On the robustness of risk-based asset allocations

Abstract

Since the subprime crisis, portfolios based on risk diversification are of great interest to both academic researchers and market practitioners. They have also been employed by several asset management firms and their performance appears promising. Since they do not rely on estimates of expected returns, they are assumed to be robust. The same argument holds for minimum variance and equally weighted portfolios. In this paper, we consider a Monte Carlo simulation, as well as an empirical global portfolio dataset, to study the effect of estimation errors on the outcomes of two recently proposed asset allocations, the equally weighted risk contribution (ERC) and the principal component analysis (PCA) portfolio. The ERC portfolio is more robust to changes in the input parameters and has a smaller estimation error than the Markowitz approaches, whereas the PCA portfolio is even more unstable than the classical approaches. In the worst-case scenario, neither approach delivers what it promises. However, in every case the resulting return–risk relationship is dominated by the Markowitz approaches.

Appendix

1.1 Detailed table (Tables 12, 13, 14) and figures (Figs. 1, 2) for the global portfolio dataset

 

Table 12 Standard deviations of the portfolio weights in % per asset

Table 13 Minimum and maximum estimated weights across the 174 empirical portfolio optimizations in % per asset

Table 14 Standard deviations of the estimated portfolio weights across 174 months in % per asset

Fig. 1

Boxplot of the estimated weights for the empirical study (174 portfolio optimizations)

Fig. 2

Boxplot of the actual risk contributions for the empirical study (174 portfolio optimizations)

1.2 Results for the CAPM estimation of the “true” returns (Tables 15, 16)

The true returns are estimated by using the following formula:

$$\begin{aligned} r_i=\alpha _i + r_f + \beta _i (r_M - r_f)+\epsilon _i, \end{aligned}$$

where $r_i$ is the return of the asset i, $r_f$ the risk free rate and $\epsilon _i$ the residual. The true covariance was calculated as the covariance of the residuals. The World-Datastream Market Index is taken as the market return and the 3-month T-Bill rate is taken as the risk free rate.

1.3 Results for the Black Litterman estimation of the “true” returns (Tables 17, 18)

The true returns are estimated by using the following formula:

$$\begin{aligned} \Pi = \lambda \Sigma w_{\text{ market}}, \end{aligned}$$

where $\Pi$ is the vector of the is the implied returns, $\lambda$ is the risk aversion parameter, $\Sigma$ the covariance matrix and $w_\mathrm{market}$ the market weights. As the portfolio consists of global diversified indices and the market capitalizations are not available for every Index, the market weights are taken as the naive weights.

Table 15 Performance statistics of the global portfolio, when the “true” returns are estimated by using the CAPM

Table 16 Performance statistics of the global portfolio in the empirical “worst case”, when the “true” returns are estimated by using the CAPM

Table 17 Performance statistics of the global portfolio, when the “true” returns are estimated by using the Black Litterman implied returns

Table 18 Performance statistics of the global portfolio in the empirical “worst case”, when the “true” returns are estimated by using the Black Litterman implied returns