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
Liquidity varies across stocks.
 More liquidity is better. Duh!
People should be willing to accept lower
returns in exchange for higher liquidity
levels.
Yale School of Management
Liquidity and Returns: Theory
Liquidity
Returns
 No: Constantinides (1986), Heaton and Lucas
(1996), Duffie and Sun (1990), Grossman and
Laroque (1990), Vayanos (1998), Vayanos and
Vila (1999)
 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
 As a cross-sectional characteristic:
 Yes: Amihud and Mendelson (1988), Brennan and
Subrahamanyam (1996), Brennan, Chordia, and
Subrahamanyam (1998), Chalmers and Kadlec (1998),
Hasbrouck (2005)
 As a time-varying risk factor
 Yes: Chordia et al. (2000), Huberman and Halka
(2001), Pastor and Stambaugh (2003), Amihud (2002),
Sadka (2004)
 No: Hasbrouck and Seppi (2001)
Yale School of Management
Asset Pricing Research
Volatility
Returns
 If the CAPM assumptions do not hold exactly, idiosyncratic risk may
be priced. Ex. Merton (1987).
 Yes, it is positively correlated with stock return
 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).
 Yes: it is negatively correlated with stock return
 Guo and Savickas (2004), Ang et al. (2005), Adrian and Rosenberg
(2005).
 No:
 Miller and Scholes (1972), Fama and Macbeth (1973), Bali et al. (2004),
Baker and Wurgler (2005).
Yale School of Management
Cross Sectional Returns & CAPM
An aside that has nothing to do with my
paper.
 Myth: The idea that only “market risk” matters
because in a diversified portfolio idiosyncratic
risk is eliminated.
 Truth: CAPM holds trivially with one stock,
but also holds for two or more stocks.
 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
Volatility
Volatility
Returns
Liquidity
Liquidity
Returns
Returns
Yale School of Management
Idiosyncratic Risk ?↔? Liquidity
 Theory implies idiosyncratic risk and liquidity
should be negatively correlated
 Asset Pricing: Merton (1998)
 Inventory Control: Ho and Stoll (1980) Spiegel
and Subrahmanyam (1995)
 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
 Market maker stands ready to buy or sell on
demand.
 Likes capital gains.
 Has a “target” inventory, dislikes ending the day away
from the target due to overnight risk.
 Can hedge out market risk.
 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?
 Idiosyncratic risk → Cross sectional return
variation.
 Liquidity → Cross sectional return variation.
 Idiosyncratic risk → Liquidity
 To what degree is the cross sectional variation in
returns due to idiosyncratic risk or liquidity?
 Do some measures explain more than others?
Yale School of Management
Answers: Acting Alone
EGARCH idiosyncratic risk strongly
positively correlated with out of sample
returns. OLS idiosyncratic risk
uncorrelated with out of sample returns.
Cost and reflective liquidity measures
correlated with returns.
 Cost
based: bid-ask spread
 Reflective: volume
Yale School of Management
Answers: Acting Together
Controlling for idiosyncratic risk cost based
liquidity measures play a minor role in cross
sectional returns
Controlling for idiosyncratic risk volume
plays a strong role in cross sectional returns.
Controlling for liquidity EGARCH
measured idiosyncratic risk plays a major
role.
Yale School of Management
Data
 Monthly from CRSP.
 Nice because fewer microstructure issues than daily
data.
 Find idiosyncratic risk is positively related to
returns. Opposite of Ang et al. finding using daily
data.
 Do not explore why the monthly and daily data produce
different answers.
 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 Idio
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
Need liquidity and idiosyncratic risk
measures.
 No kidding!
For liquidity use estimates provided on Joel
Hasbrouck’s web page.
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 (bid ask spread):
rt  cqt  ut
  cov  rt , rt 1  if cov  rt , rt 1   0
c
.
otherwise
0
Pastor and Stambaugh’s Reversal Gamma
rt  a0  a1rt 1   sign  r
e
e
t 1
V
Dollar
t 1
 t .
Yale School of Management
Forecasting Idiosyncratic Risk
OLS vs. EGARCH
OLS estimates of Idiosyncratic Risk (Idio)
Rti  R ft  i  i ,MKT  Rmkt ,t  R ft   i ,SMB SMBt  i , HML HMLt   i ,t
EGARCH estimation of Idiosyncratic Risk (Eidio)
Rti  R ft   i  i , MKT  RMKT ,t  R ft   i ,SMB SMBt  i , HML HMLt   i ,t
 i ,t  hi ,t  vt
p
q
m 1
m 1
ln hi ,t  i    i ,m ln hi ,t  m  i ,n  vt n  E vt n  i vt n 
Yale School of Management
OLS – EGARCH Contest
 Use 60 months of data (1 to 60) to estimate each
model.
 Each model forecasts month 61 idiosyncratic risk.
 OLS = Idio, EGARCH = Eidio.
 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!
 Note: “Horse Race” strongly rigged to favor OLS.
OLS
Idio
Date
Eidio
Jan-02
Jan-00
Jan-98
Jan-96
Jan-94
Jan-92
Jan-90
Jan-88
Jan-86
Jan-84
Jan-82
Jan-80
Jan-78
Jan-76
Jan-74
Jan-72
Jan-70
Jan-68
Jan-66
Jan-64
Jan-62
Forecast Error
Yale School of Management
Eidio Wins in 483 out of 505 Months
0.35
0.3
EGARCH
0.25
0.2
0.15
0.1
0.05
0
Yale School of Management
Relationship Between Idiosyncratic
Risk, Size and Liquidity
Strong relationships across these three
factors known to influence stock returns.
Yale School of Management
Size↓, Illiquidity↑ as Idiosyncratic
Risk Increases
Portfolios Sorted by Idiosyncratic Risk
25
20
15
Illiquidity (Gibbs*10^3)
10
Size (log(MktCap))
5
0
P1
2
3
4
5
6
7
8
9
P10
Yale School of Management
Size↓, Idiosyncratic Risk↑ as
Illiquidity Increases
Portfolio Sorted by Illiquidity (Gibbs)
50
40
30
20
10
0
P1
2
3
4
Size
5
6
7
8
9
P10
Eidio*10^4
Yale School of Management
Illiquidity↓, Idiosyncratic Risk↓ as
Size Increases
Portfolios Sorted by Size
0.08
0.07
0.06
0.05
0.04
0.03
Eidio
Gibbs
0.02
0.01
0
P1
2
3
4
5
6
7
8
9
P10
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
Size
-0.005***
Adjusted
R^2
0.27 Edio
0.15
explains
as much
as Size.
Yale School of Management
In Sample Test
Goal: Find high-low returns for one factor
holding the other constant.
Each month stocks are simultaneous sorted
into 10 portfolios based on each factor.
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)
Both appear to be important.
Liquidity does a somewhat better job of
sorting portfolios than does idiosyncratic
risk.
 But note the illiquid stocks have lower returns
than the liquid stocks.
 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
Portfolios Formed on Gibbs and Eidio
4.00%
3.00%
2.00%
1.00%
0.00%
-1.00%
-2.00%
-3.00%
-4.00%
-5.00%
-6.00%
Eidio10-Eidio1
Gibbs10-Gibbs1
P1
2
3
4
5
6
7
8
9
P10
For Eidio: Gibbs decile, For Gibbs: Eidio decile
Yale School of Management
Now for the Real Tests
In sample: Who really cares?
Out of sample: Potential to make actual
money!
 Everybody cares!
Yale School of Management
Portfolio Construction
Based on previous month’s Eidio form 10
portfolios
Hold this portfolio for 1 month.
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
1Low
-0.47%
-0.42%
-0.30%
-0.01%
2
0.02%
-0.02%
-0.01%
-0.05%
3
0.04%
0.14%
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%
10High
0.96%
1.06%
1.27%
-0.77%
p10-p1
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
1Low
-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%
10High
25.97%
24.78%
25.01%
p10-p1
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
1Low
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%
10High
7.99%
2.24%
1.77%
p10-p1
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
1Low
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%
10High
-0.50%
-0.32%
-0.06%
p1-p10
10.24%
3.81%
3.06%
Yale School of Management
Two Way Sorts
 Controls for one factor to see if the other factor
continues to explain cross sectional stock returns.
 Portfolios hold only stocks in the control decile
and then sort on the factor of interest.
 Three primary findings
 Controlling for Eidio the bid-ask spread has little
explanatory power.
 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
 Regression risk adjusted return (alpha) on
different characteristics
 Controls For:





Size,
Liquidity (various measures),
Idiosyncratic Risk,
Lagged returns (momentum), and
Dollar volume
 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***
retlag23
0.0089**
0.0068**
0.0134***
0.0103***
retlag46
0.0062**
0.0054**
0.0129***
0.0107***
retlag712
0.0040**
0.0029*
0.0107***
0.0091***
Nyamdvol
-0.0045***
-0.0034***
-0.0032***
-0.0022***
Nasdvol
-0.0025***
-0.0018***
-0.0018*
-0.0017***
0.0374
0.0305
0.0365
0.0313
Adjusted R2
0.2055***
Yale School of Management
Missing Factor?
Are the high idiosyncratic risk portfolio
returns due to a missing risk factor?
If so then high Eidio portfolios should have
higher volatilities.
 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)
Port1 (low)
Port2
Port3
Port4
Port5
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
Port6
Port7
Port8
Port9
Port10 (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
1 (low)
2
3
4
5
CK α
Factor ×100100
+++
1+
2−−−
3
4−−
5−
6+
-0.47%
-5.13
1.4
-1.74
0.86
0.18
-2.48
[1.19]
[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
[0.02]
[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
[0.14]
[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]
[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]
[1.23]
[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
6
7
8
9
10 (high)
Avg. fac. r
CK α
Factor ×100100
+++
1+
2−−−
3
4−−
5−
6+
0.68%
0.27
0.14
0.21
0.43
-0.2
-0.76
[2.65]
[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
[3.47]
[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
[4.08]
[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
[5.67]
[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]
[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
Are the results due to a particular time
period?
 Results by decade.
 Results by economic environment
 Expansions
vs. recessions.
 High vs. low volatility periods.
Yale School of Management
Sub-Period Analysis
Time
Ranking on Idiosyncratic Risk
Sub-Period
1 Low
10High
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
10High
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]
-0.39%
1.18%
1.57%
[-3.12]
[2.67]
[3.43]
NBER Recessions
Stable Periods
Volatile Periods
Yale School of Management
One Factor Conclusions:
Cross Sectional Returns
 OLS idiosyncratic risk estimates have little
explanatory.
 EGARCH has significant explanatory power.
Carhart alpha of 25% per year for high risk
portfolios
 Cost based measures have little explanatory
power.
 Dollar volume has significant explanatory power.
Carhart alpha of 3% for low volume stocks.
Yale School of Management
Multi-Factor Conclusions:
Cross Sectional Returns
 Liquidity and idiosyncratic Risk are negatively
correlated.
 Controlling for liquidity (however measured)
Eidio has out of sample predictive power.
 Controlling for Eidio cost based liquidity
measures have little cross sectional explanatory
power.
 Controlling for Eidio dollar volume has substantial
cross sectional explanatory power.