Pension Funds in Mexico and Chile: A Risk-Reward Comparison - Documento de Trabajo Nº 55 - Clapes UC

 
Pension Funds in Mexico
                          and Chile: A Risk-Reward
Hans Schlechter           Comparison
Bernardo K. Pagnoncelli
Arturo Cifuentes

www.clapesuc.cl           Documento de Trabajo Nº 55
Pension Funds in Mexico and Chile: A Risk-Reward Comparison
                   Hans Schlechter1 , Bernardo K. Pagnoncelli⇤2 , and Arturo Cifuentes3
                                              1
                                           CLAPES UC, Santiago, CHILE
                                       2
                                     Universidad Adolfo Ibáñez, Santiago, CHILE
                       3
                         CLAPES UC, Santiago, CHILE, and, Columbia University, New York, USA

                                                       March 2019

                                                         Abstract
            Mexico and Chile have Defined Contributions (DC) pension systems. In both cases affiliates are
        o↵ered several investment funds with, allegedly, di↵erent risk-return profiles. Analyzing actual return
        data for the April 2008-March 2018 period, and using a number of risk—and return—related metrics,
        we reach quite di↵erent conclusions in relation to such funds. In the case of Mexico, the funds delivered
        returns according to their intended risk profile, and they are consistently ranked correctly in terms of
        absolute risk, risk-adjusted returns, and cumulative returns. Chilean funds, on the other hand, exhibited
        erratic risk-return patterns, with the most conservative fund outperforming the riskiest fund in terms of
        cumulative returns. Overall our analysis is an indictment on the idea of using asset allocation limits to
        control portfolio risk (Chile), and supports the view that risk is much better controlled using an overall
        portfolio-level risk metric (Mexico). Since most pension funds still rely heavily on asset-class limits to
        manage risk, our results should serve as a serious warning against the danger of relying on this practice.
     Keywords— Pension Funds; Defined Contribution; Rank-order Metrics

     JEL classification— H55;G17;G11

1      Introduction
Mexico and Chile can be considered, arguably and with some caveats, two successful emerging market stories in
Latin America. Both countries started to implement market-oriented reforms in the 80s, based loosely on what
later became known as the Washington consensus. In short, they carried out several privatizations of government
enterprises, adopted a more flexible position on international trade and foreign investments, embraced fiscal discipline
along with a floating exchange rate, and pushed for deregulation on many fronts, among other things. Perhaps
more important, both countries enjoy the benefits of having independent central banks run by highly capable, and
internationally respected, professionals. Additionally, Mexico and Chile are the only Latin countries that belong to
the Organization for Economic Cooperation and Development (OECD). Mexico was accepted in 1994, Chile in 2010.
And today, their GDPs per capita (at PPP) are among the highest in Latin America—approximately $25, 900 in the
case of Chile, and $20, 600 in the case of Mexico (International Monetary Fund, 2018).
    Within this context, both countries have also innovated. Chile introduced in 1980 a radical modification to its
social security system; it was based on the adoption of a privately-managed defined contribution (DC) scheme, in
which, essentially, the workers are directly responsible for their retirement (Edwards, 1998). The Chilean concept also
served as a blueprint for pension-related reforms that were carried out by many emerging economies e.g. Mexico, El
Salvador, Peru, Poland, Bulgaria, Costa Rica, Hungary and Lithuania. Mexico, on the other hand, which can boast
of having the most developed and biggest fixed income market in Latin America, pioneered in the early 2000 a set
of very original collective-action clauses that were adopted by many emerging countries in relation to the issuance of
sovereign debt (Gelpern, 2003).
    ⇤ This   work was supported by Fondecyt project number 1170178.

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The Chilean pension system started its transformation in 1980 when the government introduced a mandatory DC
scheme; and it entered what we can describe as its mature phase in 2002, when the regulator implemented five di↵erent
investment funds (A, B, C, D and E). The idea behind this modification was to o↵er the affiliates di↵erent risk-return
options. The A fund (the riskiest fund) is supposed to appeal to younger workers, and ideally, deliver higher returns.
The E fund, the most conservative, is aimed at meeting the needs of older workers, close to the retirement age. Broadly
speaking, the affiliates are expected to move gradually from the A to E fund as they approach their retirement age,
in what is known as the life-cycle investment strategy (Viceira, 2008). There are, however, some restrictions based
on the age of the worker, as well as a “default” strategy for those workers who do not explicitly make a choice. The
funds are managed by private institutions known as AFPs (based on their Spanish acronym), which are exclusively
dedicated to manage pension funds (thus, cannot engage in any other activity), and are supervised by a dedicated
regulator known as Superintendencia de Pensiones (SP). Chilean workers are mandated to contribute 10% of their
salaries (this obligation stops when the salary reaches a limit of roughly $ 3,200 per month as of 2019). The AFPs
manage approximately US$ 210 billion, representing 72% of GDP (OECD, 2018). A more comprehensive description
of the Chilean pension system can be found in Superintendencia de Pensiones de Chile (2003, Chapter 4), International
Center for Pension Management (2018), OECD (2017a), Fernández (2013), and Edwards (1998).
    Mexico started the transformation of its pension system a bit later, in 1997, when it also introduced a DC scheme
inspired in large measure by the Chilean architecture. In Mexico the funds are managed by institutions known as
AFORES (again, based on their Spanish acronym). Initially, like their Chilean counterparties, they o↵ered only one
investment choice. However, the system was progressively modified and this evolution culminated in 2008 with the
introduction of five funds (known as SIEFORES 1 through 5, or more commonly SB1,. . . , SB5), which are analogous
to the five Chilean funds, except that the order is reversed: the SB1 fund is the most conservative (for workers older
than 60), and the SB5 fund the riskiest (for workers younger than 27). Later, in 2013, the system was reorganized
around four SBs (the SB4 fund and the SB5 fund were collapsed in one fund). Unlike the case of Chile, Mexican
workers contribute only 6.5% of their monthly salaries, and not all workers are affiliated with the SAR (the name by
which the previously described DC system is known). All private-sector employees who joined the workforce after July
1997, as well as most public sector workers who joined after April 2007, are enrolled in the SAR system. However,
there are still many workers covered by other state and federal programs, or some special systems available for certain
public and state university employees. In fact, this is probably the reason (in addition to the 6.5% contribution
compared to the 10% in Chile) that the AFORES manage only US$ 200 billion (approximately 15% of GDP), a figure
comparatively lower than their Chilean counterparties. AFORES, as the AFPs in Chile, are regulated by a dedicated
supervisor known as CONSAR. A more complete description of the Mexican pension system can be found in OECD
(2016a), OECD (2016b), Comisión Nacional del Sistema de Ahorro para el Retiro (2017a) and Alonso, Hoyo, and
Tuesta (2015).
    The Mexican and Chilean pension systems, partly perhaps due to the increasing global shift from defined benefits
(DB) pension schemes to DC schemes, have received a fair amount of attention, mostly from the public policy
viewpoint. The chief concerns are related to how to improve coverage, the fees associated with managing the funds
and whether there is enough competition in the sector. Moreover, there is a permanent debate on what should
the contribution rate needed to achieve an acceptable replacement rate, as well as discussions on the appropriate
retirement age and whether di↵erent mortality tables should be applied to men and women (see Comisión Nacional
del Sistema de Ahorro para el Retiro (2017b), Alonso et al. (2015), Willmore (2014), Aguila, Mejia, Pérez-Arce, and
Rivera (2013), Comisión Asesora Presidencial sobre el Sistema de Pensiones (2015), Krasnokutskaya, Li, and Todd
(2018) and López Garcı́a and Otero (2017) for more details). Fuentes, Garcı́a-Herrero, and Escrivá (2010) provide a
good summary of many pending challenges from the public policy viewpoint.
    Analysis of these pension systems from a financial vantage point is a subject that has received much less attention.
We cite a few exceptions. Fernández (2013) concluded that the Chilean asset managers exhibited significant herd
behavior and their returns, once adjusted for risk, were not better than those of index-based alternatives. Pagnoncelli,
Cifuentes, and Denis (2017) introduced a hybrid (active-passive) investment strategy for pension funds and tested
it with the Chilean system. The authors found that in most cases the managers had not done better than passive
alternatives, and they hinted that some of the investment constraints imposed by the regulator had had a negative
impact on returns. In relation to the Mexican system, Santillán-Salgado, Martı́nez-Preece, and López-Herrera (2016)
looked at all the funds returns and concluded that their volatilities showed significant variation over time and that
the returns showed autocorrelation (memory) e↵ects, a topic they considered crucial for risk management purposes.
In De la Torre, Galeana, and Aguilasocho (2018) the authors suggest a benchmark that could be useful to assess the
performance of such funds.
    With that as background, we stress that the ultimate goal of the regulators when they created di↵erent investment
funds (SB1,. . . , SB4 in Mexico; A,. . . , E in Chile) was to o↵er options with distinguishable risk-return profiles. To

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put it di↵erently, the idea was that the A and SB4 funds should deliver, in the long-term (granted, a somewhat
subjective term), higher returns than the E and SB1 funds, respectively. We believe it is then appropriate—enough
time has elapsed since they were created—to assess if the returns of these funds have been commensurate with their
intended risk profiles. We think this is important for two reasons, the most obvious is that such analysis has not been
undertaken, and thus, this study would fill an important void. The second consideration is that Mexico and Chile
have adopted a somewhat di↵erent approach to manage the risk of their pension funds (a topic we discuss in more
detail later), and thus, there are important lessons to be learned from this divergence.
    In summary, the purpose of this study is to look at the performance of the Mexican and Chilean pension funds
from a portfolio management perspective, based on di↵erent risk-return metrics. And more specifically, to examine if
the regulators were successful in creating funds with di↵erent risk-return profiles.

2     Investment Regulations in Mexico and Chile
Before we compare the investment regulation of both countries, it is important to put the matter into a broader
context: managing a pension fund is really a portfolio management problem subject to risk constraints. In the
last seventy years there has been important progress in portfolio and financial risk management. However, generally
speaking, the pension fund industry has been somewhat slow to adapt. Markowitz’s seminal work established formally
that there was a trade-o↵ between risk and return in portfolio management and proposed to use the standard deviation
as a tool to measure risk (Markowitz, 1952; Rubinstein, 2002). Notwithstanding the big conceptual innovation behind
this thought, the shortcomings of the standard deviation (e.g. it penalizes both losses as well as gains) were quickly
recognized. In due course market practitioners suggested improvements, such as the Value-at-Risk (VaR) (Linsmeier
& Pearson, 2000), and more recently the Conditional-Value-at-Risk or CVaR (Rockafellar & Uryasev, 2000, 2002).
Moreover, in the context of pensions, it has also been recognized that short-term volatility (referred to as the volatility
of journey in Smith, 2011) is irrelevant, and shortfalls (or downside risk) over long periods is what really matters (QS
Investors Research Group, 2010).
     All these improvements have been gradually incorporated into the standard practices of portfolio and risk man-
agement. In fact, Solvency II and Basel III—the insurance and banking regulations adopted after the painful lessons
of the subprime (2008) crisis—have incorporated these concepts. Pension fund regulators, however, seems to be stuck
on the idea of controlling risk via asset class limits, rather than relying on suitable metrics. In fact, an OECD
report informs that only two OECD countries (Mexico and Denmark), have investment guidelines that incorporate
quantitative risk metrics. All other countries (Chile included) still rely on old fashion asset class limits (Antolı́n et
al., 2009). Another report by the World Bank also focuses on the fact that the pension industry has been slow to
adopt the practices already embraced by the banking and insurance sector (Brunner, Hinz, & Rocha, 2008). A more
recent report (PricewaterHouseCoopers, 2016) indicates that most pension funds are still subject to guidelines based
on asset allocation limits. Berstein and Chumacero (2003) have gone a bit further and outright suggest that, in the
case of Chile, asset allocation limits have negatively impacted the returns (a suggestion that seems to agree with
Pagnoncelli et al., 2017). And finally, a 2014 report by Schroders (Schroder Investment Management, 2014), o↵ered
a rather pessimistic view of current practices, stating that “... the impact of investments risks and returns on DC
portfolios are often misunderstood.”
     In Mexico and Chile the regulators have imposed limits by asset class, but with one crucial di↵erence: in the case
of Mexico the regulator relies also on a portfolio-level risk-metric, whereas the Chilean system does not employ such
concept. Hence, these two countries can be considered a natural experiment that can be useful to explore the e↵ects
of di↵erent regulations. In what follows we describe briefly the investment guidelines prevailing in Mexico and Chile
during the 2008-2018 period, the time-frame considered in this study.

2.1     Mexico
The Mexican regulation is based on two key ideas: an upper limit by asset class for each (SB) fund, as well as a
VaR-based limit, introduced by the regulator in 2004, at the global portfolio level.
    In terms of the asset class limits, the SB4 fund has the highest upper limit for equities (45%), while the SB1 fund
has the lowest (10%). The limits for structured products, REITs and commodities are also higher for the SB4 fund
and lower for the SB1 fund. There are no lower-bound limits (except in relation to the SB1 fund, which must keep
at least 51% of its holdings invested in inflation-protected securities). In essence, the regulator does not force the
SBs to maintain a minimum exposure to any asset class. In terms of the VaR, there is a limit that increases from
the SB1 fund (0.70%) to the SB4 fund (2.10%). It refers to the maximum amount (expressed as percentage of the

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current portfolio market value) that the portfolio could lose in one day, with a 95% confidence. Its calculation relies
on market data from the previous 1,000 trading days.
    In order to control the leverage of more volatile instruments, in October 2012, the regulator introduced an
additional metric, the Conditional Value-at-Risk ( CVaR). Each AFORE needs to report, for each fund, the di↵erence
in CVaR between the complete portfolio and the portfolio excluding derivatives.

                                 Table 1: Mexican pension regulation: Key points

                   Item                                             Description
        Market and liquidity risks     Defines maximum limits for risk parameters of each SB. The VaR
                                       ranges from 0.70% for the SB1 fund to 2.10% for the SB4 fund,
                                       and the liquidity coverage ratio corresponds to 80% of the highly
                                       liquid assets.

          Risk by issuer and/or        Defines maximum limits for instruments according to their
              counterparty             medium- and long-term credit ratings, distinguishing between do-
                                       mestic and international issuers. It also limits the holdings of a
                                       single issuance for all funds to a 35% of the assets by the same
                                       fund manager.

            Asset class limits         Defines maximum limits by asset classes, allowing more exposure
                                       to variable income instruments as the intended risk-profile of the
                                       fund increases (from SB1 to SB4).

           Conflicts of interest       Limits the maximum exposure of the funds to related companies
                                       (15% for all funds), as well to companies related to the fund man-
                                       ager (5%).

         Investment vehicles and       The regulation allows all funds to use investment vehicles such as
               derivatives             mutual funds and derivatives.

        Source: Investment Regime of the SIEFORES, CONSAR.

2.2    Chile
In Chile the regulation has two main elements, upper and lower asset-class limits combined with a penalty for
underperforming with respect to its peers. In terms of the asset-class limits, similarly to the Mexican case, they are
commensurate with the intended risk profile of each fund, although the actual figures are di↵erent. The A fund, for
example, can hold up to 80% of its assets in equities, while the limit for the E fund is zero. Unlike the Mexican case,
however, the regulator imposes minimums by asset class. Thus, for instance, the D fund must have at least 5% of
its holdings in equities. Additionally, the regulator imposes a “minimum” yield test: the performance of each fund
(that is, by fund class) is compared to its peers (industry average) based on the returns of the last 36 months. A
manager whose fund underperforms must compensate the affiliates. To this end, the managers (AFPs) must maintain
a special reserve fund equivalent to 1% of the market value of such fund. This regulation has been considered the
main cause behind the herd behavior (similar portfolios) that has been detected in relation to the Chilean managers
(Chant West, 2014). Table 2 shows a summary of the regulation (a complete summary can be found in Appendix B).

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Table 2: Chilean pension regulation: Key points

                   Item                                           Description
           Minimum yield test          Fund managers are responsible for ensuring that the annualized
                                       real yield (RY) for each fund during the previous 36 months must
                                       satisfy the following condition. Let X be the real annualized yield
                                       for all the funds of the same type for the previous 36 months. And
                                                       1
                                       let Y = X       2 |X|. Then, the A and B funds must satisfy: RY
                                       must be greater or equal than min(X 4%, Y ). In the case of the
                                       C, D and E funds the condition is: RY must be greater or equal
                                       than min(X 2%, Y ).

          Risk by issuer and/or        Defines upper and lower bounds for instruments issued by a single
                 sectors               issuer. There are also di↵erent limits by economic sectors, such as
                                       the financial sector, the external sector, investment vehicles and
                                       mutual funds, and state-owned companies. Additionally, there are
                                       limits within certain sectors.

            Asset class limits         Each fund must comply with a minimum and maximum limit by
                                       asset class (maximum 80% and minimum 40% on variable income
                                       for the A fund to 20% and 5% for the E fund). Additionally, there
                                       is an upper limit for certain types of instruments.

           Conflicts of interest       Imposes limits for investment in instruments issued by companies
                                       related to the fund manager, as well as the obligation to invest re-
                                       sources from the manager’s obligatory reserve provision in shares
                                       of the fund that it manages.

         Investment vehicles and       The regulation allows operations involving derivatives for hedging
               derivatives             and investment, as well as investments in mutual funds (with some
                                       constrains) and a limited exposure to alternative assets.

       Source: Superintendencia de Pensiones de Chile.

3    The Data
The study is based on the actual—as opposed to nominal—monthly returns of each fund during the period April
2008-March 2018. In the case of Mexico, the data were downloaded from the open-data platform of the Mexican
government and corrected using the UDI, inflation-adjusted Mexican peso (Comisión Nacional del Sistema de Ahorro
para el Retiro, 2018; Banco Central de México, 2018). In the case of Chile the data were downloaded from the
website of the Chilean regulator and also corrected using the UF, inflation-adjusted Chilean peso (Superintendencia
de Pensiones, 2018). Therefore, the returns presented in this work correspond to real returns. In the case of Mexico
we considered the four basic funds (SB1, SB2, SB3 and SB4) managed by the four leading managers (based on
assets under management): XXI Banorte, Banamex, Sura and Profuturo GNP. In the Chilean case, we considered
the five funds (A, B, C, D and E), managed by all six managers: Cuprum, Habitat, Planvital, Provida, Capital and
Modelo. Tables 3 and 4 present some key statistics regarding the Mexican and Chilean system monthly returns for
the 2008-2018 period.

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Table 3: Mexican funds: monthly returns and descriptive statistics, expressed in percentage (%), April 2008
- March 2018

                            Fund    Manager              Mean     St.Dev.     Min     Max
                                    XXI Banorte          0.13      1.33      -4.52    4.25
                                    Banamex              0.13      1.37      -3.59    4.98
                            SB1
                                    SURA                 0.15      1.43      -4.27    4.22
                                    Profuturo GNP        0.11      1.45      -4.66    3.03
                                    XXI Banorte          0.17      1.70      -5.71    5.35
                                    Banamex              0.17      1.89      -6.43    7.28
                            SB2
                                    SURA                 0.21      1.94      -6.32    6.40
                                    Profuturo GNP        0.21      1.92      -6.88    6.30
                                    XXI Banorte          0.18      1.91      -6.17    5.78
                                    Banamex              0.21      2.13      -6.65    8.06
                            SB3
                                    SURA                 0.26      2.22      -6.94    7.20
                                    Profuturo GNP        0.26      2.17      -7.63    8.18
                                    XXI Banorte          0.21      2.16      -6.82    5.91
                                    Banamex              0.27      2.39      -7.14    8.42
                            SB4
                                    SURA                 0.32      2.52      -7.22    7.98
                                    Profuturo GNP        0.33      2.46      -8.41    9.15

4     Results
It is important to note that the actual pension of the future retiree is determined—when all is said and done—by
the accumulated balance at the moment of retirement. Hence, what really matters is the cumulative return over the
working life. Average monthly returns, and to some extent, monthly fluctuations of returns, are not as relevant as
long-term returns. Nevertheless, the statistics shown in Tables 3 and 4 suggest some interesting trends. Generally
speaking, Mexican funds have generated, on average, absolute returns in line with their intended risk profiles (the SB4
funds outperforming the SB1 funds). A visual inspection of the Chilean funds, however, suggest a rather di↵erent
trend (or more precisely, lack of trend), as all funds (from A through E) delivered strikingly similar results (a point
we will revisit later). This is the first sign that Mexican and Chilean funds have behaved in a very di↵erent fashion.

4.1    Cumulative Returns and Risk
Figures 1 (Mexico) and 2 (Chile) compare the 5-year cumulative returns of all funds, for each manager, considering
all possible 5-year (60-month) periods between April 2008 and March 2018. Each point in the graph represents the
cumulative return a worker would have obtained for an investment in a specific fund, made 5-year prior to the date
indicated on the horizontal axis, assuming the worker maintained the position during the entire 5-year period. For
example, the left-most points in the graphs (which correspond to March 2013) reflect the returns associated with the
April 2008-March 2013 time-window.
    These two graphs reveal some extraordinary findings. In the Mexican case, regardless of the manager, all funds
(SBs) exhibit markedly di↵erent returns; and in all cases the returns are in line with the expected risk profile of each
fund. Take Sura for example (Figure 1), the SB4 fund shows always returns superior to those of the SB3 fund; and
the SB3 fund, in turn, outperforms the SB2 fund, and so on. Thus, the funds are ranked, in terms of their cumulative
returns, for all 5-year periods, in the “correct” order. The same holds for the other three managers. In the Chilean
case, it is equally clear that in many cases the funds are ranked in a sequence which is the opposite of what it was
expected. To be precise, in many instances the A fund cumulative returns are lower than those of the B fund; and
those of the B fund, below the C fund; and so forth. Worse yet, in many periods the A fund return is below the E
fund return! This disturbing pattern occurs in 39% of the cases (that is, 24 out of the 61 time-windows considered).
    Figure 3a depicts the 5-year cumulative returns for each of the four Mexican funds (SBs), for each fund manager.
Again, each point in the graph represents the cumulative return obtained during the 5-year period ending on the
month indicated on the horizontal axis. As in Figures 1 and 2, we considered all possible 60-month periods, moving
one month at a time, starting with April 2008. Analogously, Figure 3b shows the cumulative returns for all five funds,
for each Chilean manager.

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Table 4: Chilean funds: monthly returns and descriptive statistics, expressed in percentage (%), April 2008
- March 2018

                              Fund     Manager      Mean       St.Dev.     Min    Max
                                       Cuprum       0.33        3.91     -22.44    9.13
                                       Habitat      0.36        3.84     -20.46   9.62
                                A      Planvital    0.34        3.86     -20.46   10.23
                                       Provida      0.30        3.88     -21.65    9.53
                                       Capital      0.28        3.81     -20.14    9.64
                                       Modelo       0.34        2.74      -6.23    6.53
                                       Cuprum       0.32        2.81     -15.16    6.24
                                       Habitat      0.35        2.74     -14.14   6.66
                                B      Planvital    0.31        2.74     -13.96    6.84
                                       Provida      0.29        2.77     -14.40    6.53
                                       Capital      0.27        2.75     -13.39    6.34
                                       Modelo       0.31        2.01      -4.19    4.44
                                       Cuprum       0.34        1.84      -8.84    4.04
                                       Habitat      0.37        1.76      -7.59   4.20
                                C      Planvital    0.31        1.71      -7.30    4.05
                                       Provida      0.30        1.78      -7.93    3.90
                                       Capital      0.29        1.79      -8.00    3.76
                                       Modelo       0.31        1.35      -2.55    2.91
                                       Cuprum       0.32        1.15      -4.85    2.92
                                       Habitat      0.34        1.09      -3.74   2.86
                                D      Planvital    0.29        1.04      -3.66    2.70
                                       Provida      0.29        1.11      -3.95    3.02
                                       Capital      0.29        1.12      -3.89    2.78
                                       Modelo       0.31        0.78      -1.22    2.23
                                       Cuprum       0.30        0.98      -2.59    3.57
                                       Habitat      0.33        0.96      -2.02   3.84
                                E      Planvital    0.26        0.80      -1.90    2.72
                                       Provida      0.28        0.97      -2.54    3.30
                                       Capital      0.33        0.95      -2.47    3.31
                                       Modelo       0.31        0.57      -1.11    1.90

     The message conveyed by the images is clear: while in the Mexican case there is an obvious performance di↵er-
ence by manager, in the case of Chile such di↵erence is almost negligible. All managers exhibit virtually identical
performance for their respective funds. An exception, in the Mexican case, are the SB1 funds: the discrepancies are
mild among all the managers, which is to be expected, as these portfolios are forced to be invested in highly rated
and liquid securities, which o↵er more stable and similar returns across all choices. In short, Figure 3b reinforces the
view that Chilean managers are characterized by strong herd behavior, as indicated before, and therefore selecting
the “appropriate” manager is less important than in the Mexican pension market. More precisely, the following cal-
culation stresses the point: the A fund average cumulative 5-year return, (considering the 61 time-windows permitted
by our data), for each of the Chilean managers are (according to the same order shown in Figure 2 and omitting
Modelo to avoid distorsions): 29.5%, 30.3%, 28.6%, 26.8%, and 26.4%. The corresponding mean value is 28.3% with
a standard deviation equal to 2.1%. By the same token, the corresponding figures for the four Mexican managers
and considering the the SB4 fund returns (“equivalent” to the Chilean A fund) are: 21.2%, 27.6%, 32.0%, and 29.9%.
And the corresponding mean value is 27.7% with a standard deviation equal to 5.4%. This calculation validates the
impression conveyed by Figure 3b—Chilean managers show a remarkably similar performance. Therefore, in what
follows, the Chilean results will be presented at the industry (aggregated) level as the di↵erences in performance
among managers are virtually negligible.
     Figures 4a and 4b display another curious trend. These graphs show the average cumulative return, for several

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Figure 1: Mexico: Cumulative returns by asset manager, for each fund type, for all 5-year windows, as a
function of the last month in each 5-year window (March 2013 - March 2018).

holding periods (starting with one-month and up to 60-months), for all fund managers. In Figure 4a, we see that
the Mexican funds exhibit a clearly distinguishable pattern as the periods gets longer: their returns become more
divergent. In other words, the di↵erence in performance among the four funds, for all managers, grows as the holding
period gets longer. That is not the case (Figure 4b) with the Chilean funds. The funds o↵er remarkably similar
returns. In fact, four of the five funds show returns that tend to converge over time. Considering the industry average
in Mexico for a 60-month holding period, the di↵erence in cumulative return between the riskiest funds (the SB4s)
and the most conservative funds (the SB1s) is 16.8%, The corresponding di↵erence in the case of Chile (i.e. between
the A fund and the E fund) is only a mere 3.7% (for the intermediate funds lie in between). This is rather unsettling
as it questions the benefits of actually o↵ering these five funds.
    Figures 5a and 5b show the Sharpe ratios (SRs) for all funds and all managers. (For convenience, the SRs ratios
were calculated assuming that the risk-free rate was zero.) The SR reflects the return adjusted by risk (i.e. normalized
by units of risk). Hence, one could argue that in theory, after adjusting for risk, all funds should exhibit similar SRs.
In the Mexican case, broadly speaking, the SB2, SB3 and SB4 funds do show this trend. The exception is Profuturo,
which, as of March 2015, shows a noticeable di↵erence among the SRs of these three funds. That said, what it is
interesting to notice, is that notwithstanding the similar values, the SB4 funds tend to show slightly higher SRs than
the SB2 and SB3 funds. We can say that in general investors in the SB4 funds were paid a small premium for the
risk they took. The lowest SRs associated with the SB1 funds are somehow to be expected, due, in part, to the
money market and inflation-linked securities the SB1 funds are forced to maintain. This, clearly, puts a brake on the
potential yield of the SB1 funds.

                                                           8
Figure 2: Chile: Cumulative returns by asset manager, for each fund, for all 5-year windows, as a function
of the last month in each 5-year window (March 2013 - March 2018).

    Figure 5b shows, again, another unsettling outcome in relation to the Chilean funds: investors in most funds
simply took a lot of risk while they were not compensated for it. For instance, the SR for the E fund is on average
1.6 times that of the C fund, and about 2.7 times that of the A fund. This suggests that the riskiest funds (A and B)
delivered, consistently, risk-adjusted returns that were inferiors, i.e. not in line with their risk profiles.
    In terms of looking at the risk of the di↵erent funds, to avoid the shortcomings related to the use of the standard
deviation as a risk metric, we considered two alternative metrics commonly used in risk management and financial
engineering: the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR). The VaR is the maximum loss that

                                                          9
(a) Mexico

                                                (b) Chile

Figure 3: Mexico and Chile: Cumulative returns by fund type, for each manager, for all 5-year windows, as
a function of the last month in each 5-year window (March 2013 - March 2018)

                                                   10
(a) Mexico

                                                     (b) Chile

Figure 4: Mexico and Chile: Cumulative returns by fund type, for each manager, for holding periods of
di↵erent lengths, (March 20013 - March 2018).

a portfolio can su↵er in a specific period of time, estimated with a given (normally very high) level of confidence
(Linsmeier & Pearson, 2000). If the VaR is estimated with a confidence 1 ↵, it follows that the probability of having
losses exceeding the VaR is ↵. More formally, the VaR of a random variable X with cumulative distribution function
(cdf) F (·) and with a confidence level ↵ 2 [0, 1] is defined as

                                                         11
(a) Mexico

                                                 (b) Chile

Figure 5: Mexico and Chile: Sharpe ratio by asset manager, for each fund type, for all 5-year windows, as a
function of the last month in each 5-year window (March 2013 - March 2018).

                                                    12
VaR↵ [X] := min{t|F (t)             ↵} = min{t|P (X  t)                    ↵}.

    The CVaR, which is an alternative risk metric to the VaR, considers only those losses that exceed the VaR
(Rockafellar & Uryasev, 2000). The CVaR of a random variable X with cdf F (·) and with confidence level ↵ 2 [0, 1]
is defined as
                                                                            Z   1
                                                                    1
                                          CVaR↵ [X] :=                              VaR [X]d .
                                                               1        ↵       ↵

     In short, the CVaR (also known as expected shortfall) is the expected value of the losses that exceed the VaR.
When X is a discrete random variable with support points z1 < z2 < . . . < zN and associated probabilities p1 , . . . , pN ,
it follows from (Rockafellar & Uryasev, 2002) that
                                                           "                        !                           #
                                                   1           X
                                                               k↵
                                                                                                  X
                                                                                                  N

                               CVaR↵ [X] :=                          pk         ↵       zk ↵ +             p k zk ,
                                               1       ↵
                                                               k=1                               k=k↵ +1

                                       Pk ↵               ↵
                                                               Pk       1
where ↵ 2 (0, 1) and k↵ is such that         p
                                         k=1 k
                                                  ↵ > k=1     pk .
     In this study we focused on the CVaR of the funds’ returns. Relying solely on the VaR somehow limits the scope
of the analysis, since the VaR does not fully capture the tail end of the distribution associated with lower returns.
The CVaR, which focuses on the values exceeding the VaR, does. Additionally, another shortcoming of the VaR is
that it violates the so-called subadditivity condition. The CVaR, a coherent risk metric (Artzner, Delbaen, Eber, &
Heath, 1999), does not have this limitation. In the context of pension funds we consider a fund to be riskier if its
CVaR becomes more negative.
     Figures 6a and 6b show the CVaR (90% confidence), for the Mexican and Chilean system, (again, using 5-year
shifting windows), for all fund managers. Generally speaking, in all cases the di↵erent funds exhibit risk levels in
agreement with the regulators’ intentions. An interesting trend is that in Chile, the risk associated with the D and E
funds, in recent years, seems to converge. The 90%-CVaR in the case of Chilean funds show no major trends and a
weak cyclical e↵ect; the expected loss for the A fund settles around -4.6% (conditional on incurring in losses), while
it is around -1.3% for the E fund. As for the Mexican funds, the 90%-CVaRs, present a positive trend with almost no
cyclical behavior. However, it should be noted that on average, the CVaR of the SB1 funds is around -2.6%, which
means that these funds are riskier—according to this metric—than the Chilean E funds, which show an average CVaR
of -1.29%. Additionally, the average CVaR of the SB4 funds is -3.9%, higher than its Chilean-counterparty (the A
fund is around -5.12%), meaning that in case of incurring in loses, the SB4 funds are likely to lose less money than
the Chilean A funds.

4.2     Rank-Order Metrics
Having considered di↵erent criteria to evaluate risk and return, we now examine the rank order imply by these criteria.
To this end, we focus on the two most relevant parameters, the CVaR (a more encompassing metric than the VaR)
and the 5-year cumulative return. An adequate performance of the system would mean that in terms of risk and
return the Mexican funds should displayed decreasing risk and return levels when moving from SB4 to SB1. By the
same token, the Chilean funds should also exhibit decreasing risk and returns when moving from A to E. In short, if
not always, at least most of the time the SB4, SB3, SB2 and SB1 funds should be rank-ordered as 1, 2, 3, 4 (Mexico);
and the A, B, C , D and E funds should be ranked as 1, 2, 3, 4, 5 (Chile).
    For this purpose, we consider three di↵erent rank-order metrics. Each addresses a di↵erent aspect of the departure
from the correct (desired) rank order.
  (i) Hamming distance. This metric assigns a value of 0 if a fund is in the correct position and 1 otherwise (see
      Hamming, 1950). Thus, the maximum possible value is 5, corresponding to a situation in which all the funds
      are in the “wrong” position. For example, the sequence (1, 2, 3, 4, 5) obviously results in a value equal to zero.
      However, the sequences (5, 3, 4, 2, 1) and (3, 4, 1, 5, 2) are assigned a value of 5, while (1, 2, 3, 5, 4) would
      get a 2. Let us note that the sequence (5, 4, 3, 2, 1), which in our case represents the worst-case scenario, is
      assigned a value of 4, explained by the correct position of Fund C. A shortcoming of this metric is that only
      focusses on whether a fund is in the correct position, but not the distance to its “correct” position.

                                                                    13
(a) Mexico

                                               (b) Chile

Figure 6: Mexico and Chile: CVaR based on monthly returns by asset manager, for each fund type, for all
5-year windows, as a function of the last month in each 5-year window (March 2013 - March 2018).

                                                  14
(ii) Spearman footrule. This metric considers the absolute di↵erence between the position in which a fund is,
      and the position it should have, and adds all five numerical values (see Diaconis & Graham, 1977). In short, it
      attempts to capture the magnitude of the deviation from the correct rank order as well. For example (3, 4, 1,
      5, 2) results in a value equal to |3 1| + |4 2| + |1 3| + |5 4| + |2 5| = 10. If the funds are rank-ordered
      in exactly the reverse sequence, i.e. (5, 4, 3, 2, 1), something we can describe as the worst possible situation,
      the value is 12, which is indeed the maximum possible value this metric can have.
 (iii) Kendall Tau rank distance. This metric counts the number of pairwise discrepancies between the correct
       rank order and the actual rank order (see Kendall, 1938, 1955). Since we have five funds, the possible pairwise
       comparisons are ten (1 with 2, 3, 4, and 5, and then 2 with 3, 4 and 5, and so on), and hence the maximum
       possible value is 10. The following example clarifies the calculation. Suppose the funds have been rank-ordered
       in the following sequence: (3, 4, 1, 2, 5). In this case the Kendall Tau is 4 because the pairs (3, 1), (3, 2), (4,
       1) and (4, 2) represent pairwise disagreements with respect to the desired sequence (1, 2, 3, 4, 5).
    In all three metrics, higher values are associated with higher levels of discrepancy in terms of the rank order. To
facilitate the comparisons, we normalized all metrics by their maximum value. Thus, a value of 0 reflects a perfect
rank order, that is, (1, 2, 3, 4) in the case of Mexico and (1, 2, 3, 4, 5) in the case of Chile; whereas a value of 1
indicates the maximum discrepancy with respect to the desired benchmark.
    Figures 7a and 7b show the normalized values of the three metrics applied to the CVaR, in the Mexican and
Chilean cases, as a function of time, again, using 5-year moving windows. Table 5 reports the average values. From
the plots, as well as from the tables, we can conclude that the Mexican funds are correctly rank-ordered by risk
for almost all the periods observed (with exception of Profuturo GNP), while the Chilean funds are not. When
considering a 5-year time-window, the average Hamming distance is 3.3% in the case of the Mexican funds and 22.0%
for the Chilean funds. The average Spearman distances are 1.8% and 9.0%, respectively; and the average Kendall Tau
distances are 1.0% and 5%, again, respectively. A similar tendency is detected when considering 4-year and 6-year
time-windows. The evidence indicates that the Mexican regulation was more e↵ective than the Chilean regulation in
terms of producing funds that are, in terms of their risk, rank-ordered correctly.
    Figures 8a and 8b show the normalized values of the three metrics applied to the 5-year cumulative returns, in the
Mexican and Chilean cases, as a function of time, using 5-year moving windows. Table 6 reports the average values.
The evidence is compelling. While the Mexican funds maintain very low rank-order metrics (an average Hamming
distance of 2.0%, and Spearman and Kendall Tau distances of 1.0% each for a 5-year time-window), their Chilean
counterparties present a Hamming distance of 46%, a Spearman distance of 49%, and a Kendall Tau distance of
44%. These figures suggest that the Chilean funds were in disarray for almost half of the time-period considered.
The Mexican funds, on the contrary, were ranked correctly for virtually the entire period studied (the exception is
Profuturo, which showed some anomalies starting in 2016).

5     Concluding Remarks
Mexico and Chile have structured their pension systems based on a common goal: o↵ering future retirees several
investment options, that is, funds, which in theory, should have di↵erent risk-reward profiles. These countries di↵er,
however, in the type of investment regulation they enacted to achieve this goal. Mexico structured its investment reg-
ulation around two simple concepts: maximum limits by asset classes combined with a portfolio-level risk restriction.
Obviously, the restrictions vary according to the intended risk profile of the fund. Chile, on the other hand, opted
for maximum and minimum limits by asset class, with limits dictated by the desired risk profile of each fund. Chile
does not employ any portfolio-level risk metric to control risk (such as VaR or CVaR), but it does enforce a minimum
yield test based on the industry average.
    Our analysis, based on a number of risk—and return—related metrics applied to actual fund returns during the
April 2008—March 2018 period is unequivocal: while the Mexican funds exhibited returns very much in line with
the desired risk-reward profile intended for such funds, their Chilean counterparties were characterized by an erratic
behavior, at odds with the regulator desire. To be precise: the Mexican funds exhibited, consistently, cumulative
returns in agreement with their risk profile (highest for the SB4 funds, lowest for the SB1 funds), for all 5-year
windows tested within our dataset (longer and shorter time windows yielded similar findings). Additionally, the
Mexican funds, in terms of risk (CVaR), were, again, in almost all cases ranked as expected (the SB4 funds on top,
the SB1 funds at the low end). Finally, all Mexican funds (when analyzed by asset manager) exhibited similar Sharpe
ratios (risk-adjusted returns). The exception were the SB1 funds that showed lower SRs due to their high percentage
of (mandatory) holdings in highly-rated, but low-yielding, securities.

                                                           15
(a) Mexico

                                                       (b) Chile

Figure 7: Rank-order performance metrics based on the 90%-CVaR, and a rolling 5-year time-window, March
2013-March 2018.

    The Chilean funds, however—leaving aside that all managers delivered, for the same type of fund, essentially
identical returns as it has been reported in Section 4.1—displayed some disturbing risk-reward patterns. First, 39%
of the cumulative 5-year return windows examined resulted in the A fund ranked at the bottom (in terms of returns)
while the E fund was on top, and all the other funds (B, C and D), in between, but in a sequence that was exactly
the opposite of what it was intended by the regulator. Second, in terms of risk (CVaR), the Chilean funds were,
again, unsatisfactorily ordered in about half of the cases. And third, and perhaps more troubling, the Chilean funds
exhibited consistently Sharpe ratios that were not only notoriously dissimilar among all funds, but incredibly, revealed
that the riskier funds had delivered inferior risk-adjusted returns compared to the most conservative funds. In essence,
the more risk Chilean investors took, the less they were compensated for it.
    It might be argued that perhaps our findings related to the Chilean funds are an artifact of the period considered
(April 2008-March 2008). The reason to consider this period is that Mexico implemented its multi-funds system
only in 2008. However, extending our analyses to the beginning of the Chilean multi-fund system (October 2002),
reinforces our findings, i.e., the Chilean funds keep on showing the same all undesirable patterns already identified

                                                          16
Table 5: Average rank-order metrics, based on the 90%-CVaR (March 2013 - March 2018), for di↵erent-size
time-windows.

                                                                      Window-size in years
                        Manager                   Distance            4       5          6
                                                  Hamming            0.00   0.00       0.00
                        XXI Banorte               Spearman           0.00   0.00       0.00
                                                  Kendall Tau        0.00   0.00       0.00
                                                  Hamming            0.00   0.00       0.02
                        Banamex                   Spearman           0.00   0.00       0.01
                                                  Kendall Tau        0.00   0.00       0.01
                                                  Hamming            0.00   0.00       0.00
                        SURA                      Spearman           0.00   0.00       0.00
                                                  Kendall Tau        0.00   0.00       0.00
                                                  Hamming            0.17   0.13       0.06
                        Profuturo GNP             Spearman           0.09   0.07       0.03
                                                  Kendall Tau        0.06   0.04       0.02
                                                       (a) Mexico

                                                                      Window-size in years
                        Manager                   Distance            4       5          6
                                                  Hamming            0.20   0.22       0.23
                        Industry average          Spearman           0.08   0.09       0.10
                                                  Kendall Tau        0.05   0.05       0.06
                                                        (b) Chile

in the April 2008-March 2008 time-window. To this distressing picture, we need to add two more elements: Chilean
managers (as it has been reported before) revealed a manifest herd behavior, and, for longer time periods, all Chilean
funds (from B to E) seem to converge to identical returns.
     Being mindful that correlation does not imply causation, we need to be cautious before deriving sweeping conclu-
sions. Nevertheless, one factor is salient: Mexico employs a portfolio-level risk-controlling feature, while Chile does
not. Granted, we might argue that perhaps the rationale behind a rolling daily VaR for a long-term investment vehicle
(such as a pension fund) is unwarranted. But it is certainly better than no portfolio-level risk-restriction at all. After
all, the Chilean approach (controlling risk by asset class limits) has no theoretical or empirical basis. In essence, this
form of indirect risk control is based on the notion that market conditions do not change as a function of time—a
patently false assumption.
     Moreover, we suspect that in the Chilean case, another element that has amplified the deficiencies of not having
a portfolio-level risk constraint is that the regulator forces the funds to maintain minimum positions in all assets
(we assume, in the spirit of promoting diversification). Such constraints clearly limit the ability of asset managers to
control risk properly. In short, Chilean asset managers have also been victims of the regulation.
     In summary, two things are clear. First, while the Mexicans funds behaved as intended, the Chilean funds behaved
in a manner completely at odds with their goal. And second, both countries approached the objective of creating
funds with di↵erent risk-reward profiles by means of quite di↵erent investment regulations. These two facts, taken
together, should be considered a serious warning for the many pension funds that still keep on using asset-class limits
as the preferred tool to control risk. Furthermore, the fact that the period considered included the subprime crisis
makes this situation even more alarming: a good investment policy is supposed to protect investors precisely against
the things that happen when the markets are not operating under a “normal” regime.
     Finally, a potential criticism to our analyses is that the time period considered is not long enough to derive
statistically significant conclusions. To this observation we might be able to respond noting that the data considered
to analyze the Chilean system was not a sample from a much bigger universe; it was the universe (all the data points

                                                           17
(a) Mexico

                                                      (b) Chile

Figure 8: Average rank-order metrics, based on the cumulative return (March 2013 - March 2018), for
di↵erent-size time-windows.

since the system was implemented). However, a more relevant consideration can be drawn from an entirely di↵erent
discipline that has also an important e↵ect of society’s welfare: earthquake engineering.
    When there is a quake many buildings fail. What civil engineers do then is to analyze those which fail, and see
what lessons they can derive from these failures. And then they incorporate the new insights into building codes and
construction practices. One thing civil engineers do not do after a quake, is to wait many years until a sufficiently
high numbers of building fail in order to derive “statistically significant” conclusions, or, have a “sufficiently long
observation period.” The empirical evidence is on their side: controlling by risk (i.e. magnitude of the earthquakes),
building failures have decreased over time. Had engineers waited one-hundred years to derive statistically significant
observations, earthquake engineering probably would still be in its infancy. By the same token, waiting one-hundred

                                                          18
Table 6: Average rank-order metrics, based on the cumulative returns (March 2013 - March 2018), for
di↵erent-size time-windows.

                                                                  Window-size in years
                       Manager                 Distance           4       5          6
                                               Hamming           0.07   0.03       0.00
                       XXI Banorte             Spearman          0.04   0.02       0.00
                                               Kendall Tau       0.03   0.02       0.00
                                               Hamming           0.06   0.05       0.02
                       Banamex                 Spearman          0.04   0.02       0.01
                                               Kendall Tau       0.03   0.02       0.01
                                               Hamming           0.02   0.00       0.00
                       SURA                    Spearman          0.01   0.00       0.00
                                               Kendall Tau       0.01   0.00       0.00
                                               Hamming           0.01   0.00       0.00
                       Profuturo GNP           Spearman          0.00   0.00       0.00
                                               Kendall Tau       0.00   0.00       0.00
                                                    (a) Mexico

                                                                  Window-size in years
                       Manager                 Distance           4       5          6
                                               Hamming           0.52   0.46       0.48
                       Industry average        Spearman          0.54   0.49       0.55
                                               Kendall Tau       0.48   0.44       0.51
                                                     (b) Chile

years to enjoy the benefits of a long fund-returns database—while damaging the pension prospects of several genera-
tions in the meantime—does not seem like a prudent proposition. When danger is imminent, and the consequences
are grave, one must often act with whatever information is available. And what we have observed regarding the
Chilean pension system calls for an urgent need to re-evaluate its investing regulation regime.

                                                        19
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