Implied Phase Probabilities. SEB Investment Management House View Research Group

Transcription

1 Implied Phase Probabilities SEB Investment Management House View Research Group 2015

2 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 Illustration by a Trading Strategy A Multi-State, Multi-Asset Example Estimation of Parameters...14 Conclusion...15 Editorial SEB Investment Management Sveavågen 8, SE-106 Stockholm Authors: Portfolio Manager, TAA: Peter Lorin Rasmussen Phone: peter.lorin.rasmussen@seb.se Portfolio Manager, Fixed Income & TAA: Tore Davidsen Phone: tore.davidsen@seb.dk Disclaimer This document produced by SEB contains general marketing information about its investment products. Although the content is based on sources judged to be reliable, SEB will not be liable for any omissions or inaccuracies, or for any loss whatsoever which arises from reliance on it. If investment research is referred to, you should if possible read the full report and the disclosures contained within it, or read the disclosures relating to specific companies found on Information relating to taxes may become outdated and may not fit your individual circumstances. Investment products produce a return linked to risk. Their value may fall as well as rise, and historic returns are no guarantee of future returns; in some cases, losses can exceed the initial amount invested. Where either funds or you invest in securities denominated in a foreign currency, changes in exchange rates can impact the return. You alone are responsible for your investment decisions and you should always obtain detailed information before taking them. For more information please see inter alia the simplified prospectus for funds and information brochure for funds and for structured products, available at If necessary you should seek advice tailored to your individual circumstances from your SEB advisor. Information about taxation. As a customer of our International Private Banking offices in Luxembourg, Singapore and Switzerland you are obliged to keep informed of the tax rules applicable in the countries of your citizenship, residence or domicile with respect to bank accounts and financial transactions. The bank does not provide any tax reporting to foreign countries meaning that you must yourself provide concerned authorities with information as and when required.

3 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. Introduction 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

4 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 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(xx bbbbbbbb ) = NN( 5,15) ff(xx bbbbbbbb ) = 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. Figure 1: Return Distributions of the Bull and the Bear Market Bull Market Bear Market Density Expected Return 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(XX bbbbbbbb ) + 67%ff(XX bbbbbbbb ) 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, Page 4

5 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 Density Expected Return 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

6 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 Density Expected Return To calculate the posterior probabilities (i.e. the probability of the market being in either of the two states) we use the following formula: pp kk = ππ kkff(xx μμ kk, Σ kk ) KK, kk = 1,, KK ππ kk ff(xx μμ kk, Σ kk ) kk=1 Where K denotes the number of states,, ff(xx μμ kk, Σ kk ) is the normal density of point x conditional on the state parameters, ers, and and ππis kk is the mixing probability of state k. Based on our two state distributions d the 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. Page 6

7 Figure 4: Probability Assigned to the Different States Bull Market Bear Market Probability, % 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 in the bull market and vice versa. 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. Estimation 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: KK ff(xx; θθ) = ππ kk ff kk (XX; θθ kk ) kk=1 f parameters and is the set of Where θθ is the full set of parameters and θθ kk i 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. Page 7

8 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 BBBBBBBBBB EEEEEEEEEEEEEEEE BBBBBBBB ff BBBBBBBBBB EEEEEEEEEEEEEEEE BBBBBBBB = NN , = NN , 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. 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 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

9 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

10 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. Development Over Time 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 Bull Market Bear Market Probability, % Jan-13 Jan-13 Mar-13 Apr-13 May-13 May-13 Jun-13 Jul-13 Aug-13 Sep-13 Oct-13 Nov-13 Dec-13 Jan-14 Feb-14 Mar-14 Apr-14 May-14 Jun-14 Jul-14 Aug-14 Sep-14 From the graph it is apparent that the probability of being in the bear market was high in September 2013 and late January 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

11 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. Are the Implied Probabilites Capped Figure 8: Posterior Probability of the Two Markets Based on a Window of 15 Daily Observations Bull Market Bear Market Probability, % 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 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

12 Figure 9: Univariate distributions of a bull and a bear market Bull Market Bear Market Density Expected Return 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

13 Figure 10: Return of Trading Strategy for Different Window Lengths Annualized Return, % Window Length (Trading days) 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 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. A Multi-State, Multi-Asset Example Page 13

14 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 Early Upturn Late Upturn Early downturn Late downturn 70 Probability, % Jan-13 Feb-13 Mar-13 Apr-13 May-13 Jun-13 Jul-13 Jul-13 Aug-13 Sep-13 Oct-13 Nov-13 Dec-13 Jan-14 Feb-14 Mar-14 Apr-14 May-14 Jun-14 Jul-14 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

15 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. Conclusion 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

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