Yale School of Management Cross Sectional Variation of

Yale School of Management Cross Sectional Variation of Stock Returns: Idiosyncratic Risk and Liquidity by Matthew Spiegel Xiaotong (Vivian) Wang

Yale School of Management Cross Sectional Returns via Market Microstructure l Liquidity varies across stocks. ù More liquidity is better. Duh! l People should be willing to accept lower returns in exchange for higher liquidity levels.

Yale School of Management Liquidity and Returns: Theory Liquidity Returns l No: Constantinides (1986), Heaton and Lucas (1996), Duffie and Sun (1990), Grossman and Laroque (1990), Vayanos (1998), Vayanos and Vila (1999) l Yes: Huang (2001), Homstrom and Tirole (2001), Demsetz (1968), Amihud and Mendelson (1986), Glosten and Milgrom (1987), Easley and O’Hara (2001), Easley, Hvidkjaer, and O’Hara (2001), Acharya and Pedersen (2005)

Yale School of Management Liquidity and Returns: Empirical l As a cross-sectional characteristic: ù Yes: Amihud and Mendelson (1988), Brennan and Subrahamanyam (1996), Brennan, Chordia, and Subrahamanyam (1998), Chalmers and Kadlec (1998), Hasbrouck (2005) l As a time-varying risk factor ù Yes: Chordia et al. (2000), Huberman and Halka (2001), Pastor and Stambaugh (2003), Amihud (2002), Sadka (2004) ù No: Hasbrouck and Seppi (2001)

Yale School of Management Asset Pricing Research Volatility Returns l If the CAPM assumptions do not hold exactly, idiosyncratic risk may be priced. Ex. Merton (1987). l Yes, it is positively correlated with stock return ù Linter (1965), Lehmann (1990), Xu and Malkiel (2002), Fu (2005), Goyal and Santa-Clara (2003), Ghysels, Santa-Clara, and Valkanov (2004), Jiang and Lee (2005). l Yes: it is negatively correlated with stock return ù Guo and Savickas (2004), Ang et al. (2005), Adrian and Rosenberg (2005). l No: ù Miller and Scholes (1972), Fama and Macbeth (1973), Bali et al. (2004), Baker and Wurgler (2005).

Yale School of Management Cross Sectional Returns & CAPM l An aside that has nothing to do with my paper. ù Myth: The idea that only “market risk” matters because in a diversified portfolio idiosyncratic risk is eliminated. ù Truth: CAPM holds trivially with one stock, but also holds for two or more stocks. l Diversification plays no role. (impossible with just two stocks)

Yale School of Management Conversation Between Asset Pricing and Market Microstructure? Volatility Returns Liquidity Returns Volatility Liquidity

Yale School of Management Innovation Liquidity Volatility Returns

Yale School of Management Question: Which Picture is Right? Liquidity Volatility Returns Liquidity Returns

Yale School of Management Idiosyncratic Risk ? ↔? Liquidity l Theory implies idiosyncratic risk and liquidity should be negatively correlated ù Asset Pricing: Merton (1998) ù Inventory Control: Ho and Stoll (1980) Spiegel and Subrahmanyam (1995) l Question: Are past empirical results relating returns to liquidity and idiosyncratic risk due to one, the other, or in some measure to both?

Yale School of Management Linking Liquidity & Idiosyncratic Risk l Market maker stands ready to buy or sell on demand. ù Likes capital gains. ù Has a “target” inventory, dislikes ending the day away from the target due to overnight risk. ù Can hedge out market risk. l High idiosyncratic risk stocks → lower liquidity as they leave the market maker with more risk for missing the end of day target.

Yale School of Management Questions? l Idiosyncratic risk → Cross sectional return variation. l Liquidity → Cross sectional return variation. l Idiosyncratic risk → Liquidity l To what degree is the cross sectional variation in returns due to idiosyncratic risk or liquidity? l Do some measures explain more than others?

Yale School of Management Answers: Acting Alone l EGARCH idiosyncratic risk strongly positively correlated with out of sample returns. OLS idiosyncratic risk uncorrelated with out of sample returns. l Cost and reflective liquidity measures correlated with returns. l Cost based: bid-ask spread l Reflective: volume

Yale School of Management Answers: Acting Together l Controlling for idiosyncratic risk cost based liquidity measures play a minor role in cross sectional returns l Controlling for idiosyncratic risk volume plays a strong role in cross sectional returns. l Controlling for liquidity EGARCH measured idiosyncratic risk plays a major role.

Yale School of Management Data l Monthly from CRSP. ù Nice because fewer microstructure issues than daily data. l Find idiosyncratic risk is positively related to returns. Opposite of Ang et al. finding using daily data. ù Do not explore why the monthly and daily data produce different answers. ù Good question though for my next paper!

Yale School of Management Daily vs. Monthly Data (Some clues though for the curious. ) Choose the measure working against our result Bid-Ask Spread (Gibbs) Monthly-EGARCH Idio. Risk Daily-OLS Idio. Risk Size Nyamdvol Nasdvol 0. 0015 0. 0058*** 0. 3843*** -0. 0010*** -0. 0026*** -0. 0031*** -0. 0017*** -0. 0027*** -0. 0010***

Yale School of Management Daily V. S. Monthly Data Bid-Ask bounce may contaminate the daily measure of Idiot-OLS Fama-Macbeth regression of Stock Return 0. 30*** -0. 02*** Idiot-1 -OLS Beta Size Book/Mkt -1. 59*** -0. 09 0. 44*** 0. 52*** -0. 14*** -0. 19***

Yale School of Management Effect of Bid-Ask Bounce Ask Bid Return has to go down Higher Idio. Risk Higher Spread Ask Bid Return is more likely to go down

Yale School of Management Measures l Need liquidity and idiosyncratic risk measures. ù No kidding! l For liquidity use estimates provided on Joel Hasbrouck’s web page. l For idiosyncratic risk use EGARCH model.

Yale School of Management Estimating Liquidity (Hasbrouck 2005) Hasbrouck’s Gibbs Sampler estimate of Roll’s (1984) effective Cost Pastor and Stambaugh’s Reversal Gamma

Yale School of Management Forecasting Idiosyncratic Risk OLS vs. EGARCH OLS estimates of Idiosyncratic Risk (Idio) EGARCH estimation of Idiosyncratic Risk (Eidio)

Yale School of Management OLS – EGARCH Contest l Use 60 months of data (1 to 60) to estimate each model. l Each model forecasts month 61 idiosyncratic risk. ù OLS = Idio, EGARCH = Eidio. l Use OLS model to produce an estimate of month 61 idiosyncratic risk with data from month 2 to 61. Closest forecast in average absolute value across stocks wins! ù Note: “Horse Race” strongly rigged to favor OLS.

Yale School of Management Eidio Wins in 483 out of 505 Months EGARCH OLS

Yale School of Management Relationship Between Idiosyncratic Risk, Size and Liquidity l Strong relationships across these three factors known to influence stock returns.

Yale School of Management Size↓, Illiquidity↑ as Idiosyncratic Risk Increases

Yale School of Management Size↓, Idiosyncratic Risk↑ as Illiquidity Increases

Yale School of Management Illiquidity↓, Idiosyncratic Risk↓ as Size Increases

Yale School of Management Illiquidity and Idiosyncratic Risk Table 4: Regression of Illiquidity (Gibbs) on Eidio Pooled OLS Eidio Edio 0. 006*** significant w/ or w/o 0. 009*** Size -0. 005*** Adjusted R^2 0. 27 Edio 0. 15 explains as much as Size.

Yale School of Management In Sample Test l Goal: Find high-low returns for one factor holding the other constant. l Each month stocks are simultaneous sorted into 10 portfolios based on each factor. l For a given decile on factor A calculate the return for a portfolio long in the highest factor B decile and short the lowest factor B decile.

Yale School of Management Idiosyncratic Risk vs. Liquidity (Bid -Ask) l Both appear to be important. l Liquidity does a somewhat better job of sorting portfolios than does idiosyncratic risk. ù But note the illiquid stocks have lower returns than the liquid stocks. ù Evidence (more later) that when prices rise in a stock so does trading in the stock. That is liquidity is a result of returns not a cause.

Yale School of Management In Sample Test For Eidio: Gibbs decile, For Gibbs: Eidio decile

Yale School of Management Now for the Real Tests l In sample: Who really cares? l Out of sample: Potential to make actual money! ù Everybody cares!

Yale School of Management Portfolio Construction l Based on previous month’s Eidio form 10 portfolios l Hold this portfolio for 1 month. l Repeat

Yellow: t>2. +, ++ Rank Corr. Sig. 5%, 1%. Rank CAPM Alpha++ FF-3 Alpha+ Carhart-4 Alpha++ OLS FF-3 Alpha Portfolios Sorted by Idiosyncratic Risk: Monthly Rebalance 1 Low -0. 47% -0. 42% -0. 30% -0. 01% 2 0. 02% -0. 01% -0. 05% 3 0. 04% 0. 12% 0. 01% 4 0. 06% 0. 15% 0. 19% 0. 10% 5 0. 01% -0. 01% 0. 07% 0. 05% 6 0. 12% 0. 07% 0. 08% 0. 15% 7 0. 17% 0. 11% 0. 09% 0. 14% 8 0. 08% 0. 11% 0. 15% 0. 09% 9 0. 56% 0. 60% 0. 86% -0. 03% 10 High 0. 96% 1. 06% 1. 27% -0. 77% p 10 -p 1 1. 43% 1. 49% 1. 58%

Yellow: t>2. +, ++ Rank Corr. Sig. 5%, 1%. Rank CAPM Alpha++ FF-3 Alpha+ Carhart-4 Alpha++ Portfolios Sorted by Eidio: Yearly Rebalance 1 Low -0. 17% -0. 85% -0. 89% 2 -2. 92% -0. 51% -3. 21% 3 -2. 01% -2. 48% -1. 46% 4 -1. 94% -3. 07% -0. 15% 5 -0. 17% -1. 34% -0. 20% 6 1. 26% -1. 45% -1. 87% 7 3. 79% 1. 12% 1. 57% 8 4. 98% 6. 74% 6. 96% 9 18. 57% 10. 75% 9. 87% 10 High 25. 97% 24. 78% 25. 01% p 10 -p 1 26. 14% 25. 63% 25. 90%

Yellow: t>2. +, ++ Rank Corr. Sig. 5%, 1%. Rank CAPM Alpha FF-3 Alpha Carhart-4 Alpha Panel A: Portfolios Sorted by Gibbs Sampler 1 Low 1. 83% 0. 93% -0. 81% 2 0. 07% -0. 19% 1. 51% 3 -2. 12% -2. 52% -1. 47% 4 -0. 46% -0. 34% 1. 84% 5 -2. 71% -3. 40% -1. 16% 6 -1. 22% -1. 52% 3. 25% 7 -1. 62% -4. 18% -2. 79% 8 1. 86% -2. 04% 1. 63% 9 0. 60% -2. 74% -2. 51% 10 High 7. 99% 2. 24% 1. 77% p 10 -p 1 6. 16% 1. 30% 2. 58%

Yellow: t>2. +, ++ Rank Corr. Sig. 5%, 1%. Rank CAPM Alpha- - FF-3 Alpha Carhart-4 Alpha Panel A: Sorted by $ Volume Rebalanced Annually 1 Low 9. 75% 3. 49% 2. 99% 2 6. 27% 0. 87% -0. 11% 3 6. 70% 1. 35% 1. 65% 4 6. 06% 1. 21% 1. 82% 5 3. 45% -0. 99% -0. 47% 6 2. 79% -0. 87% 0. 38% 7 3. 03% -0. 64% -0. 41% 8 2. 59% -0. 63% 0. 18% 9 1. 80% -0. 29% -0. 57% 10 High -0. 50% -0. 32% -0. 06% p 1 -p 10 10. 24% 3. 81% 3. 06%

Yale School of Management Two Way Sorts l Controls for one factor to see if the other factor continues to explain cross sectional stock returns. l Portfolios hold only stocks in the control decile and then sort on the factor of interest. l Three primary findings ù Controlling for Eidio the bid-ask spread has little explanatory power. ù Controlling for any other factor Eidio, and $ Volume continue to have significant explanatory power.

Yale School of Management Bid-Ask Spread Out of Sample Annual Alphas [t-stat][Rank Corr. Sig. ++, - -=1%; +, -=5%] Quintile Size Control Eidio Control 1 20. 86% [4. 10][++] -6. 77% [-2. 87][- -] 2 3. 83% [1. 72][++] -9. 05% [-4. 42][-] -1. 76% [-1. 21] -7. 41% [-5. 47] 4 -6. 20% [-8. 00][- -] -1. 40% [-1. 54][-] 5 -1. 11% [-1. 02][-] 34. 46% [3. 78] Equal 3. 12% [1. 84][+] 1. 96% [0. 81] 3 Most of the return is from small cap and high Eidio firms.

Yale School of Management $ Volume Out of Sample Annual Alphas [t-stat][Rank Corr. Sig. ++, - -=1%; +, -=5%] Quintile Size Control Eidio Control 1 20. 70% [4. 10][- -] 1. 45% [0. 78] [- -] 2 15. 83% [6. 17] [- -] 3. 35% [2. 30] [- -] 3 13. 32% [5. 99] [- -] 8. 91% [3. 57] [- -] 4 6. 08% [3. 70] [-] 9. 09% [3. 10] [-] 5 0. 07% [0. 05] 11. 37% [3. 95] [- -] 11. 20% [9. 51] [- -] 6. 83% [5. 15] [- -] Equal

Yale School of Management Eidio Out of Sample Monthly Alphas [t-stat][Rank Corr. Sig. ++, - -=1%; +, -=5%] Quintile Size Control Bid-Ask Control 1 2. 31% [4. 92][++] 0. 91% [2. 43] [++] 2 2. 07% [4. 97] [++] 0. 91% [2. 25] [++] 3 1. 62% [3. 26] [++] 1. 19% [4. 63] [++] 4 0. 81% [4. 59] [++] 1. 05% [4. 87] [++] 5 0. 57% [2. 18] [++] 0. 70% [4. 21] [++] Equal 1. 48% [6. 10] [++] 0. 95% [6. 78] [++]

Yale School of Management Regression Analysis l Regression risk adjusted return (alpha) on different characteristics l Controls For: ù ù ù Size, Liquidity (various measures), Idiosyncratic Risk, Lagged returns (momentum), and Dollar volume l In the table note that liquidity is not significant when Eidio is included in the regression.

Yale School of Management Fama-Macbeth Regression of Alpha on Characteristics Alpha (Bid-Ask Current Yr. ) Alpha (Bid-Ask Previous Yr. ) Eidio 0. 2496*** Bid-Ask -0. 9539*** -0. 8038*** 0. 0012 0. 1529** Lmv 0. 0016*** -0. 0002 0. 0034*** 0. 0019*** retlag 23 0. 0089** 0. 0068** 0. 0134*** 0. 0103*** retlag 46 0. 0062** 0. 0054** 0. 0129*** 0. 0107*** retlag 712 0. 0040** 0. 0029* 0. 0107*** 0. 0091*** Nyamdvol -0. 0045*** -0. 0034*** -0. 0032*** -0. 0022*** Nasdvol -0. 0025*** -0. 0018* -0. 0017*** 0. 0374 0. 0305 0. 0365 0. 0313 Adjusted R 2 0. 2055***

Yale School of Management Missing Factor? l Are the high idiosyncratic risk portfolio returns due to a missing risk factor? l If so then high Eidio portfolios should have higher volatilities. ù Missing risk factors should be correlated across the stocks and thus not diversify away.

Yale School of Management Eidio Portfolio Risk and Return (All Firms – No Liquidity Measure Restriction) Port 1 (low) Port 2 Port 3 Port 4 Port 5 mean -0. 394% 0. 086% 0. 118% 0. 377% 0. 671% s. d. 8. 926% 8. 349% 7. 627% 7. 045% 6. 398% Sharpe -0. 0442 0. 0103 0. 0154 0. 0535 0. 1049 Port 6 Port 7 Port 8 Port 9 Port 10 (high) mean 0. 753% 0. 857% 0. 825% 0. 964% 1. 077% s. d. 5. 843% 5. 191% 4. 293% 3. 681% 7. 218% Sharpe 0. 1288 0. 1651 0. 1921 0. 2617 0. 1493 Mean s. d. Sharpe Spear s. d. P-value . 458% 4. 483% 0. 1021 -0. 7455 0. 0133 Market

Yale School of Management Connor-Korajczyk Factors +++, --- = rank corr. sig. 1%, ++, -- = rank corr. sig. 5%, +, - = rank corr. sig. 10% Rank Factor × 100100 CK α +++ 5 5− 6+ -0. 47% -5. 13 1. 4 -1. 74 0. 86 0. 18 -2. 48 [3. 55] [0. 94] [1. 18] [0. 58] [0. 12] [1. 70] 0. 01% -3. 67 1. 0 -0. 01 -0. 18 0. 97 -0. 66 [3. 36] [0. 89] [0. 01] [0. 16] [0. 87] [0. 59] 0. 05% -1. 9 1. 39 0. 46 1. 77 -0. 27 0. 58 [2. 05] [1. 46] [0. 49] [1. 85] [0. 29] [0. 62] 0. 30% 0. 3 1. 47 -0. 46 2. 2 1. 63 0. 61 [0. 95] 4 4−− [0. 14] 3 3 [0. 02] 2 2−−− [1. 19] 1 (low) 1+ [0. 37] [1. 74] [0. 55] [2. 59] [1. 94] [0. 73] 0. 60% 0. 86 0. 91 0. 39 -0. 57 0. 38 -0. 51 [2. 13] [1. 26] [0. 54] [0. 80] [0. 54] [0. 72]

Yale School of Management Connor-Korajczyk Factors +++, --- = rank corr. sig. 1%, ++, -- = rank corr. sig. 5%, +, - = rank corr. sig. 10% Rank Factor × 100100 CK α +++ Avg. fac. r 6+ 0. 68% 0. 27 0. 14 0. 21 0. 43 -0. 2 -0. 76 [0. 43] [0. 22] [0. 33] [0. 66] [0. 31] [1. 18] 0. 79% 0. 21 -0. 3 0. 68 0. 61 1. 35 -0. 06 [0. 30] [0. 41] [0. 95] [0. 84] [1. 88] [0. 09] 0. 78% 2. 32 -0. 12 0. 22 -0. 37 -0. 46 0. 59 [5. 56] [0. 29] [0. 52] [0. 86] [1. 07] [1. 40] 0. 92% 0. 53 0. 26 -0. 56 -0. 31 -0. 78 0. 16 [1. 02] [0. 49] [1. 06] [0. 58] [1. 49] [0. 31] 1. 08% 0. 15 -0. 49 0. 79 -1. 84 -1. 61 -6. 36 [3. 34] 10 (high) 5− [5. 67] 9 4−− [4. 08] 8 3 [3. 47] 7 2−−− [2. 65] 6 1+ [0. 06] [0. 20] [0. 32] [0. 75] [0. 66] [2. 64] -0. 91% 2. 09% 8. 83% 6. 17% -3. 27% 2. 04%

Yale School of Management Period Specific Results l Are the results due to a particular time period? ù Results by decade. ù Results by economic environment l Expansions vs. recessions. l High vs. low volatility periods.

Yale School of Management Sub-Period Analysis Time Ranking on Idiosyncratic Risk Sub-Period 1 Low 10 High 10 -1 Jan 1962 - Dec 1970 -0. 11% 1. 10% 1. 21% [-0. 57] [1. 74] [1. 65] -0. 40% 1. 06% 1. 46% [-3. 84] [2. 76] [3. 67] -0. 42% 1. 01% 1. 43% [-3. 74] [3. 41] [3. 76] -0. 61% 0. 55% 1. 15% [-2. 31] [1. 81] [1. 78] Jan 1971 - Dec 1980 Jan 1981 - Dec 1990 Jan 1991 - Dec 2003

Yale School of Management Sub-Period Analysis Economic Environment Ranking on Idiosyncratic Risk Sub-Period 1 Low 10 High 10 -1 NBER Expansions -0. 31% 0. 77% 1. 08% [-2. 59] [1. 76] [2. 46] -0. 76% 2. 38% 3. 15% [-2. 51] [2. 25] [2. 74] -0. 78% 0. 98% 1. 76% [-2. 25] [1. 09] [2. 29] Volatile Periods -0. 39% 1. 18% 1. 57% [-3. 12] [2. 67] [3. 43] NBER Recessions Stable Periods

Yale School of Management One Factor Conclusions: Cross Sectional Returns l OLS idiosyncratic risk estimates have little explanatory. l EGARCH has significant explanatory power. Carhart alpha of 25% per year for high risk portfolios l Cost based measures have little explanatory power. l Dollar volume has significant explanatory power. Carhart alpha of 3% for low volume stocks.

Yale School of Management Multi-Factor Conclusions: Cross Sectional Returns l Liquidity and idiosyncratic Risk are negatively correlated. l Controlling for liquidity (however measured) Eidio has out of sample predictive power. l Controlling for Eidio cost based liquidity measures have little cross sectional explanatory power. l Controlling for Eidio dollar volume has substantial cross sectional explanatory power.