| | | |  | Financial Analysts Journal
Volume 63 . Number 5
.2007, CFA Institute
The Statistics of Statistical Arbitrage
Robert Fernholz and Cary Maguire, Jr.
Hedge funds sometimes use mathematical techniques to “capture” the short-term volatility of stocks
and perhaps other types of securities. This sort of strategy resembles market making and is sometimes
considered a form of statistical arbitrage. This study shows that for the universe of large-
capitalization U.S. stocks, even quite naive techniques can achieve remarkably high information
ratios. The methods used are quite general and should be applicable also to other asset classes.
M
M
arket makers in financial markets generate
profits by buying low and selling high
over short time intervals. This process
occurs naturally because, as market makers,
they offer a stock for sale at a higher price than
they are willing to pay for it and because the more
urgent buyers and sellers have to accept the market
makers’ terms. Market making, particularly that of
NYSE specialists, has been studied in the normative
context of academic finance; this approach is represented
by the seminal papers of Hasbrouck and
Sofianos (1993) and Madhavan and Smidt (1993).
High-speed trading strategies similar to market
making have putatively been used by hedge
funds in recent years. This type of strategy has
sometimes been referred to as statistical arbitrage—
or perhaps stat arb, in the abbreviated patois of the
Street. Statistical arbitrage of this nature can be
studied in the context of portfolio behavior and is
hence amenable to the methods of stochastic portfolio
theory (Fernholz 2002). In this article, we use
these methods to examine the potential profitability
of such a strategy applied to large-capitalization
U.S. stocks, but the methodology is quite general
and should be applicable also to other asset classes.
Dynamic stock portfolios can be constructed
that behave like market makers. Equal-weighted
portfolios are dynamic portfolios in which each of
the stocks has the same constant weight. In an
equal-weighted portfolio, if a stock rises in price
relative to the others, it generates a sell trade in the
stock; if the price declines, it generates a buy. Therefore,
such a portfolio will sell on upticks and buy
on downticks, the way a market maker would. Our
Robert Fernholz is chief investment officer and Cary
Maguire, Jr., is senior investment officer at INTECH,
Princeton, New Jersey.
Editor’s Note: INTECH markets unique mathematical
investment processes that attempt to capitalize on the
random movement of stock prices.
goal is to estimate the return and risk parameters
of equal-weighted portfolios and use these parameters
to determine the efficacy of statistical arbitrage
in the U.S. stock market. First, we calculate
the logarithmic return for an equal-weighted portfolio;
then, we use this calculation to analyze the
behavior of portfolios of large-cap U.S. stocks.
Finally, we propose a hedging strategy to control
the risk of the high-speed trading, and we estimate
the information ratio of the hedged strategy.
Logarithmic Return of Portfolios
Roughly speaking, the logarithmic return (log
return) of a financial asset is the change in the
natural logarithm of the asset’s value. Sometimes,
the log return yields a clearer picture of asset behavior
than is available from the usual arithmetic
return—particularly in the case of certain stock
portfolios.1 In this section, we calculate the log
return for equal-weighted portfolios and investigate
the implications regarding the behavior of
these portfolios.
First, let us recall the difference between the
arithmetic return and the logarithmic return of a
given financial asset. Consider a financial asset
represented by its value X at time t. The value X
is assumed to be positive, and it varies over time.
If a period of time dt passes and the value of the
asset moves from X to X + dX, then the arithmetic
return over the period is dX/X. The corresponding
log return is defined to be dlogX, which in this
example would be equal to log(X + dX) – logX. In
all cases, the term “log” is understood to be the
natural logarithm.
Now, let us consider an equity market and
portfolios in this market. Suppose we have a market
of n stocks represented by their prices X1, . . .,
Xn, which we think of as positive-valued, time-
dependent random processes. Because we are considering
intraday price behavior, it is reasonable to
assume that the number of shares of each company
46 www.cfapubs.org .2007, CFA Institute
The Statistics of Statistical Arbitrage
is constant and that the stocks pay no dividends. In
this case, over a period of time dt, the ith stock will
have return dXi/Xi and log return dlogXi.
Suppose we have a portfolio represented by its
value P > 0 and with weights, or proportions, p1, .. .,
pn, where pi represents the proportion of the portfolio
value invested in Xi, so that p1 + . . . + pn = 1. With
this notation, the amount that the portfolio has
invested in Xi is equal to piP. A positive pi weight
implies that P holds a long position in Xi; a negative
pi weight implies a short position. In general, the
weights are all time-dependent random processes
and there is no simple relationship among the
weights, the stock prices, and the portfolio value.
There is a simple relationship, however, among the
portfolio return, the weights, and the stock returns.
This relationship is given by
dP ndX
i
P
=Σ pi X (1)
i=1 i
(see, e.g., Markowitz 1952). A more complicated
equation for the portfolio log return exists, however,
and it is precisely the “complication” that
provides the analytical tool we use in our analysis.
For log return, Equation 1 becomes
n
*
d log P = pd log X +γ dt (2)
Σ i ip
i=1
with γ* p given by
n
2
γ* = 1 Σ p σ , (3)
p iip
2
i=1
2
where σip is the variance (rate) of Xi relative to the
portfolio (which means that σip is the tracking error
of Xi versus P). A derivation of Equation 2 and
Equation 3 can be found in Appendix A, and complete
details are available in Fernholz 2002. The
quantity γp * is the excess growth rate of P, and because
relative variances are nonnegative, for a long-only
portfolio, γ* p will be nonnegative. Equation 2 shows
that for a long-only portfolio, the log return of the
portfolio exceeds the weighted average of the log
returns of its component stocks, and the difference
is exactly the excess growth rate.
Consider an equal-weighted portfolio P in
which pi = 1/n for i = 1, . . ., n. In this case, Equation
2 becomes
n
*
d log P =Σ 1d log X +γ dt
ip
i=1 n (4)
= d log nX1 ... Xn +γ*
p dt .
Hence, the log return of P depends only on the
change in the geometric mean of the stock prices
and the excess growth of the portfolio. It follows
from Equation 3 that for the equal-weighted portfolio,
γp * is simply one-half the average of the rela
2
tive variances, σip, of the stocks.
By its nature, an equal-weighted portfolio will
sell a stock when its price rises relative to the rest
of the portfolio and buy when the price declines.
Accordingly, our plan is to use an equal-weighted
portfolio to act as a market maker and thus exploit
the short-term volatility generated by the difference
in price between buy and sell trades. To determine
the efficacy of this strategy, we must estimate
the value of γ* p in Equation 4, which we do in the
next section.
Remember that real stock portfolios cannot
have precisely constant weights because a trader
cannot trade the stocks back to the weights as fast
as the stock prices move. Instead, real portfolios are
rebalanced at discrete time intervals back to
weights that are nearly constant. Nevertheless, it is
widely known and accepted in mathematical
finance that stochastic differential equations such
as Equation 1 or Equation 2 provide a close approximation
to discrete trading strategies for real portfolios
as long as the trading intervals are short
enough (see Fouque, Papanicolaou, and Sircar
2000). In practice, portfolio strategies with trading
intervals of a month or less can be closely approximated
by continuous models of the form we are
considering (Fernholz 2002). Keep in mind, however,
that in any application, the parameters in a
continuous model must be compatible with the
trading intervals used in the strategy. We develop
this idea in the next section.
Estimation of Excess Growth Rates
In this section, we estimate the excess growth rate
for an equal-weighted portfolio of large-cap U.S.
stocks—specifically, those stocks included in either
the S&P 500 Index or the Russell 1000 Index. These
stocks are traded on the NYSE or NASDAQ, and
the data we use are the individual transaction
prices, or “prints,” of the trades that occurred on
these exchanges each day.
The estimation of the excess growth rate
depends on the estimation of relative variances
because for an equal-weighted portfolio, the excess
growth rate is simply one-half the average of the
relative variances of the stocks in the portfolio. To
estimate these relative variances, the time series of
stock prices must be sampled. To sample a time
series, a particular length of sampling interval must
be chosen, and the value of the variance estimate
may depend on this choice.2 This dependence is
particularly likely if there is trading noise being
caused by the difference between the bid and the
September/October 2007 www.cfapubs.org 47
Financial Analysts Journal
asked prices. In these circumstances, shorter sampling
intervals usually result in higher estimates of
daily variance than longer sampling intervals. A
mathematical model for trading noise is included
in Appendix B, but we make no assumption that
real stocks follow such a model.
We let VT represent the average daily relative
variance of the stocks in the equal-weighted portfolio,
estimated by sampling at intervals of length T
the time series for the log return of each stock relative
to the equal-weighted portfolio. The graph of
VT versus T is a form of variogram.3 Figure 1 shows
a variogram for the year 2005 for large-cap U.S.
stocks. As can be seen, the estimate of daily variance
decreases from 0.000273 for a 1.5-minute sampling
interval to 0.000169 for a 390-minute interval, which
is the total number of minutes the exchanges in the
United States are open each business day. By Equation
3, these variances correspond to daily excess
growth rates of about 0.0137 percent for a sampling
interval of 1.5 minutes and about 0.0085 percent for
a sampling interval of 390 minutes. To annualize
these numbers, we multiply by 250, the number of
trading days in 2005. With this multiple, we have
annual excess growth rates of about 3.41 percent for
the 1.5-minute sampling interval and about 2.11
percent for the 390-minute sampling interval. Figure
1 indicates that sampling intervals between 1.5
and 390 minutes will produce estimated excess
growth rates between these two values.
With estimates of annual excess growth that
range from 2.11 percent to 3.41 percent, which
value is appropriate for use in Equation 4? The
appropriate estimate corresponds to the sampling
interval closest to the interval at which the portfolio
will be rebalanced back to equal weights. Hence,
for a high-speed trader who trades continuously,
the 1.5 minute interval is appropriate, whereas for
a trader who trades only at the open and close of
the market, the 390-minute interval should be used.
So, the high-speed trader will generate 3.41 percent
of excess growth a year, and the open/close trader
will generate only 2.11 percent. More frequent trading
allows the portfolio to capture more volatility.
A Hedged Strategy
An investment in an equal-weighted portfolio can
be risky because, although the γ* p term in Equation
4 is always positive, the log nX1 Xn term can
vary quite a bit. If, however, we consider two
equal-weighted portfolios, each rebalanced at different
intervals, the log nX1 Xn terms should be
about equal even though, as discussed, there can
be a systematic difference in theγp * terms.
Figure 1. Estimated Average Daily VT by Minute Intervals: Large-Cap U.S.
Stocks, 2005
Variance, VT
0.00026
0.00028
0.00024
0.00022
0.00020
0.00018
0.00016
1.5 3 6 12 24 49
98 195 390
Minutes, T
48 www.cfapubs.org .2007, CFA Institute
The Statistics of Statistical Arbitrage
Consider two investments, one in equal-
weighted portfolio P1.5 that is rebalanced at
intervals of about 1.5 minutes and one in equal-
weighted portfolio P390 that is rebalanced only at
the open and close of the market. Suppose a high-
speed trader buys $100 of P1.5, sells $100 of P390
short, and combines these two investments in a
single hedged long–short portfolio. A great deal of
cancellation will occur in the holdings of the long–
short portfolio, so the total stock position will be
nowhere near $100 long and $100 short. Indeed, at
the market open, both portfolios will be equal
weighted and thus will cancel exactly, so the
hedged portfolio will have no stock holdings at all.
During the trading day, portfolio P1.5 will buy on
downticks and sell on upticks of the individual
stocks, but its total long–short holdings will remain
close to market neutral throughout the day. At the
close, both P1.5 and P390 will be rebalanced back to
equal weights, so the long and short holdings will
cancel again and the high-speed trader will be left
with just the gain, or loss, for the day.
The value of P1.5 and P390 can be approximated
by equations of the form of Equation 4, where the
*
value of will correspond to a variance estimate
γ1.5
based on a sampling interval of 1.5 minutes and
*
will correspond to a variance estimate based
γ390
on a sampling interval of 390 minutes. The result
**
is that .
3.41% but .
2.11% . So, over a year,
γ1.5 γ390
the high-speed trader, who is long $100 of P1.5 and
short $100 of P390, will accumulate about $1.30. Of
course, we must assume that the high-speed trader
is a broker/dealer; otherwise, trading commissions
would come into play and the strategy would
most likely fail.
Figure 2 shows the relative log return of the
simulated equal-weighted portfolio that was rebalanced
every 1.5 minutes versus the equal-weighted
portfolio that was rebalanced only at the open and
the close of the market each day. The simulation was
run over the year 2005, the same year used for estimating
the excess growth rates.4 Figure 2 indicates
that the relative log return was about 1.26 percent,
which is consistent with the 1.30 percent annual log
return calculated in the previous paragraph applied
to the reduced 241-trading-day year. The mean and
standard deviation for the daily log returns in Figure
2 are, respectively, about 0.0052 percent and 0.0026
percent, for a daily information ratio of about 2.0.
These data imply an annual information ratio of
about 32 for the long–short combination.
Figure 2. Simulated Cumulative Value of logP1.5 – logP390: Large-Cap
U.S. Stocks, 2005
Relative Log Return (%)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
0 50 100 150 200 250
Days
September/October 2007 www.cfapubs.org 49
Financial Analysts Journal
An annual information ratio of 32 makes high-
speed trading an attractive investment strategy—
and one that could be safely combined with considerable
leverage. If this appears to be too good to be
true, what hidden problems might there be that
could interfere with the strategy? Probably the
most difficult problem for a real high-speed trader
would be gaining access to adequate order flow. In
our simulation, we assumed that every minute and
a half we could execute a trade, but a real market
would present competition for order flow. Hence,
the performance in our simulation may be accurate
for the combined class of high-speed traders, but
each member of the class would have to compete
with the others for order flow. However, a trader
should be satisfied with an information ratio of less
than 32: Indeed, a trading strategy with an annual
information ratio of just 3 would produce a negative
year only about once every seven centuries, at
least if the log return of the strategy followed a
normal distribution. (No wonder specialists’ seats
on the NYSE were hereditary!)
What amount of trading would be needed to
generate the $1.30 profit we calculated? For a stock
with a daily relative variance of 0.000273, the corresponding
1.5-minute relative standard deviation
will be about 0.00103. With this value of σ, the
expected relative log price change over a minute and
a half will be about ±σ
2 .π
= ±0.00824. With
about 256 intervals of a minute and a half in a trading
day, the expected cumulative absolute change in the
log prices corresponding to these trades will be
about 21.1 percent. Hence, the daily trading on a
$100 equal-weighted portfolio that is traded about
every minute and a half will amount to approximately
$21.10, which gives an annual trading of
about $5,275 for a 250-day year. The annual rate of
return on trading will be 1.30/5275 = 2.5 bps.
The profit enjoyed by our high-speed trader as
market maker comes at the expense of conventional
portfolio managers, who access the market more
slowly to adjust their portfolio holdings. Hence, the
2.5 bps gained by the high-speed trader should be
comparable to the trading costs paid by these conventional
managers. The market impact (i.e., trading
cost excluding commissions) for institutional
managers of U.S. stocks has been estimated by the
Plexus Group (2006) to be about 12 bps (it is referred
to as “Broker Impact” in that publication). Why is
this cost to conventional portfolio managers greater
than the 2.5 bp gain of our high-speed trader? Several
factors contribute to this difference. First,
because the high-speed trader had to pay 2.11 percent
for the hedging portfolio, he collected only 1.30
percent a year of the 3.41 percent a year generated
by his long portfolio. The 2.11 percent hedging
expense of a high-speed trader does not revert,
however, to the conventional portfolio managers,
so they will experience the full 3.41 percent generated
by the long portfolio as trading cost. Second,
institutional equity managers are likely to make
larger trades than the average stockholder, and
larger trades are likely to have greater market
impact and, therefore, cost more to execute. Third,
if the stock price fails to revert back to its previous
value after a trade, the high-speed trader will earn
nothing, but whether or not the price reverts, all the
price changes preceding a trade will still be measured
as trading cost by the portfolio managers.
When these factors are considered, the 2.5 bps generated
by the high-speed trader appears to be reasonably
consistent with the 12 bps of market-impact
cost experienced by institutional managers.
Might there be hidden risks that were not evident
in our simulation? Probably not significant
risks, at least if our high-speed trader exercises
reasonable caution. Even in strong bull or bear
markets, the average daily drift of stock prices is a
small fraction (probably about 5 percent) of their
daily standard deviation, so normal market movement
should not have much effect. Of course, a
crash like the one on Black Monday, 19 October
1987, could be, and indeed was, a disaster for some
market makers. However, our high-speed trader is
not a true market maker and has no obligation to
maintain an orderly market: He can simply sit out
nasty days like Black Monday and let the market
fend for itself.
For simplicity, we used equal-weighted portfolios
in our example, and even with this naive
strategy, the information ratio indicates that a high
degree of leverage would be possible. A similar
analysis can be carried out for any constant-
weighted portfolio with some type of optimization
applied to the weights. For example, weights on
more volatile stocks might be increased to generate
greaterγp * or be decreased to control portfolio risk,
or the weights could be varied from day to day to
adjust for changing market conditions. Optimization
of the long–short portfolio might eliminate
much of the 2.11 percent hedging expense, and the
mean reversion evident in Figure 1 might also be
exploited in some manner (see Appendix B). Perhaps
with appropriate optimization, high-speed
trading could generate a return closer to the 12 bp
trading cost paid by institutional equity managers.
50 www.cfapubs.org .2007, CFA Institute
The Statistics of Statistical Arbitrage
Conclusion
We showed that dynamic portfolios in which the
portfolio holdings are systematically rebalanced to
maintain almost constant weights will capture in a
natural way the volatility that is present in stock
markets. These portfolios act as market makers by
selling on upticks and buying on downticks. We
showed that, in the absence of trading commissions,
such portfolios can generate remarkably high
information ratios.
We thank Jason Greene for his many helpful suggestions
and Joseph Runnels for supplying information regarding
trading costs.
This article qualifies for 1 PD credit.
Appendix A. Calculation of
Portfolio Log Return
Suppose we have a market of n stocks represented
by their prices X1, ..., Xn, which are positive-
valued, time-dependent random processes. We
assume that the number of shares of each company
is constant and that the stocks pay no dividends.
Suppose that a short period of time, dt, passes and
the price of the ith stock moves from Xi to Xi + dXi.
In this case, the (arithmetic) return of the stock over
that period is defined to be dXi/Xi. The log return
of the stock over the same period is defined as
dlogXi, and the relationship between these two
types of return is
dX σ2
i = d log X + i dt, (A1)
i
X 2
i
whereσi
2 is the variance rate of Xi at time t (which
means thatσ2i dt is the variance of logXi over the
time period dt). Equation A1 can be shown to be
exact in the infinitesimal limit as the time interval
dt tends to zero, and it continues to hold quite
accurately in practice for time intervals as long as a
month or more, at least for exchange-traded U.S.
stocks. Equation A1 is an application of Ito’s rule
from stochastic calculus, details of which can be
found in Karatzas and Shreve (1991); a thorough
development of the methods of stochastic portfolio
theory can be found in Fernholz (2002).
Suppose we have a portfolio represented by
its value P with weights p1, . . ., pn, where pi represents
the proportion of the portfolio value invested
in Xi so that p1 + ... + pn = 1. A positive pi weight
implies that P holds a long position in Xi; a negative
weight implies a short position. In general, the
weights are all time-dependent random processes
and there is no simple relationship among the
weights, the stock prices, and the portfolio value.
The classical equation for portfolio return (see
Markowitz 1952) states that
dP ndX
= pi , (A2)
P Σ iX
i=1 i
and with Equation A2 (which is Equation 1 in the
article body), we are equipped to express the log
return of P, because dlogP must satisfy an equation
corresponding to Equation A1. Hence,
dP σ2
d log P =. p dt
P 2
n σ2
dXi p
=Σ pi . dt (A3)
X 2
i=1 i
2
n 2
..
= pd log X + dt
σ p dt,
Σ i . i
σi ..
22
i=1 .
...
2
where σp is the variance rate of P. By rearranging
the terms in Equation A3, we see that the log return
of P can be written
nn .
.
1 22
d log P = pd log X + p σ .σ dt. (A4)
Σ ii ..
Σ ii p..
2
i=1 i=1
The expression
. n .
* 22
γ p = 1
.Σ piσi .σ p. dt (A5)
2
.i=1 .
is the excess growth rate of P.
The variances in Equation A5 can be replaced
by variances relative to any given portfolio. The
mathematical proof is in Fernholz (2002), but the
idea here is that the portfolio growth rate will not
depend on the choice of the numeraire asset relative
to which variances are calculated. If the variances
are measured relative to portfolio P itself, and if we
denote them by 2 , then
σip
n
γ*
p = 1 Σ piσip
2, (A6)
2
i=1
where the relative portfolio variance term in Equation
A5 measures the variance of P relative to itself,
so it vanishes. Equation A6 is Equation 3 in the
article text.
September/October 2007 www.cfapubs.org 51
Financial Analysts Journal
Appendix B. Variance
Estimation with Trading Noise
In this appendix, we develop a model for a stock
price process with trading noise. The idea is that a
“bid–ask bounce” takes place over the short term
but the price series over the long term behaves like
a random walk. The model we present here is consistent
with Figure 1, but we make no assumption
that real stock prices follow such a model. Because
of the time-dependent nature of the variance estimates,
we explicitly include the time variable for
the random processes that we consider here.
The model for stock price X with trading noise
that we propose here is
log Xt()=Yt()+Zt(),
(B1)
where Y is a random walk that satisfies
dYt σdW t(),
(B2)
()= 1
with σ > 0 constant and W1 a Brownian motion, and
where Z is an Ornstein–Uhlenbeck noise process
(see Karatzas and Shreve 1991) that satisfies
dZt()=.αZtdt+τdW t(), (B3)
() 2
where α and τ are positive constants and W2 is a
Brownian motion independent of W1. In this case,
dlog Xt αZtdt+σdW t τdW t. (B4)
()=. () 1 ()+ 2 ()
The noise process, Z, incorporates a “restoring
force,” –αZ(t), that returns the process to the origin.
The steady-state distribution for Z is normal
with mean zero, and the covariance of Z(s) and
–
t–s
Z(t) is (τ2 .
2α)eα
for s,t ≥ 0 (see Karatzas and
Shreve 1991).
In Equation B4, the variance rate for X is seen
to be σ2 + τ2, but to estimate this rate, the time series
for logX must be sampled at regular intervals.
Assume that at time 0, we start process Y at the
origin and process Z at its steady-state distribution.
If for 0 ≤
s < t we sample the time series for logX at
s and t, then we find
2
var ..
log Xt().log Xs().. = E..
log Xt().log Xs( )..
= EYt.Ys
..
. () ().
..
2
2 (B5)
+.
EZt().Zs().
..
2 α.
=σ (ts. )+τ2 .1 .e.( )ts..
α ..
..
Hence, the expected value of the estimated variance
parameter for logX with sampling intervals of
length T > 0 will be
..
VT= 1
.σ2T+τ2 (1.e.αT).
T α
.
.
.
.
(B6)
.αT
22 .1.e .
=σ +τ ,
..
αT
..
which has a limiting value of V0 = limT→0VT =
σ2 + τ2 and decreases asymptotically toward σ2
as T increases toward infinity.
Note that the shape of the variogram in Figure
1 is consistent with Equation B6, although Figure 1
is based purely on empirical data with no assumption
that stock prices follow a model of the form
given in Equations B3 and B4.
Trading noise that can be modeled by a system
of the form of Equations B3 and B4 can be predicted
to some extent—for example, by the use of Kalman
filters (see Kalman and Bucy 1961). With prediction,
one might be able to improve upon the hedging
strategy described in the article.
Notes
1.
The log return is sometimes referred to as the 3. A discussion of this type of analytical methodology can be
“geometric return” or “compound return” and perhaps found in Fouque et al. (2000).
by other names. 4. Figure 2 has only 241 days of performance because no tick
2.
See Fouque et al. (2000), for example, for a discussion of data were collected for the nine days from 24 October
the dependence of the variance estimate on the sampling through 3 November as a result of office closures resulting
interval. from Hurricane Wilma.
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