2 stock pools (US): –(1,500) = LARGE – (500,1500) = MID Calibrating Illiquidity Across Size Pools

20. « Poor man’s Markowitz » Σ -1 notoriously hard to estimate  we assume returns have a a one factor structure  closed form (no need to invert numerically)

21. ( Assume we aim for SR=.3 )

23. Backtesting algorithm 1.Estimate illiquidity λ (for the pool of stocks traded) 2.Estimate mean-reversion φ (for given strategy) 3.Set  4.Trade during 1990s and 2000s using rule: 1.Compute realized Sharpe and volatility of PNL & back to 3.  gives us an empirical capacity frontier  compute $ vol reached for Sharpe =.3

24. Backtesting results: Cash-flows in mid caps

27. Small portfolio: vol=$10m ; large portfolio: vol=$15bn

28. Back-testing capacity can differ from theoretical capacity Back-testing = 1 draw of data Also, model is approximation of data: –Signal’s process –Predictability might vanish quickly –Update Sigma matrix monthly Back-testing vs. Simulations

30. Generate confidence bands by simulating data where signal is not predictive but trader believes it is (with our estimated parameters) 1.Generate history mean-reverting signals (as we see it in data) + random returns not predicted by signal 2.Set  3.Trade using our optimal rule (i.e. trader assume signal predicts returns) 4.Compute realized Sharpe and volatility of PNL. 5.Back to 2. (to span capacity frontier) 6.Back to 1. (to simulate many such frontiers under the null)  for each size, we obtain top and bottom 5% of Sharpe Confidence bands ?

31. Back-testing “cash-lows” in mid caps

32. ROBUSTNESS

33. Assume myopic trader: For phi=0.001and SR*/SR=2, VOL(dyn)/VOL(myo)=100 Slowing trading helps a lot in preventing Sharpe deterioration Robustness 1: « does dynamic optimization really help? »

34. High sensitivity to measuring liquidity Not much sensitivity to measuring persistence Robustness 2: « what if we do not measure parameters well » True φ =.001 But Trader believes

35. OK if AR(n) with positive coefs –i.e. when true signal has more memory than trader believes Robustness 3: « what if signal dynamics is not AR(1) ? »

36. Key insights: –Illiquidity is less of a barrier for persistent signals, –Can be quantified in dynamic trading optimization, –Fundamental signals (e.g. “quality”) have substantial capacity because they are slow Rethinking the concept of “market anomaly” –can substantial arbitrage profits be made? –“Quality” is a “big” anomaly Lower “pure” Sharpe decrease is not a good criterion for anomaly disappearance: –in last 15 years, compensated by higher liquidity Strategic interactions? Aggregate vs. individual capacity? Take away

37. Supplementary slides

38. Difference in hard-to-borrow costs in large vs. mid is second- order vis-a-vis price impact concerns What About Shorting Costs? Figure 5: Cost of shorting: Hard to borrow rates by size pools