The Joint Pricing of Volatility and Liquidity
Incomplete
Claudia E. Moiseyand Je¤rey R. Russellz
September 22, 2011
Abstract
The market microstructure implications of transaction costs and price discovery are rarely
incorporated into the asset pricing literature. Our work lies at the intersection of these two
important …elds. Combining recent advances in continuous-time econometrics with no-arbitrage
arguments, we extract novel proxies for market volatility and market illiquidity from a single
time-series of high-frequency SPDRs transaction prices (SPDRs represent ownership of a trust
invested in the S&P500 index). We test the pricing ability of our proxies in the cross-section
of the monthly 25 size- and value-sorted Fama-French portfolios and 10 industry portfolios.
For our sample, we …nd the pricing performance of a model with market returns, (unexpected)
market volatility, and (unexpected) market illiquidity to be similar to that of the classical FamaFrench 3-factor model, when pricing these portfolios. While illiquidity and volatility may still
be individually negatively priced, as is shown in our data, the factor loadings associated with
volatility shocks provide a more accurate assessment of risk in a joint speci…cation. When
interpreting shocks to illiquidity and shocks to volatility as proxies for a more fundamental
distress factor, this result is suggestive of the superior robustness of the latter.
JEL Classi…cation: G12
Keywords: Market volatility, market liquidity, cross-section of stock returns, SPDRs
We thank Yakov Amihud, Torben Andersen, Federico Bandi, Rob Engle, Mark Grinblatt, John Heaton, Renè
Garcia, Petter Kolm, Dennis Kristensen, Hedi Lopes, Robert Merton, Nick Polson, Bernard Salanié, Roy Smith,
George Tauchen, Robert Whitelaw and seminar participants at Bank of Japan, Case Western Reserve, UC Santa
Barbara, Chicago GSB, Chicago FED, Columbia Economics, Duke, Edhec Business School, NYU Courant Institute,
the Federal Reserve Board, Kent State, McGill, George Mason, Texas A&M, the Latin American Meetings of the
Econometric Society in Mexico City, the Society for Financial Econometrics Inaugural Conference, the 2009 European
Finance Association Meeting, and the NYU Mini Liquidity Conference for helpful comments. We are grateful to Gene
Fama and Ken French for providing their factors and return data.
y
Weatherhead School of Management, Case Western Reserve University.
z
Booth School of Business, University of Chicago.
1
"I argue that asset pricing ignores the central fact that asset prices evolve in markets. Markets
provide liquidity and price discovery, and I argue that asset pricing models need to be recast in
broader terms to incorporate the transaction costs of liquidity and the risks of price discovery."
Maureen O’Hara, Presidential Address: Liquidity and Price Discovery, 2003
1
Introduction
Liquidity enters into asset price formation. However, it is intertemporal in nature and linked to
the trading mechanism, not to the price discovery (O’Hara, 2003). Meanwhile, price discovery
consists of incorporating new information, related to …rm fundamentals, into asset prices. While
the asset pricing literature focuses on the behavior of asset prices, it does not incorporate the
market microstructure implications. Our work lies at the intersection of these two important
…elds. Combining recent advances in continuous-time econometrics with no-arbitrage arguments,
we extract novel proxies for market volatility (the volatility of the fundamental return process) and
market illiquidity, from a single time-series of high-frequency SPDRs transaction prices.1
Two successful strands of the recent asset pricing literature have emphasized the importance
of market volatility (e.g., Ang et al., 2006, Adrian and Rosenberg, 2008, and Moise, 2011) and
market liquidity (e.g., Pástor and Stambaugh, 2003, and Sadka, 2003, Acharya and Pedersen,
2005) as systematic risk factors priced in the cross-section of stock returns. The intuition is simple.
Aggregate illiquidity and aggregate volatility are high in bad states of the world. Assets whose
returns are more positively correlated with innovations in these risk factors will provide a hedge,
thereby requiring relatively lower average returns.2
Even though market volatility and market liquidity are the result of di¤erent economic phenomena (the former being related with changes in fundamental asset values, while the latter being
the outcome of aggregate trading frictions), these factors are positively correlated (in terms of innovations, or unexpected components) and relatively higher in less favorable states of the world.
Should one take the view that volatility and illiquidity are fundamental pricing factors, as done by
the recent literature, then it is meaningful to ask whether their pricing ability is preserved, when
jointly considered. If one were to believe that they are proxies for a more fundamental factor (or
factors) varying with the state of the economy, then again it would seem relevant to ask whether
their individual explanatory power is subsumed in a model which allows for the other proxy to be
1
SPDRs represent ownership of a trust invested in the S&P500 index.
The cross-sectional relation between expected stock returns and idiosyncratic, rather than systematic, liquidity
has been investigated in numerous papers, including Amihud and Mendelson (1986), Brennan and Subrahmanyam
(1996), Datar et al. (1998), and Elaswarapu (1997). Amihud (2002), Jones (2002), and Fujimoto (2003), among
others, study the time-series properties of market excess returns and market liquidity. The e¤ect of idiosyncratic
asymmetric information in cross-sectional asset pricing is discussed in Easley et al. (2002). Amihud et al. (2006) and
Cochrane (2005) provide insightful discussions of the current state of the literature on liquidity and asset prices.
2
2
present. This is the focus of our paper.
We make two contributions to the literature. The …rst contribution is methodological. By
combining recent advances in the econometrics of high-frequency data with classical no-arbitrage
arguments, we extract novel proxies for market variance and market illiquidity from a single timeseries of high-frequency SPDRs transaction prices. The joint evaluation of the cross-sectional pricing
ability of (unexpected) market volatility and (unexpected) market illiquidity represents the second
contribution of the present paper.
Because SPDRs represent ownership of a trust invested in the S&P500 index, changes in SPDRs
fundamental value re‡ect changes in the index’s fundamental value. In addition, since SPDRs can
be redeemed for the underlying portfolio of S&P500 stocks (or created in exchange for the underlying
portfolio of assets), deviations of SPDRs transaction prices from fundamental values signal pervasive
market frictions rendering arbitrages harder to implement. We provide a method to separate the
volatility of SPDRs unobserved fundamental values (used as a proxy for market volatility) from
the volatility of the di¤erence between SPDRs transaction prices and their fundamental values
(which is a proxy for market illiquidity). The method solely requires the computation of averages
of high-frequency SPDRs transaction prices sampled at di¤erent, "optimally-selected," frequencies.
Our factors’construction hinges on classical work on market microstructure theory (O’Hara, 1995),
which says that meaningful updates to fundamental prices, which re‡ect expectation of future cash
‡ows, should be less frequent than meaningful changes to transaction prices, which are related to
the trading process.
We …nd that unexpected illiquidity and unexpected volatility are negatively correlated with
market returns, and with the returns on the size decile portfolios. Consider the size decile portfolios.
For both illiquidity and volatility, the covariances of portfolio returns with these risk factors (and the
related factor loadings) increase monotonically (while remaining negative) when moving from small
cap to large cap stocks. Hence, large cap stocks are less (negatively) exposed to both illiquidity
and volatility risks, thereby requiring lower expected returns, as empirically found in practice. We
stress that this result is not a by-product of our measure. It would hold true, as we show in the
paper, should alternative, existing measures in the literature be employed.
Importantly, while the factor loadings for both illiquidity and volatility generally align with the
average excess returns on the size (and book-to-market portfolios), thereby leading to signi…cant
pricing in the cross-section, the pricing ability varies across measures. We …nd that both of our
derived proxies are negatively individually priced in the cross-section of monthly 25 size- and
value-sorted Fama-French portfolios and 10 industry-sorted portfolios. In joint considerations,
the factor loadings associated with volatility shocks provide a more accurate assessment of risk.
When interpreting shocks to illiquidity and shocks to volatility as proxies for a more fundamental
distress factor, this result is suggestive of the superior robustness of the latter. Further, we …nd
3
the performance of a 3-factor model with market returns, (unexpected) market volatility, and
(unexpected) market illiquidity to be similar to the performance of the classical Fama-French 3factor model, when pricing the Fama-French size- and book-to-market-sorted portfolios. We stress
that such a robustness is not a by-product of the way liquidity is measured in this paper, since
results hold under alternative liquidity measures.
The remainder of the paper is structured as follows. In Section 2 we discuss the de…ning features
of SPDRs. Section 3 proposes a price formation mechanism for SPDRs transaction process. This
mechanism justi…es our identi…cation procedure for market volatility and market illiquidity. Section
4 expands on the logic of our illiquidity proxy. Section 5 evaluates the empirical properties of
the factors. Particular emphasis is placed on the relation between our derived illiquidity proxy
and known macro liquidity events. The cross-sectional pricing of illiquidity and volatility risk is
discussed in Section 6. Section 7 concludes. Technical details are in the Appendix.
2
SPDRs
Standard & Poor’s depository receipts, SPDRs (or spiders), represent shares in a trust which owns
stocks in the same proportion as that found in the S&P500 index. They trade like a stock (with the
ticker symbol SPY on the AMEX) at approximately one-tenth of the level of the S&P500 index,
and are used by large institutions and traders either as bets on the overall direction of the market
or as a means of passive management. The advantage of using SPDRs versus, for instance, S&P500
futures (SPX), is that SPDRs can be created or redeemed for the underlying basket of securities,
which is a crucial feature in our risk factors identi…cation.
SPDRs are exchange traded funds (ETFs).3 They can be redeemed for the underlying portfolio
of assets at the end of the trading day. Equivalently, investors have the right to obtain newly
issued SPDRs shares from the fund company in exchange for a basket of securities that mirrors the
SPDRs’ portfolio. This implies that SPDRs, like other ETFs, must trade at a value that is near
net asset value (NAV). If they traded above their NAV, arbitrageurs would purchase the basket of
underlying securities for a lower price and force the fund company to issue new shares. Conversely,
if they traded below their NAV, arbitrageurs would buy shares and redeem them for the underlying
portfolio of securities (see Cherkes et al., 2006, for further discussions and comparisons between
ETFs and closed-end funds). The NAV is computed at market close. During the day, an estimated
value of the portfolio called Indicative Optimized Portfolio Value (IOPV) is posted. The IOPV
is provided every 15 seconds using the most recent transaction price of each component of the
3
A growing academic literature focuses on ETFs. Among other issues, the existing work studies the dynamics of
price deviations from net asset value (Engle and Sarkar, 2002), compares the return from holding ETFs (speci…cally,
SPIDERS) to the return from holding the underlying index (Elton et al., 2002), analyzes the tax implications of
ETFs (Poterba and Shoven, 2002), investigates price discovery (Hasbrouck, 2002) and competition (Boehmer and
Boehmer, 2002) in the ETF market.
4
portfolio.
Further, SPDRs’values will not deviate much from NAVs during the day either, since the future
convergence of prices would open up the possibility for simple, immediate investment opportunities.
Assume trading prices are higher than NAVs. An arbitrageur could sell SPDRs short,4 buy the
underlying basket of security, wait for price convergence, and unwind the position for an initial
pro…t.
As Elton et al. (2002) and Engle and Sarkar (2002) point out, the process of share deletion/creation acts as an extremely e¤ective mechanism in keeping prices close to NAV and assuring
that potential di¤erences disappear quickly. Conversely, since arbitrages require acquisition of the
underlying basket of securities, the extent of deviations from NAV should signal pervasive market
frictions rendering arbitrages harder to implement.
Importantly, rather than focusing on deviations of trade prices from NAV or IOPV, we measure
deviations from the unobserved "fundamental values" of the basket of securities. Fundamental
values are only approximated by the NAVs at close and by the IOPVs during the day. This property
is important as emphasized, for example, by Engle and Sarkar (2002). The NAV is evaluated at
the closing transaction price of each of the assets. On the one hand, each closing transaction price
could be higher or lower than the individual fundamental value. On the other hand, the closing
transaction could occur earlier in the day, especially for less frequently-traded stocks.5 Similarly,
the IOPVs may be stale in that they are posted at equispaced intervals of 15 seconds and, hence,
deviate from fundamental values. More generally, it is well-known from market microstructure
theory that transaction prices (and, as said, NAVs are computed at transaction values) di¤er
from fundamental values. The size of these deviations will, again, depend on liquidity.6 Hence,
aggregate liquidity will a¤ect both the size of the deviations between SPDRs prices and NAVs (given
the previous arbitrage arguments), and the size of the deviations between the transaction prices
of the underlying basket of securities (which determine the NAVs) and unobserved fundamental
values. Lower aggregate liquidity may therefore be expected to lead to larger overall deviations.
In light of these arguments, the next section will discuss a procedure to identify the size of the
deviations between SPDRs transaction prices and the unobserved value of the underlying portfolio
of securities (and, of course, SPDRs shares). In what follows, we call the unobserved reference
value of the underlying portfolio of securities (and SPDRs shares) the "fundamental value."
We employ high-frequency transaction prices on SPDRs obtained from the Trade and Quote
4
Contrary to some individual stocks, SPDRs can be short-sold on a down tick. SPDRs are highly liquid and thus,
there are plenty of buyers willing to enter into a long position, this way ensuring that the price will rarely be driven
to unjusti…ably low levels.
5
In addition, SPDRs continue to trade 15 minutes after the NYSE closes. This is another source of error for the
posted NAV.
6
At the stock level and, by aggregation, at the index level, these deviations might also depend on asymmetric
information (see, e.g., the discussion in Stoll, 2000, and the references therein).
5
(TAQ) database in CRSP for the period February 1993 - December 2008. We use the entire
consolidated market, which comprises the following exchanges: AMEX, CBOE, NASDAQ, NYSE,
Boston, Cincinnati, Midwest, Paci…c, and Philadelphia. The cross-sectional asset pricing tests
employ monthly return data on the 25 size- and value-sorted Fama-French portfolios over the same
period. For robustness checks, we also include 10 industry portfolios among our test assets.
We now formalize our assumed high-frequency SPDRs price formation mechanism and a method
to separate the volatility of SPDRs fundamental values from the volatility of SPDRs deviations from
fundamental values, i.e., our factor proxies.
3
Extracting the volatility and illiquidity proxies
Building on the intuition laid out above, we express the logarithmic SPDRs transaction price
prevailing at the end of a trading day t of length h as
peth = pth +
t = 1; 2; :::; T;
th
where p is the unobservable fundamental value and
(1)
is an equally unobservable pricing error. As
said, higher market liquidity should lead to smaller price deviations
.
Now divide each trading day into M (equispaced, for notational simplicity) sub-periods. The
j-th intra-daily continuously-compounded return between day t
where
rej;t = pe(t
1)h+j
pe(t
1 and day t is de…ned as
j = 1; 2; :::; M;
1)h+(j 1)
(2)
= h=M is the interval over which the intra-daily returns are computed. Thus, similarly to
the observed price process, the observed return process comprises a fundamental return component,
rj;t = p(t
1)h+j
(t 1)h+(j 1)
p(t
1)h+(j 1)
, as well as a price deviation component,
j;t
=
(t 1)h+j
, i.e.,
rej;t = rj;t +
t = 1; 2; :::; T;
j;t
j = 1; 2; :::; M:
(3)
As in much recent work in high-frequency econometrics (see, e.g., the discussions in the review
papers of Bandi and Russell, 2008b, and Barndor¤-Nielsen and Shephard, 2007), we model the fundamental return process as a stochastic volatility martingale di¤erence sequence driven by Brownian
shocks, i.e.,
rj;t =
Z
(t 1)h+j
s dWs ;
(4)
(t 1)h+(j 1)
where
s
is a càdlàg spot volatility process bounded away from zero. The price deviations
are
assumed to be independent of the fundamental prices and i.i.d. with a bounded fourth moment.
6
Since the fundamental returns are uncorrelated, the structure of the return deviations carries over
to the observed high-frequency returns.
Our objects of interest are the variance of the unobserved daily fundamental returns rt =
pth
p(t
1)h
and the variance of the unobserved intra-daily price deviations
t
from equation (3).
The former will proxy for daily market-wide variance, while the latter will proxy for daily aggregate
illiquidity.
As detailed in the Appendix, the fundamental return process r has a smaller stochastic order
compared to the deviation returns. Price discreteness, as well as the existence of di¤erent prices
for buyers and sellers, for instance, justify this assumption (Bandi and Russell, 2006). Thus,
the friction returns
do not vanish at high sampling frequencies. Further, the magnitude of the
fundamental price changes decreases with the sampling interval. This assumption, which is standard
in continuous-time asset pricing, represents slow accumulation and processing of information leading
to negligible fundamental price updates over small time intervals. Exploiting this di¤erent in the
stochastic orders of
and r, one obtains
PM
2
ej;t
j=1 r
p
! Et ( 2 ) = 2Et ( 2 ):
(5)
M !1
M
The result in equation (5) hinges on the fact that the deviation process dominates the fundamental
return process at high frequencies.
Rt
We now turn to the variance of the fundamental return process, i.e., Vt = t 1 2s ds. In the
P
2 (realized variance) esej;t
absence of deviations, the sum of the squared intra-daily returns M
j=1 r
timates Vt consistently as M ! 1 (see, e.g., Andersen et al., 2003, and Barndor¤-Nielsen and
Shephard, 2002). The presence of deviations leads to an important bias-variance trade-o¤. High
sampling frequencies may determine substantial noise accumulation and biased estimates. Low
sampling frequencies may lead to (fairly) unbiased but highly volatile estimates. Therefore, we
choose every day the optimal sampling frequency
(or, equivalently, the optimal number of ob-
servations M ) for every horizon of interest. For each day in our sample, given the assumed price
formation mechanism, the (approximate) optimal number of observations Mt is de…ned as
bt
Q
bt
Mt
b t is equal to
where Q
f
M
3
PM
f
4 ,
ej;t
j=1 r
!1=3
;
(6)
the quarticity estimator of Barndor¤-Nielsen and Shephard
(2002) with returns sampled every 15 minutes, and b t is equal to
PM
2
ej;t
j=1 r
M
2
, the friction-in-
returns second moment estimator raised to the second power. The optimal number of observations
Mt can be interpreted as a signal-to-noise ratio. The higher the signal coming from the underlying
R
4 ds) relative to the size of the frictions (as represented
b t estimates t
fundamental price process (Q
t 1
s
7
by b t ), the higher the optimal number of high-frequency observations needed for realized variance
estimation. Please note that, by construction, Mt will be less than the total number of intra-daily
observations, Mt :
For each day in our sample, we estimate
PMt
2
ej;t
j=1 r
M
and
quency at which intra-daily returns are observed and
Mt
PMt
2 ;
ej;t
j=1 r
where
Mt
is the highest fre-
is an appropriately-chosen "optimal"
frequency. Note that Mt and Mt have a subscript t to make their dependence on time fully apparent. Under realistic assumptions on the unobservable components rj;t and
PMt
j;t ,
it can be shown that
2
ej;t
j=1 r
Mt
estimates consistently the variance of the intra-daily deviations (i.e., E( 2j;t )) as Mt ! 1
PMt 2
(i.e., for a large number of intra-daily observations). The quantity j=1
rej;t (realized variance)
will, in general, not identify the variance of the unobserved fundamental returns consistently. Appropriate selection of Mt (as described in the Appendix) may, however, lead to optimization of
the estimator’s mean-squared deviations from the object of interest (i.e., "fundamental return variance"). The chosen number of intra-daily returns Mt will, of course, be larger, the smaller the
price deviations
7
j;t :
While we refer the reader to the Apendix for technical details, here we …nd it important to
brie‡y emphasize the economic intuition underlying the construction of our proxies. Speci…cally,
the independence assumption between price deviations and fundamental prices hinges on classical
market microstructure theory, which states that meaningful updates to fundamental prices should
be independent of meaningful changes in transaction prices (e.g., O’Hara, 1995). That is because
the former depend on the way informed agents form expectations about future cash ‡ows and hence
hinge on potentially infrequent updates to the private information set. The latter depend on the
trading process.
Further, because the uninformed agents learn from the order ‡ow, non-negligible (discrete)
changes in transaction prices may occur regardless of the transaction frequency, i.e., even if trades
occur very close in time. This implies that the observed intra-daily returns rej;t are dominated by the
deviation component
j;t
when the trades are frequent. Conversely, they are largely dominated by
the fundamental return component rj;t when the return sampling is performed at low frequencies.
This simple intuition clari…es why sample second moments of observed returns sampled at the
highest available frequency
Mt , such as
PMt
2
ej;t
j=1 r
Mt
;identify the second moment of the return deviations
: This is due to fundamental returns that wash out at high frequency. By the same type of
PMt 2
reasoning, j=1
rej;t will give us information about the (integrated, over the trading day) volatility
of the fundamental return process if observed returns are sampled at appropiately-chosen, lower
frequencies
Mt
: Importantly, the selection of
Mt
may be conducted "optimally" based, for example,
on a mean-squared error criterion. The Appendix imposes statistical assumptions on rj;t and
7
The Appendix imposes statistical assumptions on rj;t and j;t which justify the estimators. It provides details
on the construction of Mt . It also discusses alternative approaches and potential extensions.
8
which make this intuition rigorous while fully justifying the adopted estimators. Alternative
j;t
approaches and potential extensions are discussed.
In sum, for every day in our sample we use (potentially standardized) sums of observable intradaily returns sampled at optimal (time-varying) frequencies to identify the variance components of
the returns’ unobservable components. These daily measures are subsequently aggregated to the
monthly level, as we discuss below.
PMt 2
Importantly, since j=1
rej;t is computed over a 6-hour period (from 10 a.m. to 4 p.m.), in
order to convert it into a genuinely daily measure, wePcorrect it for lack of overnight returns. We
T
1
PMt 2
Rt2
b
ej;t are the
do so by multiplying it by the constant factor = T1 Pt=1
T
j=1 r
b ; where Rt and Vt =
t=1
T
Vt
daily SPDRs returns and respectively, the (6-hour) measures over day t. This correction ensures
that the average of the corrected variance estimates coincides with the variance of the daily returns
(see, e.g., Fleming et al., 2001, 2003).
3.1
Aggregation
Since estimatesof risk factors loadings in asset pricing tests are known to be less noisy at monthly
level, we evaluate their pricing ability at this frequency. To this end, we average the deviation
variances across days in a speci…c month k to obtain a monthly measure of illiquidity:
ck = E
b k ( 2) =
IL
#Days
X
1
#Days
t=1
PM
2
ej;t;k
j=1 r
M
:
(7)
For our measure of illiquidity, we correct for the structural breaks induced by changes in tick size
in June 1997, when both Nasdaq and NYSE reduced the tick size from 1/8 to 1/16 of a dollar, and
in 2000-2001, when exchanges further moved to decimalization. We do this by using the residuals
from a regression on dummies for these break dates. That is our …nal ILk factor.
As for the fundamental return variance, we sum the daily realized variances across days in a
month to, again, obtain the corresponding monthly values:
Vbk =
#Days
X
t=1
Mt
X
j=1
2
rej;t;k
.
(8)
Finally, we consider the expected and the unexpected volatility and liquidity factors. In the price
deviation case, we have
ILk =
where ILk
1
IL
0
+
IL
1 ILk 1
is the expected illiquidity and U ILk =
In the fundamental return case, we have
9
IL
k
+
IL
k ;
is the unexpected illiquidity.
(9)
Vk =
where Vk
1
V
0
V
1 Vk 1
+
is the expected volatility and U Vk =
V
k
+
V
k;
(10)
is the unexpected volatility.
To summarize, we hypothesize that (i ) unexpected SPDRs fundamental price variance re‡ects
innovations in market variance and (ii ) unexpected SPDRs friction variance re‡ects changes in aggregate illiquidity. As discussed, the former hypothesis is justi…ed by the basket nature of SPDRs.
The latter hypothesis relies on no-arbitrage arguments. Put it di¤erently, even though SPDRs trade
like any other stock, we expect innovations in SPDRs frictions to be a much less noisy measure of
innovations in overall market liquidity than innovations in any individual stocks’frictions. As generally argued in the industry (see, e.g., Gastineau, 2001, and Spence, 2002), the liquidity properties
of an ETF should re‡ect the liquidity properties of the underlying portfolio of securities.8
4
More on the logic of the liquidity measure
Using the notation from the previous section, the price deviation (with respect to net asset value)
for trading a share of stock s in the S&P500 basket at time j on day t can be written as
where
s;j
pes;j
ps;j =
s;j
= (1
2Isell )
s;j ;
(11)
0 is the cost of buying or selling, the subscript j is short for (t
a sell indicator taking on the value 0 for a buy order and 1 for a sell
order.9
1)h + j , and Isell is
The price deviation for
trading the S&P500 portfolio at time j is a (value-weighted) average of price deviations expressed
as
500
X
ws
s;j
= (1
2Isell )
s=1
where 0 > ws > 1 and
P500
s=1 ws
500
X
ws
s;j ;
(12)
s=1
= 1: By the nature of SPDRs, as discussed earlier, the price
di¤erence for trading a SPDRs share at time j, i.e., SP DR;j , should approximately be equal to
P500
10 Hence, the variance of
SP DR , our object of interest, represents the variance of the
s=1 ws s;j .
portfolio’s price deviations (associated with buy or sell orders) about the portfolio’s net asset value.
This variance is small if, of course, individual stocks trade near neat asset value. More explicitly,
8
On Yahoo Finance, for example, we read: "Some investors appear to believe that the liquidity of an ETF is
dependent on the fund’s average trading volume, or the number of shares traded per day. However, this is not the
case. Rather, a better measure of ETF liquidity is the liquidity of the underlying stocks in the index."
9
As is customary, we assume that buy orders occur at prices above fundamental values whereas sell orders occur
at prices below fundamental values (see, e.g., Roll, 1984).
10
Here, we are, of course, assuming that SPDRs prices roughly coincide with NAVs. Deviations of SPDRs prices
from NAVs would introduce an additional liquidity-related contamination whose contribution is, as said, measurable
given our approach.
10
E(
2
SP DR )
0
= E @(1
2Isell )2
500
X
ws
s=1
s
!2 1
0
A = E@
500
X
ws
s=1
if the sell indicator and the individual price deviations
s
!2 1
A=
500
X
ws wu E(j
s;u=1
s j j u j);
(13)
are independent. Should the price
deviations be cross-sectionally independent and identically distributed, then
E(
2
SP DR )
= (E(j
2
s j)) ;
(14)
and our assumed measure would capture the common (across stocks) squared expected absolute
price deviation from net asset value. Should the price deviations be cross-sectionally independent
but not identically distributed, then
E(
2
SP DR )
500
X
=
ws E(j
s=1
!2
s j)
;
(15)
and the measure would be a squared weighted average of expected absolute price deviations from
net asset value. Finally, should the
s be cross-sectionally dependent (as likely the case in the
presence of aggregate liquidity shocks) and not identically distributed, then
E(
2
SP DR )
=
500
X
s=1
ws E(j
!2
s j)
+
500
X
s;u=1
ws wu Cov(j
s j j u j):
(16)
Generally speaking, the larger the individual price deviations from net asset value, the larger the
measure. Importantly, when the price deviations are cross-sectionally correlated, as suggested
by the existence of commonality in illiquidity, the pair-wise correlations ought to be taken into
account. Simply averaging value-weighted …rm-speci…c estimates might drastically underestimate
the variance of the aggregate price deviations (and, therefore, the extent of market frictions) if
these covariances are on average positive. This observation, in turn, suggests that …rst applying
the methods to individual stocks (rather than to an index) and subsequently value-weighing the
…rm-speci…c estimates might lead to a misleading measure if the ability to buy and sell an aggregate
portfolio near fundamental values is the object of interest, as in our case. In this sense, using an
index (and straightforward no-arbitrage reasoning) provides a meaningful solution to the empirical
issues that would be posed by the (hardly tractable) computation of moments and cross-moments
of individual price deviations for a broad array of stocks using high-frequency data.
11
5
A look at the factors
5.1
Market volatility
The literature has been extensively employing the 5- and 20-minute frequencies to estimate realized
market variance. These frequencies are usually employed to reduce the impact of market microstructure noise ( , in our notation) on the estimates of the fundamental price variance. Choosing …xed
intervals is, of course, ad-hoc and generally sub-optimal from a …nite sample mean-squared-error
standpoint (see Appendix). Not surprisingly, we …nd that the optimally-sampled realized estimates
appear better behaved than the estimates obtained by sampling at …xed intervals11 . As said, we
focus on optimally-sampled realized variances in what follows.
For comparison, Fig. 1 Panel A) plots the annualized values of our monthly realized variance
estimate constructed using sums of daily realized variance estimates, as per equation (8) above, and
the annualized implied volatility (VIX) from the options market, downloaded from CBOE. VIX is
known as the fear index in the …nancial markets. Both variances spike during well-known …nancial
crises like the 1997 Asian crisis, the 1998 LTCM crisis and Russian debt default, the Sept. 2001
attack, the recent 2007-2009 …nancial market turmoil, etc.
5.2
Market liquidity
Fig. 1 Panel B) plots our monthly illiquidity estimates: IL from equation (7), before corrrecting
it for the changes in tick size. The graph suggests a general decline in illiquidity with spikes corresponding again to known (il-)liquidity events, such as the Asian crisis (October 1997), the LTCM
collapse and Russian debt default (October/November 1998), the 9/11 terrorist attack, the recent
…nancial crisis, and so on. In agreement with what is expected from a proper liquidity measure,
the documented decline mirrors well-known downward trends in the average bid-ask spreads across
stocks. Also, the graph re‡ects the increase in liquidity associated with changes in tick size which
were introduced in 1997 and 2000.
For comparison, we also employ in our tests established liquidity factors. One is the innovations
in the Pástor and Stambaugh liquidity measure.12 Pástor and Stambaugh’s measure is a price
reversal measure. The idea underlying it is that less liquid stocks should have larger price reversals
following signed order ‡ow than more liquid stocks. For stock i in month k, liquidity is de…ned as
the least-squares
estimate from the regression
e
ri;t+1;k
=
i;t
+
i;k ri;t;k
+
e
i;k sign(ri;t;k )vi;t;k
11
+ "i;t+1;k ;
(17)
In the interest of brevity, results using the 5- and 20-min frequencies are not reported here; they are available
upon request.
12
Data on Pástor and Stambaugh’s liquidity measure are downloaded from CRSP.
12
where r is a stock return, re is an excess stock return, and v is dollar volume. Pástor and Stambaugh
expect
to be negative in general (the price impacts of trades get reversed in the future) and larger
in magnitude for less liquid stocks. To construct innovations in aggregate liquidity, they scale the
di¤erences in the monthly liquidity measures by relative market size at k and average the di¤erences
across stocks with data available in consecutive months, i.e.,
bk =
mk
m1
Subsequently, they run the regression
bk = a + b bk
Nk
1 X
Nk
i;k
:
(18)
+ uk :
(19)
i;k 1
i=1
1
+c
mk 1
mk
bk
1
Finally, innovations in (or unexpected) aggregate (il-)liquidity are measured by P Sk =
uk
100 .
We
will denote this factor UPS. The correlation between UIL (our illiquidity proxy) and UPS is -0.07
(positive innovations signal possible illiquidity events in our case, while negative innovations signal
illiquidity events in the case of Pástor and Stambaugh’s measure). The correlation between UIL and
market returns is -0.20, i.e., increases in illiquidity are often associated with market downturns (the
correlation between market returns and Pástor and Stambaugh’s illiquidity proxy is 0.32). Next
subsection expands on this …nding. The correlation between UIL (UPS ) and UV (unexpected
realized variance) is 0.36 (-0.39). This looks natural to us, as we expected a positive relation
between market volatility and market thinness, or illiquidity. In our sample, our illiquidity proxy
has a larger correlation with SMB (i.e., the di¤erence in returns between small and large …rms)
than UPS does (-0.12 versus 0.07) (see Table I).
Alternative aggregate liquidity measures have, of course, been proposed. Amihud (2002) uses the
P#Days jri;t;k j
1
so-called "illiquidity ratio." For each stock i and each month k, he computes #Days
t=1
vi;t;k :
An aggregate measure can then be de…ned by averaging across stocks for each month k:
Nt
1 X
Ak =
Nk
i=1
#Days
X jri;t;k j
1
#Days
vi;t;k
t=1
!
;
(20)
where r i;t;k and v i;t;k represent the return and dollar volume (measured in millions), respectively,
of stock i on day t of month k.
As earlier, this measure can be re-scaled by
mk
m1
. Amihud’s "illiquidity ratio" looks directly at
price impacts. Periods of illiquidity are periods during which small volumes determine large price
moves. The correlations between U IL and unexpected Amihud (UA) is equal to -0.29.
13
5.3
Expected and unexpected premia
To make sure that the asset pricing results are not a by-product of our way of measuring volatility
and illiquidity, we …rst perform the tests using alternative, existing measures from the literature.
To this end, using daily stock market data downloaded from CRSP for the period January 1963 to
December 2000, we follow the approach in Anderson et al. (2003) for building monthly volatility.
We sum up squared daily market returns within a month, and then take the square root of this
quantity:
v
u
uX
2
Vt = t
Rm;t+i
:
(21)
i=0
represents the number of trading days within a month. As in equation (10), the residual from an
AR(1 ) model …t to the market volatility series, V t , is the unexpected volatility component, UV t .
To capture liquidity risk, we use the unexpected component of either the Pastor-Stambaugh factor
(UPS) or of the Amihud factor (UA) (see previously descriptions in Section 5.2).
Next, we …t autoregressive models to the monthly volatility and liquidity series. Using the
notation from Section 3.1 (with the only di¤erence being that the subscript t refers to the frequency
of interest, which is monthly), a …rst-order autoregression of Vt on Vt
Vt =0:011+ 0:74 Vt
(4:04)
(13:18)
as per equation (10), gives
1 +UV t ;
(22)
with an R2 = 55:7%; while a …rst-order autoregression of ILt on ILt
ILt = 0:000242 + 0:71 ILt
(4:46)
1,
(11:77)
1
1,
as per equation (9), yields
+ U ILt ;
(23)
with an R2 = 50:1%: Both volatility and illiquidity are, as expected, highly persistent. The numbers
reported in parentheses below the estimates are their t-statistics.
Subsequently, we analyze di¤erent versions of the following time series model:
e
Ri;t
=
i
+
EV
i Vt 1
+
UV d
i UVt
+
EL
i Lt 1
+
UL d
i UILt
+ ut :
(24)
First, we perform our tests using the return on the market portfolio on the left-hand side of equation
e is replaced by Re ). In the …rst model speci…cation, we regress excess market
(24) (thus Ri;t
m;t
returns only on lagged V and unexpected V (i.e., the residuals from the volatility autoregression
(22)). Then, we perform the same exercise using lagged IL and unexpected IL as regressors (from
equation (23)). Finally, we use all four regressors from equation (24). Estimates for a regression
e
of Rm;t
(the monthly excess return on the market) on Vt
1
and the estimated residuals from the
previous volatility autoregression are reported in Table II Panel A). Estimates of the regression of
14
e
Rm;t
on ILt
1
and the estimated residuals from the illiquidity autoregression are reported in Table
II Panel B), as well as estimates from joint considerations (see Panel C).
Table II shows that (i ) expected market returns are a statistically-signi…cant and decreasing
function of unexpected volatility and unexpected illiquidity, (ii ) when volatility and illiquidity are
considered jointly, both are strongly statistically signi…cant in the time series dimension, (iii ) both
expected liquidity and expected illiquidity hardly a¤ect excess market returns. This last result is
of course consistent with the well-established inability to …nd robust risk-return trade-o¤s for the
market at the monthly frequency.
Regressions (23) and (24) are in the spirit of Amihud (2002). He …nds that expected market
returns are an increasing function of expected illiquidity (measured by virtue of (20)) and a decreasing function of unexpected illiquidity. These …ndings can be justi…ed. Being persistent, higher
current illiquidity will translate into higher expected illiquidity. If higher expected illiquidity translates into higher expected returns, then higher unexpected illiquidity should lead to a drop in prices
and, hence, lower realized returns.
Our results are qualitatively similar to those in Amihud (2002). However, the statistical significance of only the (negative) coe¢ cient on ILt
1
indicates that unexpected illiquidity has a more
signi…cant e¤ect on realized stock market returns than expected liquidity. Using several aggregate
liquidity measures in the literature, Fujimoto (2003) argues in favor of the same conclusion. We
also show that, when allowing for unexpected volatility, the statistical signi…cance of unexpected
illiquidity is unchanged.
Next, we run the same regressions using excess returns on the Fama-French size-sorted decile
portfolios (see Table III). Consider …rst the regressions of excess portfolio returns on expected
and unexpected market volatility. The coe¢ cients on Vt
1
are negative but insigni…cant across
portfolios. The coe¢ cients on unexpected volatility have a negative sign, as in the market case,
and are consistently signi…cant. More importantly for our purposes, these coe¢ cients decrease (in
absolute value) monotonically when moving from small cap stocks to large cap stocks. This result
is in line with the …ndings in Moise (2011). In other words, smaller stocks have more exposure to
unexpected volatility risk than larger stocks. This exposure may be priced in equilibrium, as we
show in the next section.
Implementing the same regressions on expected and unexpected illiquidity using the Amihud
factor yields the same result (see Table IV). The absolute values of the coe¢ cients on unexpected
illiquidity decreases virtually monotonically in the size dimension (going from small stocks to large
stocks). As earlier in the market case, however, the joint consideration of unexpected illiquidity
and unexpected volatility leads to the coe¢ cients on both factors being signi…cant in the time series
dimension (see Table V). This said, the e¤ects of the two factors on excess returns remains roughly
monotonic with smaller stocks being more correlated with unexpected volatility and illiquidity than
15
larger stocks.
We emphasize that none of these results depends on the proposed illiquidity proxy. Using the
U P S measure would again yield factor loadings on unexpected illiquidity which increase with size
(or equivalently, factor loadings on unexpected liquidity which decrease with size), thereby calling
for a negative risk premium in the size dimension.
In sum, the loadings with respect to illiquidity surprises and volatility surprises align with the
returns on the size-sorted portfolios, suggesting the potential for negative risk prices. We now turn
to this issue.
6
Cross-sectional asset pricing
We consider a classical intertemporal asset-pricing model as in Merton (1973). Denote excess
e : Assume the existence
returns on a generic asset i by Rie ; and excess returns on the market by Rm
of p state variables Fs . Equilibrium expected excess returns are expressed as linear combinations of
the (conditional) beta of the asset returns with the market return,
of the asset returns with the state variables,
E(Rie ) =
s
i,
m
i ,
and the (conditional) betas
namely
m
m i
+
p
X
s
s i:
(25)
s=1
The lambdas have the usual interpretation in terms of prices of risk. Speci…cally,
of market risk and
s
m
is the price
is the price of risk associated with the generic s factor.
We use the Fama-French 25 size- and value-sorted portfolios as test assets and focus on 2- and
e and either volatility
3-factor models. In the 2-factor model case, p = 1 and the two factors are Rm
e , UV and
measure (UV or UIL). In the 3-factor model case, p = 2 and the three factors are Rm
e , SMB and HML. The Re and Re represent excess returns
UIL, or the Fama-French factors: Rm
m
i
on the asset i and the market portfolio, respectively.
The regressors in equation (25) are the slope coe¢ cients in the return generating process:
e
Ri;t
=
i+
m e
i Rm;t
+
p
X
s
i Fs;t
+
i;t ;
i = 1; 2; :::; n;
(26)
s=1
with n being the number of test assets. In order to compare our results with the existing literature,
we assume time invariant betas in equation (26).
6.1
The pricing of liquidity and volatility risk
The asset pricing tests are run using a two steps estimation approach. First, we estimate the betas
for each portfolio, according to equation (26). Given the factor loadings, the prices of risk are
16
estimated in a second step by regressing cross-sectionally the portfolios’average excess returns on
the factor loadings as implied by equation (25).
Fig. 2 Panel A) plots the monthly average excess returns of the 25 Fama-French portfolios
e and UV ). Panel B)
as well as the volatility factor loadings obtained from a 2-factor model (Rm
presents the unexpected volatility and unexpected illiquidity loadings (using Amihud’s factor) from
e , UV, and UIL). The average excess returns have a familiar pattern: they
a 3-factor model case (Rm
tend to increase in the growth-value dimension (i.e., going from low book-to-market to high bookto-market stocks) and decrease in the size dimension (i.e., going from small stocks to large stocks).
In the 2-factor and in the 3-factor model speci…cations, both the UV and the UIL factor loadings
decrease with value and increase with size, albeit sometimes not monotonically. The inverse relation
between size and factor loadings is generally stronger, and is in line with the …ndings on the relation
between size and volatility documented by Moise (2011). This result is, of course, fully consistent
with our results in the previous section. Further, since the relation between excess returns and
factor loadings is largely negative across size- and value-sorted portfolio, volatility and illiquidity
risks might be priced with a negative sign in this sample.
We test this implication by regressing average excess returns on factor loadings as described by
equation (25). We use the 25 size- and book-to-market-sorted portfolios as test assets. Table VI
reports the factors risk premiums, using both the Fama-MacBeth (1973) t-statistics as well as the
Shanken (1992) corrected t-statistics, the latter correcting for the errors-in-variables property of
e and UV, which assumes that
the generated regressors. In a 2-factor model speci…cation with Rm
volatility is the additional source of systematic risk besides the market portfolio, the volatility risk
is indeed priced. Its coe¢ cient is negative and statistically signi…cant. In a 2-factor model with
e and liquidity, UPS is positively priced with a signi…cant price of risk, while UA has a negative
Rm
coe¢ cient which is less signi…cant. We interpret the negative prices of risk for both volatility
and illiquidity as investors paying a premium for holding assets that pay o¤ when volatility and
illiquidity are high, which are characteristic of recessions and …nancial crises, as captured by the
graphs presented in Figure 1.
e , UV and liquidity in a 3-factor model, we …nd that liquidity loses its
When combining Rm
explanatory power. Thus, in our sample market liquidity appears to carry a price of risk that is less
signi…cant than the one for market volatility. More interestingly, we …nd the pricing performance of
a model with market returns, (unexpected) market volatility, and (unexpected) market illiquidity
to be similar to that of the classical Fama-French 3-factor model, when pricing these portfolios
(R2adj of 74.59% or 70.67%, versus 72.24% for the Fama-French model).
For robustness checks, we repeat the asset pricing exercise by including returns on the 10
industry portfolios downloaded from Ken French’s website. Table VII presents the results. We …nd
a robustly negative price of risk for volatility risk, associated by a marginally signi…cant (for UPS )
17
or insigni…cant (for UA) price of liquidity risk. Again, the factor loadings associated with volatility
shocks provide a more accurate assessment of risk in a joint speci…cation. Also, the 3-factor model
with market returns, volatility and illiquidity continues to perform as well as the Fama-French
3-factor model.
Having tested out hypothesis using existing measures of volatility and liquidity from the literature, we now turn our attention to the pricing of our volatility and illiquidity factors, UV-HF and
UIL, respectively, built using high-frequency data. We repeat the above exercises by …rst employing
the 25 size and book-to-market sorted portfolios over the period January 2000 - Decemebr 2008.
e and UV-HF, the
We report the results in Table VIII. In a 2-factor model speci…cation with Rm
volatility risk is signi…cantly priced. The volatility coe¢ cient is negative, again supporting volatile and liquidity,
ity as a hedging instrument in the …nancial markets. In a 2-factor model with Rm
our illiquidity factor, UIL, has also a signi…cantly negative price of risk. In constrast, the UPS
factor does not exhibit explanatory power over this sample. In a 3-factor model speci…cation, our
illiquidity factor loses its power, result in line with the above …ndings. When we also inlcude the
10 industry portfolios among the tests assets, results are robust (see Table IX).
Table X presents the contribution of volatility risk premium to the expected returns on the test
assets. There is a larger contribution to the expected returns on small …rms for both samples, as
captured by Panels A) and C) (the …rst three columns), accompanied by a larger contribution to the
expected returns on value …rms over the most recent period, as seen in Panel C). Among the industry portfolios, non-durables and manufacturing have a consistently positive volatility contribution
across the two sample periods (Panels B) and D)).
TO BE COMPLETED.
7
Conclusions
Market volatility and market liquidity have received much attention in recent work on cross-sectional
asset pricing. As there is substantial correlation between macro volatility and macro illiquidity
events, the subject of the present work is to investigate the joint pricing of these risk factors. We
do so by extracting novel volatility and illiquidity proxies from high-frequency SPDRs transaction
data. In particular, aggregate illiquidity is measured by the volatility of the di¤erence between
observed SPDRs prices and unobserved SPDRs fundamental values. Market volatility is measured
by the volatility of SPDRs fundamental values.
We show that innovations in our derived illiquidity proxy correlate with macro illiquidity events
in important ways. We also show that, when individually considered in the context of classical
asset pricing paradigms, market volatility and market illiquidity are negatively priced in the crosssection of stock returns. In joint considerations, the factor loadings associated with volatility shocks
18
provide a more accurate assessment of risk. When interpreting shocks to illiquidity and shocks to
volatility as proxies for a more fundamental distress factor, this result is suggestive of the superior
robustness of the latter. In our sample, the performance of a 3-factor model with market returns,
(innovations in) market volatility, and (innovations in) market illiquidity appears to be similar
to the performance of the Fama-French 3-factor model when pricing the Fama-French size- and
book-to-market-sorted portfolios.
Liquidity and volatility are admittedly more fundamental economic variables than factors built
based on …rm-level characteristics like SMB and HML, which have been routinely used as pervasive risk proxies by the literature. Should one believe that investors "fundamentally" care about
changes in equilibrium asset values (aggregate volatility) as well as about market frictions (aggregate illiquidity), then our …ndings would point to the joint statistical signi…cance of correlated
factors previously analyzed in isolation. We do not necessarily take this view but emphasize that,
even when volatility and liquidity are solely interpreted as proxies for one or more fundamental
factors, their joint consideration might be bene…cial for the purpose of cross-sectional asset pricing.
19
8
Appendix
This Appendix provides a discussion of the assumptions imposed on the fundamental return process r and
price deviations justifying our identi…cation approach.
As in much recent work in high-frequency econometrics (see, e.g., the discussions in the review papers
of Bandi and Russell, 2008b, and Barndor¤-Nielsen and Shephard, 2007), we model the fundamental return
process as a stochastic volatility martingale di¤erence sequence driven by Brownian shocks, i.e.,
rj;t =
Z
(t 1)h+j
s dWs ;
(27)
(t 1)h+(j 1)
where s is a càdlàg spot volatility process bounded away from zero. Spot volatility can therefore display
jumps, diurnal e¤ects, high-persistence (possibly of the long memory type), and nonstationarities. The
price deviations
are assumed to be independent of the fundamental prices and i.i.d. with a bounded
fourth moment. In consequence, the return deviations follow an MA(1) model with a negative …rst-order
autocovariance equal to 2E( 2 ). Since the fundamental returns are uncorrelated, the structure of the return
deviations carries over to the observed high-frequency returns. The empirical autocorrelation properties of
the high-frequency SPDRs returns strongly supports this simple speci…cation.
Importantly, the deviations are modeled as having a stochastic order of magnitude Op (1). Price discreteness, as well as the existence of di¤erent prices for buyers and sellers, for instance, justify this assumption
(Bandi and Russell, 2006). Thus, the friction returns do not vanish at high sampling frequencies. Technically, = Op (1).
p Di¤erently from the deviation returns, the fundamental return process r has an order of
magnitude Op ( ) over any sub-interval of length , thereby implying that the magnitude of the fundamental
price changes decreases with the sampling interval. This assumption, which is standard in continuous-time
asset pricing, represents slow accumulation and processing of information leading to negligible fundamental
price updates over small time intervals.
Exploiting the di¤erent stochastic orders of and r, Bandi and Russell (2006, 2008) show that sample
moments of high-frequency return data sampled at the highest frequencies at which information arrives identify the deviation moments. Availability of high sampling frequencies is represented here by an asymptotic
design which lets the distance between observations go to zero in the limit or, equivalently, lets the number
of observations M go o¤ to in…nity for every trading day. In the second moment case, one obtains
PM
2
ej;t
j=1 r
M
p
! Et (
M !1
2
) = 2Et (
2
):
(28)
The result in equation (28) hinges on the fact that the deviation process dominates the fundamental
return process at high frequencies. More explicitly, when computing sample moments of the observed return
p
data, the fundamental return component r washes out asymptotically since its stochastic order, Op ( ),
is smaller than the stochastic order of the frictions , Op (1). Hence, the moments of the observed returns
consistently estimate the deviation moments at high frequencies. Importantly, this consistency result would
not be a¤ected by the presence of infrequent news arrivals leading to discrete changes in the fundamental
price process. In other words, one could easily allow for the presence of a Poisson jump component in the
fundamental prices or returns r (see Bandi and Russell, 2005). The estimator, and its consistency properties,
would not change. These arguments justify using the estimator in equation (28). Because we employ a large
number of high-frequency returns per trading day (the average number of intra-daily returns is about 3,000)
we expect the consistency result in equation (28) to be fairly accurate and the corresponding estimator to
be informative.
Rt
We now turn to the variance of the fundamental return process, i.e., Vt = t 1 2s ds. In the absence of
PM 2
deviations, the sum of the squared intra-daily returns j=1 rej;t
(realized variance) estimates Vt consistently
as M ! 1 (see, e.g., Andersen et al., 2003, and Barndor¤-Nielsen and Shephard, 2002). The presence of
deviations leads to an important bias-variance trade-o¤. High sampling frequencies may determine substantial noise accumulation and biased estimates. Low sampling frequencies may lead to (fairly) unbiased but
20
highly volatile estimates. Bandi and Russell (2006, 2008) provide a simple rule-of-thumb to optimize this
trade-o¤ and choose the optimal sampling frequency (or, equivalently, the optimal number of observations
M ) for every horizon of interest. For each day in our sample, given the assumed price formation mechanism,
the (approximate) optimal number of observations Mt is de…ned as
Mt
b t is equal to
where Q
f
M
3
PM
f
4
ej;t
,
j=1 r
bt
Q
bt
!1=3
;
(29)
the quarticity estimator of Barndor¤-Nielsen and Shephard (2002) with
returns sampled every 15 minutes, and b t is equal to
PM
j=1
M
2
r
ej;t
2
, the friction-in-returns second moment
estimator raised to the second power. The optimal number of observations Mt can be interpreted as a signalb t estimates
to-noise ratio. The higher the signal coming from the underlying fundamental price process (Q
Rt
4
ds) relative to the size of the frictions (as represented by b t ), the higher the optimal number of
t 1 s
high-frequency observations needed for realized variance estimation.
Fig. 3, Panel A) reports a histogram of the optimal sampling intervals and corresponding descriptive
statistics. The average interval is about 29 minutes, the median value is about 14 minutes. Fig. 3, Panel
B) presents a time-series plot of the optimal sampling intervals. The intervals display an obvious downward
trend. This trend is due to deviation second moment estimates being relatively higher in the …rst part of the
sample. According to the ratio in equation (29), in order to achieve deviation reduction, a higher relative
deviation component should lead to a smaller optimal number of return observations and, thus, a lower
optimal sampling frequency.
Identi…cation of both the deviations second moments and the return variance can be generalized. We
could allow for virtually unrestricted dependence in the frictions along similar lines as in Bandi and Russell
(2005). As discussed above, this extension is empirically unimportant for our data given its clear MA(1)
structure. Consistent (in the presence of price deviations) estimates of the fundamental return variance may
be obtained by using kernel estimators such as those proposed by Zhang et al. (2005) and Barndor¤-Nielsen
et al. (2008). Experimentation with these alternative estimators did not lead to di¤erent results.
21
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24
Panel A) Market Volatility
Panel B) Market Illiquidity
Figure 1. Monthly illiquidity and volatility factors. Panel A) plots our annualized optimally sampled volatility and
the implied volatility (VIX) series. Panel B) plots our illiquidity factor. Data cover the period January1993 – December
2010.
25
Panel A) Monthly Excess Returns and (-) Volatility Loadings
Panel B) (-) Volatility Loadings and (-) Illiquidity Loadings
Figure 2. Monthly average excess returns, volatility and illiquidity loadings. Panel A) plots portfolio average
excess returns and volatility loadings (with a minus sign) for the 25 Fama-French size- and value-sorted portfolio from
a 2-factor model that also includes the excess return on the market portfolio. Panel B) plots the volatility and Amihud’s
illiquidity factor loadings (both with a minus sign) from a 3-factor model that also includes the excess return on the
market portfolio. The return data is collected for the period January1963 – December 2000.
26
Table I
Cross-Correlations
The variables of interest are: Rem (the excess market return), UV (monthly unexpected market volatility built using
daily data), UA (the unexpected Amihud illiquidity factor), UV  HF (our monthly unexpected market volatility built
using high-frequency data), SMB (the Fama-French size factor), HML (the Fama-French value factor), UIL (our
monthly unexpected illiquidity factor built using high-frequency data), and UPS (the unexpected Pastor and
Stambaugh factor). Data cover the period January 1963 to December 2000 in Panel A, and January 2000 to December
2008 in Panel B. ***, ** and * denote significance at the 1%, 5% and 10% levels, respectively.
Panel A) January 1963-December 2000
Rem
UV
UA
UPS
SMB
HML
Rem
1
-0.30***
-0.47***
0.32***
0.30***
-0.40***
UV
UA
UPS
SMB
HML
1
0.49***
-0.47***
-0.33***
0.13***
1
-0.29***
-0.38***
0.17***
1
0.16***
-0.11**
1
-0.29***
1
Panel B) January 2000-December 2008
Rem
UV-HF
UIL
UPS
SMB
HML
Rem
1
-0.55***
-0.20*
0.25**
0.27**
-0.35***
UV-HF
UIL
UPS
SMB
HML
1
0.40***
-0.42***
-0.18
0.02
1
-0.07
-0.12
0.07
1
-0.00
0.05
1
-0.50***
1
27
Table II
Time Series Regressions of Market Excess Returns on Expected and Unexpected Volatility and Liquidity
1/ 2

Rm2 ,t i 

 i 0


We compute monthly volatility using daily data as follows: V  
t

, where Rm, t denotes daily return on the
stock market portfolio, and Δ represents the number of trading days in a given month. We then run time series regressions
of monthly excess market returns on expected and unexpected volatility, and expected and unexpected illiquidity. Expected
volatility is the last month volatility, while unexpected volatility is the residual from an AR(1) model fitted to the volatility
series. We build the expected and unexpected illiquidity variables in the same way. For liquidity we consider the Amihud’s
illiquidity factor and the Pastor and Stambaugh liquidity factor. In Panel A) we run regressions of excess market returns on
expected
volatility,
Vt 1 ,
and
unexpected
volatility,
 tV ,
namely,
Rme ,t 1   mV   mEV Vt 1   mUV  tV  uVt ,
where  tV  Vt  ˆ0V  ˆ1V Vt 1 . In Panel B) we run regressions of excess market returns on expected liquidity, Lt 1 , and
unexpected liquidity,  tL , namely, Rme ,t 1   mL   mEL Lt 1   mUL tL  utL , where  tL  Lt  ˆ0L  ˆ1L Lt 1 . In Panel C) we use
all factors. We use the superscripts EV, UV, EL, UL to denote expected volatility, unexpected volatility, expected liquidity
and unexpected liquidity, respectively. Data cover the period January 1963 – December 2000. T-statistics are reported in
2
are reported in percentages.
parentheses. Radj
Panel A) Expected and Unexpected Volatility
-0.11
 mEV
(-1.03)
 mEL
 mUV
-0.88
(-6.86)
2
Radj
9.03
Panel B) Expected and Unexpected Liquidity
Amihud factor
Pastor and Stambaugh factor
-0.18
-1.84
(-0.59)
(-0.41)
 mUL
-6.64
(-11.27)
44.58
(9.59)
2
Radj
21.68
16.54
28
Panel C) Expected and Unexpected Volatility and Liquidity
Amihud factor
Pastor and Stambaugh factor
-0.11
0.14
EV
m
(-0.88)
(1.00)
 mUV
 mEL
 mUL
2
Radj
-0.30
(-2.17)
-0.29
(-1.98)
0.09
(0.22)
3.26
(0.57)
-5.95
(-8.83)
39.25
(6.88)
22.17
17.31
29
Table III
Time Series Regressions of Excess Returns on Expected and Unexpected Volatility –
Size-Sorted Portfolios
We run time series regressions of monthly excess portfolio returns on expected and unexpected volatility. We use monthly
value-weighted return data collected for the period January 1963 – December 2000 for the 10 Fama-French portfolios sorted
2
on size. We present results for the 2nd, 4th, 6th, 8th, and 10th decile portfolios. T-statistics are reported in parentheses. Radj
are
reported in percentages.
2nd
4th
6th
8th
10th
 mEV
-0.11
(-0.74)
-0.12
(-0.86)
-0.06
(-0.49)
-0.01
(-0.06)
-0.17
(-1.54)
 mUV
-1.53
(-8.58)
-1.40
(-8.54)
-1.18
(-7.86)
-1.03
(-7.19)
-0.67
(-5.34)
2
Radj
13.69
13.57
11.73
10.08
5.44
30
Table IV
Time Series Regression of Excess Returns on Expected and Unexpected Liquidity–
Size-Sorted Portfolios
We run time series regressions of monthly excess portfolio returns on expected and unexpected illiquidity. As aggregate
liquidity proxy, we use the Amihud’s factor in Panel A), and the Pastor and Stambaugh factor in Panel B). We use monthly
value-weighted return data collected for the period January 1963 – December 2000 for the 10 Fama-French portfolios sorted
2
on size. We present results for the 2nd, 4th, 6th, 8th, and 10th decile portfolios. T-statistics are reported in parentheses. Radj
are
reported in percentages.
Panel A) With the Amihud Factor
2nd
4th
6th
8th
10th
 mEL
-0.42
(-0.99)
-0.42
(-1.08)
-0.24
(-0.67)
0.10
(0.28)
-0.39
(-1.30)
 mUL
-10.79
(-13.38)
-10.08
(-13.68)
-8.99
(-13.31)
-7.69
(-11.67)
-5.35
(-9.10)
2
Radj
28.23
29.18
27.94
22.81
15.02
Panel B) With the Pastor and Stambaugh Factor
2nd
4th
6th
8th
10th
 mEL
-1.73
(-0.26)
-1.44
(-0.24)
-3.56
(-0.65)
-6.42
(-1.25)
0.65
(0.15)
 mUL
58.28
(8.62)
55.78
(9.04)
51.83
(9.28)
48.57
(9.22)
40.39
(8.96)
2
Radj
13.74
14.93
15.65
15.62
14.34
31
Table V
Time Series Regression of Excess Returns on Expected and Unexpected Volatility and Liquidity –
Size-Sorted Portfolios
We run time series regressions of monthly excess portfolio returns on expected and unexpected volatility and liquidity. As
aggregate liquidity proxy, we use the Amihud’s factor in Panel A), and the Pastor and Stambaugh factor in Panel B). We
use monthly value-weighted return data collected for the period January 1963 – December 2000 for the 10 Fama-French
portfolios sorted on size. We present results for the 2nd, 4th, 6th, 8th, and 10th decile portfolios. T-statistics are reported in
2
parentheses. Radj
are reported in percentages.
Panel A) With the Amihud Factor
2nd
4th
6th
8th
10th
 mEV
-0.04
(-0.23)
-0.05
(-0.32)
-0.01
(-0.08)
-0.04
(-0.25)
-0.12
(-0.94)
 mUV
-0.60
(-3.18)
-0.52
(-3.03)
-0.39
(-2.43)
-0.39
(-2.49)
-0.17
(-1.22)
 mEL
-0.22
(-0.42)
-0.22
(-0.45)
-0.14
(-0.31)
0.24
(0.54)
-0.14
(-0.37)
 mUL
-9.34
(-10.18)
-8.82
(-10.52)
-8.05
(-10.43)
-6.76
(-8.98)
-4.99
(-7.39)
2
Radj
29.54
30.32
28.60
23.55
15.01
Panel B) With the Pastor and Stambaugh Factor
2nd
4th
6th
8th
10th
 mEV
0.22
(1.11)
0.20
(1.11)
0.23
(1.38)
0.23
(1.51)
0.09
(0.68)
 mUV
-0.94
(-4.41)
-0.81
(-4.15)
-0.57
(-3.23)
-0.42
(-2.49)
-0.09
(-0.59)
 mEL
8.64
(1.05)
7.79
(1.04)
5.25
(0.77)
1.85
(0.29)
3.50
(0.62)
 mUL
40.09
(4.92)
40.17
(5.39)
41.25
(6.07)
41.22
(6.41)
39.14
(7.04)
2
Radj
17.82
18.52
18.06
17.24
14.17
32
Table VI
Risk Premiums for Size and Book-to-Market Portfolios
January 1963 - December 2000
We
report
the
estimated
factors’
risk
premiums
associated
with
the
asset
pricing
model
p
E ( Rie )  0  m  im   s  is . The left-hand side variable is the vector of mean excess returns on the Fama-French 25
s 1
portfolios sorted on size and book-to-market. The betas represent factors’ risk loadings estimated from the corresponding
time-series model: Rie,t   i   im Rme ,t 
p

s 1
s
i
Fs ,t   i ,t . The risk factors Ft, are: UV (unexpected market volatility),
SMB (the Fama-French size factor), HML (the Fama-French value factor), and unexpected liquidity (proxied by either
UA , the unexpected Amihud’s factor, or by UPS , the unexpected Pastor and Stambaugh factor). The time period is
January 1963 to December 2000. Estimation is conducted using the Fama-MacBeth procedure. The Fama-MacBeth tstatistics are reported in parentheses, while the Shanken (1992) corrected t-statistics are reported in brackets.
̂0
̂m
̂UV
1.38
(3.32)
[3.29]
-0.60
(-1.29)
[-1.28]
2.10
(5.11)
[4.29]
-1.45
(-3.34)
[-2.90]
1.88
(4.65)
[4.16]
-1.28
(-2.95)
[-2.70]
1.37
(3.27)
[2.35]
-0.69
(-1.53)
[-1.16]
2.20
(5.51)
[3.92]
-1.48
(-3.44)
[-2.60]
-1.10
(-3.84)
[-2.78]
2.07
(4.78)
[4.01]
-1.42
(-3.13)
[-2.71]
-0.69
(-2.68)
[-2.27]
1.16
(3.26)
[3.19]
-0.60
(-1.45)
[-1.43]
̂UA
̂UPS
̂SMB
̂HML
2
Radj
17.28
-0.68
(-2.55)
[-2.16]
71.96
-0.07
(-1.73)
[-1.57]
59.08
0.05
(3.19)
[2.31]
55.88
0.05
(1.23)
[0.90]
74.59
0.01
(0.74)
[0.62]
70.67
0.14
(0.88)
[0.88]
33
0.45
(3.26)
[3.26]
72.24
Table VII
Risk Premiums for Size and Book-to-Market, and Industry Portfolios
January 1963 - December 2000
Our test assets are the 25 size and book-to-market sorted portfolios and the 10 industry portfolios. The Fama-MacBeth tstatistics are reported in parentheses, while the Shanken (1992) corrected t-statistics are reported in brackets.
̂0
̂m
̂UV
0.85
(2.60)
[2.60]
-0.14
(0.36)
[0.36]
1.41
(4.55)
[4.08]
-0.79
(-2.19)
[-2.03]
1.09
(3.64)
[3.49]
-0.49
(-1.37)
[-1.33]
1.05
(3.15)
[2.51]
-0.40
(-1.02)
[-0.86]
1.76
(5.68)
[3.80]
-1.04
(-2.93)
[-2.18]
-1.27
(-4.25)
[-2.89]
1.35
(4.61)
[3.83]
-0.73
(-2.11)
[-1.86]
-0.64
(-2.51)
[-2.11]
0.72
(2.78)
[2.75]
-0.15
(-0.44)
[-0.44]
̂UA
̂UPS
̂SMB
̂HML
2
Radj
-1.30
-0.60
(-2.28)
[-2.06]
38.06
-0.06
(-1.45)
[-1.40]
20.14
0.04
(3.08)
[2.48]
29.96
0.07
(1.62)
[1.12]
49.91
0.03
(2.07)
[1.73]
41.45
0.15
(0.97)
[0.97]
34
0.32
(2.33)
[2.33]
43.32
Table VIII
Risk Premiums for Size and Book-to-Market Portfolios (Using High-Frequency Data)
January 2000 - December 2008
Out tests assets are the 25 size and book-to-market sorted portfolios. UV  HF refers to our monthly unexpected
market volatility built using high-frequency data, while UIL is our monthly unexpected illiquidity factor built using
high-frequency data. The Fama-MacBeth t-statistics are reported in parentheses, while the Shanken (1992) corrected tstatistics are reported in brackets.
̂0
̂m
1.12
(1.24)
[1.22]
-0.91
(-0.90)
[-0.89]
0.30
(0.38)
[0.30]
-0.37
(-0.39)
[-0.32]
1.30
(1.46)
[0.87]
-1.20
(-1.21)
[-0.77]
1.00
(1.92)
[1.89]
-0.81
(-1.12)
[-1.11]
0.69
(1.03)
[0.60]
-0.75
(-0.87)
[-0.56]
0.60
(1.59)
[1.49]
-0.93
(-1.55)
[-1.50]
̂UV  HF
̂UIL
̂UPS
̂SMB
̂HML
2
Radj
16.03
-1.12
(-2.72)
[-2.25]
43.56
-0.02
(-2.98)
[-1.81]
45.92
0.00
(0.11)
[0.11]
-1.23
(-2.82)
[-1.76]
12.52
-0.02
(-2.56)
[-1.52]
56.14
0.41
(1.01)
[1.01]
35
0.87
(2.35)
[2.34]
75.75
Table IX
Risk Premiums for Size and Book-to-Market, and Industry Portfolios (Using High-Frequency Data)
January 2000 - December 2008
Our test assets are the 25 size and book-to-market sorted portfolios and 10 industry portfolios. UV  HF refers to our
monthly unexpected market volatility built using high-frequency data, while UIL is our monthly unexpected illiquidity
factor built using high-frequency data. The Fama-MacBeth t-statistics are reported in parentheses, while the Shanken
(1992) corrected t-statistics are reported in brackets.
̂0
̂m
0.87
(1.46)
[1.46]
-0.80
(-1.07)
[-1.06]
-0.03
(-0.05)
[-0.04]
-0.09
(-0.12)
[-0.11]
0.49
(0.71)
[0.59]
-0.46
(-0.56)
[-0.49]
0.52
(1.19)
[1.11]
-0.49
(-0.75)
[-0.72]
-0.01
(-0.02)
[-0.02]
-0.08
(-0.11)
[-0.09]
0.64
(1.84)
[1.73]
-0.95
(-1.67)
[-1.63]
̂UV  HF
̂UIL
̂UPS
̂SMB
̂HML
2
Radj
17.12
-1.01
(-2.54)
[-2.15]
49.47
-0.01
(-2.24)
[-1.90]
44.83
0.03
(1.33)
[1.25]
-1.03
(-2.55)
[-2.02]
24.82
-0.01
(-1.75)
[-1.39]
56.90
0.42
(1.03)
[1.03]
36
0.76
(2.03)
[2.02]
67.67
Table X
The Contribution of Volatility Risk Premium to Expected Returns for Size and Book-to-Market, and
Industry Portfolios
The contribution of volatility risk premium to the expected returns, while controlling for market risk premium, is
computed for each portfolio as the product between the volatility risk premium and its loadings on returns. Results are
reported in percentage, on a monthly basis. Panels A) and B) refer to volatility built using daily data, while Panels C)
and D) refer to volatility built using high-frequency data
Panel A) 25 Size and Book-to-Market Sorted Portfolios (1963-2000)
Book-to-Market Equity (BE/ME) Quintiles
Size
Quintiles
Low
2
3
4
High
Small
2
3
4
Big
NoDur
Durbl
0.06
-0.05
Manuf
0.02
0.36
0.27
0.19
0.08
-0.10
0.37
0.25
0.17
0.06
-0.06
0.31
0.25
0.15
0.04
-0.05
0.30
0.22
0.09
0.01
-0.18
0.34
0.26
0.15
0.06
-0.10
Panel B) 10 Industry Portfolios (1963-2000)
Portfolio
Enrgy
HiTec
Telcm
Shops
Hlth
Utils
Other
-0.14
-0.15
0.00
-0.04
-0.09
0.05
-0.06
Panel C) 25 Size and Book-to-Market Sorted Portfolios (2000-2008)
Book-to-Market Equity (BE/ME) Quintiles
Size
Quintiles
Low
2
3
4
High
Small
2
3
4
Big
NoDur
0.34
Durbl
0.23
Manuf
0.44
-0.02
-0.14
0.05
-0.13
-0.24
0.13
0.34
0.47
0.48
0.27
0.42
0.44
0.51
0.30
0.42
Panel D) 10 Industry Portfolios (2000-2008)
Portfolio
Enrgy
HiTec
Telcm
Shops
Hlth
Utils
1.00
-1.01
0.18
0.26
0.04
0.42
0.01
-0.27
37
0.17
0.32
0.31
0.62
0.06
-0.13
0.19
0.79
Other
-0.06