1. Introduction and literature review

Among the most relevant tasks in risk measurement and portfolio management activities, is the proper expected return and risk exposure parameters. This has been done since the seminal proposal of ^{Markowitz (1959}, ¹⁹⁵²⁾. In order to measure the portfolio’s expected return, several methods have been used, such as the arithmetic or exponential mean and even some more robust factor mean methods (^{Alexander, 2002}, ^{Ang & Bekaert, 2002a}, ^{Sharpe, 1963}, ¹⁹⁶⁴⁾. In these last models, the expected return is not a fixed, but time-varying parameter.

Some other extensions to the expected mean can be found with the use of the ARMA⁴ or ARMAX models, in which the stochastic process that generates the return time series ( rt ) is determined given the past (lagged) values of the returns and also the lagged values of the residuals ( εt ). Also, if it is necessary to the analyst, in this model can be included the values of other non-lagged exogenous factors such as social or Economic variables ( xk,t ):

As a methodological note, in the present paper we will follow the Financial Econometrics and asset pricing practice of measuring the returns at t ( rt ) with the continuous time method as follows:

Given the expected return ( μ ) calculation, either with an unconditional mean or a conditional one, such as a factor, ARMAX or the ARMA one in (1), it is of interest for the investor to proxy the risk exposure. A departing method is the “conventional” or constant variance method ( σt2=∑t=1T‍(rt-μ)⋅N-1 ) with a fixed value at t. Following this method and as an extension to the rationale of the ARMA models, such as (1), ^{Engle (1982)} and ^{Bollerslev (1987)} made a significant breakthrough with their time varying GARCH⁵ model:

In the previous expression, the variance ( σt2 ) value at t is determined with the lagged squared residuals of the mean value (ARCH term) and the lagged values of the variance ( σt−q2 ) in (3). This last term is known as the GARCH term. Departing from this, (3) is an ARCH model when its functional form is σt2=σ0+∑p=1P‍βp⋅εt2 or a GARCH one when all the terms in (3) are used.

Given the time varying features of the GARCH model, several improvements in asset pricing and risk measurement were developed in the Financial industry. This is in order to determine either a proper asset pricing valuation for investment purposes or the proper risk measurement for capital adequacy practices. This type of model lead to more reliable and fitted values of the risk exposure during distress time periods. Departing from the symmetric GARCH rationale of (3), several extensions for asymmetric or leverage effects have been made. Among the most used and well-known, we can mention the EGARCH model of ^{Nelson (1991)} or the GJR-GARCH of ^{Glosten, Jaganathan and Runkle (1993)}. Also, the extensions have been made in the log-likelihood function (LLF) by using other probability density functions such as the t-Student, the asymmetric t-Student or the generalized error distribution.

An interesting result that departs from (4) if the fact that the sum ∑p=1P‍βp+∑q=1Q‍γp lead to a concept known as “persistence”. This concept means that, if ∑p=1P‍βp+∑q=1Q‍γp≈1 , the high volatility levels observed at t , will persist in longer future time periods, an issue that is proper of the behavior of the return time series of financial assets. Among the most relevant explanations to this issue, we mention the works of ^{Dueker (1997)}, ^{Lamoureoux and Lastrapes (1990)}; ^{Hamilton and Susmel (1994)}, ^{Klaassen (2002)} and ^{Hass, Mitnik and Paolella (2004)} who suggest that the high persistence is due to the fact that the stochastic process must be modeled with multimodal probability density functions in which there are =1,2, … ,S regimes or states of nature, also in which there are s location (expected return) and s scale (risk exposure or variance) parameters.

Departing form the seminal proposal of the Markov-Switching (henceforth MS) models of ^{Hamilton (1989}, ¹⁹⁹⁴⁾, the expected return (mean) and risk exposure (variance) can be estimated for S states. Departing from this rationale, an ARMA model, such as (1), can be expressed as a MS-ARMA model as follows:

Given the parameter vector θ=[αs,βs,γs] , the MS model has some extra parameters or outputs in θ , such as the smoothed probability ξs,t of being in the S at t . This along with the transition probabilities ( πi,j=Pst=i|st-1=j,θ,rt,θ=[αs,βs,γs,σs2,ξs,t] ) of transiting from one regime s=i at t , to another one s=j in t+1 . As noted, this behavior is one of a latent or unobservable Markovian chain (first order Markovian chain) and these transition probabilities can be arranged in a transition probability matrix Π :

One limitation of the MS model in (5) is that it assumes that the variance in each regime is fixed trough time. Departing from this assumption, ^{Hamilton y Susmel (1994)}, ^{Klaassen (2002)} and ^{Haas, Mitnik and Paolella (2004)} propose to extend the GARCH model in (4) to a s regime version. That is, they propose and developed the mathematical foundations of a Markov-Switching GARCH (MS-GARCH) model as the general form of the next expression:

Given their dynamic nature, their smoothed probabilities of being in a given s regime at t and their transition probabilities, MS models have been used in several applications. Among these, we find the crisis time periods identification and spillover effects in financial markets. With this rationale, we found the works of ^{Ang and Bekaert (2002b}, ^2002c), ^{Kritzman, Page and Turkington (2012)}, ^{Klein (2013)}, ^{Areal, Cortez and Silva (2013)}, ^{Zheng y Zuo (2013)}, and the one of ^{Hauptmann et. al. (2014)} (among others) who test the use of MS models to estimate the performance of stock markets in developed countries such as France, Germany, Japan, Switzerland, the U.K., or the U.S. In their results, they found evidence in favor of using two or three regime MS models. In other type of securities, ^{Alexander and Kaeck (2007)}, ^{Castellano and Scacia (2014)} and ^{Ma, Deng and Ho (2018)} study the appropriateness of the use of MS models in credit default swaps (CDS) and their spillover effect in FX, stock or oil markets.

For the review of the use of MS models in Emerging countries’stock markets, we mention the work of ^{Zhao (2010)}, ^{Walid et. al. (2011)}, ^{Walid and Duc Khuong (2014)}, the one of ^{Rotta and Valls-Pereira (2016)}, ^{Mouratidis et. al. (2013)}, ^{Miles and Vijverberg (2011)}; ^{Lopes and Nunes (2012)}, ^{Kanas (2005)}, ^{Álvarez-Plata and Schrooten (2006)}, ^{Parikakis and Merika (2009)}, ^{Girdzijauskas (2009)}, ^{Dubinskas and StungurienÄ- (2010)}; ^{Kutty (2010)}, ^{Dufrénot, Mignon and Péguin-Feissolle (2011)} y ^{Ahmed et. al. (2018)}. These Works found appropriate the use of MS models in the stochastic process of these markets, along with the application of these to measure the spillover effect in FX (given the monetary policy decisions) and stock markets.

Complementary to the previous works and of special interest for the present paper, we found the works of ^{Brooks and Persand (2001)}, ^{Ang and Bekaert (2002b}, ^2002c), ^{Kritzman, Page and Turkington (2012)}, ^{Hauptmann et. al. (2014)} and ^{De la Torre, Galeana and Álvarez-García (2018)}. These papers study the application of MS models in active portfolio management decisions, against a buy and hold or passive investment (index replication) strategy.

The rationale behind the investment decision with the use of MS models is found in ^{Brooks and Persand (2001)} and ^{Ang and Bekaert (2002b}, ^2002c). These authors use the active investment strategy in stock indexes of developing countries such as the U.S. and the U.K., along with the use of these in internationally diversified portfolios, given the use of a MS covariance matrix.

Following these two works, we found the results of ^{Kritzman, Page and Turkington (2012)}, ^{Hauptmann et. al. (2014)} and ^{De la Torre, Galeana and Álvarez-García (2018)} closely related to the present paper. The last work test, in part, the application of MS models in the investment process of emerging stock markets (Mexico).

With this brief literature review, we found the next research topics of interest to develop herein:

Departing from these reviews of interest, we want to extend these and their results about the use of MS-GARCH models for investment purposes in these countries (the Andean ones). Our main interest is to probe that the use of either MS-ARCH or MS-GARCH models, from a U.S. dollar-based investor perspective, generates alpha or extra returns against a buy and hold strategy in the three countries of the Morgan Stanley Capital International (MSCI) Andean index. In order to make this proof, we will test the next active investment strategy:

We are using the MSCI index family by the fact the it is a well-known and widely used benchmarking company for investment purposes (^{Bodie, Kane, Marcus, 2014}, ^{Maggin, Tuttle, Pinto, & McLeavey, 2007}). The Global Industry Classification Standard used by ^{MSCI (2018b)}, leads to a proper country and industry classification and aggregation method for the management of internationally diversified portfolios. Also, we decided to test the use of MS-GARCH models in the countries of the Andean region because these countries have shown an over-performance in relation to the other countries of the Emerging markets MSCI index. Another reason that lead us to make our tests in these countries is the aforementioned fact that little has been written about the use of MS models in that regions.

In order to perform our tests, we will run a weekly discrete event simulation from January 1st , 2000 to January 30th , 2019 (a total of 996 weeks) in which a theoretical investor will execute the aforementioned investment strategy.

Departing form these motivations and given our main interest herein, we structured the paper as follows: in the next section we will briefly discuss the MS-GARCH model and we will describe how has been used in the investment process. In the third section we will present a description and Statistical summary of the input data, along with the description of the pseudocode of our simulations. In the fourth section we present a review of our results and finally, in the fifth section, we present our conclusions and guidelines for further research.

2. The MS-GARCH model and its use in active investing

The MS-GARCH model used herein is the one with the functional form given in (6). In order to estimate it in a feasible way and as ^{Haas, Mitnik y Paolella (2004)} suggest, the MS-GARCH model must be inferred only from the residuals. Departing from the fact that the results in the present paper are among the first tests of the use of MS-GARCH models in the Andean stock markets, we assumed that the residuals must be calculated by detrending the return time series with the arithmetic mean ( εt=rt−μ ). From the estimated MS models, we filtered and then used the smoothed probabilities (Gaussian and t-Student) of being in the s regime at t . These were estimated as follows ( νs are the Markov-Switching degrees of freedom):

These smoothed probabilities, determined from the filtered probabilities (7) and (8) with ^{Kim’s (1994)} algorithm, led us to a stable probability or mixing law πs of the next Log-Likelihood function to maximize in the estimation process⁶:

In order to estimate the parameter vector θ in (9), we used the ^{Viterbi (1967)} maximization algorithm and from all the smoothed probabilities, we are interested in the observed values of ξs,t at t=T (the last date) and in the transition probabilities matrix Π . With these parameters, we estimated (forecasted) the smoothed probabilities ξs,t+1 of being in each regime at t+1 :

Given these probabilities and by following the related literature in time series analysis (^{Ang & Bekaert, 2002b}, ^{Brooks & Persand, 2001}, ^{Hamilton, 1989}, ¹⁹⁹⁰, ¹⁹⁹⁴, ^{Hauptmann et al., 2014}, ^{Kritzman et al., 2012}), we focused our attention in the probability of the high-volatility regime ( ξs=2,t+1 ). With it, the investor was able to determine if she will be in the high-volatility regime at t+1 , by using the next indicator function:

This function will be used in the pseudo algorithm that we will describe in the next section.

3. Simulation method description

4. Simulations results

As a departing point, we show, in Table 4, the performance summary of a passive investment made in each stock index and in the risk-free asset. As shown, the risk-free asset leads to a 38.38% accumulated return (a yearly 2.11%). Also, the three stock indexes paid higher accumulated returns (alpha) with a 145.20% (7.99 yearly) paid by the Chilean index, 730.94% (40.23%) in Colombia, and 969.06

Of special interest are the cases of the risk exposure metrics. Either the conditional value at risk (CVaR) and the max drawdown show significant potential losses in the three stock indexes. This is a result that we expect to reduce by performing active trading activities with the simulation pseudo algorithm of section 2.1.

In order to test this last statement, we present in Table 5 the results of the active trading strategy made (with the each of the 6 MS models of interest) in the Chilean stock index. As noted, the use of the t-Student constant MS and the Gaussian MS-GARCH variance models led to alpha generation in comparison to the passive strategy in the same stock market. From these two models, the use of the Gaussian MS-ARCH model led to the highest accumulated return with a 351.44% (19.34% yearly) versus a 145.20% (7.99%) of the buy and hold strategy. Also, and as theoretically expected, the risk exposure is considerably lower than the passive strategy, being the use of the Gaussian MS-GARCH the case that lead to the lowest risk exposure with a -6.88% of potential loss (measured with the Gaussian MS-ARCH model). Complimentary to this, the Sharpe ratio shows improvements against the one observed in the passive strategy. This is noted in practically all the six MS models.

By accepting that the Gaussian MS-GARCH model is the best for the active investing of a U.S. dollar-based investor in Chile, we believe that the observed improvements are because the MS-GARCH model is more accurate to forecast the probability of the high-volatility regime. With this, the fourth step in the simulation’s pseudo algorithm lead to more suitable investment decisions at t and to a better performance and risk exposure values in the simulated portfolio.

In order to proof this statement, we present in Figure 1 the historical performance of the six simulated active portfolios versus the passive or buy and hold strategy (shaded area). In the lower panel of this Figure we also present the historical investment level in the risky asset (stock index). As noted, in the most recent financial crisis episode (July 2007 to January 2009)⁸ the simulation led to a better performance in the Gaussian MS-GARCH scenario. This happened by the fact that the simulated investor sold the index in most of that time period and bought the risk-free asset, leading to a lower risk exposure. This result is also observed during the European debt crisis of 2013 and the commodity price fall in 20013 and 2014.

In a similar fashion to the previous country, we present in Table 6 the performance results for the Colombian stock market. In this case the Gaussian MS-ARCH model is the one that led to better performance results with an accumulated return of 1,236.37% (68.03% yearly). This return is higher than the 730.94% (40.23% yearly) of the passive strategy and also the use of this MS model led to better risk exposure metrics and mean-variance efficiency (Sharpe ratio). This is result is contrary to the observed in Table 3 in which the use of the Gaussian MS-GARCH (and not MS-ARCH) is better for time series fitting.

The performance result in the Colombian case is due, similar to the Chilean one, to the fact that the Gaussian MS-ARCH model is more sensitive to forecast the probability of being in the high volatility regime at t+1 .

This is shown in Figure 2, in which the historical performance of the Gaussian MS-ARCH model shows a more stable and continuous trend during the simulations and an interesting performance during the 2015-2019 period. A period in which the passive strategy had a contraction. As noted, during those weeks, the simulated portfolio with the Gaussian MS-ARCH model was rebalanced from the risky asset to the risk-free one.

As methodological note that will apply to the next stock index (Peru) it is noted in Table 5 that the t-Student constant variance MS model was not feasible in some weeks, given the estimation algorithm. Departing from this, we excluded this index from the simulations and show the legend “not feasible” in that Table.

Finally, we present in Table 6 the results of the simulations made in the Peruvian stock market. For this specific case, none of the simulated portfolios lead to a better result or alpha generation than the passive one. Given this result, we observe that it is better, for a U.S. dollar-based investor, to perform a passive investment strategy in this specific stock market.

Among the potential causes of this result is the fact that the Peruvian stock market is highly exposed to the performance of commodity prices (basic and precious metals) and also to the local financial industry. Therefore, the MS analysis should incorporate other exogenous factors, different to the historical return value of the stock index. As a corollary of our results, we observed that the use of an active investment strategy leads to alpha generation if the U.S. dollar-based investor uses a Gaussian MS-GARCH model in a weekly rebalanced portfolio in the Chilean market and if she does it with a Gaussian MS-ARCH model in the Colombian one. Also, the use of the MS, MS-ARCH or MS-GARCH models does not generate alpha or extra return from a passive (buy and hold) strategy in the Peruvian case.

5. Conclusions

Markov-Switching (MS) models were proposed by ^{Hamilton (1989}, ¹⁹⁹⁰, ¹⁹⁹⁴⁾, in order to model time series in which the stochastic process has an s≥2 number of regimes or states of nature. That is, the stochastic process is not generated with a unimodal or single-regime probability density function but with a s multimodal one that has s location (usually mean) and s scale (usually variance) parameters.

At the time of writing this paper, several applications of these models have been proposed and tested. Among the most related examples are the forecast of the probability that a given Economy will be in a recession regime at t+k. This along with the proper modelling of multivariate MS models for spillover effects among economies and financial markets. In a parallel way, there are tests about the use of MS models in the performance of Credit Default Swaps (CDS) and the contagion effect that their high-volatility regime has in other credit, stock or currency markets.

The Econometric application of these studies has been done in several developed countries such as the European Monetary Union countries or in specific countries such as France, Germany, Japan, Switzerland, the United Kingdom and the United States. Also there have been several applications and tests in developing countries such as Argentina, Brazil, China, Mexico, some Arab countries and South Africa. Departing from this bibliometric review, little has been written about the application of MS models in the stock markets of the Andean region. These countries are Emerging economies, given the current Morgan Stanley Capital International (MSCI) classification method, and had received no attention for MS modelling applied in investing.

The application of MS models in investment decisions is also an issue that has had little review in the current literature. The first works that study this use are the pioneering works of ^{Brooks and Persand (2001)} and ^{Ang and Bekaert (2002a)} or the extensions and particular tests of ^{Kritzman, Page and Turkington (2012)}, ^{Hauptmann et. al. (2014)} or ^{De la Torre, Galeana and Álvarez-García (2018)}. These last works test the use of MS investment strategies in portfolios and stock indexes of developed countries and only the work of ^{De la Torre, Galeana and Álvarez-García (2018)} makes a test in a developing stock market: Mexico.

Departing from this scant literature about the use of MS models for investment purposes in Emerging stock markets, we want to extend the current literature to the test of the use of two-regime MS models in the three countries that are members of the MSCI Andean stock index: Chile, Colombia and Peru (^{MSCI Inc., 2012}). We also want to test these three countries by the fact that these are members also of the well-known MSCI Emerging markets index (^{MSCI Inc., 2018a}). An benchmark that is also a member of the MSCI world index (^{MSCI Inc., 2018b}). Our main interest herein was to test, from a U.S. dollar-based perspective, if the use of two-regime MS models in these three markets is appropriate for active investment.

Complementary to our main interest, we want to extend the literature about the use of MS models in active investing. We did this by using a time-varying variance with either the use of a MS autoregressive conditional heteroskedastic (MS-ARCH) variance or the use of a MS generalized autoregressive conditional heteroskedastic (MS-GARCH) one.

In order to test the usefulness of MS-ARCH or MS-GARCH models for active investing in these three countries, we assumed a two-regime (Gaussian or t-Student distributed) stochastic process in the three stock indexes. Departing from this, we tested the next investment strategy:

By performing weekly discrete event simulations from January 2000 to January 2019 (a total of 996 weeks), we estimated each week (with data from June 1998) a MS, MS-ARCH or MS-GARCH model at t (with Gaussian or t-Student probability density function). With this estimation, we forecasted the probability of being in the high-volatility regime at t+1 and executed the previous investment strategy. Our results suggest that the use of the Gaussian MS-GARCH model in the Chilean stock market leads to the generation of alpha or extra returns and lead to the best performance in the simulations. Also, the Gaussian MS-ARCH model lead to the best performance and alpha generation in the Colombian market but, for the case of the Peruvian one it is preferable to perform a passive investment strategy, by the fact that the use of any of the MS models tested herein leads to a poorer performance than a buy and hold method.

Given the assumptions used in our simulations and the limitations found in these, we suggest extending the present research in the next aspects:

With the results that we present in this paper (the test about the use of MS-GARCH models) we hope to contribute to the literature about the use of MS models for investment decisions, along with our review of the benefits of investing actively in the stock markets of the Andean region.