How the use of Markov-Switching Sharpe Ratios can improve Mexican Pension Funds Investment Decisions

* Facultad de Contaduría y Ciencias Administrativas Universidad Michoacana de San Nicolás de Hidalgo oscar.delatorre.torres@gmail.com, Orcid: 0000-0001-9281-974X ** EGADE Business School, Tecnológico de Monterrey. Roberto.santillan@itesm.mx, ORCID: 0000-0001-5162-1403 ** Facultad de Contaduría y Administración Universidad Nacional Autónoma de México Francisco_lopez_herrera@yahoo.com.mx, Orcid: 0000-0003-2626-9246 AbstRAct Towards the end of the 20th century, as part of the encompassing structural reforms that modernized the Mexican economy after decades of an obsolete imports-substitution model, overregulated economic sectors, and state-controlled productive sectors, the creation of an individual savings account pension system to replace the “pay-as-yougo” anachronic pension system prevalent since 1943, was necessary given the country’s demographic trends. The analysis presented in this paper uses a Markov-switching model to obtain the Sharpe ratio of different SIEFOREs portfolios for different subperiods and volatility regimes (normal and crisis). The results confirm that not all SIEFOREs are good (or bad) URL: estocastica.azc.uam.mx


Introduction
D uring the last three decades of the 20 th century, the Mexican economy experienced significant structural reforms. There was a complete overhaul of the state-owned productive sector during the 1980s, that reduced its direct participation in the production of goods and services; the economy was opened to foreign trade and investment (Mexico joined the General Agreement on Trade and Tariffs in 1985); and, there was an encompassing deregulation and privatization of different economic sectors (transportation, mining, communications, etc.). These structural reforms, aimed to the modernization of the economy, faced severe headwinds due to the collapse of oil prices at a time when exports of that product represented a large percentage of Mexican Exports (80% in 1982) and a very significant component of the government tax revenues (30 %), in addition to a devastating earthquake in Mexico City and other nearby cities which destroyed critical urban infrastructure, added to the environmental uncertainty and resulted in very high inflation and very slow economic growth.
During the last decade of the 20 th century the government realized that the traditional pay-as-you-go pension system faced an increasing long-term sustainability challenge due to changing demographics. The demographic growth rate had decreased from 3.5% annually during the 1960s to 2.4% by the early 1980s, 2% at the beginning of the 1990s, and only 1.5% in the first years of the 21 st century. 1 Without a major reform, an increasingly smaller base of contributors would support a larger aging population. That is, as the base of the pyramid was shrinking in size, it was gradually being transformed into an "inverted" demographic pyramid.
After careful analysis of the different alternatives to face that challenge, the Mexican government opted for a system similar to Chilean pension scheme developed during the 1980s, based on individual savings accounts for workers, and managed by private entities. In 1997, the Mexican government took a step forward to reorient the pension system, beginning a transition from a non-funded, defined benefit pensions system, towards a defined contribution system in which the workers, the Mexican government, and the employer make monthly contributions to a individual retirement savings accounts. These resources are invested in different assets¸ and these along with market returns obtained are continuously reinvested until retirement. The vehicle through which these portfolios are managed, is known as SIEFOREs. This paper explores what would be the implications of having better informed decision-makers who advise Mexican workers on who to use SIEFOREs. To achieve this goal, a quantitative method that evaluates the performance of SIEFOREs during periods of relative stability and compares it to periods of high volatility is proposed.
Calderón-Colín, Domínguez and Schwartz (2009) suggest that future pensioners select a SIEFORE subject to confusion, due to the noisy information they receive, and prove that the selection of SIEFORE is a response to non-related to performance factors, such as management companies expenditure in marketing, 2 the fact that funds are managed by a well known financial group or insurance company, and the number of sales representatives that these companies employ. In sharp contrast to a market where investors benefit from the competition among suppliers of services, the increasing number of competitors in the Mexican pension funds industry has had a limited impact in the reduction of financial costs to savers.
Currently, the only performance measure published by the industry's supervisor, the CONSAR, 3 is the percentage variation of the stock value of each SIEFORE, and accordingly, investors have very limited information to choose among different SIEFOREs. That information does not reveal the quality of SIEFOREs' management nor how much risk is undertaken by their portfolios during periods of high and low volatility. So, there is a clear need for more detailed information about SIEFOREs' performance, management and riskiness, which should also be accompanied by an improved financial education of the population. Otherwise, the objective to empower citizens and let them take control of their pension fund savings is not likely to be attained. Better educated and better informed savers would allocate their savings to the best performing SIEFOREs, enhancing the demand elasticity for their services, and represent an important motivation for SIEFOREs' managers to continuously improve their performance (reduce their costs and reduce their fees), resulting in better pensions for the population.
This paper contributes to a better understanding of the characteristics of the industry that can enhance the long-term benefits enjoyed by future pensioners by studying the performance of SIEFOREs during periods of 2 The acronym for the Mexican asset management company of a SIEFORE: Administradora de Fondos para el Retiro, AFORE. It is literally translated as "Retirement fund management company". 3 The acronym of Mexican regulatory entity for the Mexican pension system, specially for the supervision and regulation of SIEFORE: Comisión Nacional del Sistema del Ahorro para el Retiro or "National Retirement Savings Commission". 63 stability (normal periods) and periods of high volatility (crisis periods), with the use of Markov Switching econometric model that confirms that not all SIEFOREs are good (or bad) performes all the time. Good performers during normal periods are not so during periods of exacerbated turbulence and vice-versa. Market conditions convey information that can support rational decisions (when to change from one SIEFORE to another), to minimize risk and maximize returns.
The following section presents a brief literature review on Private Pension Fund Systems, with an emphasis on the recent evolution of the AFOREs/ SIEFOREs system in Mexico. Section 1 presents a review of the literature on private pension fund systems, Section 2 introduces and describes the SIEFOREs mechanism and its recent evolution. Section 3 presents the performance evaluation methodology proposal, and the performance evaluation results. Finally, the last section concludes the analysis, summarizes the findings and suggests some guidelines on how decisions makers may improve the SIEFOREs system considering an information availability perspective.

A brief review of the literature on private pension funds systems
While the literature on pension systems in different parts of the world is extensive, this research is focused on Latin American defined benefit pension fund systems. Albo et al. (2007) make actuarial projections to estimate the replacement rate and financial perspectives of Mexico's 1997 new pension system. Their methodology and results later inspired the studies of Alonso, Hoyo and Tuesta (2014)limited the growing fiscal cost of the previous pay-as-you-go scheme. Sixteen years on from its creation, the Retirement Savings System (SAR, who improved the analysis by incorporating the impact of educational levels in Mexico. 4 Following Albo et al. (2007), Fuentes et al. (2010) make another review to the reforms to the Mexican pension system and talk about the fiscal benefits that the new system brought. While the various reforms implemented since 1997 had different fiscal and economic benefits, they reiterate the need to increase the replacement rate to attain international standards, and to Estocástica: Volumen 10, número 1, enero -junio, 2020 pp. 5-26 D. C. Martínez Vázquez, C. Bucio Pacheco y H. A. Olivares Aguayo Introducción E l impacto de las reclamaciones en una compañía de seguros puede desequilibrar la estabilidad de la misma. Por esta razón, es fundamental una adecuada administración, evaluación y previsión de la siniestralidad dentro de un horizonte de tiempo finito; particularmente considerando las condiciones económicas y sociales de los asegurados, para garantizar un correcto nivel de reservas y cálculo de primas (cumpliéndose el principio de ganancia neta dentro del seguro). 1 El modelo colectivo de riesgo, describe el agregado de reclamaciones como un fenómeno adverso para el patrimonio de una aseguradora, que puede presentarse durante un período de tiempo [0,T]. Uno de los supuestos, que generalmente se considera por comodidad, es que existe independencia entre el número de reclamaciones y el monto de las mismas, lo que contrapone lo estipulado por la teoría del modelo colectivo de riesgo. Clasificación JEL: G22, D81, C15. Palabras clave: modelo colectivo de riesgo, seguros, cópula, reclamaciones dependientes.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found. O.V. de la Torre Torres, R.J. Santillán-Salgado, F. López-Herrera introduce a universal pension system with portable rights across pension systems.
By focusing in the competitiveness of the pension fund management process in Mexico and other Latin American countries (Chile, Colombia, Costa Rica, El Salvador, Peru and Uruguay), Guillén (2011) uses Data Envelopment Analysis (DEA) to study the relative and absolute competitiveness of pension fund managers in those countries. He also performs a regression analysis with OLS, with fixed parameters and fixed time and country factors. The reported findings suggest that the Mexican case has an acceptable relation between its relative and absolute competitiveness, but policy makers must make legal improvements to enhance competitiveness. 5 His results lead to the conclusion that even when pension funds could have an influence in financial markets given their relative size, they are exposed to systemic risks. Therefore, a more intense competition among SIEFOREs could lead them to achieve better performance and to reduce their exposure to financial markets' volatility. Alonso et al. (2014), followed the steps of Albo et. al. (2007) and extend the latter by incorporating the "educational dimension". Their results suggest that, in order to increase the replacement rate and obtain better retirement conditions for individuals, the Mexican government should increase the whole mandatory contribution from 6.5% to 11.5% in 2017. If this happened, they suggest that the fiscal impact of pension payment could be reduced by 2.9% every year. They also suggest a periodically adjusted contribution system, (i.e., to change the total contribution), in response to changing economic conditions, and the creation of a single governmentsponsored manager of pension funds. Finally, they argue, independent workers should be allowed to contribute for their retirement. This study is, by the lenght and depth of its proposals, a major contribution to the literature on modern pension fund systems.
Martínez-Preece and Venegas-Martínez (2014) study the performance of the Type 1 and Type 2 SIEFOREs with an equally weighted performance benchmark of each SIEFORE Type, 6 using ARIMA-GARCH models. They do 5 A related final result that we hope to achieve with the implementation of our proposal. 6 Even though they didn't claimed the proposal of a performance benchmark as in De la Torre, Galeana and Martinez (2015) or De la Torre, Galeana and Aguilasocho (2015), they used and studied, as a methodological solution, an equally weighted performance benchmark of these two Type of SIEFOREs. this in two different time periods: June, 1997to August, 2004and September 2004to December 2010. They show that, in terms of the Sharpe ratio, the Type 2 SIEFORES underperformed the most conservative Type 1. This happened due to a higher volatility (as expected) and to the asymmetry of the GARCH process inherent to the time series. In their conclusions, they make some recommendations that are in line with proposal of this paper, such as the need of a better performance measure for the management of the SIEFOREs and to inform the pension fund savers when they are in a scenario of higher volatility and potential loss. Santillán-Salgado et al. (2016) made a similar review by studying the performance of the SIEFOREs in different subperiods (1997-2012, 2004-2012 and 2008-2012), and found that the investment policy and life cycle profile of the SIEFOREs has experienced changes. With the use of ARMA-FIGARCH models, the interpretation of their results concludes the presence of fractional integration in the returns time series. These authors again insist on the need of having more transparent public information that would allow savers to make more informed decisions regarding the SIEFORE in which they keep their savings.
Several papers mentioned in the brief review presented in this section consistently suggest that more informed decisions by investors would result in an increased competitiveness of SIEFOREs. What this work intends to prove is that, had Mexican pension-savers during the period of analysis, had more complex and detailed information about the performance of different SIEFOREs, and had there existed the administrative flexibility to allow them to switch from one SIEFORE to another in response to the quality of management and the conditions of the market, savers would have allocated their savings in a wiser and more profitable way. And, in the end, those conditions would have improved the performance of SIEFOREs' managers through the pressure of competition.

Some Background on SIEFOREs
The SIEFOREs are mutual funds where the pension savings of Mexican workers, affiliated to the public social security system known as IMSS, 7 are invested. There are five Types of SIEFORES: the basic Type 0 SIEFORE or SB0, and four additional Types. The higher the number in the name of 7 IMSS is an abreviation of Instituto Mexicano del Seguro Social, or Mexican Social Security Institute.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found. O.V. de la Torre Torres, R.J. Santillán-Salgado, F. López-Herrera the SIEFORE, the younger are the potential investors, e.g., SB4 or SIEFORE Type 4 is target to pension savers or investors with age under 36, SB3 has an investment policy for investors between 37 and 45 years old; SB2 focuses on workers between 46 and 59 years old, SB1 is for investors that are 60 years old or more and SB0 is for retired workers. The authorized investment policy for each Type of SIEFORE is given in Table 1. As mentioned, saving in SIEFOREs follows a life cycle investment policy. 1/ Only financial assets with a mxA or haigher credit score. 2/ Only asset with an A+ or higher credit quality. 3/ Only through benchmarks allowed in the Appendix M of the CONSAR (2016) rules 4/ In the present paper the commodities will be assumed as local assets even though they are US denominated.
Source: Based in CONSAR (2016). The SIEFOREs covered in this analysis for each of the four investment Types are presented in Table 2.
These are the only SIEFOREs for which daily information for the period from November 30, 2008 to December 30, 2014 is available.
To protect savers, there are legal restrictions that forbid investors to change from one SIEFORE to another. Workers must hold the same SIEFORE for, at least, a twelve-month period. As mentioned by Colín, Domínguez and Schwartz (2009), there is a lot of noise and, as a consequence, confusion in the selection of the best SIEFORE.

A performance metrics proposal and performance results
The main contribution of this work is the introduction of more refined measures to evaluate the performance of SIEFOREs beyond the simple metrics published by CONSAR. In order to achieve that end, Sharpe ratios for "normal" and "crisis" periods for each SIEFORE Type are estimated. The empirical analysis ranks the SIEFOREs' performance during normal and crisis periods, compared with the actual ranking method of the net return. In order to measure the performance in "normal" and "crisis" time periods, historical daily public mutual fund price of each SIEFORE is shown in table 2.
The percentage price variation reflects the net return paid by the SIEFORE between period The percentage price variation reflects the net return paid by the SIE and period 1 t  .
Nota 2, pág. 68 It is important to remember that the observance of state k in t is m probability matrix that contains the probability of being in regime k to regime k j  . This transition probability is denoted by , . These public prices are published daily by the Mexican Stock Exchange and can be found in CONSAR (2016). In order to make the performance analysis, the historical prices from November 30, 2008 to December 31, 2014 are used. Even though the historical prices collected by CONSAR are given since 1997, it is important to mention that the actual investment policy authorized by CONSAR started in March of 2008 with five Types of SIEFORES but in February they were reduced to four. Therefore, November 30 2008 is chosen for two reasons. First, to start the analysis with nine months of historical data for a better fit of the initial values in the model and, second, CONSAR published the historical stock prices for each SIEFORE incorporating the value of the mergers and splits among SIEFORES since 2008. It is well known that, in 2008, Mexico had five Types of SIEFORES available to workers according to their age. 8 In February 2013, two SIEFORES were merged, and the age range dis-8 As it is in a life cycle investment style briefly described in the introduction section.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found. O.V. de la Torre Torres, R.J. Santillán-Salgado, F. López-Herrera tribution changed, leaving only four Types of SIEFOREs. The prices reported by CONSAR consider the merger effect in historical prices due to the aforementioned changes.
In order to analyze the data, the prices of each SIEFORE are compounded: 11 among SIEFORES since 2008. It is well known that, in 2008, Mexico had five Types of SIEFORES available to workers according to their age 8 . In February 2013, two SIEFORES were merged, and the age range distribution changed, leaving only four Types of SIEFOREs.
The prices reported by CONSAR consider the merger effect in historical prices due to the aforementioned changes.
In order to analyze the data, the prices of each SIEFORE are compounded: Hamilton's (1989) filter is used to calculate observed daily mean returns and their standard deviations for two regimes: a low volatility regime   1 k  and a high volatility regime   2 k  , also identified as "normal" and "crisis" periods, respectively. Normal and crisis periods are a latent process that can be modeled with a two state Markovian process with a 8 As it is in a life cycle investment style briefly described in the introduction section.
(1) Hamilton's (1989) filter is used to calculate observed daily mean returns and their standard deviations for two regimes: a low volatility regime 11 ng to their age 8 . In February 2013, two SIEFORES ion changed, leaving only four Types of SIEFOREs.
der the merger effect in historical prices due to the f each SIEFORE are compounded: late observed daily mean returns and their standard tility regime   1 k  and a high volatility regime d "crisis" periods, respectively. Normal and crisis odeled with a two state Markovian process with a escribed in the introduction section. and a high volatility regime 11 were merged, and the age range distribution changed, leav The prices reported by CONSAR consider the merger e aforementioned changes.
In order to analyze the data, the prices of each SIEFORE Hamilton's (1989) filter is used to calculate observed da deviations for two regimes: a low volatility regime  k   2 k  , also identified as "normal" and "crisis" period periods are a latent process that can be modeled with a t 8 As it is in a life cycle investment style briefly described in the intro , also identified as "normal" and "crisis" periods, respectively. Normal and crisis periods are a latent process that can be modeled with a two state Markovian process with a probability of being in regime of normal times probability of being in regime of normal times   1 k  or crisis times   2 k  denoted with 1  and 2  respectively.
In order to determine these two parameters, the filter i was implemented in a Gaussian twostate Markovian process (each state for each regime -normal or crisis): It is important to remember that the observance of state in is modeled with a fixed transition probability matrix that contains the probability of being in regime in time and transiting to regime . This transition probability is denoted by : Once the mean and standard deviation were estimated for each of the ten SIEFOREs in the four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: In the previous expression, k i   and k i   are as previously defined in (2) and (3) and In order to determine these two parameters, the filter i was implemented in a Gaussian twostate Markovian process (each state for each regime -normal or crisis): It is important to remember that the observance of state in is modeled with a fixed transition probability matrix that contains the probability of being in regime in time and transiting to regime . This transition probability is denoted by : Once the mean and standard deviation were estimated for each of the ten SIEFOREs in the four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: In the previous expression, k i   and k i   are as previously defined in (2) and (3) and In order to determine these two parameters, the filter i was implemented in a Gaus state Markovian process (each state for each regime -normal or crisis): It is important to remember that the observance of state in is modeled wit transition probability matrix that contains the probability of being in regime and transiting to regime . This transition probability is denoted by : Once the mean and standard deviation were estimated for each of the ten SIEFOR four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: and probability of being in regime of normal times   1 k  or crisis times   2 k  deno 1  and 2  respectively.
In order to determine these two parameters, the filter i was implemented in a Gauss state Markovian process (each state for each regime -normal or crisis): It is important to remember that the observance of state in is modeled with transition probability matrix that contains the probability of being in regime in and transiting to regime . This transition probability is denoted by : Once the mean and standard deviation were estimated for each of the ten SIEFORE four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: respectively. In order to determine these two parameters, the filter i was implemented in a Gaussian two-state Markovian process (each state k for each regime -normal or crisis):

Nota 1, pág 67
The percentage price variation reflects the net return paid by the SIEFORE between p and period 1 t  .
Nota 2, pág. 68 It is important to remember that the observance of state k in t is modeled with a fixed probability matrix that contains the probability of being in regime k i  in time t and to regime k j  . This transition probability is denoted by , i j p :

Nota 4, pág. 69
In the previous expression, k i   and k i   are as previously defined in (2) an is the expected return for the risk free asset 1 in the regime 2 .
Nota 5, pág. 72 It is important to remember that the observance of state Nota 1, pág 67 The percentage price variation reflects the net return paid by the SIEFORE betwe and period 1 t  .
Nota 2, pág. 68 It is important to remember that the observance of state k in t is modeled with a probability matrix that contains the probability of being in regime k i  in time to regime k j  . This transition probability is denoted by , It is important to remember that the observance of state k in t is modeled with a probability matrix that contains the probability of being in regime k i  in time t to regime k j  . This transition probability is denoted by , i j p :

Nota 4, pág. 69
In the previous expression, k i   and k i   are as previously defined in (2 is the expected return for the risk free asset 1 in the regime 2 � � k'th is modeled with a fixed transition probability matrix that contains the probability of being in regime flects the net return paid by the SIEFORE between period t ember that the observance of state k in t is modeled with a fixed transition at contains the probability of being in regime k i  in time t and transiting his transition probability is denoted by , i j p : pression, k i   and k i   are as previously defined in (2) and (3) and the expected return for the risk free asset 1 in the regime 2 .
� � k'th and transiting to regime Nota 1, pág 67 The percentage price variation reflects the and period 1 t  .
Nota 2, pág. 68 It is important to remember that the observ probability matrix that contains the probab to regime k j  . This transition probabilit

Nota 4, pág. 69
In the previous expression, k i is the expected return for Nota 5, pág. 72 The analysis presented in this paper uses M . This transition probability is denoted by 67 tage price variation reflects the net return paid by the SIEFORE between period t 1 t  .
. 68 ant to remember that the observance of state k in t is modeled with a fixed transition matrix that contains the probability of being in regime k i  in time t and transiting j  . This transition probability is denoted by , i j p : . 69 vious expression, k i   and k i   are as previously defined in (2) and (3) and  1 k  is the expected return for the risk free asset 1 in the regime 2 .
. 72 s presented in this paper uses Markov-switching models to calculate the Sharpe ratio In order to determine these two parameters, the filter i was implemented in a Gaussian twostate Markovian process (each state for each regime -normal or crisis): It is important to remember that the observance of state in is modeled with a fixed transition probability matrix that contains the probability of being in regime in time and transiting to regime . This transition probability is denoted by : Once the mean and standard deviation were estimated for each of the ten SIEFOREs in the four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: In the previous expression, k i   and k i   are as previously defined in (2) and (3) and is the expected return for the risk free asset 9 in the regime 10 .
Once the mean and standard deviation were estimated for each of the ten SIEFOREs in the four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: probability of being in regime of normal times   1 k  or crisis times   2 k  denoted with 1  and 2  respectively.
In order to determine these two parameters, the filter i was implemented in a Gaussian twostate Markovian process (each state for each regime -normal or crisis): It is important to remember that the observance of state in is modeled with a fixed transition probability matrix that contains the probability of being in regime in time and transiting to regime . This transition probability is denoted by : Once the mean and standard deviation were estimated for each of the ten SIEFOREs in the four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: In the previous expression, k i   and k i   are as previously defined in (2) and (3) and is the expected return for the risk free asset 9 in the regime 10 . In the previous expression, In the previous expression, k i   and k i   are as previously defined in (2) and (3) an is the expected return for the risk free asset 9 in the regime 10 .
Using the above mentioned methods, a ranking determined by the expected return an Sharpe ratios of each SIEFORE in each investment style or Type was estimated.
The historical performance of each Type of SIEFORE is presented in Tables 3 to in terms of accumulated returns, along with daily mean return, standard deviation, an 9 The observed mean percentage variation of daily prices, provided by Valmer's CETES benchmark. 10 The percentage variation of the 28 day CETEs benchmark provided by Valmer and Economatica is used.
� � k'th and 12 four investment regimes, a Markov-switching Sharpe ratio is estimated as follows: In the previous expression, k i   and k i   are as previously defined in (2) and (3) an is the expected return for the risk free asset 9 in the regime 10 .
Using the above mentioned methods, a ranking determined by the expected return an Sharpe ratios of each SIEFORE in each investment style or Type was estimated.
The historical performance of each Type of SIEFORE is presented in Tables 3 to in terms of accumulated returns, along with daily mean return, standard deviation, an 9 The observed mean percentage variation of daily prices, provided by Valmer's CETES benchmark. 10 The percentage variation of the 28 day CETEs benchmark provided by Valmer and Economatica is used.
� � k'th are as previously defined in (2) and (3) and Nota 4, pág. 69 In the previous expression, k i   and k i   are as previously defined in (2) and is the expected return for the risk free asset 1 in the regime 2 .
Nota 5, pág. 72 The analysis presented in this paper uses Markov-switching models to calculate the Sh   , i k S of SIEFOREs for different sub periods, identifying two volatility regimes (nor 1 k  and crisis with 2 k  ).
1 The observed mean percentage variation of daily prices, provided by Valmer's CETES benchm 2 The percentage variation of the 28 day CETEs benchmark provided by Valmer and Economati used.

� � k'th
is the expected return for the risk free asset 9 in the k´th regime. 10 Using the above mentioned methods, a ranking determined by the expected return and Sharpe ratios of each SIEFORE in each investment style or Type was estimated.
The historical performance of each Type of SIEFORE is presented in Tables 3 to 6, in terms of accumulated returns, along with daily mean return, standard deviation, and minimum and maximum values. The performance ranking in terms of net accumulated return of each SIEFORE 11 is shown in the last column. The three best SIEFOREs are shaded in grey.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found.   Tables 3 to 6 show a similar net return performance analysis to that of CONSAR. In all cases SURA, GNP and Banamex had the best results and Inbursa had the worst.
Tables 7 to 9 present the results of the expected return, risk and Sharpe ratios observed in each SIEFORE by using (2) and (4) i.e. the normal and crisis periods. The same Tables show the ranking of the ten SIEFOREs in each investment Type, given their observed Sharpe ratio in normal and crisis times. The new ranking is compared with the returns previously shown in tables 3 to 6. As expected, the best performing SIEFOREs with the original method are not always the best when one differentiates between normal and crisis periods. For example, SURA is a middle rank SIEFORE in normal times, but the second best performer in crisis times. As expected, Inbursa is the worst performer in normal times, but the most stable during crisis periods, suggesting the possibility of active investment management, as long as investors have the necessary information.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found.  This table presents the normal and crisis periods expected return, risk and Sharpe ratios of Type 1 by using Hamilton's (1998) filter. SIEFOREs. * means significant that the expected return is significant with 10% of probability, ** at 5% probability and *** at 1% probability. This table presents the normal and crisis periods expected return, risk and Sharpe ratios of Type 3 by using Hamilton's (1998) filter. SIEFOREs. * means significant that the expected return is significant with 10% of probability, ** at 5% probability and *** at 1% probability.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found.  Table 9. Performance of Type 4 SIEFOREs during normal and crisis periods. This table presents the normal and crisis periods expected return, risk and Sharpe ratios of Type 4 by using Hamilton's (1998) filter. SIEFOREs. * means significant that the expected return is significant with 10% of probability, ** at 5% probability and *** at 1% probability.
Source: prepared by authors with data of CONSAR (2014)..  (4). In table 7, the expected return in all Type 1 SIEFOREs is significantly different from zero during normal periods, and only Inbursa has an expected return different from zero during crisis periods. This is due to the more stable historical performance that leads the SIEFOREs to underperform their investment policy benchmark. Panel b) of the same Table presents the Sharpe ratio results for each SIEFORE in each volatility regime, and presents the ranking of each SIEFORE according to their Sharpe ratios for each volatility regime.

Conclusion
A defined benefit pension fund system should encourage competition among pension fund managers to motivate a search for better performance. Calderón-Colín, et al. (2009) and Guillen (2011) mention that there is low competition among Mexican pension funds (SIEFOREs) due to informational asymmetry (noisy or uninformed decisions), and the absence of legal incentives.
The analysis presented in this paper uses Markov-switching models to calculate the Sharpe ratio Nota 2, pág. 68 It is important to remember that the observance of state k in t is modeled w probability matrix that contains the probability of being in regime k i  in to regime k j  . This transition probability is denoted by , i j p :

Nota 4, pág. 69
In the previous expression, k i   and k i   are as previously defined is the expected return for the risk free asset 1 in the re Nota 5, pág. 72 The analysis presented in this paper uses Markov-switching models to calcu   , i k S of SIEFOREs for different sub periods, identifying two volatility re 1 k  and crisis with 2 k  ). 1 The observed mean percentage variation of daily prices, provided by Valmer's CE 2 The percentage variation of the 28 day CETEs benchmark provided by Valmer and used.

Conclusion
A defined benefit pension fund system should encourage com managers to motivate a search for better performance. Calde Guillen (2011) mention that there is low competition amo (SIEFOREs) due to informational asymmetry (noisy or uni absence of legal incentives. The analysis presented in this paper uses Markov-swit Sharpe ratio ( ) of SIEFOREs for different sub periods, iden (normal with 1 k  and crisis with 2 k  ). The results confir good (or bad) performers all the time. In some cases, good perf are not so during crisis periods and vice-versa. This evidenc market conditions conveys information that can support rationa from good performers during normal times to good performer this is a preliminary finding, it opens the door for further analys investment decision rules.  Hamilton's (1998) filter. SIEFOREs. * means significant that the expected probability, ** at 5% probability and *** at 1% probability. Source: Own elaboration with data of CONSAR (2014).

Conclusion
A defined benefit pension fund system should encourage com managers to motivate a search for better performance. Cald Guillen (2011) mention that there is low competition am (SIEFOREs) due to informational asymmetry (noisy or un absence of legal incentives. The analysis presented in this paper uses Markov-swit Sharpe ratio ( ) of SIEFOREs for different sub periods, iden (normal with 1 k  and crisis with 2 k  ). The results confir good (or bad) performers all the time. In some cases, good perf are not so during crisis periods and vice-versa. This evidenc market conditions conveys information that can support ration from good performers during normal times to good performer this is a preliminary finding, it opens the door for further analy investment decision rules.

� � S i ,k
). The results confirm that not all SIEFOREs are good (or bad) performers all the time. In some cases, good performers during normal periods are not so during crisis periods and vice-versa. This evidence suggests that awareness of market conditions conveys information that can support rational decisions to change savings from good performers during normal times to good performers during crisis periods. While this is a preliminary finding, it opens the door for further analyses that should lead to optimal investment decision rules.
The current Mexican legislation allow changes from one SIEFORE to another only every twelve months, so the implementation of a more informed and flexible investment framework requires the liberalization of the current legislation. However, the findings reported in this study give support to such an initiative, and should be regarded as evidence that the current legislation is too rigid, frustrating the opportunity that well informed rational investors improve their retirement savings returns in the long run.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found. O.V. de la Torre Torres, R.J. Santillán-Salgado, F. López-Herrera More research is needed to fully understand the multiple areas of potential improvement to the current Mexican Pension Funds system. This is a first contribution in that direction, and we expect that others will follow. While the adoption of a modern Pension Funds System opens numerous possibilities, the main objective, which is to guarantee the living standards of future pensioners should be the highest priority. CONSAR (2016 Introducción E l impacto de las reclamaciones en una compañía de seguros puede desequilibrar la estabilidad de la misma. Por esta razón, es fundamental una adecuada administración, evaluación y previsión de la siniestralidad dentro de un horizonte de tiempo finito; particularmente considerando las condiciones económicas y sociales de los asegurados, para garantizar un correcto nivel de reservas y cálculo de primas (cumpliéndose el principio de ganancia neta dentro del seguro). 1 El modelo colectivo de riesgo, describe el agregado de reclamaciones como un fenómeno adverso para el patrimonio de una aseguradora, que puede presentarse durante un período de tiempo [0,T]. Uno de los supuestos, que generalmente se considera por comodidad, es que existe independencia entre el número de reclamaciones y el monto de las mismas, lo que contrapone lo estipulado por la teoría del modelo colectivo de riesgo. Clasificación JEL: G22, D81, C15. Palabras clave: modelo colectivo de riesgo, seguros, cópula, reclamaciones dependientes.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found. In this Appendix is presented the decision-making algorithm that a theoretical group of pension savers would have followed if they have had access to the Markov-Switching Sharpe Ratios such as the ones presented in table 3 and tables 7 to 9, along with the probability of being in "normal" or "crisis" periods. The assumption stated is that even if the investors had access to this information, they suffer the impact of some externalities such as the fact that their SIEFORE is managed by a division of a big financial institution, or that the marketing efforts of their SIEFORE lead them to decisions not entirely informed. It is also assume that the impact of the legal restriction to move retirement proceedings only once a year has an impact in the investment levels. Therefore, in order to simulate the partially informed scenario portfolio the next algorithm is followed:

Definitions:
being in "normal" or "crisis" periods. We induce the assumption that even if the investo had access to this information, they suffer the impact of some externalities such as the fa that their SIEFORE is managed by a division of a big financial institution, or that th marketing efforts of their SIEFORE lead them to decisions not entirely informed. We als assume that the impact of the legal restriction to move retirement proceedings only once year has an impact in the investment levels. Therefore, in order to simulate the partial informed scenario portfolio we followed the next algorithm: Definitions: being in "normal" or "crisis" periods. We induce the assumption that even if the investo had access to this information, they suffer the impact of some externalities such as the fa that their SIEFORE is managed by a division of a big financial institution, or that t marketing efforts of their SIEFORE lead them to decisions not entirely informed. We al assume that the impact of the legal restriction to move retirement proceedings only once year has an impact in the investment levels. Therefore, in order to simulate the partial informed scenario portfolio we followed the next algorithm: Definitions: , i j S = Sharpe ratio of the i-th SIEFORE at time time, given the k-regime. being in "normal" or "crisis" periods. We induce the assumption that even if the investo had access to this information, they suffer the impact of some externalities such as the fa that their SIEFORE is managed by a division of a big financial institution, or that t marketing efforts of their SIEFORE lead them to decisions not entirely informed. We al assume that the impact of the legal restriction to move retirement proceedings only once year has an impact in the investment levels. Therefore, in order to simulate the partial informed scenario portfolio we followed the next algorithm: Definitions: , i j S = Sharpe ratio of the i-th SIEFORE at time time, given the k-regime. The expected return, observed at t, of the Max Sharpe portfolio, given the k volatility regime.
being in "normal" or "crisis" periods. We induce the assumption that even if the investo had access to this information, they suffer the impact of some externalities such as the fa that their SIEFORE is managed by a division of a big financial institution, or that t marketing efforts of their SIEFORE lead them to decisions not entirely informed. We al assume that the impact of the legal restriction to move retirement proceedings only once year has an impact in the investment levels. Therefore, in order to simulate the partial informed scenario portfolio we followed the next algorithm: Definitions: , i j S = Sharpe ratio of the i-th SIEFORE at time time, given the k-regime.
= The probability that the performance or investment policy of th SIEFOREs in a given investment style or Type is in regime 2 or "crisis", given the pas The expected risk or standard deviation, observed at t, of the Max Sharpe portfolio, given the k volatility regime.
being in "normal" or "crisis" periods. We induce the assumption that even if the investo had access to this information, they suffer the impact of some externalities such as the fa that their SIEFORE is managed by a division of a big financial institution, or that th marketing efforts of their SIEFORE lead them to decisions not entirely informed. We als assume that the impact of the legal restriction to move retirement proceedings only once year has an impact in the investment levels. Therefore, in order to simulate the partial informed scenario portfolio we followed the next algorithm: Definitions: = The probability that the performance or investment policy of th SIEFOREs in a given investment style or Type is in regime 2 or "crisis", given the pas The expected return, observed at t, of the i-th SIEFORE, given the k volatility regime.
being in "normal" or "crisis" periods. We induce the assumption that even if the investo had access to this information, they suffer the impact of some externalities such as the fa that their SIEFORE is managed by a division of a big financial institution, or that t marketing efforts of their SIEFORE lead them to decisions not entirely informed. We al assume that the impact of the legal restriction to move retirement proceedings only once year has an impact in the investment levels. Therefore, in order to simulate the partial informed scenario portfolio we followed the next algorithm: Definitions:   2 % MaxSharpe P k I   = The probability that the performance or investment policy of th SIEFOREs in a given investment style or Type is in regime 2 or "crisis", given the pas The expected risk or standard deviation, observed at t, of the i-th SIEFORE, given the k volatility regime.

22
being in "normal" or "crisis" periods. We induce the assumption that even if the investo had access to this information, they suffer the impact of some externalities such as the fa that their SIEFORE is managed by a division of a big financial institution, or that t marketing efforts of their SIEFORE lead them to decisions not entirely informed. We al assume that the impact of the legal restriction to move retirement proceedings only once year has an impact in the investment levels. Therefore, in order to simulate the partial informed scenario portfolio we followed the next algorithm: Definitions: , i j S = Sharpe ratio of the i-th SIEFORE at time time, given the k-regime.   2 % MaxSharpe P k I   = The probability that the performance or investment policy of th SIEFOREs in a given investment style or Type is in regime 2 or "crisis", given the pas being in "normal" or "crisis" periods. We induce the assumption that eve had access to this information, they suffer the impact of some externalitie that their SIEFORE is managed by a division of a big financial instit marketing efforts of their SIEFORE lead them to decisions not entirely i assume that the impact of the legal restriction to move retirement procee year has an impact in the investment levels. Therefore, in order to sim informed scenario portfolio we followed the next algorithm: Definitions: , i j S = Sharpe ratio of the i-th SIEFORE at time time, given the k-regime. = The probability that the performance or investm SIEFOREs in a given investment style or Type is in regime 2 or "crisis", g The probability that the performance or investment policy of the SIEFOREs in a given investment style or Type is in regime 2 or "crisis", given the pas observed data of the benchmark. 22 cess to this information, they suffer the impact of some externalities such as the fact eir SIEFORE is managed by a division of a big financial institution, or that the ing efforts of their SIEFORE lead them to decisions not entirely informed. We also that the impact of the legal restriction to move retirement proceedings only once a as an impact in the investment levels. Therefore, in order to simulate the partially ed scenario portfolio we followed the next algorithm: ions: harpe ratio of the i-th SIEFORE at time time, given the k-regime.  k = The expected risk or standard deviation, observed at t, of the i-th SIEFORE, he k volatility regime.

AbstRAct
The collective risk model is defined in the actuarial literature as an important risk distribution analysis tool for insurance companies. Actuarial textbooks assume an independent behavior between the number of claims and their amount. The main objective of this paper is to show that under certain circumstances evidence of dependency between the variables studied may be found. O.V. de la Torre Torres, R.J. Santillán-Salgado, F. López-Herrera By running this, we used a base-100 value at January 2010 in each SIE-FORE Type by using the SIEFORES described in table 2.