The author wants to thank Rosa Hilda Posadas and Antonio Delgado García former General Managers of Pensiones Civiles del Estado de Michoacán and Gustavo Arias, the actual Finance director, for the support and interest given to the present paper. Any Grammar, Semantics or style error remain as the author’s responsibility. El autor agradece al CONACyT por apoyar la publicación del presente artículo, al amparo de la convocatoria de Apoyos Complementarios para la Consolidación Institucional de Grupos de Investigación (Retención, 2012)".

This paper presents an assesment of an active portfolio management process with a t-Student orthogonal GARCH (O-GARCH) covariance matrix, in order to achieve a 7.5% actuarial target return and to formulate alpha in defined benefit pension funds for Dirección de Pensiones Civiles del Estado de Michoacán. To test this, three discrete event simulations were performed using, in the first one, a passive portfolio management process with a target position rebalancing discipline and, in the other two, an active portfolio management with range portfolio rebalancing where an equally weighted covariance and a t-Student O-GARCH covariance matrix are used. The results suggest that the O-GARCH matrix is suitable for active portfolio management in this sort of pension funds.

En este artículo se evalúa la utilidad de un proceso de administración activa de portafolios empleando una matriz de covarianzas GARCH ortogonal (O-GARCH) con función de verosimilitud t-Student, al aplicarlo en la reserva técnica de fondos de pensiones de beneficio definido de la Dirección de Pensiones Civiles del Estado de Michoacán. Esto tanto para lograr el objetivo de 7.5% anual de rendimiento establecido en su estudio actuarial como para definir

There are several pension fund schemes in Mexico. Among them, the defined benefit and the defined contribution are the most common. In the former scheme, a percentage of the salary earned over the last year as an employee at the time of retirement. In the latter, the employee retires only with the amount of money saved during the accumulation period, ^{1}

In each state of Mexico, there are several pension funds that manage the retirement savings for their public servants. Among these, a defined benefit pension fund, which is owned by the public servants of Michoacán and known as “Dirección de Pensiones Civiles del Estado de Michoacán (DPCEM)", is the focus of this paper.

The DPCEM covers all the public servants in the State of Michoacán (about 15,000). Among the most observable legal liabilities in the pension plan is that all the beneficiaries retire with the 100% of its salary paid during the last year in service. In order to fund this, the Government and the employees save an amount equal to the 4% of the employees' monthly wages, considering an actuarial yearly wage increase of 6.5%. By assuming a life expectancy of 81 years for Mexico and due to the fact that an employee in this pension fund can retire after 30 years of service (an average 45-50 years of age at retirement) and a 4% theoretical inflation rate, it is necessary for DPCEM either to achieve a nominal 7.5% yearly return in the investment policy of its Technical Reserve (TR) ^{2}

This paper focuses on the first proposal and its aim is to show the usefulness of an active portfolio investment process that uses a t-Student log likelihood O-GARCH covariance matrix in the TR of DPCEM. The main focus is to test an active portfolio management process with the historical asset allocation in the six different markets shown in the investment policy statement (IPS) presented in _{bmk}) related to it.

Why an active portfolio management process? With the advent of the

Due to several economical, financial and behavioral circumstances, the aggregate optimality (as a proxy definition of ^{3}

In some cases (such as _{bmk}). In contrast, the active management practice, which seeks to outperform a benchmark or a market index, could be executed with two rebalancing disciplines (among the most used) known as _{bmk}; the latter consists of discretionary investment proportions that must follow upper and lower asset or market type limits, stated in an IPS like the one presented in

From several strategies widely used as rebalancing disciplines in the portfolio management practice and from those previously mentioned, DP- CEM decided to test an RPR discipline using the IPS shown in

In order to asses whether a passive Target Position (TP) or an active Range Portfolio Rebalancing (RPR) portfolio management strategy is more suitable to the fund, three discrete event simulations were performed. One was performed for the passive portfolio management case with a TP regime and two for the RPR active portfolio management that use two different co- variance matrixes: 1) a constant or equally weighted covariance and 2) an O-GARCH with a t-Student log likelihood one.

Once established the aim of the present paper (to test the use of active portfolio management with O-GARCH matrixes in the Technical Reserve of DPCEM and similar ones) the results will be presented as follows: In a second section, a brief explanation of the Markowitz-Tobin-Sharpe-Lintner model is given along with a review of the O-GARCH covariance matrix model. Following this, the assumptions and general structure of the algorithm used in the three discrete event simulations are presented along with the results obtained. After this, the document ends with the concluding remarks.

The Markowitz-Tobin-Sharpe-Lintner or MTSL model is a theoretical proposal that incorporates a risk-free asset in the asset allocation, a drawback that the original _{i})] ⋯E(_{n})]’ an l x n asset or market grouping matrix D and an l x l minimum or maximum limits vector d established in the IPS:^{4}

Subject to:

w*’l = 1

w* ≥ 0

The second step is given by the proportion in the total investment budget co in ^{5}

Once ω is determined, the final optimal portfolio is a linear combination of rf and the risky asset w * :

The portfolio selection model in (3) is the so-called Markowitz-Tobin-Sharpe- Lintner model or MTSL (^{6}

With the earliest proposals of

As a starting point for this paper, the model departs from the assumption that the returns vector _{i} of the i^{th} asset is either^{7}_{1}...,r_{i}.]. With this, the log likelihood problem can be solved through two functions. ^{8}

Subject to:

When ^{9}

Subject to:

gl ≥ 2

The expression in (6) will be used by the DPCEM based on the theoretical assumption that the t-Student matrix is more suitable to model sample probabilities and more appropriate to measure the characteristic fat tails of the financial data used.

For the multivariate case, one of the first proposals is the one made by

Despite its low computational efficiency, this model does not take into account the correlation clustering effect. A model that solves this situation is the BEKK GARCH of ^{10}_{
c
} that leads to the definition of a n x n matrix of eigenvectors E anda n spectrum

The computational efficiency of the O-GARCH model is based on the variance (eigenvalues) of the principal components (_{1}...,_{i}.]) in the diagonal elements of

With the definition of

Why use this specific multivariate GARCH model? There are two reasons: a) computational efficiency b) its practical usefulness in financial risk modeling to calculate high dimension matrixes with low latency data.

The computational efficiency of the O-GARCH model can be compared against the

Several papers can be mentioned related to the practical usefulness of the O-GARCH, ^{11}^{12}

Another practical use is reviewed by ^{13}^{14}

Following

Also,

Therefore, because of its computational efficiency, O-GARCH model's ability to measure the correlation clustering effect even if the historical data is not latent, ^{15}^{16}

Now that the calculation of the MTSL model and the O-GARCH covariance matrix have been reviewed as parts of the optimizer used in the portfolio management process, the assumptions of the three discrete event simulations performed are presented, noting that a proof of the presence of volatility and correlation clustering in the six benchmarks of the investment universe is shown in

Given a time frame from January 2^{nd}, 2002, to December 31^{st}, 2010, 470 weekly interval simulations are performed for each simulated portfolio, using each of the benchmarks presented in Appendix one as financial assets.^{17}

These financial assets or benchmarks were valued at Mexican pesos (MXN) at a December 29^{th}, 2000 base 100 value and incorporated currency impact. The length of each time series (_{i}) is T =52 weeks and it is assumed that these represent the behavior of zero tracking error Exchange Traded Funds (ETF’s) that replicate them.

A quantitative analysis algorithm that performed the entire portfolio selection process (analysis, rebalancing and mark to market valuation) was programmed in MATLAB and, among the most relevant ones, the following assumptions and parameters were considered:

The theoretical^{18}

The financial data sources are Bloomberg™, Reuters™ and Infosel^{MR}.

In order to incorporate the impact of financial costs, a 0.25% fee is assumed in each trade either in the ETF’s or in the FX market (noting that an institutional investor such as DPCEM can get access to a lower transaction cost). This fee will be used in order to measure a higher impact in the final turnover results.

The risk-free asset rf used is the weekly secondary market curve rate of 28-day-maturity Mexican treasury certificates (CETES). This rate was published on 2012 by Banco de México.

Only an MXN bank account and two investment contracts (one in US dollars and another in MXN pesos) will be used. When a foreign asset position (USD valued) is sold, the amount is turned into Mexican pesos by selling USD using the current FX rate. When the opposite happens, the US dollar amount is funded from the Mexican bank account.

The expected values in the return vector r are shown in the following expression:

In order to calculate the O-GARCH matrix with (10) using (6) as the log likelihood function, the algorithm selected the best GARCH (p, q) model for eachmain component by using different ARCH lag terms truncated at the value of five and different GARCH lag terms truncated at the value of two. The goodness of fit of the best GARCH model in each principal component is measured by the Bayesian information criterion of

For the passive management (TP) portfolio simulation, the main assumption is that all the
investment balance is allocated in the risky asset given by the benchmark asset
allocation (w* ^{=} w_{bmk}) shown in ^{19}^{nd}, 2002,
base 100.

The historical value of the simulated portfolios and their accumulated turnover is presented in chart one and summarized in table one. It is shown that the three simulated portfolios and the benchmark had a better performance than a theoretical financial asset that paid the 7.5% target return (light area). As shown, the simulated portfolios using the O-GARCH covariance matrix lead to a superior turnover than the benchmark, the passively managed and the constant parameter covariance matrix portfolios.

In order to confirm this result and to follow the portfolio management performance evaluation practices, a quality chart of the difference between the observed weekly return of each simulated portfolio and the benchmarks is presented in Chart two. As noted, the O-GARCH portfolio showed the highest positive alpha against the benchmark, suggesting a better performance if an O-GARCH matrix is used in the active portfolio management.

Portfolio or benchmark
Acumulated turnover
Yearly effective return
7.5% actuarial target return
105.05%
11.67%
Benchmark
205.34%
22.82%
Passive management: Target Position
172.12%
19.12%
Active management: constant covariance matrix. Active management: Gaussian OGARCH
131.13%
14.57%
covariance matrix
210.14%
23.35%
Active management: t-Student OGARCH
212.22%
23.58%

A more detailed examination of the results obtained during the three simulations is presented in Chart three where the historical allocation between the risk-free asset rf and the risky portfolio w * can be observed. The reader should note that the portfolios simulated with an O-GARCH covariance matrix (specifically the t-Student one) were more sensitive in the risk-free asset investment proportion during the dates where the financial crisis was acute (e.g. the Lehman Brothers Chapter eleven filling in the September-October 2008 period). This is due to the fact that the volatility and correlation clustering effect^{20}

Another perspective of these results is shown with a complete historical asset allocation in Chart four. In the case of the O-GARCH covariance matrix portfolio, the optimizer manages more accurately the investment in riskier markets such as the Mexican equity (IPC index) or the foreign equities proxied with the MSCI Global Gross equity. This historical behavior is summarized in the box plots of chart five that shows the different investment levels in each asset type for each portfolio.

It should be noted that the passive portfolio and the active one that use the equally weighted covariance matrix were highly concentrated in the Mexican government bond and international treasury bond markets (especially the former), suggesting that even though the IPS presented in suggests the presence of home bias in the asset allocation, the O-GARCH matrix handles, this drawback better thanks to an active and a proper management of risky assets during high volatility and correlation clustering periods.

With the results shown in charts four and five, two questions could be posed: Given the historical asset allocation resumed in chart five, does this higher active investment proportions in risky assets explain the better performance in the O-GARCH simulated portfolio? And, do the financial, political, and economic events have an impact on the behavior of the simulated portfolios, leading to a better performance with the use of a t-Student O-GARCH matrix? In order to answer the first question, Chart six presents the historical performance of the six markets in the IPS of

As noted, the best performers were the Mexican equity, Mexican sovereign bonds, and Mexican treasury markets. If this historical performance is compared with the investment proportions of Chart four and

The second question "Do the financial, political and economic events have an impact in the behavior of the simulated portfolios, leading to a better performance with the use of a t-Student O-GARCH matrix?” is answered in Chart seven where the historical behavior of the three simulated portfolios is compared with the financial and economic events shown in Chart six. The most notable period depicted in this chart is when the Lehman Brothers’ chapter 11 filing took place. During this time period, the volatility and correlation clustering effect was more observable.^{21}

Columns
43.6680667
3
14.5560222
1.84961698
13.61501%
Error
14393.7718
1829
7.86974947
Total
14437.4399
1832

Now that it has been shown that the portfolio management process using the t-Student O-GARCH matrix outperforms a 7.5% annual target return, it is necessary to know the behavior of the risk exposure and turnover relation (financial efficiency) observed, by using this active portfolio management process and covariance matrix to answer the following question: Do we have a higher financial efficiency if an active portfolio management process with a t-Student O-GARCH covariance matrix is used?

To answer this question, the efficiency of the portfolio management process is measured with the historical

Chart eight presents the historical values observed in each simulated portfolio along with a boxplot comparison. Table three presents the results of a one-way ANOVA test in the historical SR levels, suggesting, along with the results of the boxplot in Chart eight, that the use of a t-Student O-GARCH matrix leads to a better and more stable risk-return trade-off than both the passive and equally wieghted covariance matrix portfolios.

Given the IPS of Appendix one and from the results observed in the three simulations performed, it is concluded that the range portfolio rebalancing discipline with a t-Student O-GARCH matrix in an active portfolio management process is the most suitable for the Technical Reserve of the defined benefit pension fund of interest in this paper (Pensiones Civiles del Estado de Michoacán) and similar ones. This conclusion is supported by the achievement and outperforming of the 7.5% actuarial target return and by a higher turnover than the benchmark (alpha), the passively managed and the equally wieghted covariance matrix portfolios.

As noted in the results obtained, the use of a t-Student O-GARCH matrix leads to a more suitable asset allocation in the simulated portfolios. This remark is confirmed by the fact that it managed, in a better fashion, the investment proportion in the risk-free asset rf given the presence of volatility and correlation clustering. Also of interest is that the actively managed O-GARCH portfolio was more sensitive to the influence of financial, political, and economic events. This can be observed by using a softer portfolio performance and a more appropriate asset allocation in the risky asset w * during critical time periods.

As a final remark, it is noted that even though the use of a t-Student 0- GARCH Matrix could lead to a higher exposure to risk given the higher return, the observed financial efficiency (risk-return trade off) is higher in this case, supporting the use of this kind of active portfolio management process with this type of covariance matrix.

Before retirement.

A trust created to support the pension plan when the outflows are higher than the inflows (about 2032 with the current scheme).

Index tracking means that the manager must replicate the behavior or (if possible) the conformation of a market benchmark or index. This practice could lead to some limitations such as the impact of financial costs (trade fees, market timing, tax impact or liquidity) therefore the enhanced index tracking discipline tries to achieve higher gross returns than the replicated benchmark in order to compensate the impact of financial costs.

Such as the one described in

In the case of Pensiones Civiles del Estado de Michoacán, a value of A = 4 , related to a “neutral risk aversion investor” is set.

This in an almost parallel approach to

Given the information set It−1 that makes rt conditional.

Please see

An equally weighted covariance matrix is given by:

A property observed in fixed income assets or futures term structures.

I.e. that the number of assets could lead to a flatter log likelihood function.

Please refer to note 10.

Also proposed in Alexander (2002).

I.e. the financial asset price does not change due to a lack of liquidity.

At this point the out of sample and robust estimation scenario is set aside.

Assuming that these values represent the behavior of zero tracking error ExchangeTraded Funds (ETF’s) invested in each benchmark.

The original value of the pension fund was modified to MXN$ 10 million due to confidentiality issues.

As noted, this is an index tracking passive portfolio management practice.

In order to confirm that the level of volatility clustering was high in certain time periods like the aforementioned one, please refer to the historical ARCH test results shown in Appendix two.

It is also when the optimization problem given in (2) leads to the highest concentration in the risk free asset. Please refer to chart four in comparison with chart six to confirm this and to Appendix three for the proof of the presence of volatility clustering in those periods.

Using a T=52 return time series length from t to t-51.

1/ Even though Mexico has sovereign debt in USD, EUR, JPYand GBP, VALMER values the benchmark in MXN. 2/ Even though the country members quote theirassets in their local currencies, the price vendorturn theirvalue to US dollars in orderto calculate the benchmark's value. Therefore this benchmark incorporates currency impact.

Market
CURRENCY
Index or benchmark
benchmark ticker used in the present paper
Price vendor
Target investment level
investment level allowed
Maximum investment level allowed
Investment level by currency exposure
Mexican government debt market (fixed, float, real and quasi-sovereign debt)
MXN
Valmer México Gubernamental
MEX_GUBERNAMENTAL
Bolsa Mexicana de Valores S.A.B. de C.V. trough VALMER
67.061%
51%
100%
MXN exposure
The most traded stocks i n the Mexican stock market (IPC index members)
MXN
índice de Precios y Cotizaciones
IPCB100
Bolsa Mexicana de Valores S.A.B. de C.V.
23.471%
0%
35%
30%-100%
Mexican sovereign debt (UMS)
MXN/1
Valmer UMS
MEXJJMS
Bolsa Mexicana de Valores S.A.B. de C.V. trough VALMER
2.367%
0%
20%
USD exposure
United States Treasury bills and bonds markets
USD
EFFA-Bloomberg US treasuries index
EFFAUSB100
Bloomberg Inc. and European Federation of Financial Analists (EFFA)
2.367%
0%
20%
World treasury bonds markets of the 24 main developed and 24 developing economies (ex US) according to Standard & Poors
USD/2
Standard & Poors - Citigroup international treasury bond index ex-US
S&P-CITB100
S&P Stock Indexes and Citigroup Inc.
2.367%
0%
20%
0%-70%
World equity markets from the 24main developed and 24main deveolping countries according to MSCI
USD/2
MSCI Global Gross Equity Index USD
MSCIWORLDGBIOO
MSCI Inc.
2.367%
0%
20%
Total invested in the index
100.000%

Outflows (
Inflows (
Date
Amount (MXN)
Date
Amount (MXN)
Date
Amount (MXN)
18/01/2002
$
985,516.29
16/07/2004
$
641,630.06
02/07/2009
$ 89,559,398.71
22/02/2002
$
3,285,054.31
10/09/2004
$
8,212,635.78
18/12/2009
$ 66,358,097.13
22/03/2002
$
2,628,043.45
28/01/2005
$
82,662,629.71
12/02/2010
$ 289,453.33
A monthly $356.4284 outflow
19/04/2002
$
3,285,054.31
08/04/2005
$
33,659.65
paid the last week of each
14/06/2002
$
2,628,043.45
29/04/2005
$
474,292.10
month. The outflow is the
09/08/2002
$
1,642,527.16
06/05/2005
$
3,360.22
payment of custodial bank
06/09/2002
$
1,642,527.16
19/05/2006
$
3,285,054.31
services for the assets (ETF's
20/12/2002
$
679,476.88
09/06/2006
$
4,927,581.47
that replicate the six
10/01/2003
$
6,570,108.63
24/08/2007
$
22,995,380.20
benchmakrs) in the managed
02/05/2003
$
3,285,054.31
07/09/2007
$
8,212,635.78
portfolios.
01/08/2003
$
4,927,581.47
25/01/2008
$
19,710,325.88
05/09/2003
$
5,256,086.90
28/03/2008
$
6,570,108.63
19/09/2003
$
4,927,581.47
25/04/2008
$
6,570,108.63
09/01/2004
$
6,570,108.63
02/05/2008
$
218,473.54
07/05/2004
$
4,927,581.47
05/06/2009
$:
163,404,217.91

Source: Pensiones Civiles del Estado de Michoacán. The real numbers were changed due to confidenciality. This numbers reflect the behavior of the original magnitudes.

This appendix presents the evidence of the volatility and correlation clustering in the six markets (benchmarks) of the investment policy in Table one. In order to test the presence of the volatility clustering, the Engle (1982) ARCH test was performed in each asset and in each of the weekly dates used in the discrete event simulations. A 95% confidence level is used to test the next hypothesis:

Where R^{
2
} is the coefficient of determination of the next auxiliary regression given

This test was performed on each weekly date used for the simulation from January 2, 2002, to December 31, 2010.22^{22}

As noted in Chart A.2, not all the dates presented an ARCH effect, suggesting that not in all of them an O-GARCH matrix should be used in the MTSL model. In order to accept a general use of GARCH models in all the dates, a Poisson probability function hyphotesis test is used with a mean of λ and a 95% confidence level given by + (95% ⋅ λ = 28.10. With these parameters, the number of dates that report the presence of the ARCH effect were compared, and if this number was higher than 28.10, the presence of the ARCH effect was accepted for all the dates by assuming that the number of dates is high enough to generalize the presence of this phenomenon in each asset.

The results of these hypotheses tests are presented in the right panels of Chart A.2 and Table A.3. It can be shown that almost all the benchmarks (excepting the US treasuries -EFFAUSB100- that is not conclusive] lead to the acceptance of the ARCH effect for all dates.

Benchmark
Poisson critical value
Number of dates with ARCH effect
Conclusion
VLMR-MEX-GUBERN AMENTAL
28.10529586
100
This asset has ARCH efect
VLMR-MEX-UMS
28.10529586
55
This asset has ARCH efect
IPCB100
28.10529586
31
This asset has ARCH efect
S&P-CITB100
28.10529586
51
This asset has ARCH efect
EFFAUSB100
28.10529586
28
The test is not conclusive
MSCIWORLDGBIOO
28.10529586
60
This asset has ARCH efect

Once the evidence of the ARCH effect in the six benchmarks is presented, it is necessary to demonstrate the usefulness of an O-GARCH covariance matrix by testing the presence of the correlation clustering effect. In order to do so, the return time series _{i} in each asset was divided into two time groups by using the following distance suggested by Chow, Kritzman & Lowry (1999):

Where _{t} is a 6x1 vector with the observed return at date t in each asset, _{t} in each asset and _{0} the covariance matrix for the same data:

Each date t or returns vector r_{t} from that date was included in the "unusual dates” set Θ by following the rule:

Where v are the degrees of freedom related to the number of assets included in the covariance matrix C_{0}. Once Θ and Θ^{c} are defined with (16), two correlation matrixes were calculated for the usual and unusual date sets and the correlation of Θ^{c} was compared with Θ, leading to the results shown in

As noted, the correlation observed in "unusual times” increased in eight of 15 pairs of assets (or markets), suggesting the presence of correlation clustering in turbulent or unusual times. This can also be observed in the difference of the effective correlation (determinant) value observed in both matrixes. I

Asset type
VLMR_MEX_UMS
EFFAUSB100
SP500TRB100
MSCIWORLDGBIOO
VLMR_MEX_GUBERNAMENTAL
IPCB100
VLMR_MEX_UMS
1
EFFAUSB100
0.408287007
1
SP500TRB100
-0.340421318
-0.417051245
1
MSCIWORLDGBIOO
-0.352979265
-0.354964665
0.91810741
1
VLMR_MEX_GUBERNAMENTAL
-0.04527618
0.196305779
0.099137752
0.110340901
1
IPCB100
-0.010496348
0.05721501
-0.052602401
0.008871604
-0.019101146
1
Efectlve correlation (Determinant)
0.089648924
"
Asset type
VLMR_MEX_UMS
EFFAUSB100
SP500TRB100
MSCIWORLDGBIOO
VLMR_MEX_GUBERNAMENTAL
IPCB100
VLMR_MEX_UMS
1
EFFAUSB100
-0.117425925
1
SP500TRB100
-0.369673719
-0.286097744
1
MSCIWORLDGBIOO
-0.416630382
-0.255187292
0.9557821
1
VLMR_MEX_GUBERNAMENTAL
0.255163266
-0.057908116
0.481861901
0.49041005
1
IPCB100
-0.143888126
0.365550275
-0.08956764
-0.049046676
-0.151424226
1
Efectlve correlation (Determinant)
0.023188107
Asset type
VLMR_MEX_UMS
EFFAUSB100
SP500TRB100
MSCIWORLDGBIOO
VLMR_MEX_GUBERNAMENTAL
IPCB100
VLMR_MEX_UMS
0
EFFAUSB100
0.525712932
0
SP500TRB100
0.0292524
-0.130953501
0
MSCIWORLDGBIOO
0.063651118
-0.099777373
-0.03767469
0
VLMR_MEX_GUBERNAMENTAL
-0.300439446
0.254213895
-0.382724149
-0.380069149
0
IPCB100
0.133391778
-0.308335266
0.036965239
0.05791828
0.132323079
0