Portfolio optimization with sparse multivariate modeling

Abstract

Portfolio optimization approaches inevitably rely on multivariate modeling of markets and the economy. In this paper, we address three sources of error related to the modeling of these complex systems: 1. oversimplifying hypothesis; 2. uncertainties resulting from parameters’ sampling error; 3. intrinsic non-stationarity of these systems. For what concerns point 1. we propose a $L_0$-norm sparse elliptical modeling and show thatsparsification is effective. We quantify the effects of points 2. and 3. by studying the models’ likelihood in- and out-of-sample for parameters estimated over different train windows. We show that models with larger off-sample likelihoods lead to better performing portfolios only for shorter train sets. For larger train sets, we found that portfolio performances deteriorate and detaches from the models’ likelihood, highlighting the role of non-stationarity. Investigating the out-of-sample likelihood of individual observations we show that the system changes significantly through time. Larger estimation windows lead to stable likelihood in the long run, but at the cost of lower likelihood in the short term: the “optimal” fit in finance needs to be defined in terms of the holding period. Lastly, we show that sparse models outperform full-models and conventional GARCH extensions by delivering higher out of sample likelihood, lower realized volatility and improved stability, avoiding typical pitfalls of conventional portfolio optimization approaches.

Appendices

Appendix A. Elliptical distributions

Consider an n-dimensional vector of multivariate returns ${\varvec{X}}=(x_{1},x_{2},...,x_{n})$. If ${\varvec{X}}$ is elliptical distributed, then its probability density function is defined as:

$$\begin{aligned} f_{X}({\varvec{X}}) = c_n|{\varvec{J}}|^{1/2} g_n \left[ ({\varvec{X}}-{\varvec{\mu }}) {\varvec{J}} ({\varvec{X}}-{\varvec{\mu }})^T\right] , \end{aligned}$$

(A.1)

where ${\varvec{\mu }}\in {\mathbb {R}}^{1\times n}$ is the vector of location (mean) parameters and $c_n$ is a normalization constant. The matrix, ${\varvec{J}} = {\varvec{\Omega }}^{-1} \in {\mathbb {R}}^{n\times n}$ is the generalized precision matrix, a positively defined matrix which is the inverse of the dispersion matrix ${\varvec{\Omega }}$. When the covariance is defined (as we assume in this paper) then ${\varvec{\Omega }}= (-\psi '(0))^{-1}{\varvec{\Sigma }}$, that is, ${\varvec{\Omega }}$ is proportional to the covariance matrix and the proportionality factor is the inverse of the first derivative of the characteristic generator evaluated at 0. The function, $g_n(\cdot )$ is called density generator.

Also, let us stress that $({\varvec{X}}-{\varvec{\mu }}) {\varvec{J}} ({\varvec{X}}-{\varvec{\mu }})^T$ - i.e., the generalized, square Mahalanobis distance - is a quadratic term and hence a nonnegative quantity provided that the matrix ${\varvec{\Omega }}$ is positive definite. To ease the notation, for the remaining of the paper we shall refer to the generalized Mahalanobis distance as $d^2$:

$$\begin{aligned} d^2 = ({\varvec{X}}-{\varvec{\mu }}) {\varvec{J}} ({\varvec{X}}-{\varvec{\mu }})^T. \end{aligned}$$

(A.2)

For dfferent density generators $g_n(\cdot )$ we obtain different distributions of the elliptical family. It is easy to see, for example, that the normal distribution is obtained by using:

$$\begin{aligned} g(u) = e^{-u/2}, \end{aligned}$$

(A.3)

and ${\varvec{\Omega }}= {\varvec{\Sigma }}$.

Similarly the Student-t distribution is obtained by using:

$$\begin{aligned} g_{n}(u) = \left( 1+\frac{u}{v}\right) ^{-\frac{n+v}{2}}, \end{aligned}$$

(A.4)

where v is the degrees of freedom, and ${\varvec{\Omega }}= \frac{\nu -2}{\nu } {\varvec{\Sigma }}$.

The validity of the mean–variance framework for elliptical distributions has long been established in literature (Owen and Rabinovitch 1983). This proposition is derived easily from two properties of the elliptical distributions. First, for every elliptical distribution with defined mean and variance, the distribution is completely specified by them ( Owen and Rabinovitch (1983) or Chamberlain (1983)), with all the higher moments being either zero or proportional to the first or second moment. Second, any linear combination of multivariate elliptically distributed variables is also an elliptically distributed variable. In the case of normal distribution and Student-t distribution they also have the same density generator function. Further details on these properties are provided in Appendix B.

It follows that, if asset returns have a multivariate elliptical distribution ${\varvec{X}} \sim {\mathcal {E}}_n({\varvec{\mu , \Omega }}, g_n)$, then the portfolio expected return and dispersion are given by, respectively, ${\mathbb {E}}[r_p]={\varvec{W\mu }}^T$ and $\sigma _{p}={\varvec{W \Omega W^T}}$, matching the optimization framework outlined in ''Modern portfolio theory'' section.

With respect to our likelihood analysis, considering distributions with probability density function of the form specified in Eq. A.1, the corresponding likelihood function is of the form:

$$\begin{aligned} {\mathcal {L}}_{ED}({\varvec{\theta ; X}}) = \left| {\varvec{J}}\right| ^{1/2} \; g_n\left( d^2 \right) \;. \end{aligned}$$

(A.5)

where ED denotes the general Elliptical Distributions and we omitted the constant of integration.

To stress the general validity of our analysis for other elliptical distributions, we repeated the experiments discussed in ''Methodology'' section considering the Student-t generator.

Assuming a Student-t distribution of the log returns, the log-likelihood (Eq. A.5) is:

$$\begin{aligned} \ln {\mathcal {L}}_\mathrm{Student} = \frac{\ln |{\varvec{J}}|}{2} - \frac{n+\nu }{2} \ln \left( 1+ \frac{d^2}{\nu -2} \right) \end{aligned}$$

(A.6)

where n is the sample size and $\nu$ is the degree of freedom. Figure 10a reports the likelihood comparison for the same resamplings as in Fig. 3 but using a Student-t log-likelihood as in Eq. A.6. Here we used $n=500$ observations (i.e., the out-of-sample size) and $\nu =3$. We verified that this findings are robust across different degrees of freedom in the range $\nu =[2.1,4]$.

Fig. 10

Out-of-sample log-likelihood computed using the maximum likelihood and the TMFG covariances. These results are coherent with the findings related to the normal distribution presented in ''Likelihood comparison'' section

Appendix B. Properties of elliptical distributions

In this section we recall some useful properties of Elliptical Distribution which we referred to in our discussion and particularly in Section A.

Property 1

(Distribution Definition) Consider an n-dimensional random vector ${\varvec{X}}=(X_1,...,X_n)$. ${\varvec{X}}$ has a multivariate elliptical distribution with location parameter ${\varvec{\mu }}$ and dispersion parameter $\mathbf \Omega$,written as ${\varvec{X}} \sim {\mathcal {E}}({\varvec{\mu , \Omega }} )$ if its characteristic function $\phi$ can be expressed as:

$$\begin{aligned} \phi _X({\varvec{w}}) = {\mathbb {E}}(e^{i {\varvec{w}} {\varvec{X}}})=e^{i {\varvec{w}} {\varvec{\mu }}}\psi \left( {\frac{1}{2}{\varvec{w}} {\varvec{\Omega }} {\varvec{w}}^T} \right) , \end{aligned}$$

(B.1)

for some location parameter ${\varvec{\mu }} \in {\mathbb {R}}^{1\times n}$, positive-definite dispersion matrix ${\varvec{\Omega }} \in {\mathbb {R}}^{n\times n}$ and for some function $\psi (\cdot ):[0,\infty ) \rightarrow {\mathbb {R}}$ such that $\psi \left( \sum _{i=1}^{n}w^2_i \right)$ is a characteristic function, which is called characteristic generator. If ${\varvec{X}} \sim {\mathcal {E}}({\varvec{\mu , \Omega }} )$ and if its density $f_X({\varvec{X}})$ exists, it is of the form defined in Eq. A.1.

Property 2

(Density Generator) The function $g(\cdot )$ defined in Section Appendix A is guaranteed to be density generator if the following condition holds:

$$\begin{aligned} \int _{0}^{\infty } x^{n/2-1}g_n(x)\mathrm{d}x < \infty . \end{aligned}$$

(B.2)

Property 3

(Affine Equivariance) If ${\varvec{X}}=(X_1,...,X_n)$ is an n-dimensional elliptical random variable with location parameter ${\varvec{\mu }}$ and dispersion parameter ${\varvec{\Omega }}$ so that ${\varvec{X}} \sim {\mathcal {E}}_X({\varvec{\mu , \Omega }} )$, then for any vector ${\varvec{a}} \in {\mathbb {R}}^{1\times m}$ and any matrix ${\varvec{B}}\in {\mathbb {R}}^{m\times n}$ the following affine equivariance holds:

$$\begin{aligned} {\varvec{Y}} = {\varvec{a + BX}}\sim {\mathcal {E}}_Y({\varvec{a+B\mu , B\Omega B}}). \end{aligned}$$

(B.3)

In other words, any linear combination of multivariate elliptical distributions is another elliptical distribution.

In the special cases of normal, Student-t and Cauchy distributions, the induced density generators are m-dimensional version of the original generator of ${\varvec{X}}$.

For the proof of Properties 1, 2 and 3, we refer to Fang et al. (1990).

This implies that any portfolio $Y = \beta _1 X_1+...+\beta _n X_n$ of elliptically distributed variables is distributed accordingly with a (univariate) elliptical distribution, which is a location-scale distribution. Furthermore, for any univariate elliptical distribution all moments can be obtained from the first and second moments (if defined). In particular, for centered variables with zero mean ($\mu _Y=0$), the resulting distribution of Y is symmetrical around zero and it has all odd moments equal to zero and all even moments given by:

$$\begin{aligned} \mu _{2m} = c_m \mu _2^m, \end{aligned}$$

with

$$\begin{aligned} c_m = \frac{(2m)!}{(2^mm!)}\frac{\psi ^{(m)}(0)}{(\psi ^{(1)}(0))^m}. \end{aligned}$$

Where $\psi ^{(m)}(0)$ indicated the $m^\mathrm{th}$ derivative of $\psi (\omega )$ computed at $\omega =0$.

As an example, in the normal (0,1) case, $\mu _2=1$, $c_m=0$ for all $m=1,2,...$, the kurtosis is $\mu _{(4)}=\frac{4!}{2^4 4!}=3$, and $\mu _{(2m)}=\frac{(2m)!}{(2^m m!)}$. For the proof we refer to Berkane and Bentler (1986), which derived this property by succesive differentiation of $\phi (\cdot )$, and to Maruyama and Seo (2003), which attained the same result by expressing the elliptical distribution in terms of a random vector with uniform distribution on the unit sphere.

Therefore, the mean–variance optimization is of general applicability and relevance for any portfolio generated from multivariate elliptically distributed variables.

Appendix C. Orthogonal GARCH estimation and further results

In this section, we report details on the O-GARCH estimation discussed in Section 6. In particular, Tables 3 and 4 report the average AIC and BIC statistics across the 100 resamplings we considered in the experiment. As discussed in ''O-GARCH comparison'' section, both the AIC and BIC criteria support a GARCH(1,1) specification.

Table 3 Average AIC information criterion for different GARCH specifications across 100 resamplings for different estimation windows

Table 4 Average BIC information criterion for different GARCH specifications across 100 resamplings for different estimation windows

Table 5 below reports Median, 5th and 95th percentiles all the statistics obtained. The table shows that the GARCH(1,1) specification delivered the lowest AIC and BIC statistics for each of the main percentiles considered. In other words, the GARCH(1,1) specification selected in our experiment is the preferred specification according to the AIC and BIC criteria in all cases, and not only in mean across resamplings.

Table 5 AIC and BIC information criteria corresponding to different GARCH specifications

Having selected the GARCH(1,1) specification, for each resampling we estimated the model parameters (Table 6 reports the estimated parameters and corresponding p-Values), forecasted one steps ahead principal components and then reconstruct the covariance matrix as in Eq. 5.

Table 6 Average parameters and corresponding p-values for the GARCH(1,1) model estimated for the principal components across 100 resamplings

Having selected the O-GARCH(1,1) specification we performed the same experiments discussed in ''Impact of precision matrix estimate on optimal portfolios'' section to compare the optimal minimum variance portfolio features when the O-GARCH covariance is used in the optimization. Figure 11 reports the number of active long and short positions obtained with the O-GARCH covariance matrix. As discussed in ''O-GARCH comparison'' section, the O-GARCH positions closely track the full covariance behaviour leading to the same conclusions and instability problems.

Fig. 11

Buy/sell active positions. O-GARCH comparison

Appendix D. Turnover comparison

In ''Backtest'' section we compared the performances and stability of daily rebalanced portfolios, constructed based on OGARCH and TMFG precision matrices. Particularly relevant for the purpose of this paper is the stability of portfolios to new observations that, from a practitioner standpoint, results in lower turnover and transaction costs. We reported the average turnover in Fig. 9, while Fig. 12 reports a histogram of all daily turnovers computed across 100 resamplings and comparing TMFG to OGARCH for each estimation window. The histograms show that, other than delivering a significantly lower turnover on average as noted in ''Backtest'' section, TMFG portfolios’ turnover is heavily positively skewed, with a pick in the distribution around zero, supporting and strengthening our claim of superior stability of TMFG portfolios.

Fig. 12

Average daily turnover across 100 resamplings for different estimation window lengths