Minimum risk versus capital and risk diversification strategies for portfolio construction

Abstract

In this paper, we propose an extensive empirical analysis on three categories of portfolio selection models with very different objectives: minimization of risk, maximization of capital diversification, and uniform distribution of risk allocation. The latter approach, also called Risk Parity or Equal Risk Contribution (ERC), is a recent strategy for asset allocation that aims at equally sharing the risk among all the assets of the selected portfolio. The risk measure commonly used to select ERC portfolios is volatility. We propose here new developments of the ERC approach using Conditional Value-at-Risk (CVaR) as a risk measure. Furthermore, under appropriate conditions, we also provide an approach to find a CVaR ERC portfolio as a solution of a convex optimization problem. We investigate how these classes of portfolio models (Minimum-Risk, Capital-Diversification, and Risk-Diversification) work on seven investment universes, each with different sources of risk, including equities, bonds, and mixed assets. Then, we highlight some strengths and weaknesses of all portfolio strategies in terms of various performance measures.

Appendix: The HFB Approach

The Historical Filtered Bootstrap (HFB) approach (Barone-Adesi et al, 1999; Brandolini et al, 2001; Zenti and Pallotta, 2000; Marsala et al, 2004) consists of a mixed procedure in which one represents the market returns using, e.g., an ARMA-GARCH model to filter the time series, and then computes the empirical standardized residuals from the data without assuming any specific probability distribution. Below we provide a step-by-step description of the HFB procedure implemented in our work.

• 1. We filter the returns time series of each asset by an univariate ARMA-EGARCH model. More precisely, for the observed returns of asset i we compute the Maximum Likelihood estimators of the following AR(1)-StudT-EGARCH(1,1) model:

  $$\begin{aligned} &{\text{ AR(1)} }{:}\,r_{i,t}=a_{i}+b_{i} r_{i,t-1}+ \eta _{i,t} \\ &{\text{ StudT-EGARCH(1,1)}}{:}\,\log \sigma _{i,t}^{2} = \delta _{i} + \gamma _{i} \log \sigma _{i,t-1}^{2} + \alpha _{i} \left( |z_{i,t-1}|- E(|z_{i,t-1}|) \right) +\beta _i z_{i,t-1} \\ &\eta _{i,t} = \sigma _{i,t} z_{i,t} \end{aligned}$$

  where $z_{i,t} = \sqrt{\frac{\nu _i -2}{\nu _i}} T_{\nu _i}$, $T_{\nu _i}$ follows a Student's t-distribution with $\nu _i$ degrees of freedom, and $\hat{\theta }_{i}=\left\{ \hat{a}_{i}, \hat{b}_{i}, \hat{\alpha }_{i}, \hat{\beta }_{i}, \hat{\gamma }_{i}, \hat{\delta }_i, \hat{\nu }_{i} \right\}$ are Maximum Likelihood estimators obtained on 500 daily data.

• 2. Using the estimators $\hat{\theta }_{i}$ of all n assets available in the market, we compute from data the standardized residuals $\hat{z}_{i,t}=\hat{\eta }_{i,t}/\hat{\sigma }_{i,t}$ with $t=1,\,\ldots,\,T$ and $i=1,\,\ldots,\,n$, where $\hat{\eta }_{i,t}$ are the empirical residuals and $\hat{\sigma }_{i,t}$ are their estimated volatilities.

• 3. We bootstrap in a parallel fashion the matrix of the empirical standardized residuals $\hat{Z}=\left\{ \hat{z}_{i,t}\right\}$ with $t=1,\,\ldots,\,T$ and $i=1,\,\ldots,\,n$. More precisely, we randomly sample with replacement the rows of the matrix $\hat{Z}$, thus allowing to capture the multivariate shocks of the entire system.

• 4. The bootstrapped standardized residuals $\hat{Z}^{\mathrm{boot}}=\left\{ \hat{z}_{i,s}^{\mathrm{boot}}\right\}$, with $s=1,\,\ldots,\,S=10{,}000$ and $i=1,\,\ldots,\,n$, are then used as multivariate innovations in the (univariate) AR(1)-StudT-EGARCH(1,1) models to simulate the assets returns.

For more details, see Barone-Adesi et al (1999), Brandolini et al (2001), Cesarone and Colucci (2016), Zenti and Pallotta (2000), and Marsala et al (2004).