AlternativeEdge® Note
alternative investment solutions
Nov. 14, 2012
It’s the autocorrelation, stupid
Galen Burghardt
Drawdown – the money you’ve lost since reaching your most recent high water mark – could well
be the single most discussed aspect of risk in the world of investing – especially in the world of active asset management. It represents money that once was within the investor’s grasp but now is
gone. The active manager can be in drawdown 80 percent of the time. It causes active managers
to wonder if their approaches to trading are flawed. It causes investors to wonder if their active
managers will change the way they trade in an effort to protect themselves. And, several years ago,
when we polled the investors at one of our early gatherings about what they considered the most
important measure of risk, drawdown topped the list. In fact, it was the results of this informal poll
that led us to publish Understanding drawdowns in 2004, and the model we developed there has
served us well for eight years.
Prime Clearing Services | Advisory Group
Galen Burghardt
[email protected]
Ryan Duncan
[email protected]
Alex Hill
[email protected]
Chris Kennedy
[email protected]
Lauren Lei
[email protected]
Lianyan Liu
[email protected]
Jovita Miranda
[email protected]
James Skeggs
[email protected]
Brian Walls
[email protected]
Tom Wrobel
[email protected]
newedge.com
Even so, the work of trying to understand drawdowns can reveal astonishing insights into the way
we think about risk. In this case, we have learned that failing to account for autocorrelated returns
can lead to serious biases in
Exhibit 1
our estimates of return volaNet asset values for two return series
tilities. Here is a case in point.
(annualized mean = 5%, annualized volatility = 15%)
4,000
While doing some work on
pension fund investments,
3,500
World Equity
CTA Index
we raised the question about
3,000
why drawdowns in equities
2,500
can be so much deeper and
last so much longer than one
2,000
would expect given their vol1,500
atility. In the upper panel of
1,000
Exhibit 1, we have drawn two
net asset value series – one
500
Source: Barclay Hedge, Bloomberg, Newedge Alternative Investment Solutions
for global equities and one
0
for CTAs – in a way that im1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
parts to each a mean return
Distribution of maximum drawdown depths for return series with
of 5% and a return volatility
non-zero autocorrelations
of 15% based on monthly reWorld equity - positive ACFs
turns. (See the appendix for a
CTA Index - negative ACFs
description of how these two
series were derived.)
Probability density function
Lianyan Liu
[email protected]
Net asset value
[email protected]
It is clear to the eye,
though, that these two series are very different and
Source: Newedge Alternative Investment Solutions
that equities exhibit much
more risk. The difference, as
we will show in this note, can
be explained by the fact that
-100%
-90%
-80%
-70%
-60%
-50%
-40%
-30%
-20%
-10%
equity returns tend to exhibMaximum drawdown depth
it positive autocorrelation
while CTA returns tend to exhibit negative autocorrelation. As a result, the standard square root
of time rule that the industry uses to translate daily, weekly, or monthly volatilities into annualized
volatilities is wrong and produced biased results.
0%
2
it ’s the autocorrelation, stupid
The effects of autocorrelation can be huge. In the lower panel of Exhibit 1, we have drawn two maximum drawdown distributions. Underlying assumptions about length of track record, mean return and
return volatility are the same in both cases, but the expected maximum drawdown with the positive
autocorrelation shown by equities is nearly double that for an asset with the negative autocorrelation
shown by CTAs.
The main conclusions of this round of work are these:
qCTA index returns in general, and the returns of trend following CTAs in particular, seem to exhibit negative autocorrelations that are both statistically significant, persistent through time,
and persistent across trend following CTAs.
qFor trend following CTAs, our failure to incorporate autocorrelation into our volatility measures
has produced estimates that are much too high. The 15% return volatility for CTAs in Exhibit 1
really should be closer to 9.4%. (And the 15% return volatility for global equities should be closer
to 17.9%.)
qBy incorporating autocorrelation estimates into our drawdown model, we can produce much
better estimates of the kinds of drawdowns we should expect for all liquid hedge fund strategies, not just trend following CTAs.
qAs long as one does it with care, one can use autocorrelation estimates as an effective diagnostic tool to uncover influences on return volatilities that otherwise could take decades to find.
In the note that follows, we
qDescribe the data sets that we used to establish the presence and persistence of autocorrelation in CTAs’ returns
qPresent a drawdown puzzle involving the 67 CTAs who had ever appeared in the Newedge CTA
Index that brought the importance of autocorrelated returns to our attention
qShow how autocorrelated returns provide the key to unlocking the puzzle and the effect autocorrelation has on the way we should translate single-period volatilities into multi-period volatilities.
qReport on what we found when we examined other data sets of CTA returns.
qReturn to the global equity/CTA comparison and conclude the note with an analysis of how autocorrelation affects risk and biases our measures of risk-adjusted returns in a potentially big way.
The data sets we used to establish the presence and persistence of
autocorrelated returns
When we first presented our findings on autocorrelated returns and their importance for drawdowns
and risk measures, the data we used were mainly monthly returns for the 67 CTAs who had ever appeared in the Newedge CTA Index. Because these are among the largest and most successful CTAs in
the industry, this was a respectable data set. Most of the questions, though, focused on whether the
results would hold up in the face of other data and on whether the results were stable through time.
Here are descriptions of the data we’ve used.
A concatenated CTA index (January 1990 through July 2012)
This index – modified to produce a 5% mean return and a 15% return volatility – appears in Exhibit 1. To
construct this index, we chained together the Barclay CTA index from 1990 through December 1999 and
the Newedge CTA index from January 2000 through July 2012. While the Barclay CTA index comprises
a much broader set of CTAs than does the Newedge CTA index, they correlate well with one another
and are both free of most of the biases that affect indexes of self-reported returns.
CTAs that have appeared in the Newedge CTA Index
This was a natural data set for us to use. The index is based on returns of the 20 or so largest CTAs that are
open for business and willing to provide daily return data. The index went live January 2000 and since
then 67 different CTAs have been used in the calculation of the index. Because the number of CTAs is
Newedge alternative investment solutions
3
it ’s the autocorrelation, stupid
70
Exhibit 2
Tallies for Newedge CTA Index constituents
Source: Barclay Hedge, Newedge Alternative Investment Solutions
60
Number
50
Active
Add
Drop
For each CTA, we used the full track record available
to us beginning with January 1977. Exhibit 2 shows
how the number of CTAs in this set reporting returns
during any given year varied through time. As the exhibit shows, the number of CTAs increased more or less
without interruption until 2006 and 2007. A history of
adds and drops is provided as well.
40
30
20
10
All CTAs with $50 million and a 3-year track
record
0
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Exhibit 3
Tallies for larger CTA dataset
600
Source: Barclay Hedge, Newedge Alternative Investment Solutions
500
Number
400
Active
Add
Drop
To determine whether what we found with the 67 CTAs
in the previous data set would hold up in a broader
data set, we pulled together a data set that comprised
the returns of all CTAs who ever had $50 million under
management and a three-year track record. A summary
tally of the population of this set is provided in Exhibit
3. Altogether, the set includes the returns of 783 CTAs.
A summary tally of how this data set evolved is provided in Exhibit 3.
A complete description of the work that went into
assembling these data will be provided in a separate
note. For now, it is enough to note that special care was
taken to avoid duplications and to isolate what would
be the lead program for each CTA.
300
200
100
0
relatively small in this case and because we know them
so well, it was easy for us to create two subsets – one
for trend followers and one for non-trend followers. Our
classification was based chiefly on reputation and confirmed correlation.
A mini trend index
To mimic the approach we used to identify trend and
non-trend managers in the Newedge CTA Index (ie. reputation and correlation), we needed to create a trend-following index with a much longer track record
than that of the Newedge Trend Sub-Index. In doing so, we identified five well-known trend followers
with continuous monthly returns from February 1988 through July 2012. From these five return series,
we constructed an equally weighted index that we rebalanced at the end of each year.
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
While this index helped us to sort the larger data set based on correlation it was also used to check
for persistence in autocorrelation though time.
The Newedge Trend Indicator
The last data set we have is not a CTA, but the Newedge Trend Indicator, which uses a 20/120 moving average trend following model to trade 55 markets that fall into four sectors – equities, currencies,
interest rates and commodities. The construction of this index is described in Two Benchmarks for Momentum Trading.
Our findings
The focus of this section is on the importance of autocorrelated returns for the ways we think of risk
and how we can use estimates of autocorrelation to improve our expectations for drawdowns and
other measures of risk.
Newedge alternative investment solutions
4
it ’s the autocorrelation, stupid
A second empirical puzzle
This work was inspired by two empirical problems. The first, we described in the introduction. That is,
why could two return series – one for equities and one for CTAs – with identical means and volatilities
produce such wildly different drawdown experiences.
Exhibit 4
Expected and observed maximum drawdowns for components
of the Newedge CTA Index
Expected maximum drawdown
-100%
-90%
-80%
-70%
-60%
-50%
-40%
-30%
-20%
-10%
0%
0%
Trend
non-trend
equal
-10%
-20%
-30%
-40%
-50%
-60%
-70%
-80%
-90%
Source: Barclay Hedge, Newedge Alternative Investment Solutions
-100%
Observed maximum drawdown
100%
Exhibit 5
p-values of observed maximum drawdowns for CTA Index trend
components
Source: Barclay Hedge, Newedge Alternative Investment Solutions
90%
80%
P-value
70%
60%
50%
40%
30%
20%
10%
0%
23 9 25 12 7
4 17 19 21 11 20 3 13 8 10 6 15 22 18 24 5 14 16 2
1
Trend CTA index
100%
Exhibit 6
p-values of observed maximum drawdowns for CTA Index
non-trend components
Source: Barclay Hedge, Newedge Alternative Investment Solutions
90%
80%
P-value
70%
60%
50%
40%
30%
20%
10%
0%
25 34 13 20 29 11 39 32 9 12 42 8 24 35 28 23 21 37 30 38 31 27 18 16 36 14 10 33 40 19 2 17 7 26 41 1 6 4 15 5 22 3
A second was what we found when we used
our drawdown model to predict what various CTAs’
maximum drawdowns should have been. For this
exercise, we used all CTAs who had ever been part
of the Newedge CTA Index anytime from 2000
through April 2012. In all, there were 67 CTAs of
whom 25 would generally be recognized as trend
followers. For each of these CTAs, we used the entire
recorded track record irrespective of when it was included in the index. In each case, we calculated the
CTA’s mean return and return volatility and used the
length of the CTA’s track record to reckon his expected maximum drawdown.
What happened when we did this is shown in
Exhibit 4 where we plot each CTA’s expected maximum drawdown against the realized maximum
drawdown. The upper right hand corner in this
exhibit represents 0, and drawdowns are shown
in negative numbers with expected drawdowns
measured along the vertical axis and realized drawdowns measured along the horizontal axis. The line
that runs from the lower left hand corner to the
upper right hand corner shows where the two are
equal. For any observation below the line, the realized maximum drawdown was smaller than the
model predicted. For any observation above the line,
the realized drawdown was larger than predicted.
The puzzle that leaps off the page most clearly is
that with only two exceptions, the trend following
CTAs’ realized maximum drawdowns were smaller
than the model predicted. One doesn’t really need
much more convincing that something is up, but
in Exhibit 5, we show the p-values for the trend following CTAs. If our drawdown model were doing
well, these p-values would follow a more or less
straight line from the lower left hand corner to the
upper right so that the values would be uniformly
distributed from 0.00 to 1.00 with two or three CTAs’
results falling in each 0.10 band. But what we see is
that all but two of the values were less than 50% and
most of these were less than 20%. (The numeric labels along the horizontal axis in this exhibit and the
next represent the CTAs’ places on our original list
and bear no relationship to any ordering related to
name, return, volatility, or age.)
Actually, a less obvious but related puzzle in-
Non-trend CTA index
Newedge alternative investment solutions
5
it ’s the autocorrelation, stupid
0.15
volves our results for non-trend CTAs. In this case, the
dots are at least scattered around the line in Exhibit 4,
which is encouraging. But some of the dots are too far
away from the line. The evidence for this is shown in
Exhibit 6, which shows the non-trend CTAs’ p-values
for this exercise. What stands out here is that while
the p-values seem to be stretched out along the diagonal from lower left to upper right, four of the 42 CTAs’
had p-values that were 100% or only slightly less. And
that’s too many.
Exhibit 7
Autocorrelations for trend CTA #9
(Sum of correlations = -0.42)
Estimated autocorrelations
0.10
0.05
0.00
-0.05
-0.10
Estimates of autocorrelation
-0.15
-0.20
Source: Barclay Hedge, Newedge Alternative Investment Solutions
1
2
3
4
5
Autocorrelation lag
1.0
Exhibit 8
Autocorrelations for trend components
Estimated autocorrelations
0.8
Lag-5
Lag-4
Lag-3
Lag-2
Lag-1
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Source: Barclay Hedge, Newedge Alternative Investment Solutions
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Trend CTA index
0.40
Estimated autocorrelations
0.30
0.20
0.10
0.00
Exhibit 9
Autocorrelations for non-trend CTA #3
(Sum of correlations = 0.42)
To be honest, autocorrelated returns were not on our
original list of things that might explain what’s going
on in these two puzzles. More likely, we thought, the
cause would be in variable volatility or something else
that was not stationary. But none of what we tried
produced any useful results. And so we turned to autocorrelation and seem to have found a plausible answer.
For example, consider the autocorrelation pattern for Trend follower #9 (TF9), whose p-value was
next to lowest. We chose #9 partly because of its low
p-value and partly because this CTA’s maximum drawdown was furthest from its expected value in Exhibit
4. When we estimated the correlations between one
month’s returns and returns from one to five months
earlier (say, TF9’s June return with TF9’s May, April,
March, February, and January returns), the pattern
of estimates we found was what you see in Exhibit
7. The first correlation was positive, but the rest were
all negative. To provide some idea of statistical significance, we have shown 2-standard-deviation lines
for the length of track record we used for this CTA. As
you can see, the negative values were all significant or
close to significant.
[Note: In what we report in this note, we limited our
estimates to five lags, mainly because for most of the
CTAs we examined, the size and significance of estimates beyond five lags tended to be small.]
What we found when we estimated autocorrelation structures for the other trend followers is shown
in Exhibit 8. The vertical bars comprise shorter bars
-0.20
Source: Barclay Hedge, Newedge Alternative Investment Solutions
that represent the value of the autocorrelation value
-0.30
for each of the five lags. The graphic is useful in two
1
2
3
4
5
Autocorrelation lag
ways. First, it shows that the patterns of positives and
negatives could vary across CTAs. Second, and more
importantly, it is easy to see that the net autocorrelation values – that is, the sum of the negatives and positives – were mainly
negative for the trend following CTAs.
-0.10
In contrast to Trend follower #9, consider what we found for Non-trend follower #3 (Non-TF3) in Exhibit 9. The p-value for
this CTA was 1.00, which means that this CTA’s realized maximum drawdown was far greater than our theoretical maximum
drawdown distribution would have suggested was possible. And, when we estimate autocorrelations for Non-TF3, we came
Newedge alternative investment solutions
6
it ’s the autocorrelation, stupid
Exhibit 10
Autocorrelations for non-trend components
1.0
0.8
What we found when we estimated autocorrelation structures for the 42 non-trend followers is shown
in Exhibit 10. In this case, there is no systematic bias,
at least in terms of sign. Some are positive. Some are
negative. And so the fact that realized drawdowns for
the non-trend set were scattered around the expected
drawdown line in Exhibit 4 makes sense.
Estimated autocorrelations
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
up with the pattern shown in Exhibit 9. In this case, the
first two correlations appear to be both positive and
statistically significant.
Lag-5
Lag-4
Lag-3
Lag-2
Lag-1
1 2 3 4 5 6 7 8
The scandal of the square root of time rule
At the heart of this note is the industry’s failure to take
autocorrelation into account when converting esti9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
mated
daily, weekly, or monthly volatilities into the
Non-trend CTA index
annualized volatilities that are used for risk assessment.
In practice, the convention is to calculate, say, a daily price or return volatility (that is, the standard deviation of daily price changes or returns) and multiply the result by the square root of the number of
business days in a year – usually something in the neighborhood of 2561/2. If one is using weekly price
changes or returns, one would use the square root of 52. And with monthly returns, the multiplier
would be the square root of 12.
Source: Barclay Hedge, Newedge Alternative Investment Solutions
5 5 5 This square
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!!! = !!!!
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!
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in –this
of research,
standard
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thatround
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in this
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.
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values ofor
Exhibit
11, we
compare
the
that
one
would
get usingcalculations
an
estimated
deviation
of
! .. In
Consider
what
happens
to
annualized
volatility
calculations
for
different
values
of
!
In Exhibit
11, we
!
compare
the
volatilities
that
one
would
get
using
an
estimated
monthly
volatility
or
standard
deviation
of
size(which
(n) gets is
large
relative
to number
of
autocorrelation
lags (k). In practice, each of the autocorrelation
4.33%
15%
divided
by
the
square
root
of
12).
compare
the volatilities
that one
would
get using
an12).
estimated monthly volatility or standard deviation of
4.33%
(which
is
15%
divided
by
the
square
root
of
values
is multiplied
a value by
equal
(n-i)/n,root
where
the number of the lag (in our case, somewhere
4.33%
(which
is 15%bydivided
thetosquare
ofi is
12).
The size of the wedge that autocorrelation drives between the conventional and correct volatility
between
1 and
5). With
5 monthly
lags, the effect
ofbetween
this weighting
is to reduce the
value
of 2.0volatility
to 1.75
The
size of the
wedge
that
autocorrelation
drives
the
conventional
and
correct
calculations
arewedge
shownthat
in Exhibit
11 as thedrives
autocorrelation
adjustment
to volatility,
which
we have
The
size
of
the
autocorrelation
between
the
conventional
and to
correct
volatility
if
one
has
24
months
of
data.
If
one
has
60
months
of
data,
the
2.0
would
be
reduced
1.90.
But
with
calculations
are
shown
in
Exhibit
11
as
the
autocorrelation
adjustment
to
volatility,
which
weofhave
calculated
by
plugging
the
various
sums
of
autocorrelations
into
the
expression
(1
+
2
x
sum
calculations
are
shown in
Exhibit
11
as the
autocorrelation
adjustment
to volatility,
which
weofhave
calculated
by
plugging
the
various
sums
of
autocorrelations
into
the
expression
(1
+
2
x
sum
autocorrelations)
and
taking
the
square
root.
The
0.00
adjustment
ratio
that
goes
with
a
correlation
of -0.5
the autocorrelation
values
we havesums
foundof
in autocorrelations
this round of work,into
the standard
square root
of
time
ruleof
calculated
by
plugging
the
various
the
expression
(1
+
2
x
sum
autocorrelations)
and
taking
the
square
root.
The
0.00
adjustment
ratio
that
goes
with
a
correlation
of
-0.5
highlights
the
fact
that
the
theoretical
lower
limit
when
autocorrelations
is
negative
is
-0.5.
Notice
that
produces annualized
volatility
that are
either
much
too high or
much
toogoes
low. with
Consider
what
autocorrelations)
and
theestimates
squarelower
root.
The
0.00
adjustment
ratio
that
a correlation
of
-0.5
highlights
thefor
factwhich
thattaking
the
theoretical
limit
when
autocorrelations
is negative
is -0.5.
Notice
that
the
only
case
the
conventional
and
correct
volatility
calculations
are
the
same
is
the
one
for
highlights
the
fact
that the
theoretical
lower
when
autocorrelations
is negative
is -0.5.
Notice
that
In Exhibit
11,the
we same
compare
the
happens
tofor
annualized
volatility
calculations
for
different
values
of
ρi.the
the
only
case
which
the
conventional
andlimit
correct
volatility
calculations
are
is the
one
for
which
thecase
sum
ofwhich
the autocorrelation
values
is
zero.
This
is when
independence
assumption
holds,
so
the
only
for
the
conventional
and
correct
volatility
calculations
are
the
same
is
the
one
for
which
the sum
ofstandard
the autocorrelation
values
isrule
zero.
This is when the independence assumption holds, so
this
is
when
the
square
root
of
time
works.
which
the sum
the autocorrelation
values
isrule
zero.
This is when the independence assumption holds, so
this is when
theofstandard
square root of
time
works.
Newedge
alternative investment solutions
this is when the standard square root of time
rule works.
The table explains why the volatility calculation for TF9 was too high. In this case, the correlations
The
table explains
why the
volatility
calculation
for TF9
was too high. Inwould
this case,
the correlations
summed
-0.42, which
means
that the
correct value
for TF9’s
be less
0.45 of what
The
tabletoexplains
why the
volatility
calculation
for TF9
was toovolatility
high. In this case,
thethan
correlations
it ’s the autocorrelation, stupid
7
Exhibit 11
The wedge that autocorrelation drives between conventional
and correct calculations of annualized volatility
volatilities that one would get using an estimated monthly
volatility or standard deviation of 4.33% (which is 15% divided by the square root of 12).
Monthly
volatility
4.33%
4.33%
4.33%
4.33%
4.33%
4.33%
4.33%
4.33%
4.33%
4.33%
4.33%
Sum of
autocorrelations
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Autocorrelation
adjustment to
volatility
0.00
0.45
0.63
0.77
0.89
1.00
1.10
1.18
1.26
1.34
1.41
Annualized
volatility
without AC with AC
15%
0.00%
15%
6.71%
15%
9.49%
15%
11.62%
15%
13.42%
15%
15.00%
15%
16.43%
15%
17.75%
15%
18.97%
15%
20.12%
15%
21.21%
The size of the wedge that autocorrelation drives between the conventional and correct volatility calculations
are shown in Exhibit 11 as the autocorrelation adjustment to
volatility, which we have calculated by plugging the various
sums of autocorrelations into the expression (1 + 2 x sum of
autocorrelations) and taking the square root. The 0.00 adjustment ratio that goes with a correlation of -0.5 highlights
the fact that the theoretical lower limit when autocorrelations is negative is -0.5. Notice that the only case for which
Source: Barclay Hedge, Newedge Alternative Investment Solutions
the conventional and correct volatility calculations are the
same is the one for which the sum of the autocorrelation
values is zero. This is when the independence assumption holds, so this is when the standard square
root of time rule works.
The table explains why the volatility calculation
for TF9 was too high. In this case, the correlations
summed to -0.42, which means that the correct
value for TF9’s volatility would be less than 0.45 of
what the standard square root of time rule would
yield. In other words, the standard approach produces a volatility that is more than 2 times too large.
Exhibit 12
Maximum drawdown distributions for trend #9
(track record = 321 months, mean = 17.9%, volatility = 34.4% , observed
max DD = -43.61%)
Source: Barclay Hedge, Newedge Alternative Investment Solutions
On the other hand, the sum of the autocorrelation factors for Non-TF3 was +0.42. In this case,
the adjustment would be a little over 1.34, which
means that the correct volatility estimate for this
CTA should be more than 34% larger than what is
produced by the standard approach.
Actual ACFs
ACFs = 0
Second puzzle resolved
-100%
-90%
-80%
-70%
-60%
-50%
-40%
-30%
-20%
-10%
0%
Maximum Drawdown
Exhibit 13
Maximum drawdown distributions for Non-trend CTA #3
(track record = 108 months, mean = 7.5%, volatility = 5.3% , observed max
DD = -14.8%)
Source: Barclay Hedge, Newedge Alternative Investment Solutions
Actual ACFs
ACFs = 0
-30%
-25%
-20%
-15%
Maximum Drawdown
-10%
-5%
0%
We are now ready to apply what we have learned
about autocorrelation to our drawdown model.
Exhibits 12 and 13 show the effect of including autocorrelation factors in our models.
In the first instance, we have drawn two maximum drawdown distributions for TF9 – one without
allowing for autocorrelation and one in which autocorrelation is taken into account. Our original
drawdown model would use length of track record
(321 months), mean return (17.9%), and annualized
volatility of returns (34.4%) as its inputs. And with
these values, the maximum drawdown distribution is centered around 60%. When we incorporate
negative autocorrelation, though, the drawdown
distribution is shifted quite a bit to the right and is
now centered around a value slightly larger than
40%. And with this distribution, the observed maximum drawdown of -43.6% appears to be more
comfortably in the middle.
When we work with Non-TF3, we find that the
Newedge alternative investment solutions
8
it ’s the autocorrelation, stupid
Exhibit 14
Expected and observed maximum drawdowns for components of
the Newedge CTA Index with allowance for aucorrelations
-90%
Expected maximum drawdown
with allowance for autocorrelations
-100%
-80%
-70%
-60%
-50%
-40%
-30%
-20%
-10%
-10%
-20%
-30%
trend
non-trend
equal
-40%
-50%
-60%
-70%
-80%
-90%
Source: Barclay Hedge, Newedge Alternative Investment Solutions
Observed maximum drawdown
15%
0%
0%
-100%
Exhibit 15
Correlation between Newedge Trend Subindex and mini trend
index
(January 2000 through July 2012, rho = 94.7%)
We also find that the empirical puzzle presented
in Exhibit 4 is mainly corrected when we use the new
drawdown model to reckon expected maximum
drawdowns. In Exhibit 14, we have plotted expected
versus realized maximum drawdowns for the original
set of CTAs, but this time we have used their respective autocorrelation estimates when calculating the
expected maximum drawdown for each.
This approach produces two good outcomes. First,
the trend followers’ values are now scattered above
and below the line rather than almost entirely under
the line. Second, the non-trend followers’ values are
now scattered closer to the line. In both cases, the
resulting differences between experience and expectation exhibit more reasonable variability.
10%
Mini trend index
presence of autocorrelation shifts the maximum
drawdown distribution noticeably to the left. Using
inputs of track record (108 months), mean return
(7.5%), and annualized return volatility (5.3%), we
find that the distribution is center somewhere close
to 5%. When we allow for positive autocorrelation,
however, the distribution shifts out and is centered
over something closer to 8%. And this manager’s
maximum drawdown of -14.8%, while it is toward
the upper end of the distribution, is more probable
than it would have been with the original distribution.
5%
0%
-5%
-10%
Extending the work to a broader data set
Source: Newedge Alternative Investment Solutions
-15%
-15%
0.10
-10%
-5%
To address the question of whether what we found
0%
5%
10%
15%
20%
in the first round of work was an accident of the data,
Newedge Trend Sub-Index
we tackled the problem of working with a much
more comprehensive set of data for a total of 783 CTAs.
Working with this set posed a special challenge because we could not know each of the 783 CTAs
as intimately as we knew the 67 larger, well established CTAs who had been part of the Newedge CTA
Index. At the same time, it allowed us a chance to approach the problem of classification in a slightly
different way. And, as a result, we uncovered more
Exhibit 16
useful
insights into the relationship between trend
Autocorrelations for the mini trend index and its components
following and autocorrelated returns.
0.05
0.00
Autocorrelation
-0.05
-0.10
Lag-5
Lag-4
Lag-3
Lag-2
Lag-1
-0.15
-0.20
-0.25
-0.30
-0.35
Source: Barclay Hedge, Newedge Alternative Investment Solutions
-0.40
Mini Sub Index Manager 1
Manager 2
Manager 3
Manager 4
Manager 5
The first step in working with this data set was to
construct what we call a “mini trend index” that could
serve as a benchmark for trend following behavior. To
do this, we identified five CTAs who have always been
known as trend followers and whose track records
were available for an extended history. As it was, we
were able to start this mini trend index in February
1988 and run it through July 2012, which is the last
month used in this work. As a reasonableness check
on whether this mini trend index could serve as a legitimate proxy for trend followers, we plotted monthly
returns for the mini trend index against monthly re-
Newedge alternative investment solutions
9
it ’s the autocorrelation, stupid
Exhibit 17
Sum of first five autocorrelations for CTAs
1.0
5.000
Source: Barclay Hedge, Newedge Alternative Investment Solutions
0.8
4.000
0.6
3.000
Autocorrelation
0.2
0.0
2.000
-0.2
1.000
-0.4
-0.6
-0.8
-1.0
0.000
We then calculated the correlation of each of the 783
CTAs’ monthly returns with those on the mini trend inCTA Index (order from high correlation to low correlation with mini trend index)
dex and ordered them from highest to lowest. The next
step was to use each CTA’s own track record to calculate
autocorrelation factors. The results of these two exercises are overlaid on each other in Exhibit 17. The
sum of the first five autocorrelation factors is measured on the left vertical axis. The correlation of each
CTA with the mini trend index is measured on the right vertical axis.
1
18
35
52
69
86
103
120
137
154
171
188
205
222
239
256
273
290
307
324
341
358
375
392
409
426
443
460
477
494
511
528
545
562
579
596
613
630
647
664
681
698
715
732
749
766
Correlation
0.4
turns for our Newedge Trend Sub-index and found a
correlation of 94.7%. The scatter is shown in Exhibit 15
and shows monthly returns from January 2000, which
is the first month for which the trend subindex data
were available. And, out of curiosity, we calculated the
autocorrelation factors for the five CTAs used in the
mini trend index and came up with the results shown
in Exhibit 16. For the index itself and for two of the component CTAs, all of the autocorrelation factors were
negative. For three of the CTAs, one of the five lagged
correlations was positive. But on balance, the autocorrelations for all five were negative.
-1.000
The results are really quite astonishing. The CTAs with the highest correlation to the mini trend index
are on the left and consistently exhibit negative autocorrelations. This seems to be true for the first 200
7 sum of a CTA’s autocorreor so CTAs on the left. Then, as correlation with the mini trend index falls, the
lation factors tends to rise. A visual scan of the results suggests that the next 200 CTAs are distributed
or less
evenly around the
zeroThen,
autocorrelation
line. Then,with
whenthe
correlation
buted more or less evenly around more
the zero
autocorrelation
line.
when correlation
mini with the mini trend inindex falls to 0.25 and below, thedex
sums
are mainly factors are mainly positive.
fallsof
to the
0.25remaining
and below, CTA’s
the sumsautocorrelation
of the remainingfactors
CTA’s autocorrelation
ve.
A quick comment about the sizes of the autocorrelation values is probably in order. The negative
look smaller
than is
theprobably
positive numbers,
in fact
there is a kind of asymmetry here. First,
ck comment about the sizes of thenumbers
autocorrelation
values
in order. but
The
negative
ers look smaller than the positiveasnumbers,
but in
there istoa know
kind of
here.
First,
as for autocorrelations is -0.5.
noted above,
it isfact
important
thatasymmetry
the theoretical
lower
bound
above, it is important to know that
the
theoretical
lower
bound
for
autocorrelations
is
-0.5.
Anything smaller than this would imply a negative variance, which might be interesting but is not posing smaller than this would imply a negative variance, which might be interesting but is not
sible.for
So an
roughly
-0.25
the CTAs
would
be grouped
ble. So an average of roughly -0.25
theaverage
CTAs of
who
would
be for
grouped
aswho
trend
followers
is as trend followers is roughly
between
andifthe
And ifusing
we apply this number using
ly halfway between zero and the halfway
theoretical
limit.zero
And
wetheoretical
apply thislimit.
number
!!!
=
!!!!
1+2
!
!!!
!!
ould find that the variance (not the
would
just half
the normal
expansion
wevolatility)
would find that
thebe
variance
(notof
thewhat
volatility)
would be
just half offor
what the normal expansion for
-period to multi-period would produce.
single-period to multi-period would produce.
e positive side, the theoretical upperOn
limit
the sum
autocorrelation
factors
+2.5
the of
positive
side,for
thefive
theoretical
upper limit
of thewould
sum forbe
five
autocorrelation factors would
ch would represent a correlation of
eachwould
of therepresent
five factors.
As it is,ofthere
some
stray
be 0.5
+2.5for
– which
a correlation
0.5 forare
each
of the
five factors. As it is, there are some
values, some of which exceed the theoretical upper limit. These can be the result of sampling error
large
some ofbewhich
theoretical
limit. These can be the result of samiable volatility in the underlying stray
series,
butvalues,
they cannot
usedexceed
in anythe
applied
workupper
or simulations.
pling
error or variable
volatility
in theindex
underlying
series,
cannot be used in any applied work
ally, however, for those CTAs with
correlations
to the
mini trend
below
0.25,but
thethey
average
was in the neighborhood of +0.50.
And if we Generally,
apply thishowever,
number,forwe
findCTAs
thatwith
thecorrelations
variance would
or simulations.
those
to the be
mini trend index below 0.25,
e that we would get without considering
autocorrelations.
So
at
least
in
variance
terms,
values
of
the average value was in the neighborhood of +0.50. And if we apply this number, we find that the
nd double are mirror images of each other.
variance would be double that we would get without considering autocorrelations. So at least in vari-
orrelation and the Newedge Trend
anceIndicator
terms, values of half and double are mirror images of each other.
ewedge Trend Indicator is not a Autocorrelation
CTA, but it doesand
represent
a trendTrend
following
model that correlates
the Newedge
Indicator
with the returns of well recognized
trend
followers.
So,
as
a
final
piece
of
evidence,
we calculated
The Newedge Trend Indicator is not a CTA, but it does represent a trend following model that correlates
tocorrelation factors for this model’s returns. The results of this exercise are shown in Exhibit 18.
well with
the returns
well recognized
followers.
So, as
a final piece
ugh none of the factors is significantly
negative,
theofpattern
of mainlytrend
negative
values
suggests
that of evidence, we calculated
rce is at work here as well.
the autocorrelation factors for this model’s returns. The results of this exercise are shown in Exhibit 18.
uestion of persistence
Newedge alternative investment solutions
10
it ’s the autocorrelation, stupid
Exhibit 18
Autocorrelations for Newedge Trend Indicator
(January 2000 through July 2012)
Estimated autocorrelations
0.20
Although none of the factors is significantly negative,
the pattern of mainly negative values suggests that the
force is at work here as well.
0.15
The question of persistence
0.10
As it is, we have found evidence of significant autocorrelation in all of the data we have been able to
assemble. There remains, however, the question of
whether it is persistent. To address this question, the
best data set we have comprises the returns of the
five trend followers that we used to construct the mini
trend index. The advantage of this set is it length and
consistency. We have uninterrupted return series for all
five going back to 1988, and so we don’t have to worry
about changes in the composition of our data.
0.05
0.00
-0.05
-0.10
-0.15
-0.20
Source: Barclay Hedge, Newedge Alternative Investment Solutions
1
2
3
4
5
Autocorrelation lag
Exhibit 19
60-month rolling autocorrelations for mini trend index components
1.0
0.8
Manager 1
Manager 2
Manager 3
Manager 4
Manager 5
average
0.6
Autocorreelation
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Source: Barclay Hedge, Newedge Alternative Investment Solutions
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Exhibit 20
Net asset values for two return series
(annualized mean = 5%, annualized vol = 15%)
4,000
World Equity
3,500
CTA Index
The choice of a 60-month rolling estimation period
was influenced in part by the time it takes to detect
autocorrelation with any kind of statistical reliability.
The standard deviation of a single correlation estimate
when the true correlation is zero is roughly one over the
square root of the number of observations, or n-1/2. With
60 months, the standard deviation would be about 0.13
[ = 60-1/2 ], which means that five years or 60 months is
about the amount of time needed to detect an overall
correlation of 0.25.
Why investors should care about autocorrelation
3,000
Net asset value
For these CTAs, we used 60-month rolling periods
to calculate the average value of each autocorrelation
factor from lag 1 to lag 5, and from these we also calculated the sums. These six time series are charted in
Exhibit 19. One notices at least two things in these histories. First, the estimated values can vary quite a lot
over time, and, in some instances can be positive for an
individual manager. Second, and most important, the
sum of the five individual sums has never been positive
and seems to have been roughly stable in the area of
-0.20, plus or minus, from the late 90s on.
Perhaps the single most important reason to pay attention to autocorrelated returns is that if we do, we
can get better ideas of the riskiness of various assets in
our portfolios. If we return to the normalized net asset
value series for world equities and CTAs shown in upper panel of Exhibit 1, which we have reproduced here
as Exhibit 20, we can reconsider the way we compare
the two series.
2,500
2,000
1,500
1,000
500
The eye tells you that these two series represent
entirely different kinds of risk. And if we estimate the
autocorrelation structure for the two, what we find is
shown in Exhibit 21. The autocorrelation values for world equities are generally positive and sum to
+0.21. In contrast, the autocorrelation values for CTAs are generally negative and sum to -0.31.
Source: Barclay Hedge, Bloomberg, Newedge Alternative Investment Solutions
0
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Newedge alternative investment solutions
11
it ’s the autocorrelation, stupid
Exhibit 21
Autocorrelations for world equities and the CTA index
(Autocorrelation sum = +0.21 for world equities and -0.31 for the CTA index)
0.15
0.10
World Equity
CTA Index
0.00
-0.05
-0.10
-0.15
-0.20
It also makes sense that we should revise the way
we calculate risk-adjusted returns. As shown in Exhibit
23, the ratios of return to risk would be 0.33 for both
equities and CTAs if we ignore autocorrelation. But
Source: Barclay Hedge, Bloomberg, Newedge Alternative Investment Solutions
we
see that the correct volatility estimate for equilag-1
lag-2
lag-3
lag-4
lag-5
Autocorrelation lag
ties is really just over 18%, while the correct volatility
estimate for CTAs is just under 10%. As a result, if we
calculate the return/risk ratios taking this into account, we find that the ratios are far from the same. For
equities, the ratio is now 0.28, while for CTAs, the ratio is 0.53 – almost double that of equities.
Another way to think of the riskiness of the two assets is reflected in Exhibit 24, which shows the
results of running the following experiment, which would be relevant for a defined benefit plan. In
each case, we started with $1,000 and then withdrew $50 a year (or really $4.17 a month) to pay a fixed
cash obligation. The $50 corresponds to the 5% reExhibit 22
turn
assumed for each asset. We then simulated the
Distributions of maximum drawdown depths for return series
with non-zero autocorrelations
distribution of outcomes for these two assets over
a
horizon of 270 months, which corresponds to the
Source: Barclay Hedge, Bloomberg, Newedge Alternative Investment Solutions
histories we have used in this note.
Probability density function
Autocorrelations
0.05
And so, even though the standard square root
of time rule produces annualized volatilities of 15%
for both series when applied to monthly volatilities,
we know that the correct volatilities are higher for
equities and lower for CTAs. Knowing this, our expectations for drawdowns are much different for the two.
As shown in Exhibit 22, it is now perfectly plausible
that equities could produce 50% drawdowns as they
did in the past decade.
World equity - positive ACFs
CTA Index - negative ACFs
-100%
-90%
-80%
The two distributions are instructive, in part because they drive home the point that if insolvency
or bankruptcy is a possibility, then standard deviations alone are not an adequate measure of risk. In
this exercise, insolvency occurs whenever the value
of the asset drops below $4.17 and the fund cannot
meet its obligation.
We also see that an investor’s evaluation of risk
requires some estimate of the costs of insolvency. In
-70%
-60%
-50%
-40%
-30%
-20%
-10%
0%
the equities example, the probability of insolvency
Maximum drawdown depth
was 21.7% [ = 100% - 78.3%, which is the probability of survival]. For CTAs, the probability of insolvency was 4.0% [ = 100% - 96.0% ]. On the other hand,
the expected value of the asset if it is still solvent at the end of the period was $2,519 for equities but
only $1,224 for CTAs. The difference between these two expectations may be part of the reason that
so many defined benefit public employ pension plans are invested so heavily in equities.
Exhibit 23
Effects of autocorrelation on performance measures
Without autocorrelation Series A
corrections
Series B
With autocorrelation
Series A
corrections
Series B
Annualized Annualized Return/risk
mean
volatility
ratio
5%
15.0%
0.33
5%
15.0%
0.33
5%
17.9%
0.28
5%
9.4%
0.53
Source: Barclay Hedge, Bloomberg, Newedge Alternative Investment Solutions
Using autocorrelation as a diagnostic
For those investors who are interested in having the best available measures of risk at their disposal, the possibility of using
autocorrelation estimates to detect biases in standard volatility
measures is good news. At least it is fairly good news. The challenge is that detecting autocorrelation takes time, and the time
it takes may be longer than many risk evaluation horizons allow.
On the other hand, the alternative is in some sense worse. One need not, in principle, use autocorrelation to get an unbiased estimate of an asset’s riskiness. One could, instead, calculate independent,
Newedge alternative investment solutions
12
it ’s the autocorrelation, stupid
Exhibit 24
Probability distributions for ending net asset values
(5% mean, 15% vol, 270 months)
– autocorrelation / 96.0% survival / mean = $1,224
Probability density function
+ autocorrelation / 78.3% survival / mean = $2,519
Source: Newedge Alternative Investment Solutions
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Net asset value ($)
non-overlapping n-period returns (where n is long
enough to include all the correlation lags). But the
time it would take would be much, much longer
than is required to work with correlations. To see
why, consider a simple 1-period lag. In such a case,
one should be calculating 2-month returns. If they
are to be non-overlapping, then one would get 6
observations per year instead of 12. But for them to
be independent, one would have to skip a month
between each 2-month return period. If you do
this, you are down to 4 observations a year. And in
our case, where the lags seem to stretch out to five
months, one could really only get one independent
observation a year. In which case, it would take a
few decades to learn what would be available in
only a few years if one were to use autocorrelations
to augment the standard measures of volatility.
Where do we go from here?
We are persuaded at this point that the presence of autocorrelated returns helps to resolve the two
empirical puzzles that prompted this work in the first place. At least one natural extension of this work,
then, will be to amend our manager evaluation reports to reflect the new drawdown model and to include some information about autocorrelation in each manager’s returns.
Timing investments But there are a number of questions that will require some attention. For one,
we examined the question of whether one could time one’s investments in CTAs using past returns,
found generally that the answer was no, and published our findings in Every drought ends in a rainstorm.
The main conclusions were that CTAs’ returns appeared to be uncorrelated through time. Returns conditioned on past returns appear to be the same as unconditional returns. And the numbers of runs
of gains and losses appeared to be thoroughly consistent with randomness. The data set was small,
though, and we may arrive at different findings with the broader data sets we have used in this work.
Drawdown control If returns are autocorrelated, it is possible to improve one’s risk-adjusted returns
using what is loosely called drawdown control. For return series with positive autocorrelation, a rule
that reduces position sizes when losing money and increasing position sizes when making money will
improve risk-adjusted returns. If returns are negatively autocorrelated, the opposite should be true, in
which case drawdown control would increase position sizes when losing money and would decrease
position sizes when making money.
Where does autocorrelation come from? And why is it negative for trend followers? These
are thorny questions but important to explore. In early conversations about this work, one of the questions raised was whether autocorrelation was evidence of inefficiency – of money being left on the
table, or money lost unnecessarily. For that matter, part of the reasoning we used in Every drought ends
in a rainstorm to explain the absence of conditionality in returns was that active trading, if well done,
would wring returns from price series that exhibited positive or negative momentum and produce return series that were free from autocorrelation.
Newedge alternative investment solutions
13
it ’s the autocorrelation, stupid
Postscript on statistical influences on drawdowns
In Understanding drawdowns, we found that the three most influential variables were length of track
record, return volatility and mean return. Skewness, which indicates whether upside returns are larger
or smaller than downside returns, didn’t seem to have much effect. Neither did excess kurtosis, which
measures the likelihood of unusually large positive or negative returns.
The reason is most likely the power of the central limit theorem, which states that the sum of enough
independent random variables will be normally distributed, no matter how each of the individual variables is distributed. So one can add up the oddest, most peculiar, random variables, and the central
limit theorem pounds them all into a fairly uniform, normally distributed paste. In fact, skewness and
kurtosis probably do matter for CTAs with short track records, but only because the central limit theorem has not had time to work its erosive magic.
In contrast, autocorrelation does not disappear as the track record lengthens. If it’s there, it’s there,
and becomes more noticeable with the passing of time, not less. The wedge it drives between estimates
of volatility based on returns for single periods that are shorter than the length of the autocorrelated
lags and of the asset’s multi-period volatility simply becomes clearer and more pronounced.
Acknowledgements
We want to thank Mark Carhart of Kepos Capital for his contribution to this piece of work. At our September 2011 manager/investor forum in San Francisco, he presented some work on drawdowns that
explored the possibility of using drawdown control to improve risk adjusted returns if, as is the case
for certain classes of hedge funds, returns are positively autocorrelated. He also encouraged the use
of drawdown per unit of volatility when comparing one manager with another. It was the latter suggestion that produced the empirical puzzle that you find in Exhibit 4. And it was his thinking about
autocorrelated returns that led us to think of autocorrelation as the solution to the puzzle. In addition,
he has met with us regularly over the past year to discuss the ways in which we might use drawdowns
as a diagnostic tool when evaluating expectations about a manager’s future and about the appropriateness of drawdown control as a risk management tool.
Any questions about the derivations behind the transformation shown on page 6, the theoretical
limits on positive and negative autocorrelations, and the calculation of significance bands for autocorrelation estimates should be directed to Lianyan Liu.
Newedge alternative investment solutions
14
it ’s the autocorrelation, stupid
Appendix: Normalizing world equity and CTA returns so
that they have equal means and volatilities
The actual
net asset
value seriesworld
for World
Equity
CTAsreturns
are those
in Exhibit
25. Over
this and volatili
Appendix:
Normalizing
equity
andand
CTA
so shown
that they
have equal
means
period,
CTA were Normalizing
the superior asset
if compared
solely
on returns
(higher
than
those
on equal
equities)
and and volatili
Appendix:
world
equity and
CTA
returns
so that
they
have
means
The (lower
actualthan
net asset
series
for World
Equity
and CTAs are on
those
shown behavior,
in Exhibit 25. Over t
volatility
that ofvalue
equities).
To isolate
the effect
of autocorrelation
drawdown
period,
CTA
superior
asset
ifidentical
compared
solely
on and
returns
(higher
thanin
those
The
actual
netwere
asset
value
series
for
World
Equity
and
CTAs
are
those
shown
Exhibit
25. Overan
t
however,
we
created
two the
new
series
that
have
mean
returns
return
volatilities
but
thaton equities)
volatility
(lower
than
of equities).
To isolatesolely
the effect
of autocorrelation
on drawdown
behav
period, CTA
were
thethat
superior
asset if compared
on returns
(higher than those
on equities)
an
preserve
the pattern
or than
sequence
of
each.
That
is, whatever
autocorrelation
would
find
however,
we created
two
seriesinthat
have
identical
mean
returns
andone
return
volatilities
butbehav
that
volatility
(lower
that new
of returns
equities).
To
isolate
the effect
of
autocorrelation
on
drawdown
in thepreserve
original series,
one would
also
in
modified
series. That
the
ortwo
sequence
ofthe
returns
in each.
is, whatever
autocorrelation
one would
fi
however,
wepattern
created
newfind
series
that
have
identical
mean
returns and
return volatilities
but that
the
original
would also
in the
modified
series.
preserve
theseries,
patternone
or sequence
offind
returns
in each.
That
is, whatever autocorrelation one would fi
the original series, one would also find in the modified series.
ToTo
do do
thisthis
requires
only two
first isThe
to normalize
original, actual
series so itactual
has mean
= 0so it has mea
requires
onlysteps.
twoThe
steps.
first is tothe
normalize
the original,
series
and
volatility
=1 as only
follows:
To do
this
two steps. The first is to normalize the original, actual series so it has mea
and volatility
= 1requires
as follows:
and volatility =1 as follows:
R2t = (R1t − µ1 ) /σ 1
R2t = (R1t − µ1 ) /σ 1
where
R1t original
is the original
return
fort, month
µ1 is the
average
return
for series
and σ1 is its volatili
where
R1t is the
return for
month
µ1 is thet,average
return
for series
1, and
σ1 is its1,volatility
standard
of returns.
resulting
of Rreturn
have a1,mean
zero
and a
where
R1tdeviation
is the original
returnThe
for month
t, µdistribution
for series
and σof
its volatili
2 will now
1 is the average
1 is
will now
aany
mean
of zero and athat
or standard
deviation
of returns.
The
distribution
of R2 will
standard
deviation
of 1.0,
butresulting
the
ofdistribution
returns
aucorrelation
oneand
woul
returns.
Thesequence
resulting
ofpreserve
R2 have
will now
have
a mean of zero
a
standard
deviation
ofseries.
1.0, but
the sequence
of returns
preserve
any
aucorrelation
that one
wouldthat one
in
the
original
The
second
step
simply
requires
that
the
second
be converted
to awoul
third
standard
deviation
of 1.0,
but
the sequence
ofwill
returns
will
preserve
anyseries
aucorrelation
as the
follows:
in
the
originalseries.
series.
second
simply
requires
that
the second
series
be converted
to a third
find in
original
TheThe
second
step step
simply
requires
that the
second
series be
converted
to a
as
follows:
third series as follows:
R3t = µ3 + σ 3R2t
R3t = µ3 + σ 3R2t
Where µ and σ can be any values you choose. In this note, we arbitrarily chose a mean of 5%, whi
Where
and σthe
candifference
be any values
youthe
choose.
In this
note,
we arbitrarily
a meanEquities,
of 5%, whi
roughlyµsplits
between
observed
mean
returns
for CTAschose
and World
and
Where µ and σ can be any values you choose. In this note, we arbitrarily chose a mean of 5%, which
roughly
the difference
between
the equity
observed
mean returns
and World
volatilitysplits
of 15%,
which looks
more like
volatility.
Thesefor
areCTAs
the series
that weEquities,
show in and
Exh
roughly splits the difference between the observed mean returns for CTAs and World Equities, and a
volatility
15%,
looks
more
likethat
equity
volatility.
These means
are theand
series
that we show
and 20. Inofthe
end,which
we have
two
series
would
have identical
volatilities
(and, in
as Exh
a re
volatility
of
which
looks
equity
Thesehave
are the
series that
we show
in Exhibits (and, as a re
and
20.15%,
In
the end,
wemore
havelike
two
that difference
would
means
and volatilities
identical
risk-adjusted
returns),
soseries
thatvolatility.
any
inidentical
drawdown
behavior
can be attributed only
1 andidentical
20. In the
end,
we have two
series that
would
identicalin
means
and volatilities
(and,
a rereturns),
so that
anyhave
difference
drawdown
behavior
canasbe
attributed only
pattern
ofrisk-adjusted
returns.
sult, pattern
identicalof
risk-adjusted
returns. returns), so that any difference in drawdown behavior can be attributed
only to the pattern of returns.
Exhibit 25
Net asset values for two return series
4,500
4,000
World Equity (mean=3.9%, vol = 16.2%)
CTA Index (mean =6.2%, vol=9.1%)
Net asset value
3,500
3,000
2,500
2,000
1,500
1,000
500
Source: Barclay Hedge, Bloomberg, Newedge Alternative Investment Solutions
0
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Acknowledgements
Acknowledgements
Newedge alternative investment solutions
it ’s the autocorrelation, stupid
15
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Newedge alternative investment solutions