Equilibrium Asset Pricing in Directed Networks

May 22, 2021

Equilibrium Asset Pricing in Directed Networks
Nicole Branger, Patrick Konermann, Christoph Meinerding, Christian Schlag
Review of Finance, Volume 25, Issue 3, May 2021, Pages 777–818, https://doi.org/10.1093/rof/rfaa035

The existence of network linkages between firms, industries, or countries can turn microeconomic shocks into aggregate fluctuations. Taking this as given, the question arises what such a potential amplification implies for asset prices. In this paper, we develop a tractable consumption-based equilibrium asset pricing model that allows us to trace risk premia back to the core input of any network model, namely the individual entries of the connectivity matrix.

The fact that links in a network usually have a direction, i.e., it makes a difference whether a link goes from node i to node j or the other way around, turns out to be of first-order importance for expected excess returns. We propose an equilibrium asset pricing model, in which negative cash flow shocks in some assets can increase the probability of subsequent cash flow shocks in other assets. The direction and the magnitude of this ”timing of shocks” characterize the network in our model. Based on a series expansion of the closed-form solution of our model, we prove for arbitrary networks that directed links between cash flows affect the cross-section of risk premia through three channels:

  1. Shocks that can propagate through the economy command a higher market price of risk (”spreading channel”). The more an asset loads on such shocks, the higher is thus its risk premium.
  2. Shock-receiving assets load on systematic risk factors more than resilient assets. They earn an extra premium for spillover risk because their valuation ratios drop upon shocks in connected assets (”receiving channel”).
  3. A hedge effect pushes risk premia down: when a shock propagates through the economy, an asset that is unconnected becomes relatively more attractive and its valuation ratio increases (”hedging channel”). This is because the entire economy becomes riskier, but the unconnected asset is unaffected and therefore, in relative terms, less risky as compared to the economy as a whole. Put differently, an unconnected asset is the best hedging device against the propagation of shocks.

The first two channels increase risk premia, while the third one pushes them down, so that the overall impact of directed shock propagation on risk premia depends on the tradeoff between these channels.

As our main theorem for arbitrary directed networks reveals, each asset’s risk premium is affected by all three channels in general networks. The spreading and the receiving channel are determined only by the direct linkages from and to a particular asset. In contrast, the hedging channel is driven by all other linkages in the network. This has three important implications. First, in a given network, the hedging channel operates through all assets except the ones which cannot spread their shocks anywhere. Second, it is not possible to construct a network in which the hedging channel is shut down completely. Third, the risk premium of an asset also depends on the existence of cash flow linkages in very remote or unconnected parts of the economy.