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 Markowitz (1952)standard model couldn’t solve. In the MTSL, the optimal portfolio selection is now conceived as a linear combination of a riskfree asset rf and a risky portfolio given by w *. This selection is performed in a two-step problem that starts with the solution of the next optimization problem given an n x 1 investment proportion vector w, an x covariance matrix C, an nx 1 expected return vector r = [E(r _i)] ⋯E(r _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

w*=argmaxw⁡w'.r-(rf.1)'.w.C.w-1/2 (1)

Subject to:

w*’l = 1

w* ≥ 0

D.w* = d

The second step is given by the proportion in the total investment budget co in w * and the proportion (1 - ω) in rf determined, following Levy and Markowitz (1979), with a quadratic utility function in the next expression:⁵

ω=argmaxω⁡ω∙(w'∙r)-ω2∙0.5∙Aw'∙C∙w (2)

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

w**=ω∙w*+(1-ω)∙rf (3)

The portfolio selection model in (3) is the so-called Markowitz-Tobin-Sharpe- Lintner model or MTSL (Markowitz, 1987, p. 5) and is one of the most widely used in the portfolio management industry (as is the case in this paper) because it incorporates a risk-free asset in the asset allocation step. Although the rationale of the MTSL is a very straightforward one, its main drawback is observed in the calculation of r and C due to the presence of estimation error; its computational inefficiency and also the presence of volatility and correlation clustering. As a potential solution for the computational efficiency problem, Sharpe (1963) proposed an alternative calculation of such parameters that led to the proposal of the CAPM model (Sharpe, 1966).⁶ This valuation model is the theoretical foundation of several heuristics and alternative portfolio selection models that have, as a central concept, the covariance of assets now proxied through the covariance with a common factor or set of factors. Although the CAPM model is a straightforward rationale for asset pricing, and setting aside the theoretical critiques made against it, the presence of volatility and correlation clustering and the potential presence of estimation error are some of its main drawbacks as is the general case in the models of the Modern Portfolio Theory. Therefore, a pure mean-variance framework was adopted that led to choose the MTSL as the optimizer for the asset allocation step in the portfolio management process.

Best and Grauer (1991) suggested that the optimal portfolio selection is sensitive to the sample magnitude observed in r and C, the aim of this paper is to asses the performance of an active portfolio management process using the t-Student Orthogonal GARCH (O-GARCH) matrix, in defined benefit pension funds such as DPCEM. Given this, it is necessary to review this multivariate volatility model and the need for a t-Student log likelihood function.

With the earliest proposals of Engle (1982) and Bollerslev (1986), the calculation of short term volatilities that incorporate the volatility clustering effect, allowed the financial practice to forecast risk levels more appropriately. The univariate GARCH(p,q) model follows an expression shown by:

σt2=κ+∑i=1pβi∙εt-i2+∑i=1qγi∙σt-i2 (4)

As a starting point for this paper, the model departs from the assumption that the returns vector r _i of the i^th asset is either⁷ r i | I ~ N ( r ¯ i , σ r i 2 ) or r 1 | I ~ t ( r ¯ i , σ r i 2 , g l ) , leading to the more general assumption of multivariate elliptic probability functions in the set of returns time series X = [r₁...,r_i.]. With this, the log likelihood problem can be solved through two functions. r i | I ~ N ( r ¯ i , σ r i 2 ) , the vector of parameters θ i = κ , β i , γ i leads to⁸ σ t 2 = σ t 2 ( θ i ) in (4) and the optimal set of parameters 6* is shown with the solution of the next optimization problem:

argmaxσt2(θi)⁡=12∑t=1Tlogσt2+εi,tσt2 (5)

Subject to:

κ , β i , γ i ≥ 0

∑ i = 1 p β i + ∑ j = 1 q γ j ≤ 1

When r 1 | I ~ t ( r ¯ i , σ r i 2 , g l ) , the solution is shown by the next log likelihood function maximization⁹:

arg maxσt2(θi)⁡=-∑t=1Tlog⁡σt2+gl+12∙log1gl-2∙εtσt2+T∙logΓgl+12gl-2∙π1/2∙Γgl2 (6)

Subject to:

κ , β i , γ i ≥ 0

∑ i = 1 p β i + ∑ j = 1 q γ j ≤ 1

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 Bollerslev (1990), which starts with the use of a constant correlation matrix H and a diagonal matrix S defined by univariate GARCH variances:

Ct=St∙Ht∙St (7)

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 Engle and Kroner (1993), but, in some cases, this represents a hard computational task given the log likelihood maximization problem inherent to it. As a solution for this situation, a model known as Orthogonal GARCH, O-GARCH(p,q) or simply O-GARCH is proposed in Alexander and Chibumba (1996), Alexander (2002), and a generalized version in Van der Weide (2002). This model departs from the spectral decomposition of an equally weighted¹⁰ covariance matrix C _c that leads to the definition of a n x n matrix of eigenvectors E anda n spectrum Λ :

Cc=E∙Λ∙E (8)

The computational efficiency of the O-GARCH model is based on the variance (eigenvalues) of the principal components (P = X.E = [p ₁...,p _i.]) in the diagonal elements of Λ . Once this matrix is defined, a selection of principal components, eigenvalues and eigenvectors, is made by sorting the eigenvectors and principal components from the highest (h) to the lowest eigenvalue. Following this, the next selection criteria is applied, given a total variance explanation level v (percentage) previously fixed:

λi∈Λ*,ei∈E*,pi∈P*⟺∑h=1iλhtrace(Λ)≤v (9)

With the definition of Λ *, the calculation of a univariate GARCH volatility is made in each main component in P* by using the log likelihood function given in (6). This will lead to the definition of a GARCH spectrum Ω and to the next matrix composition of the expected O-GARCH covariance matrix:

COGARCH≈E*∙Ω∙E*' (10)

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 Engle and Kroner (1993) BEKK-GARCH model and a previous and more general proposal of Bollerslev (1986) known as Vech-GARCH. In the latter case, the number of parameters to be estimated is given by the n number of factors with the following expression: [n +1)(n(n +1) +1)] / 2. In the BEKK-GARCH it is necessary to calculate n(n + 5)/2 parameters and in the O-GARCH model, as noted in (9) and (10), it is necessary to calculate at most n • p • q parameters where n is the number of main components selected with (9), p the maximum ARCH lag term allowed in the univariate GARCH of each main component model and q the maximum GARCH lag term.

Several papers can be mentioned related to the practical usefulness of the O-GARCH, Alexander (2002) that presents the calculation of large covariance matrixes with different kinds of assets like currencies, UK gilt bonds term structure, English equities, and the oil futures term structure, noting that the limitation of quality and amount of data¹¹ and the “dimensionality course"¹² can be avoided thanks to the main components analysis inherent to this multivariate GARCH model.

Another practical use is reviewed by Bredin and Hyde (2004) who test several covariance models, such as the equally weighted¹³, the exponentially weighted; the O-GARCH and the O-EWMA, ¹⁴ in the Irish FX market previous to Ireland's integration to EMU. Their results supported the O-GARCH model as the best one for capital reserve purposes and the O-EWMA for compliance.

Following Bredin and Hyde, Cifarelli and Paladino (2004, 2006) use the O-GARCH model to test the contagion of credit default events in the behavior of sovereign bond term structures in Latin America and Asia. They found, thanks to the O-GARCH model ability to capture the correlation and volatility clustering effect with the lack of data, that there is a weak integration in the Asian and Latin American sovereign bond markets in low volatility time periods, but this tends to disappear with the presence of contagion after specific market shocks such as the 2001 Argentinean default.

Also, Chen et al (2008) and Kearney and Muckley (2008) use this multivariate model to quantify either the Chinese stock market Value at Risk or the pegging effect (high and stable correlation and volatility) in the main Asian markets and currencies.

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 and its several practical applications such as the usefulness to quantify a measure of financial markets integration, to identify possible currencies that could potentially be pegged, and to quantify large covariance matrixes of heterogeneous financial assets by avoiding the "dimensionality course”, it is the most appropriate for measuring the volatility of a portfolio invested in the six financial markets stated in the IPS given in Appendix one. In fact, its usefulness is still researched nowadays in extensions such as the one made by Sharifi et. al. (2012) where M-estimators are tested to calculate more robust GARCH parameters with less stringent moment conditions¹⁶.

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 Appendix three.

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 (r _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 theoretical¹⁸ starting value of the four simulated portfolios is 10,000,000.00 MXN, using the inflows and outflows presented in Appendix 2.

∀Eri∈r,Eri=∑i=1Tr¯i∙n-1 (11)

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 Schwarz (1978).

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 Appendix one. In order to rebalance from the actual investment proportion, a 0.25% financial cost is also incorporated and the algorithm shown in Figure one is performed.¹⁹ In order to execute the two discrete event simulations in the active portfolio management simulations, the algorithm of figure two was also used. Finally, with the results obtained, the three historical simulated portfolios were valued in a January 2^nd, 2002, base 100.

Figure 1

Figure 2

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.

Chart 1

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.

Chart 2

Table 1

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 effect²⁰ was measured more accurately in this period, leading to a higher concentration in the risk-free asset for longer time periods in comparison with the other two portfolios.

Chart 3

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.

Chart 4

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 Appendix one along with the historical accumulated value of the 7.5% annual target rate (light area).

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 Chart 5, it should be noted that the highest investment proportions in the O-GARCH models are in these three markets. When inspecting Chart four more closely, the performance of the portfolio analysis in the O-GARCH cases suggests a more sensitive asset allocation in the presence of volatility and correlation clustering, i.e. these two portfolios were better diversified during the most uncertain time periods in the financial markets.

Chart 5 Portfolio code (box plot): 1= Target portfolio -passive-, 2= Constant covariance -active, 3= t-Student O-GARCH covariance -active-

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 For this reason and because of their statistical properties, the O-GARCH portfolio had a softer behavior than the benchmark and the equally weighted covariance matrix portfolio when the financial crisis was acute.

Chart 6

Chart 7

Table 2

Source	SS	df	MS	F	Prob>F
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 Sharpe ratio (1966) of each simulated portfolio. If there’s either no difference or a lower level of Sharpe ratio (SR) observed in the value of the active O-GARCH portfolio against the other two cases, this portfolio management process is accepted as the best option.

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.