Externalities as Arbitrage
How can we assess whether macro-prudential regulations are having their intended effects? If these regulations are optimal, their marginal benefit of addressing externalities should equal their marginal cost of distorting risk-sharing. These risk-sharing distortions will manifest as trading opportunities that constrained intermediaries are unable to exploit. Focusing in particular on arbitrage opportunities, I construct an "externality-mimicking portfolio" whose returns track the externalities
... the externalities that would rationalize existing regulations as optimal. I conduct a revealedpreference exercise using this portfolio and test whether the recovered externalities are sensible. I find that the signs of existing CIP violations are inconsistent with optimal macro-prudential policy. unable to exploit (due to the capital controls). If the planner's optimal policy is to limit borrowing by domestic agents, the on-shore rate will be higher than the off-shore rate; if instead the optimal policy encourages borrowing, the reverse will be true. That is, the sign of the arbitrage is determined, under optimal policies, by the direction of the externality. Moreover, the magnitude of the arbitrage measures the size of the distortion created by the capital controls, and hence under an optimal policy is determined by the magnitude of the externalities being addressed. By observing the arbitrage, we can therefore infer the nature of the externalities that would justify the capital controls that create the arbitrage. This example illustrates the basic idea behind the exercise. Reality, of course, is more complicated. Real-world macro-prudential policies such as leverage constraints on financial intermediaries have complicated effects, and it is not a priori obvious how they redistribute wealth between agents across states of nature and over time. Consequently, it is not clear whether or not these policies alleviate or exacerbate externalities. The framework I develop in this paper examines the arbitrages created by macro-prudential policies to assess whether or not these policies are having their intended effects. I begin by outlining a general equilibrium with incomplete markets (GEI) framework that distinguishes between two classes of agents, "households" and "intermediaries." In this framework, I show that optimal policy equates the marginal benefits of addressing externalities with the marginal cost of distorting risk-sharing (as in Farhi and Werning (2016) ). I then show that, under some additional assumptions about how policy is implemented, these risk-sharing distortions will manifest themselves as arbitrage opportunities. To clarify the underlying mechanism and provide a concrete example, I develop the example of capital controls in a simple model, building on Fanelli and Straub (2019) . The central contribution of the paper uses the relationship between arbitrages and externalities to construct what I call the "externality-mimicking portfolio." The returns of this portfolio are the projection of the externalities onto the space of returns. The portfolio can also be thought of as representing the minimum difference between the household and intermediary SDFs necessary to explain observed arbitrages (an analog of Hansen and Richard (1987)), or as the portfolio that maximizes what I call the "Sharpe ratio due to arbitrage" (an analog of Hansen and Jagannathan (1991)). Using data on interest rates, foreign exchange spot and forward rates, and foreign exchange options, I construct an externality-mimicking portfolio. The weights in this portfolio are entirely a function of asset prices; no estimation is required. If policy is optimal, this portfolio's returns track the externalities the social planner perceives when consider-2 ing transfers of wealth between the households and intermediaries in various states of the world. When its returns are positive (negative), the planner perceives positive (negative) externalities when transferring wealth from intermediaries to households. In "bad times," we would expect this portfolio to have negative returns, consistent with the idea that the planner would like to encourage intermediaries to hold more wealth in these states. I consider two definitions of "bad times." First, intuitively, bad times can be defined as times in which the intermediaries have a high marginal utility of wealth. Using this definition, I show that it is sufficient to study the expected returns of the externality-mimicking portfolio, and test if they are positive. Second, I define "bad times" using the stress test scenarios developed by the Federal Reserve. I argue that these tests are statements about when the Fed would like intermediaries to have more wealth, and as a result the returns of the externality-mimicking portfolio should be negative in the stress test scenarios. However, I find that the expected return of the externality-mimicking portfolio is generally negative, and that its returns in the stress tests are often positive. This implies that the externalities that would justify current regulation are positive in bad times, which appears inconsistent with intuition and suggests that regulations are not having their desired effect. The basic issue is that some CIP violations (e.g. AUD-USD and JPY-USD) have the wrong sign. That is, because JPY appreciates and AUD depreciates vs. USD in bad times, optimal policy should encourage intermediaries to be long JPY and short AUD (i.e. short the carry trade). But the signs of the CIP violations are such that they encourage intermediaries to be long the carry trade, taking on more macro-economic risk. I speculate that this issue arises from an interaction between leverage constraints (which do not consider the "sign" of a trade) and demand from customers for carry trade risk, as suggested by Du et al. (2018). My theoretical framework builds on the GEI framework of Geanakoplos and Polemarchakis (1986) and Farhi and Werning (2016). My example of capital controls resembles both Fanelli and Straub (2019) and example 5.4 of Farhi and Werning (2016). The framework I develop specializes the standard GEI model in several respects. First, I assume that there are two classes of agents, households and intermediaries, who have different degrees of access to markets, in the spirit of Gromb and Vayanos (2002). Second, a key difference between this paper and the work of Farhi and Werning (2016), and also the discussion of pecuniary externalities in Dávila and Korinek (2017), is my focus on an implementation of the constrained efficient allocation using quantity constraints, rather than agent-state-or agent-state-good-specific taxes. Studying this implementation is both realistic, in the sense that regulation on banks takes this form, and it enables the empirical exercise that follows. Externalities and Arbitrage In this section, I describe the connection between externalities and arbitrage under optimal policy in a GEI framework. First, I discuss the marginal-cost vs. marginal-benefit tradeoff facing the planner, building on the existing literature. Next, I introduce financial intermediation into the GEI framework, describe how the planner can implement optimal policies, and show that implementing the optimal policies creates apparent arbitrage opportunities.