Implied Phase Probabilities
SEB Investment Management
House View Research Group
2015
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Market and Gaussian Mixture Models . . . . . . . . . . . . . . . . . . . . . . . 4
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Development Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Are the Implied Probabilities Capped . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Illustration by a Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
A Multi-State, Multi-Asset Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Estimation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
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Editorial
SEB Investment Management
Sveavågen 8,
SE-106 Stockholm
Authors:
Portfolio Manager, TAA: Peter Lorin Rasmussen
Phone: +46 70 767 69 36
E-mail: [email protected]
Portfolio Manager, Fixed Income & TAA: Tore Davidsen
Phone: +45 33 28 14 25
E-mail: [email protected]
Is it possible to deduct the implied probability of the marginal investor
being either bullish or bearish? If so, can such a probability measure be used
actively in an asset allocation context? These are the questions which we
attempt to answer in this paper.
The paper presents a methodology to assign probabilities of the market
being in a predefined set of different states. These states can be defined ad
hoc, by an asset allocation model, or be estimated directly from the market.
In the remainder of the paper we focus on a simplified asset allocation model
defined by two states: A bull and a bear market. The return on equities and
bonds in the two states are conditioned on the change in one of the most
popular leading indicators in the market: The OECD amplitude adjusted leading indicator. Each state is defined by a set of unique returns, covariances
and a frequency by which it appears over time. By using our proposed methodology, it is possible to assign probabilities of the current market being in
either of the two states. That is, based on the observed returns over a given,
short, period of time we are able to say which state the current market most
likely is in. As an example of how this information can be used in practice,
assume that we strongly believe that we are in a bear market (with equities
underperforming bonds) but the market (through the methodology of this
paper) tells us that we are in a bull market (equities outperforming bonds).
We should then consider selling equities into strength. Alternatively the model can simply be used to identify turning points in the investment cycle. For
example identifying points in time where the probability of the market being
in the bull state rises, while the probability of the bear state diminishes.
Mathematically we focus on cluster models, and in particular Gaussian Mixture models. We consider it our prerogative to keep a fairly detailed view
throughout the paper, as we believe the subject in itself is mostly interesting
to the mathematically oriented reader. That being said, we do focus on the
intuition rather than the mechanics.
Page 3
Introduction
models. We consider it our prerogative to keep a fairly detailed view
throughout the paper, as we believe the subject in itself is mostly interesting to
the mathematically oriented reader. That being said, we do focus on the
intuition rather than the mechanics.
statedwe
wefocus
focuson
onGaussian
Gaussian Mixture
Mixture models.
models. To illustrate
illustrate the
the intuition
intuition
The Market AsAsstated
The Market and
behindthese,
these, we
we start
start by
by looking
looking at
at aa single
single synthetic
synthetic asset
asset class;
class; call
call itit
and Gaussian behind
Gaussian Mixture
equities
if
you
like.
We
pretend
that
the
“market”
can
be
in
one
of
two
separate
Mixture Mod- equities if you like. We pretend that the “market” can be in one of two sepaModels
states: a bull market with a positive expected return and low volatility, and a
rate states: a bull market with a positive expected return and low volatility,
high and
volatility.
beara market
with a with
negative
expectedexpected
return and
and
bear market
a negative
return
high Assuming
volatility. the
Asreturnsthe
to be
normally
distributed,
two statesthe
aretwo
defined
as:are defined as:
suming
returns
to be
normallythe
distributed,
states
𝑓𝑓(𝑋𝑋𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ) = 𝑁𝑁(−5,15)
𝑓𝑓(𝑋𝑋𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ) = 𝑁𝑁(+10,10)
expected negative
negative return
Thatis,is,ininthe
thebear
bearmarket
marketthe
theasset
assetdelivers
delivers an
an expected
That
returnof
of
5%
and
in
the
bull
market
the
asset
delivers
an
expected
positive
return
of
5% and in the bull market the asset delivers an expected positive return of
10%.The
Thereturn
return distributions
distributions ofofthe
areare
graphed
in Figure
1. 1.
10%.
thetwo
twomarkets
markets
graphed
in Figure
Figure 1: Return Distributions of the Bull and the Bear Market
0.050
Bull Market
Figure 1: Return Distributions
of the Bull and the Bear Market
0.045
Bear Market
0.050
0.040
0.045
0.035
Bull Market
Bear Market
0.040
0.030
0.035
DensityDensity
els
0.025
0.030
0.025
0.020
0.020
0.015
0.015
0.010
0.010
0.005
0.005
0.000
0.000-80
-80
-60
-60
-40
-40
-20
0
-20
0
20
Expected
Expected
Return Return
20
40
40
60
60
Now
the bull
bullmarket
marketisistwice
twiceasaslikely
likely
observed
Nowassume
assume that the
to to
bebe
observed
overover
timetime
as
bear
market.
ThatThat
is, if is,
weifobserve
the market
over a very
timelong
horizon
asthethe
bear
market.
we observe
the market
overlong
a very
time
we should
the bull
in 2/3
of theinperiods
bear market
the
horizon
wesee
should
seemarket
the bull
market
2/3 of and
the the
periods
and thein bear
remaining
1/3.remaining
Put differently,
33%differently,
of the time33%
we are
thetime
bearwe
market,
market
in the
1/3. Put
ofinthe
are inand
the
the other
67%and
of the
we67%
are inofthe
market.
This
weighted
market
bear
market,
thetime
other
thebull
time
we are
intime
the bull
market.
This
is
a
Gaussian
mixture,
defined
as:
time weighted market is a Gaussian mixture, defined as:
𝑓𝑓(𝑋𝑋) = 33%𝑓𝑓(𝑋𝑋𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ) + 67%𝑓𝑓(𝑋𝑋𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 )
The probabilities denoting the frequencies of the different states are
The probabilities denoting the frequencies of the different states are mathemathematically termed the mixing probabilities.
matically termed the mixing probabilities.
On a technical note, the mixing probabilities must lie between 0 and 1, and
sum to one. Naturally the mixing probabilities can be adjusted freely if it is
On
a technical
note, the
probabilities
more
likely to observe
one mixing
market over
the other. must lie between 0 and 1,
The resulting mixed distribution is graphed in Figure 2. This can be interpreted
as the distribution of the market, observed over a long time horizon. To stress
the4point further, if you observe the returns of the asset class over many years
Page
(for instance 20 years) the resulting histogram will resemble Figure 2.
and sum to one. Naturally the mixing probabilities can be adjusted freely if
it is more likely to observe one market over the other.
The resulting mixed distribution is graphed in Figure 2. This can be interpreted as the distribution of the market, observed over a long time horizon.
To stress the point further, if you observe the returns of the asset class over
many years (for instance 20 years) the resulting histogram will resemble Figure 2.
Figure 2: Return Distribution of the Full Market
0.035
0.030
0.025
Density
0.020
0.015
0.010
0.005
0.000
-80
-60
-40
0
-20
Expected Return
20
40
60
Note that the Gaussian mixture appears to be non-normal, as it is skewed
to the left and appears to have “fat” tails. The fact that a Gaussian mixture
is not necessarily normally distributed, or even unimodal, should not be a
cause of concern. In fact, it can be reassuring to note that many of the stylized facts of finance – skewed and fat tailed return distributions – can be
explained in an underlying Gaussian setting; if you believe the market to
exist in a series of discrete states. As a purely qualitative observation, most
investors would probably agree that the change from a bull market to a bear
market often happens quite violently, which supports the state space theory
(i.e. that markets shift instantly from one state to another as opposed to a
gradual shift between states).
To summarize: When we look at the market over a long time horizon we see
the return distribution of Figure 2. We, therefore, believe the market to be
non-normal (fat tails and skewed) but in reality it is merely the weighted
average of two distinct markets: A bear market and a bull market. These
Page 5
two markets are in themselves normally distributed. Based on this simple
intuition the purpose of this note is to show how to assign probabilities of
the market being in either one of the two states.
To illustrate how these probabilities can be calculated, assume that a 3%
return is observed on the asset class described above. We would like to assess how probable it is, that this return is generated by either the bull market
or the bear market. This probability is called the posterior probability. Figure
3 graphs the full return distribution with the observed return of 3% as the
red line.
Figure 3: Return Distribution of the Full Market and an Observed Return
0.040
0.035
0.030
0.040
0.040
0.040
0.025
0.035
Density
0.035
0.035
0.030
0.020
0.030
0.030
0.025
0.020
Density
Density
Density
0.015
0.025
0.025
0.020
0.020
0.010
0.015
0.0150.015
0.005
0.010
0.0100.010
0.000
0.005
-80
0.005
0.005
0.000
-80
0.000
0.000
-80 -80
-60
-60
-60 -60
-40
-40
-40 -40
20
0
-20
Expected Return
-20
0
-20
0 0
-20 Return
Expected
Expected
Return
Expected
Return
20
20 20
40
40 40
40
60
60
60 60
ToTocalculate
posteriorprobabilities
probabilities
probability
the market
calculate the
the posterior
(i.e.(i.e.
thethe
probability
of theofmarket
being
To To
calculate
thethe
posterior
probabilities
(i.e.(i.e.
thethe
probability
of the
market
being
calculate
posterior
probabilities
probability
of the
market
being
being
in either
thestates)
two states)
the following
in either
of theoftwo
we usewe
theuse
following
formula:formula:
in either
of the
twotwo
states)
wewe
useuse
thethe
following
formula:
in either
of the
states)
following
formula:
𝜋𝜋𝑘𝑘 𝑓𝑓(𝑥𝑥|𝜇𝜇𝑘𝑘 , Σ𝑘𝑘)) )
𝑓𝑓(𝑥𝑥|𝜇𝜇
𝑘𝑘 , Σ𝑘𝑘𝑘𝑘, Σ𝑘𝑘 , 𝑘𝑘 = 1, … , 𝐾𝐾
𝑘𝑘 𝑓𝑓(𝑥𝑥|𝜇𝜇
𝑝𝑝𝑘𝑘 = 𝐾𝐾𝜋𝜋𝑘𝑘𝜋𝜋
𝐾𝐾, 𝐾𝐾
𝑝𝑝𝑘𝑘 𝑝𝑝=𝑘𝑘 =
𝑘𝑘 1,
=…
1,,…
∑𝐾𝐾𝑘𝑘=1𝐾𝐾 𝜋𝜋𝑘𝑘 𝑓𝑓(𝑥𝑥|𝜇𝜇𝑘𝑘 , Σ𝑘𝑘 ), 𝑘𝑘 ,=
∑𝑘𝑘=1
∑𝑘𝑘=1
𝜋𝜋𝑘𝑘𝜋𝜋
𝑓𝑓(𝑥𝑥|𝜇𝜇
Σ𝑘𝑘 )
𝑘𝑘 , Σ𝑘𝑘𝑘𝑘, )
𝑘𝑘 𝑓𝑓(𝑥𝑥|𝜇𝜇
)
Where 𝐾𝐾 denotes the number of states, 𝑓𝑓(𝑥𝑥|𝜇𝜇𝑘𝑘 , Σ𝑘𝑘) is) the normal density of
the
normal
density
of
Where
denotes
thenumber
number
states,
𝑓𝑓(𝑥𝑥|𝜇𝜇
is the
normal
density
Where
𝐾𝐾 denotes
the
number
of
states,
𝑓𝑓(𝑥𝑥|𝜇𝜇
Where
K𝐾𝐾denotes
the
ofofstates,
is the
normal
density
ofof
𝑘𝑘 , Σ𝑘𝑘𝑘𝑘, Σ𝑘𝑘
point 𝑥𝑥 conditional on the state parameters, and 𝜋𝜋𝑘𝑘 is the mixing probability of
mixing
probability
of of
point
conditional
the
state
parameters,
and
𝜋𝜋𝑘𝑘𝜋𝜋is𝑘𝑘the
isisthe
mixing
probability
point
𝑥𝑥 conditional
state
parameters,
and
point
x𝑥𝑥conditional
on on
thethe
state
parameters,
and
the
mixing
probabistate 𝑘𝑘. Based on our two state distributions and the observation of a return of
state
𝑘𝑘. Based
on on
ouron
two
state
distributions
andand
thethe
observation
of aofreturn
state
𝑘𝑘. k.
Based
our
two
state
distributions
observation
a return
lity
of
state
Based
our
two
state
distributions
and
the observation
ofofaof
3%, we find that there is a 63% probability that the market is in the bull state
3%,
we
find
that
there
is
a
63%
probability
that
the
market
is
in
the
bull
state
3%,
we
find
that
there
is
a
63%
probability
that
the
market
is
in
the
bull
state
return of 3%, we find that there is a 63% probability that the market is in the
and a 37% probability that the market is in the bear state.
andand
a 37%
probability
thatthat
thethe
market
is inismarket
the
bear
a 37%
market
in the
bear
bull
state
and
aprobability
37% probability
that
the
isstate.
instate.
the bear state.
Asan
analternative
alternative representation
of of
thethe
posterior
probabilities,
Figure Figure
4 graphs4
AsAs
representation
posterior
probabilities,
an an
alternative
representation
of the
posterior
probabilities,
Figure
4 graphs
As
alternative
representation
of the
posterior
probabilities,
Figure
4 graphs
the outcome
of a range
ofrange
point of
estimates.
graphs
the
outcome
of aof
point estimates.
thethe
outcome
of aofrange
point
estimates.
outcome
a range
of point
estimates.
Figure 4: Probability Assigned to the Different States
Figure
4: Probability
Assigned
to the
Different
States
Figure
4: Probability
Assigned
to the
Different
States
100
100 100
90
90 90
80
80 80
70
70 70
60
60 60
,%
, %%
ity,
Page 6
Bull Market
Bull Market
Bull Market
Bear Market
BearBear
Market
Market
Figure 4: Probability Assigned to the Different States
100
Bull Market
Bear Market
90
80
Probability, %
70
60
50
40
30
20
10
0
-5
5
0
10
Return
The figure should be read as follows: For each observed return there is a
corresponding probability to be in either state. The probabilities correspond
to the line dividing the two areas. A low observed return results in a low probability
to be inprobability
the bull market
vicestate.
versa.
corresponding
to be inand
either
The probabilities correspond to
corresponding
probability
to
either
state.
The
probabilities
correspond
the line
dividing
the
two
areas.
Ainlow
observed
return
results
in a low
corresponding
probability
to be
in be
either
state.
The probabilities
correspond
to to
the dividing
line
dividing
the market
two areas.
A observed
low
observed
probability
to be
inthe
the two
bull
and
vice
versa.
the
line
areas.
A low
returnreturn
resultsresults
in a in
lowa low
probability
to
be
in
the
bull
market
and
vice
versa.
probability to be in the bull market and vice versa.
Instead
thethe
mixing
probabilities
andand
state
distribution
paInsteadofofmerely
merelysetting
setting
mixing
probabilities
state
distribution
mation
they
canmerely
be
estimated
directly
from
the
That
is, provided
provided
Instead
of
setting
the
mixing
probabilities
and
distribution
parameters
they
can
be
estimated
directly
from
themarket.
market.
Thatstate
is,
Instead
of
merely
setting
the mixing
probabilities
and state
distribution
Estimation rameters
mation
you
have
a
set
of
actual
observed
returns,
it
is
possible
to
determine
which
parameters
they
can
be
estimated
directly
from
the
market.
That
is,
provided
you have athey
set of
observeddirectly
returns,from
it isthe
possible
to That
determine
which
parameters
canactual
be estimated
market.
is, provided
parameters
–
mixing
probabilities
included
–
best
describe
the
data.
Folloyou have
aofset
of
actual
observed
returns,
itdescribe
is possible
to determine
parameters
– mixing
probabilities
included
–itbest
data.
Following
you
have
a set
actual
observed
returns,
is possible
tothe
determine
whichwhich
wing
this
approach,
you
are
actually
conducting
a
full
cluster
analysis.
parameters
–
mixing
probabilities
included
–
best
describe
the
data.
Following
this approach,
you are
actually conducting
analysis.
parameters
– mixing
probabilities
included a– full
bestcluster
describe
the data. Following
this approach,
youactually
are actually
full cluster
analysis.
thisforth
approach,
you are
conducting
a full acluster
analysis.
So
the parameters
are to
be conducting
estimated
directly
from
the market, you
So forth the parameters are to be estimated directly from the market, you need
need to maximize the likelihood function of the Gaussian mixture. That is
So forthparameters
the likelihood
parametersfunction
are to be
directly
from That
the market,
you need
to forth
maximize
the
ofestimated
the Gaussian
mixture.
is you need
So
you
needthe
to maximize: are to be estimated directly from the market, you need
to
maximize
the
likelihood
function
of
the
Gaussian
mixture.
That
is
you
tomaximize
maximize:the likelihood function of the Gaussian mixture. That is you need need
to
𝐾𝐾
to maximize:
to maximize:
𝐾𝐾 𝜋𝜋 𝐾𝐾 𝑓𝑓 (𝑋𝑋; 𝜃𝜃 )
𝑓𝑓(𝑋𝑋; 𝜃𝜃) = �
𝑘𝑘 𝑘𝑘
𝑘𝑘
(𝑋𝑋;
=𝜋𝜋�
(𝑋𝑋;
) 𝜃𝜃𝑘𝑘 )
𝑘𝑘=1
𝑓𝑓(𝑋𝑋; 𝑓𝑓(𝑋𝑋;
𝜃𝜃) =𝜃𝜃)
�
𝑘𝑘 𝑓𝑓𝜃𝜃
𝑘𝑘 𝑘𝑘
𝑘𝑘 𝑓𝑓𝑘𝑘𝜋𝜋
Where 𝜃𝜃 is the full set of parameters𝑘𝑘=1
and 𝜃𝜃
is
the
set
of parameters for state 𝑘𝑘.
𝑘𝑘
𝑘𝑘=1
of of
parameters
for state
𝜃𝜃 isfull
the
parameters
and
𝜃𝜃𝑘𝑘 isset
ofset
parameters
for state
𝑘𝑘.
WhereWhere
𝜃𝜃 isisthe
ofset
parameters
and 𝜃𝜃
Where
the
fullsetfull
set
of of
parameters
and
isthe
the
set
parameters
for 𝑘𝑘.
𝑘𝑘 is the
In
practice
this
should
only
be
optimized
by
the
EM
algorithm.
If
you
insist
on
state k.
In itpractice
thisbyshould
only
be
optimized
thealgorithm.
EM algorithm.
Ifinsist
yougood
insist
directly
numerical
optimization,
need
a very,
Indoing
practice
this should
only be
optimized
by thebyyou
EM
If youvery
on on
Indoing
practice
this
should
onlynumerical
be true
optimized
by the
algorithm.
Ifvery
youvery
insist
doing
it directly
by
younumber
need
a states
very,
This
being
if optimization,
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Page 7
separation,
it suffices
for illustrative
purposes.
separation,
but it but
suffices
for illustrative
purposes.
Estimation
Where 𝜃𝜃 is the full set of parameters and 𝜃𝜃𝑘𝑘 is the set of parameters for state 𝑘𝑘.
In practice this should only be optimized by the EM algorithm. If you insist on
doing it directly by numerical optimization, you need a very, very good
optimizer. This being especially true if you increase the number of states or the
of yourwe
system.
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present a practical example based on total returns of
An Example
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the S&P 500 index and a generic US 10Y government bond (constant time
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of 10 we
years).
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maturity of 10 years). We define two separate states of the market on the basis
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and a rising
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market. Note that we are well aware that this is a rather rudimentary
separation, but it suffices for illustrative purposes.
Tostart
start the
the analysis
analysis we
and
covariances
of the
two two
assets
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weestimate
estimatethe
thereturns
returns
and
covariances
of the
asin each
of the
two two
states.
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at least
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sets
in each
of the
states.
In order
to gain
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on
assumption
that
the
separate
states
are
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fact
normally
distributed,
we
use
the assumption that the separate states are in fact normally distributed, we
log-returns:
use
log-returns:
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
+6.00 −0.11
+1.15
= 𝑁𝑁 ��
�,�
��
𝑓𝑓 �
�
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
−0.11 +14.30
−0.62
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
+0.03
+5.50 +0.14
= 𝑁𝑁 ��
�,�
��
𝑓𝑓 �
�
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
+1.97
+0.14 +9.70
The“covariance”
“covariance”matrix
matrixis isspecified
specifiedas as
correlation
matrix,
the
The
thethe
correlation
matrix,
withwith
the stanstandard
deviations
in the
diagonal.
expected,equities
equitiesoutperform
outperform bonds
dard
deviations
in the
diagonal.
AsAs
expected,
bondsinin
the
bull
market
and
underperform
in
the
bear
market.
Note
that
equities
and
the bull market and underperform in the bear market. Note that equities
bonds
are negatively
correlated
only inonly
the bear
market.
and
bonds
are negatively
correlated
in the
bear market.
regard to
to the mixing
mixing probabilities,
is in
thethe
bullbull
market
51.5%
of
InInregard
probabilities,the
themarket
market
is in
market
51.5%
time
andand
in the
bearbear
market
48.5%
of theoftime.
ofthethe
time
in the
market
48.5%
the time.
Now, assume that the return distributions for each state and the mixing probabilities are a true representation of the full market. The first thing we want
to look at are the posterior probabilities for a range of different returns. Note
5
that this is an extension of the example given in the previous chapter, since
we now have two asset classes. Figures 5 and 6 show the posterior probabilities for the bull and bear market respectively, calculated for an appropriate
set of returns for equities and bonds.
Page 8
Figure 5: Posterior Probabilities of the Bull Market
Figure 6: Posterior Probabilities of the Bear Market
Note that with only two states, Figures 5 and 6 are mirror images of each
other. The figures show – not surprisingly – that the probability of the market being bullish increases with the return on equities.
Page 9
In spite of the apparent complexity of the figures, they serve to show that
any market can be assigned a probability when the number of states, the
mixing probabilities, and the state distributions are known (and well defined). In practice many investment professionals have their own view or
model specifying exactly these parameters. It should be noted that this approach helps you identify the implied probabilities of the market being in
either one of the states regardless of whether the parameters are estimated
or simply assumed.
As an utilization of our method, consider the following example. Calculate
15 day rolling returns and project these onto a monthly horizon. Compute
the implied probability of these being generated by either the bull market or
bear market. The results are presented in Figure 7.
Figure 7: Posterior Probability of the Two Markets Based on a Window of 15
Daily Observations
100
Bull Market
Bear Market
90
80
70
Probability, %
60
50
40
30
20
Sep-14
Jul-14
Aug-14
Jun-14
Apr-14
May-14
Feb-14
Mar-14
Jan-14
Dec-13
Oct-13
Nov-13
Sep-13
Jul-13
Aug-13
Jun-13
May-13
Apr-13
May-13
Jan-13
0
Mar-13
10
Jan-13
Development Over
Time
From the graph it is apparent that the probability of being in the bear market was high in September 2013 and late January 2014. These were periods
where equities underperformed bonds. They were also periods of increased
uncertainty around monetary policy and growth (in September 2013 uncertainty around the tapering of QE3 and in January 2014 uncertainty around
US growth due to weakening macroeconomic momentum).
In practice this information can be used to reduce risk if the probability of
being in a bear market increases. Obviously this resembles a momentum
strategy, but since it also incorporates the covariances of the system it is
a different signal than simply using first order moments (i.e. buying when
returns are positive and vice versa).
Page 10
A purely visual inspection of the Figure 7 seems to indicate that the bull
market probability is capped at around 80%. One might question why that
is, since some of the rolling returns are extremely positive. To illustrate and
discuss this point further, Figure 8 shows the same information as Figure 7,
just over a longer time horizon.
A purely visual inspection of the Figure 7 seems to indicate that the bull
market probability is capped at around 80%. One might question why that
is, since some of the rolling returns are extremely positive. To illustrate and
discuss this point further, Figure 8 shows the same information as Figure 7,
just over a longer time horizon.
Figure 8: Posterior Probability of the Two Markets Based on a Window of 15
Daily Observations
100
90
Bull Market
Bear Market
80
Probability, %
70
60
50
40
30
20
10
Q1-09
Q2-09
Q2-09
Q3-09
Q4-09
Q1-10
Q2-10
Q3-10
Q4-10
Q1-11
Q2-11
Q3-11
Q4-11
Q1-12
Q2-12
Q3-12
Q4-12
Q1-13
Q2-13
Q3-13
Q4-13
Q1-14
Q2-14
Q3-14
0
As can clearly be visualized, even over this new sample – covering the last
five years – the bull market never obtains a posterior probability in excess
of 80%.
The intuitive explanation hereof is that very large positive returns of equities are actually more likely in the bear market than in the bull market! This
being an artifact of the larger volatility in the bear market.
The same point can be stressed mathematically by looking once more at the
distributions of a bear and a bull market; Figure 9. For the point of illustration, we have increased the volatility of the bear market.
Page 11
Are the Implied
Probabilites Capped
Figure 9: Univariate distributions of a bull and a bear market
0.050
0.045
Bull Market
Bear Market
0.040
0.035
Density
0.030
0.025
0.020
0.015
0.010
0.005
0.000
-80
-60
-40
-20
0
Expected Return
20
40
60
If we look at the density functions conditioned on a positive return of 40%,
we see that the bear market is more probable than the bull market. This is
because the posterior probabilities are constructed by weighting the point
probabilities, very large positive and negative returns become more likely to
be assigned to the high volatility market. The latter which for practical purposes usually is the bear market. Hence, a cap in the posterior probability for
the bull market will exist in practice.
Illustration by a
Trading Strategy
To put our money where our mouths are we will test a trading strategy based
on implied probabilities. The approach is best described by the following
three rules:
1. If the probability of being in a bear market is above 30% the portfolio
consists of only bonds
2. If the probability of being in a bear market is below 30% the portfolio
consists of only equities
3. When the signal changes (i.e. the implied probability changes from above 30% to below 30% or vice versa) the portfolio composition is changed the next morning (so as to be sure to trade only on data available at
the time of trading).
Figure 10 shows the annualised return from this trading strategy, based on
data from January 1990 to September 2014, for a range of different window
lengths (i.e. the rolling window of observations used to compute the signal).
The red line indicates the return obtained by a long-only investment in equities (i.e. always fully invested in equities regardless of the signal).
Page 12
Figure 10: Return of Trading Strategy for Different Window Lengths
30
Annualized Return, %
25
20
15
10
5
0
0
5
10
15
20
Window Length (Trading days)
25
30
It appears that the optimal window length is between 10 to 15 days. Using
a window of this size seems to generate a reliable signal and generate an
excess performance. However, data inspection shows that the result is to a
large extend dominated by data from 2001 and 2008.
We also see that using a short window length appears not to be optimal.
This is likely due to a high level of noise in the daily observations and, therefore, a much more uncertain signal.
Before concluding, it should be noted that the results are very stylized. We
utilize only two asset classes (of which one is generic and therefore not
directly tradable) and we do not incorporate trading costs (e.g. bid-offer
spreads). Nevertheless, the results show that there may be some validity in
the approach of segregating the markets into a bull state and a bear state.
The examples above are perhaps a little simplified. Most investors have
multistate asset allocation models and, naturally, invest in more assets than
just equities and bonds. To illustrate the method in such a setting, Figure 11
shows the rolling posterior probabilities, based on a window length of 15
trading days, for a universe with four assets: bonds, equities, Investment
Grade bonds and High Yield bonds.
Page 13
A Multi-State,
Multi-Asset
Example
Furthermore, we divide the investment cycle into four states based on the
OECD Leading indicator:
• Early upturn: CLI is below 100 and rising
• Late upturn: CLI is above 100 and rising
• Early downturn: CLI is above 100 and falling
• Late downturn: CLI is below 100 and falling
Figure 11: Posterior Probability of the Four State Model Based on a Window
of 15 Daily Observations
100
Early Upturn
Late Upturn
Early downturn
Late downturn
90
80
Probability, %
70
60
50
40
30
20
Jul-14
Jun-14
May-14
Apr-14
Mar-14
Jan-14
Feb-14
Dec-13
Oct-13
Nov-13
Sep-13
Aug-13
Jul-13
Jul-13
Jun-13
May-13
Apr-13
Feb-13
Jan-13
0
Mar-13
10
As can be visualized, the figure is somewhat more complicated, but the story is in essence the same as the two state models. If nothing else the Figure
illustrates that the methodology can easily be expanded to multiple states
and asset classes.
Estimation of
Parameters
As a final example, say we want to estimate the mixing probabilities and
state distributions on the basis of the returns directly. That is, we disregard
the simplified asset allocation model we have made, and merely look at
what the data tells us.
Assuming there to be two states, we find that the bull market has a mixing
probability of 75%. In this market, equities deliver a positive monthly return
of 1.4% and bonds deliver a positive return of 0.5%. In the bear market the
return on equities is -1.3% and 0.76% on bonds.
Page 14
This paper shows how it is possible to derive information from the historical
market – or a view of the market – indicating the implied probabilities of the
market currently being in a predefined set of market states. The advantage
of this approach compared to other methods, employing only correlations,
is that we obtain a more intuitive interpretation of the results. More specifically, we restrict the resulting probability space to being strictly positive and
sum to one.
Additionally, the paper illustrates how a trading strategy based on this concept proves that an excess return may be obtained using only a very simply
trading rule.
Page 15
Conclusion