Alternative risk premia: contagion and portfolio choice

Abstract

Portfolio managers regard contagion as the death of diversification. The simultaneous jump to worst decile returns for most investments in a portfolio is hard to offset by diversification alone. Our results find substantial contagion across ARP strategies, which is difficult to predict. We derive the optimal asset allocation for an ARP portfolio under contagion risk and show that the investor’s best defence is to take less portfolio leverage. In addition, he should shy away from assets that perform poorly in contagion states.

Appendices

Appendix 1: Data

The ARP strategy data in this paper consist of excess (vs cash) returns for 27 strategy return streams across styles and asset classes for the period 1 June 2008 to 31 January 2019. All strategies are briefly described in Table 5 and stem from a global investment bank. Our observation frequency is daily. Given the commoditization of ARP strategies, similar return profiles can be obtained from most providers and we are confident that our results generalize across all providers. Table 4 displays the risk return characteristics of our raw data. All calculations have been performed with strategy series rescaled to a 10% per annum volatility such that difference is portfolio allocations are more easily to interpret. Returns for equities, bonds and commodities are represented by the following indices: MSCI AC, Barclay/Bloomberg Multiverse and GSCI. The unconditional correlation structure between all ARP series as well as asset classes (bonds, equities and commodities) and volatility (VIX) are illustrated in Fig. 5. Panel A provides a hierarchical clustering for all series, where the similarity between strategies is expressed as distance, i.e. 1 minus correlation. The distance between strategies is typically around one as most strategies show zero correlation, which can be seen from the histogram of strategy correlations (lower triangular part of the correlation matrix) in panel (b). In general, correlations increase as strategies share the same universe or apply the same style. Examples of this are either trend following strategies (FX and Cross-Asset trend following) are close or commodity strategies (COCurve and COTS). From 465 unconditional correlations, 90% are between − 0.14 and + 027. The highest correlation is between IR carry and bonds (0.8). The second highest is between equities and commodities. Correlations with returns on equities, bonds and percentage changes in the VIX are displayed in Fig. 6. In accordance with the literature, we see that a large fraction of strategies show positive correlation with equities (i.e. loose value if equities fall) and negative correlation with volatility (i.e. loose value if volatility rises). However, in each group we also see defensive strategies with the opposite behaviour. Not surprisingly, equity strategies still show some exposures to equities, bond strategies to bonds and volatility strategies to volatility (Table 5).

Table 4 Risk return characteristics

Fig. 5

Correlation structure

Fig. 6

Exposure to equities, bonds and VIX

Table 5 Strategy descriptions

Appendix 2: Theory

The investors value function at time t as a function of its state variable, $W_{t}$, i.e. investors wealth is given by

$$ V\left( {W_{t} ,t} \right) = \mathop {\hbox{max} }\limits_{{\left\{ {w_{n} } \right\}}} E\left( {\tfrac{{W_{t}^{1 - \gamma } }}{1 - \gamma }} \right) $$

(B1)

The investor needs to find the proportion of wealth invested into $n = 1, \ldots ,N$ risky assets subject to the dynamics of wealth given by

$$ \tfrac{{{\text{d}}W_{t} }}{{W_{t} }} = \left[ {{\mathbf{w}}^{{\prime }} {\varvec{\upalpha}} + r} \right]{\text{d}}t + {\mathbf{w}}^{{\prime }} \left[ {{\mathbf{\sigma dZ}}_{t} } \right] + {\mathbf{w}}^{{\prime }} {\mathbf{J}}_{{\mathbf{t}}} {\text{d}}Q\left( \lambda \right) $$

(B2)

where w is the vector of asset weights, ${\varvec{\upalpha}}$ and ${\varvec{\upsigma}}$ are the vector of excess returns over cash and asset volatilities, r is cash. The vector of diffusion shocks is given by ${\mathbf{dZ}}_{t}$, while rand jump amplitudes are denoted by ${\mathbf{J}}_{{\mathbf{t}}}$ and occur according to a Poisson process, dQ, with intensity $\lambda$. The Hamilton–Jacobi–Bellman equation looks like

$$ 0 = \mathop {\hbox{max} }\limits_{{\left\{ w \right\}}} \left\{ {\tfrac{\delta V\left( \cdot \right)}{\delta t} + \tfrac{\delta V\left( \cdot \right)}{\delta W}W_{t} \left[ {{\mathbf{w}}^{{\prime }} {\varvec{\upalpha}} + r} \right] + \tfrac{1}{2}\tfrac{{\delta^{2} V\left( . \right)}}{{\delta W^{2} }}W_{t}^{2} {\mathbf{w}}^{{\prime }} {\mathbf{\varSigma w}} + \lambda E\left[ {V\left( {W_{t} + W_{t} {\mathbf{wJ}}_{t} ,t} \right) - V\left( \cdot \right)} \right]} \right\} $$

(B3)

where the last term accounts for jumps. Guessing the solution to look like

$$ V\left( {W_{t} ,t} \right) = A\left( t \right)\tfrac{{W_{t}^{1 - \gamma } }}{1 - \gamma } $$

(B4)

This changes the Hamilton–Jacobi–Bellman equation to:

$$ 0 = \mathop {\hbox{max} }\limits_{{\left\{ w \right\}}} \left\{ {\frac{1}{A\left( t \right)}\frac{{{\text{d}}A\left( t \right)}}{{{\text{d}}t}} + \left( {1 - \gamma } \right)\left[ {{\mathbf{w}}^{{\prime }} {\varvec{\upalpha}} + r} \right] - \frac{{\left( {1 - \gamma } \right)\gamma }}{2}{\mathbf{w}}^{{\prime }} {\mathbf{\varSigma w}} + \lambda E\left[ {\left( {1 + {\mathbf{wJ}}_{t} } \right)^{1 - \gamma } - 1} \right]} \right\}\sum {} $$

(B5)

Differentiating with respect to w, we arrive at

$$ {\varvec{\upalpha}} - \gamma {\mathbf{\varOmega w}} - \lambda E\left( {{\mathbf{J}}\left( {1 + {\mathbf{w}}^{{\prime }} {\mathbf{J}}} \right)^{ - \gamma } } \right) = 0 $$

(B6)

This equates with the equation in the main text.