implied phase probabilities · the bear market. that is, if we observe the market over a very long...

2015

Implied Phase ProbabilitiesSEB Investment Management House View Research Group

Editorial

SEB Investment ManagementSveavågen 8,SE-106 Stockholm

Authors:

Portfolio Manager, TAA: Peter Lorin RasmussenPhone: +46 70 767 69 36E-mail: [email protected]

Portfolio Manager, Fixed Income & TAA: Tore Davidsen Phone: +45 33 28 14 25 E-mail: [email protected]

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Disclaimer

Table of Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3The Market and Gaussian Mixture Models . . . . . . . . . . . . . . . . . . . . . . .4Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8Development Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Are the Implied Probabilities Capped. . . . . . . . . . . . . . . . . . . . . . . . . . . 11Illustration by a Trading Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12A Multi-State, Multi-Asset Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Estimation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Editorial

IntroductionIs 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 lea-ding 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 met-hodology, 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 mo-del 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 Mix-ture 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.

The Market and Gaussian Mixture Models

As stated we focus on Gaussian Mixture models. To illustrate the intuition behind these, we start by looking at a single synthetic asset class; call it equities if you like. We pretend that the “market” can be in one of two sepa-rate states: a bull market with a positive expected return and low volatility, and a bear market with a negative expected return and high volatility. As-suming the returns to be normally distributed, the two states are defined as:

Introduction 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.

The Market and Gaussian Mixture Mod-els

As stated we focus on Gaussian Mixture models. To illustrate the intuition behind these, we start by looking at a single synthetic asset class; call it equities if you like. We pretend that the “market” can be in one of two separate states: a bull market with a positive expected return and low volatility, and a bear market with a negative expected return and high volatility. Assuming the returns to be normally distributed, the two states are defined as:

𝑓𝑓(𝑋𝑋𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) = 𝑁𝑁(−5,15) 𝑓𝑓(𝑋𝑋𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) = 𝑁𝑁(+10,10)

That is, in the bear market the asset delivers an expected negative return of 5% and in the bull market the asset delivers an expected positive return of 10%. The return distributions of the two markets are graphed in Figure 1.

That is, in the bear market the asset delivers an expected negative return of 5% and in the bull market the asset delivers an expected positive return of 10%. The return distributions of the two markets are graphed in Figure 1.

Figure 1: Return Distributions of the Bull and the Bear Market

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Bull MarketBear Market

Now assume that the bull market is twice as likely to be observed over time as the bear market. That is, if we observe the market over a very long time horizon we should see the bull market in 2/3 of the periods and the bear market in the remaining 1/3. Put differently, 33% of the time we are in the bear market, and the other 67% of the time we are in the bull market. This time weighted market is a Gaussian mixture, defined as:

2

Figure 1: Return Distributions of the Bull and the Bear Market

Now assume that the bull market is twice as likely to be observed over time as the bear market. That is, if we observe the market over a very long time horizon we should see the bull market in 2/3 of the periods and the bear market in the remaining 1/3. Put differently, 33% of the time we are in the bear market, and the other 67% of the time we are in the bull market. This 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 mathematically 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 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

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The probabilities denoting the frequencies of the different states are mathe-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 more likely to observe one market over the other.

The resulting mixed distribution is graphed in Figure 2. This can be inter-preted 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 Fi-gure 2.

Figure 2: Return Distribution of the Full Market

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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 sty-lized 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

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 as-sess 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

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To calculate the posterior probabilities (i.e. the probability of the market being in either of the two states) we use the following formula:

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𝑝𝑝𝑘𝑘 =𝜋𝜋𝑘𝑘𝑓𝑓(𝑥𝑥|𝜇𝜇𝑘𝑘 ,Σ𝑘𝑘)

∑ 𝜋𝜋𝑘𝑘𝑓𝑓(𝑥𝑥|𝜇𝜇𝑘𝑘 ,Σ𝑘𝑘)𝐾𝐾𝑘𝑘=1

, 𝑘𝑘 = 1, … ,𝐾𝐾

Where 𝐾𝐾 denotes the number of states, 𝑓𝑓(𝑥𝑥|𝜇𝜇𝑘𝑘 ,Σ𝑘𝑘) is the normal density of point 𝑥𝑥 conditional on the state parameters, and 𝜋𝜋𝑘𝑘 is the mixing probability of state 𝑘𝑘. Based on our two state distributions and the observation of a return of 3%, we find that there is a 63% probability that the market is in the bull state and a 37% probability that the market is in the bear state. As an alternative representation of the posterior probabilities, Figure 4 graphs the outcome of a range of point estimates. Figure 4: Probability Assigned to the Different States

The figure should be read as follows: For each observed return there is a

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Where K denotes the number of states,

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, 𝑘𝑘 = 1, … ,𝐾𝐾



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is the normal density of point x conditional on the state parameters, and

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, 𝑘𝑘 = 1, … ,𝐾𝐾



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is the mixing probabi-lity of state k. Based on our two state distributions and the observation of a return of 3%, we find that there is a 63% probability that the market is in the bull state and a 37% probability that the market is in the bear state.

As an alternative representation of the posterior probabilities, Figure 4 graphs the outcome of a range of point estimates.

EstimationInstead of merely setting the mixing probabilities and state distribution pa-rameters they can be estimated directly from the market. That is, provided you have a set of actual observed returns, it is possible to determine which parameters – mixing probabilities included – best describe the data. Follo-wing this approach, you are actually conducting a full cluster analysis.

So forth the parameters are to be estimated directly from the market, you need to maximize the likelihood function of the Gaussian mixture. That is you need to maximize:

5

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 in the bull market and vice versa.

Estimation Instead of merely setting the mixing probabilities and state distribution parameters they can be estimated directly from the market. That is, provided you have a set of actual observed returns, it is possible to determine which parameters – mixing probabilities included – best describe the data. Following this approach, you are actually conducting a full cluster analysis. So forth the parameters are to be estimated directly from the market, you need to maximize the likelihood function of the Gaussian mixture. That is you need to maximize:

𝑓𝑓(𝑋𝑋;𝜃𝜃) = �𝜋𝜋𝑘𝑘

𝐾𝐾

𝑘𝑘=1

𝑓𝑓𝑘𝑘(𝑋𝑋;𝜃𝜃𝑘𝑘)

Where 𝜃𝜃 is the full set of parameters and 𝜃𝜃𝑘𝑘is 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 dimension of your system.

An Example In the following, we present a practical example based on total returns of the S&P 500 index and a generic US 10Y government bond (constant time to maturity of 10 years). We define two separate states of the market on the basis of the OECD amplitude adjusted leading indicator for the US. A falling indicator defines a bear market, and a rising indicator defines a bull market. Note that we are well aware that this is a rather rudimentary separation, but it suffices for illustrative purposes. To start the analysis we estimate the returns and covariances of the two assets in each of the two states. In order to gain at least some tractability on the assumption that the separate states are in fact normally distributed, we use log-returns:

𝑓𝑓 � 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐵𝐵�𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵

= 𝑁𝑁��+1.15−0.62� , �+6.00 −0.11

−0.11 +14.30��


= 𝑁𝑁��+0.03+1.97� , �+5.50 +0.14

+0.14 +9.70��

The “covariance” matrix is specified as the correlation matrix, with the standard deviations in the diagonal. As expected, equities outperform bonds in the bull market and underperform in the bear market. Note that equities and bonds are negatively correlated only in the bear market. In regard to the mixing probabilities, the market is in the bull market 51.5% of the time and in the bear market 48.5% of the time.

Where

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𝐾𝐾

𝑘𝑘=1





= 𝑁𝑁��+1.15−0.62� , �+6.00 −0.11

−0.11 +14.30��


= 𝑁𝑁��+0.03+1.97� , �+5.50 +0.14

+0.14 +9.70��


is the full set of parameters and

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𝐾𝐾

𝑘𝑘=1





= 𝑁𝑁��+1.15−0.62� , �+6.00 −0.11

−0.11 +14.30��


= 𝑁𝑁��+0.03+1.97� , �+5.50 +0.14

+0.14 +9.70��


is the set of parameters for state k.

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 dimension of your system.

Figure 4: Probability Assigned to the Different States

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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 pro-bability to be in the bull market and vice versa.

In the following, we present a practical example based on total returns of the S&P 500 index and a generic US 10Y government bond (constant time to maturity of 10 years). We define two separate states of the market on the basis of the OECD amplitude adjusted leading indicator for the US.

A falling indicator defines a bear market, and a rising indicator defines a bull market. Note that we are well aware that this is a rather rudimentary separa-tion, but it suffices for illustrative purposes.

To start the analysis we estimate the returns and covariances of the two as-sets in each of the two states. In order to gain at least some tractability on the assumption that the separate states are in fact normally distributed, we use log-returns:

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𝐾𝐾

𝑘𝑘=1





= 𝑁𝑁��+1.15−0.62� , �+6.00 −0.11

−0.11 +14.30��


= 𝑁𝑁��+0.03+1.97� , �+5.50 +0.14

+0.14 +9.70��


The “covariance” matrix is specified as the correlation matrix, with the stan-dard deviations in the diagonal. As expected, equities outperform bonds in the bull market and underperform in the bear market. Note that equities and bonds are negatively correlated only in the bear market.

In regard to the mixing probabilities, the market is in the bull market 51.5% of the time and in the bear market 48.5% of the time.

Now, assume that the return distributions for each state and the mixing pro-babilities 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 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 probabi-lities for the bull and bear market respectively, calculated for an appropriate set of returns for equities and bonds.

An Example

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 mar-ket being bullish increases with the return on equities.

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

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From the graph it is apparent that the probability of being in the bear mar-ket 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 uncer-tainty 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).

Development Over Time

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 de-fined). In practice many investment professionals have their own view or model specifying exactly these parameters. It should be noted that this ap-proach 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.

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.

Are the Implied Probabilites Capped

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

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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 equi-ties 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 illustra-tion, we have increased the volatility of the bear market.

Figure 9: Univariate distributions of a bull and a bear market

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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 pur-poses usually is the bear market. Hence, a cap in the posterior probability for the bull market will exist in practice.

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 abo-ve 30% to below 30% or vice versa) the portfolio composition is chan-ged 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 equi-ties (i.e. always fully invested in equities regardless of the signal).

Illustration by a Trading Strategy

Figure 10: Return of Trading Strategy for Different Window Lengths

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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, there-fore, 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.

A Multi-State, Multi-Asset

Example

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.

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

0

10

20

30

40

50

60

70

80

90

100

Prob

abili

ty, %

Jan-

13

Feb-

13

Mar

-13

Apr-

13

May

-13

Jun-

13

Jul-1

3

Jul-1

3

Aug-

13

Sep-

13

Oct

-13

Nov

-13

Dec

-13

Jan-

14

Feb-

14

Mar

-14

Apr-

14

May

-14

Jun-

14

Jul-1

4

Early UpturnLate UpturnEarly downturnLate downturn

As can be visualized, the figure is somewhat more complicated, but the sto-ry 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.

ConclusionThis 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 specifi-cally, 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 con-cept proves that an excess return may be obtained using only a very simply trading rule.

implied phase probabilities · the bear market. that is, if we observe the market over a very long...

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