«Abstract In this paper we examine the performance of the Markov Switching model with intra-regimes changes such as the bull market correction and ...»
Asset allocation under multiple regimes
Master’s Thesis in Quantitative Finance
In this paper we examine the performance of the Markov Switching model with
intra-regimes changes such as the bull market correction and bear market rallies. We
accommodate this short time rehearsals by imposing restrictions on the transition probability matrix. We compare the model with classic mean-switching and dynamic VAR
models in an asset allocation problem with diﬀerent number of regimes, initial states choices and asset distributions used in the estimation process. In an out-of-sample and bootstrap veriﬁcation we give evidence that the constrained model outperforms other models in terms of risk-adjusted returns in the long horizon above 2 years.
Keywords: regime switching, transition probability matrix, asset allocation 1 Introduction Investors and academics agree on the existence of short term trends of opposite direction to the main trend in the stock market. During bear or bull markets the prices of stock can increase or respectively decrease for a short period of time without necessarily meaning a change in the state of the economy. These phenomena called bull corrections and bear rallies might mislead an investor by implying a change in the economy main trend and impact its investments. We want to verify that by accommodating these short time rehearsals through component states of the bear and bull markets in our models, we can improve his allocation choices. We believe that an investor incorporating this knowledge can signiﬁcantly improve its results. It is in fact the case as we ﬁnd in an out-of-sample evaluation that this model outperforms classic regime-switching models in terms of risk adjusted returns for almost all tested horizons, especially in the long term (above 2 years).
Regime switching models have been within the scope of ﬁnancial academic interest for quite some time now. In the past most studies assumed linear dependencies between regressors and dependent variables such as asset returns. More recently, it has been proven that such dependencies are not linear and they do vary across time (Ang and Bekaert (2002a,b, 2004), Ang and Chen (2002), Garcia and Perron (1996), Guidolin and Timmermann (2005a,b, 2006, 2007), Guidolin and Hyde (2012) to cite a few). This is most clearly visible in stock returns where bull markets follow bear markets and so on, but it has been shown that it aﬀects not only equity returns, but also interest rates data (Gray (1996)) or macroeconomic fundamentals (Hamilton (1989)). These diﬀerent states of economy not only impact expected returns but also returns characteristics such as higher correlation in bear markets or higher correlation Erb et al. (1994), Campbell et al. (2002).
Although the method has been widely propagated due to Hamilton (1989), Ang and Bekaert (2002a) were the ﬁrst to use a simple 2 regime model to solve an asset allocation problem showing signiﬁcant improvement compared to the non regime dependent model. Ever since, many authors tried to augment the number of regimes for a better ﬁt of models for the data. Guidolin and Timmermann (2005a) investigate the economic implication of 3 regimes
- bear, bull and normal market - on UK stocks and bonds. The eﬀect is particularly large at short investment horizons and if ignored, the presence of such regimes leads to important welfare costs. Finally, Guidolin and Timmermann (2007) show that 4 regimes are required to capture the joint distribution of both equities and bonds. They are characterized as crash, slow growth, bull and recovery states. The authors additionally perform an out of sample forecast test to demonstrate the beneﬁts of accounting for the presence of these regimes in asset returns.
Other authors explore the presence of predictability in asset returns under regime switches.
The topic has been widely covered in a linear speciﬁcation. Authors like Barberis (2000) or Campbell and Thompson (2008) show the evidence for predictability of stock premium by variables such as the dividend yield, term spread or net equity issuance. On the other hand Guidolin and Ono (2006) include many macroeconomic fundamentals in a VAR(1) component and show that it is not regime dependent. The authors explain it by the fact that linkages between the macroeconomy and ﬁnancial markets are stable over time.
Maheu et al. (2012) is an important paper as it takes a diﬀerent approach to switching model extensions. Instead of augmenting the number of regimes, they point for the existence of bull market corrections and bear market rallies. Such high-frequency reversals could be modelled by imposing restrictions on the transition probability matrix. Thanks to these modiﬁcations they could get a better speciﬁcation of the 2 general regimes - the bear and bull market. We use a similar model in our work. Although the results of the paper prove that the model ﬁts the data well and gives accurate distribution forecasts the authors do not extend their analysis to any economic evaluation.
A ﬁrst signiﬁcant contribution of this paper is to show how the use of a model accommodating intra-regime changes can inﬂuence asset allocation framework and then to compare out-ofsample its performance with a set of classic MS models. Second, we test the presence of intra-regimes in monthly data instead of weekly time-frames adopted in Maheu et al. (2012).
We believe that monthly data are most frequently used by long term investors. Third, we are sceptical about the priors imposed in the Bayesian estimation of Maheu et al. (2012). Instead, we prefer to let the data speak and estimate the model in a frequentist approach. Finally, we use a joint distribution of stocks and bonds to estimate the regime distribution. This approach is then compared to classic stock only distribution and the distribution including the dividend yield. We perform a thorough analysis estimating 4 models with mean and covariances dependent of regimes and 4 regimes additionally including a VAR(1) component with the dividend yield.
Our work diﬀers from Kole and Dijk (2016) in several aspects. The authors perform a comparison of a set of regime dependent models both with constant and time-varying transition probabilities, but they limit themselves to models with just two and three regimes. In our opinion it is interesting to include 4 regime models into the analysis as Guidolin and Timmermann (2007) prove that they ﬁt the data very well. What is more, the authors only mention the speciﬁcation of Maheu et al. (2012) but do not include it in their evaluation, this is in fact something we want to explore the most. Next, the authors use rule based semi-parametric (they set a value for minimal changes in prices for the regime to switch) and parametric regime switching models, whereas we use a joint-distribution of both stocks and bonds. Finally, in their analysis the investor can choose from stocks and a risk free rate only, we expand the range of assets with long-term bonds.
We ﬁrst perform a static allocation based on the model estimates from the whole sample.
This test’s purpose is to answer the question of many long term investors ”what assets should I invest into?”. We set an allocation framework similar to the famous paper of Barberis (2000) results of which became a point of reference both for academia and investors. We compare the results with other regime switching models as well. The results are in opposition to the main results of Barberis. In fact, it is not always optimal to increase allocation to stocks with time, even when predictability is taken into account. In a linear framework, predictability from variables such as the dividend yield lowers risk in longer horizons, increasing the allocation to stocks. Regime switching has an opposite eﬀect, return innovations and future expected returns have a negative correlation leading to a reduced allocation. In other words, we know that the economy will not always remain in a given state. Even in a strong bull trend, we should be aware that the economy might ﬁnally fall into a bear state. Both eﬀects have an impact on asset allocation resulting in diﬀerent shapes of the allocation schemes depending on the initial state and time horizon.
We implement these ﬁndings into an economic evaluation in order to determine the best model. We choose between 10 models in total. The classical IID model, the linear non switching VAR model, models with regime switching mean returns and volatility in 4 regime conﬁgurations - 2,3 and 4 regimes and 4 regimes with bull rallies and bear market corrections and, ﬁnally, models in the same regime conﬁgurations but with explanatory variables in a VAR(1) conﬁguration.
We ﬁnd that the performance of regime dependent models often depend on the investment horizon. Adding regimes does not necessarily lead to better results, especially depending on the time horizon. Models with just 2 regimes perform better in short horizon than models with 3 or 4 regimes, which on the other hand give better results in the medium and long term. The main ﬁnding of the out-of-sample evaluation is that an elaborate extension of the model, with 4 regimes but the transition matrix modiﬁed to accommodate short timerehearsals, gives the best results of all the models. Our model not only manages to ﬁt the data very well, but also gives a great advantage in asset allocation. In short horizon the investor incorporates brief changes in asset return trends, whereas in the long term, the model give a better speciﬁcation of the two main regimes - the bull and bear market. Thanks to that, the allocation in this horizon is more stable, especially because of the fact that in the long run a good prediction of returns distribution dominates the model market timing abilities. Interestingly, even in the longest period the asset allocation diﬀer from the investor who ignores regime.
Despite these realistic implications the autoregressive component does not give economic advantages. Mean-switching models dominate their autoregressive counterparts both in testing and in the real life test of out-of-sample veriﬁcation. Keeping the number of variables at an acceptable level, these models prove to be suﬃciently complex to ﬁt the data and are simple enough to be robust on estimation errors.
A common question when increasing the number of regimes is whether the model does not overﬁt the data. The same issue has been raised by Guidolin and Timmermann (2007). In order to address that problem and perform a ﬁnal check on our constrained model, we use a historical bootstrap to compare 4 models - the IID model, a 2MS and 4 MS model and the constrained 4 regime model. Unlike the paper of Guidolin and Timmermann (2007) who use a parametric bootstrap based on the estimates of the 4 regime model, we use a historical bootstrap from Politis and Romano (1994) as it does not bias the results in favor of that model.In order to include dependence in the data we use a block bootstrap where the optimal block length is chosen based on the method described in Politis and White (2004).
The results show a good performance of both 4 regime models in the short run, however in the long run all models are outperformed in the long run by the constrained 4 regime model, which proves our model to allow a good data ﬁt resulting in a good return distribution forecast.
This paper is organized as follows. Section 2 describes the data. Section 3 covers the methodology of model construction, testing procedures and portfolio calculations. Section 4 presents the results of model estimation, their tests results and the resulting static allocation values. Section 5 evaluates the out-of-sample performance of the models. Section 6 extends the model veriﬁcation with a bootstrap. Section 7 concludes.
Our analysis concentrates on a US investor considering three classes of assets: stocks, bonds and cash. In our research we use the popular data set furnished by Goyal & Welch covering a wide variety of variables. For stock we use the S&P500 Index end of month values from the Center for Research in Security Press. The stock returns are continuously compounded returns. For bond we use the long term return on bonds which is made from a portfolio of long term bonds from Ibbotson’s Stocks, Bonds, Bills and Inﬂation Yearbook - it allows to maintain perpetuity even when some bonds were not issued for some periods of time. Finally for cash and the risk free rate used to obtain excess returns we use the ex post real T-bill rate calculated as the diﬀerence between the log return from 3-month T-bill and log inﬂation.
The T-bill rates have been taken from Bloomberg, whereas inﬂation values from the Federal Reserve Economic Data. We replace that variable from Goyal & Welch, as they do not give precise information on the assets used for their riskfree variable. Stock and Bonds returns are excess returns calculated over the T-bill rate. The dividend yield or the D/P ratio is calculated as the log of the dividend paid by companies from the index through the last 12 months (sum of dividends from t − 11 to t) divided by the index price. In order for an increased convergence, in the estimation process the D/P ratio is additionally divided by
100. Following the literature we use data after the Treasury Accord from 1951. Therefore our data set covers the period from January 1954 until December 2014 - the latest available update of Goyal & Welch. It gives us a sample o 732 observations. For the out of sample analysis we use a period of 30 years, from January 1985 to December 2014, covering among others the last ﬁnancial crisis. The table below presents the summary statistics of the data used calculated on the full sample. Data summery statistics are reported in Table 1 below.
Table 1. Summary statistic
The table presents the average, standard deviation, minimum, maximum and the ﬁrst order correlation of the ex post real T-bill returns, stocks excess returns, long-term bonds excess returns and the D/P ratio. The statistics are calculated on the full sample period from January 1954 to December 2014 and are reported in monthly units.