Economic and financial risk factors, copula dependence and risk sensitivity of large multi-asset class portfolios

Abstract

In this paper we propose a flexible tool to estimate the risk sensitivity of financial assets when exposed to any sort of risks, including extreme ones, from the financial markets and the real economy. This tool works with observations and a priori views. Our contribution is threefold. First, we combine copulas and factorial structures which allow us to capture the whole dependencies between the returns of a large number of assets of multiple classes. We build what we call a Cvine Risk Factors (CVRF) model, which can disentangle financial and explicitely economic like activity, inflation, emerging, etc, and more generally speaking real sphere related risks. Second, this model provides the way to extend the well known linear multibeta relationship in a non-linear version and to assess the exposures of any asset to several factorial risk directions in the cases where the risks are extreme. The exposure measures are relevant Cross Conditional Values at Risk (Cross-CVaR). Third, as an application of the methodology, we solve an optimization program to find portfolios that are the most diversified in capital while being immunized to extreme shocks to a given risk factorial direction. Varying the immunization constraint, we recover the portfolio strategies which are the most widely used today. For example, adopting the ERC (Equal Risk Contribution) rule insures optimal capital based diversification and immunization against inflation risk. Accordingly, we propose a unified view and a rationalization ex post of several current portfolio strategies that appear different at a first glance.

Appendices

A Presentation of the data

See Table 3.

Table 3 Description of data set

B Dependence structure

See Tables 4, 5.

Table 4 Dependence structure matrix $M^s$

Table 5 Kendall’s tau

A value 1 (resp. 0) in cell (i, j) means that return of asset i depends on ( resp. is independent from) return of asset j, conditionally on the returns of the previous assets $j-1$, $j-2$, ...if $j>1$, and unconditionally otherwise.

All diagonal entries are equal to 1 since the return of each asset is obviously linked with itself; imposing that all elements of the three first columns are equal to 1 means that the returns of all assets depend on the returns of the three main indexes: World Inflation Linked Bonds and World Government Bonds which capture interest rate risk and MSCI World which account for the so-called market risk.

The value 0 in the (15, 4)-cell indicates that the returns of the US High Yields bonds and Euro Sovereign bonds indexes are independent, once conditioned by the returns of the three indexes associated with the first three columns ans so on.

A negative coefficient between equity and real rate factor ($-\,0.21$) is consistent with the low inflation and flight-to-quality periods observed through our sample.

C Algorithm for conditional simulations

D Diffusion of extreme shocks to fixed income indexes

The results reported in next table show how a shock to a fixed income index spreads out to the other assets and how the responses to this shock can be decomposed into the contributions of the different risk factors.

This shock may be to a global factor.

For example, the shock is to the World Inflation Linked Bonds index, associated with the real rate risk, that is a drop in the return of the index by $-6.3 \%$; thus the column named “Rear rate” gives the responses of all assets to this shock. For example, the response of the MCI index is an increase of its return by $4.4\%$.

The shock may be more specific.

For example, a shock to the Sovereign Bond index, that is a drop in its return by $-4.6\%$ (in sub-table “IBoxx Euro Sovereigns”) can be decomposed into the contributions of Real rate risk by $-1.4\%$ Inflation risk by $-1.1\%$ , Market risk by$-0.1\%$ Eurodebt risk by $-2\%$ .

In the same way and for the same shock, the response of the MSCI index can be decomposed into a partial response to real rate with magnitude $3.1\%$, $5.6\%$ to inflation and $-4.3\%$ to market risk (Table 6).

Table 6 Decomposition of total exposures to extreme shocks to fixed income indexes

E Risk-based diversification

The Effective Number of Bets approach to risk parity proposed by Meucci et al. (2014) builds on one decomposition of the portfolio return as a combination of bets, or uncorrelated factors:

$$\begin{aligned} R_P= \sum \limits _{k=1}^K b_k{\overline{F}}_k \end{aligned}$$

that are obtained by a specific de-correlating transformation, the minimum torsion linear transformation, that least disrupts the original factors.

Once specified the relative contributions to total risk from each bet:

$$\begin{aligned} b_k=\frac{Var(b_k{\overline{F}}_k)}{Var(R_P)} \end{aligned}$$

with ${\overline{F}}_k$ denoting bet k, the risk-based diversification measure is defined in a similar way to the weight-based one and gives the number of bets, between 1 and K for the respective case of maximal concentration and maximal diversification:

$$\begin{aligned} N_K=exp\left( -\sum \limits _{k=1}^K{b_klnb_k}\right) \end{aligned}$$

F Sensitivity analysis

1.1 F.1 Sensitivities of the baskets

Table 7 Risk factor sensitivity of the baskets

As expected, the baskets composed of Government bonds and Other bonds respectively are mainly exposed to real rate and inflation/and or nominal rate risks, while the baskets composed of equities or commodities are especially affected by market and commodity risks.

1.2 F.2 Portfolio optimization

Table 8 Factor robust asset allocations and their risk and diversification features

1.3 F.3 Risk factor sensitivities of standard benchmark portfolios

Table 9 Risk and capital based asset allocations and their risk and diversification features