Externalities as Arbitrage∗

Benjamin Hébert†

June 26, 2020

Abstract

How can we assess whether macro-prudential regulations are having their intended

effects? If these regulations are optimal, their marginal benefit of addressing external-

ities is equal to their marginal cost of distorting risk-sharing. Under plausible assump-

tions about implementation, these risk-sharing distortions will manifest as arbitrage

opportunities that constrained intermediaries are unable to exploit. Measuring arbi-

trage opportunities allows us to construct an “externality-mimicking portfolio” whose

returns track the externalities that would rationalize existing regulations as optimal.

We can then conduct a sort of revealed-preference exercise and ask whether the re-

covered externalities are sensible. Conducting this exercise, I find that the signs of

existing CIP violations are inconsistent with optimal macro-prudential policy.

∗Acknowledgements: I would like to thank Eduardo Dávila, Darrell Duffie, Emmanuel Farhi, Gary Gor-ton, Denis Gromb, Valentin Haddad, Barney Hartman-Glaser, Bob Hodrick, Jens Jackwerth, Anton Korinek,Arvind Krishnamurthy, Hanno Lustig, Matteo Maggiori, Semyon Malamud, Ian Martin, Tyler Muir, JesseSchreger, Ludwig Straub, Jeremy Stein, Dimitri Vayanos, and seminar participants at the MIT Sloan Ju-niors Conference, Yale SOM, EPFL Lausanne, Stanford, UCLA, UC Irvine, Columbia Juniors Conference,Duke Macro Jamboree, Gersenzee ESSFM, Barcelona FIRS, and LSE for helpful comments, and in partic-ular Peter Kondor for a helpful discussion. I would like to thank Wenxin Du for help understanding datasources, Michael Catalano-Johnson of Susquehanna Investment Group for help with bounds on Americanoption prices, Colin Eyre and Alex Reinstein of DRW for help understanding SPX and SPY options, andJonathan Wallen and Christian Gonzalez-Rojas for outstanding research assistance. All remaining errors aremy own.†Stanford University and NBER. bhebert@stanford.edu

1 Introduction

Following the great financial crisis (GFC), apparent arbitrage opportunities have appearedin financial markets. These arbitrage opportunities, such as the gap between the federalfunds rate and the interest on excess reserves (IOER) rate, or violations of covered interestrate parity (CIP), are notable in part because they have persisted for years after the peak ofthe financial crisis. Many authors (e.g. Du et al. (2018)) have argued that macro-prudentialregulations put in place after the GFC have enabled these arbitrages to exist and persist.

If these arbitrages are caused by macro-prudential regulations, does that imply thatthere is something wrong with the regulations? More generally, what can be learned fromthe patterns of arbitrage across assets induced by regulation? Can we use these patterns toassess whether macro-prudential regulations are having their intended effects?

In this paper I show that, under optimal policy, there is a close connection between theexternalities policy is meant to address and the arbitrages policy creates. The core point issimple: if a policy is optimal, its marginal costs equal its marginal benefits. The marginalcosts of macro-prudential policies manifest themselves as arbitrage opportunities. Conse-quently, if policy is optimal, we can infer the marginal benefits (addressing externalities)from observed arbitrage opportunities. This enables a revealed-preference exercise. First,I calculate the externalities that would rationalize existing policy. Second, I ask whetherthese revealed externalities are reasonable. If they are not reasonable– I will argue manyCIP violations have the wrong sign– then we can infer that policy is far from optimal.

Macro-prudential policies, such as restrictions on leverage or capital controls, are de-signed to reallocate wealth between agents across states of nature and over time. For ex-ample, a capital control that limits borrowing from foreigners ensures that in bad times,domestic agents have more wealth and foreigners have less wealth, relative to what wouldhave occurred in the absence of capital controls. If there is an externality that can be ad-dressed by reallocating wealth from foreign to domestic agents in bad times (e.g. if the realexchange rate affects the welfare of domestic households that do not participate in finan-cial markets, Fanelli and Straub (2018)), then capital controls might be an optimal policy.However, capital controls have a cost– they distort risk-sharing and inter-temporal tradebetween foreign and domestic agents. A social planner must weigh this cost against thebenefits of capital controls.

Moreover, using capital controls can create arbitrage. If the government implementsoptimal policy by raising the on-shore cost of borrowing, this will create a wedge between

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the on-shore and off-shore cost of borrowing. That is, it will create an apparent arbitragethat agents are unable to exploit due to the capital controls. If the optimal policy is to limitborrowing by domestic agents, the on-shore rate will be higher than the off-shore rate; ifinstead the optimal policy encourages borrowing, the reverse will be true. That is, the signof the arbitrage is determined, under optimal policies, by the direction of the externality.Consequently, by observing the arbitrage, we can infer the nature of the externalities thatwould justify the capital controls that create the arbitrage.

This example illustrates the basic idea behind the exercise. Reality, of course, is morecomplicated. Real-world macro-prudential policies such as leverage constraints on finan-cial intermediaries have complicated effects, and it is not a priori obvious how they redis-tribute wealth between agents across states of nature and over time. Consequently, it isnot clear whether or not these policies alleviate or exacerbate externalities. The frameworkI develop in this paper allows us to examine the arbitrages created by a macro-prudentialpolicy and assess whether or not the policy is having its intended effect.

I begin by outlining a general equilibrium with incomplete markets (GEI) frameworkthat distinguishes between two classes of agents, “households” and “intermediaries.” In thisframework, I show that optimal policy equates the marginal benefits of addressing exter-nalities with the marginal cost of distorting risk-sharing (as in Farhi and Werning (2016)). Ithen show that, under some additional assumptions about how policy is implemented, theserisk-sharing distortions will manifest themselves as arbitrage opportunities. To clarify theunderlying mechanism and provide a concrete example, I develop the example of capitalcontrols in a simple model, building on Fanelli and Straub (2018).

The central contribution of the paper uses the relationship between arbitrages and ex-ternalities to construct what I call the “externality-mimicking portfolio.” The returns ofthis portfolio are the projection of the externalities onto the space of returns. The portfo-lio can also be thought of as representing the minimum difference between the householdand intermediary SDFs necessary to explain observed arbitrages (an analog of Hansen andRichard (1987)), or as the portfolio that maximizes what I call the “Sharpe ratio due toarbitrage” (an analog of Hansen and Jagannathan (1991)).

Using data on interest rates, foreign exchange spot and forward rates, and foreign ex-change options, I construct an externality-mimicking portfolio. The weights in this port-folio are entirely a function of asset prices; no estimation is required. The returns of thisportfolio represent an estimate of the externalities the social planner perceives when con-sidering transfers of wealth between the households and intermediaries in various states of

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the world. When the returns are positive (negative), the planner perceives positive (nega-tive) externalities when transferring wealth from intermediaries to households. As a result,in “bad times,” we would expect this portfolio to have negative returns, consistent with theidea that the planner would like to use regulations to encourage intermediaries to hold morewealth in these states.

I consider two definitions of “bad times.” First, intuitively, bad times can be defined astimes in which the intermediaries have a high marginal utility of wealth. Using this defini-tion, I show that it is sufficient to study the expected returns of the externality mimickingportfolio, and test if they are positive. Second, I define “bad times” using the stress testscenarios developed by the Federal Reserve. I argue that these tests are statements aboutwhen the Fed would like intermediaries to have more wealth, and as a result the returns ofthe externality-mimicking portfolio should be negative in the stress test scenarios.

However, I find that the expected return of the externality mimicking portfolio is gener-ally negative, and that its returns in the stress tests are often positive. This implies that theexternalities that would justify current regulation are positive in bad times, which appearsinconsistent 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 wrongsign. That is, because JPY appreciates and AUD depreciates vs. USD in bad times, optimalpolicy should encourage intermediaries to be long JPY and short AUD (i.e. short the carrytrade). But the signs of the CIP violations are such that they encourage intermediaries to belong the carry trade, taking on more macro-economic risk. I speculate that this issue arisesfrom an interaction between leverage constraints (which do not consider the “sign” of atrade) 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 Polemar-chakis (1986) and Farhi and Werning (2016). My example of capital controls resemblesboth Fanelli and Straub (2018) and example 5.4 of Farhi and Werning (2016). The frame-work I develop specializes the standard GEI model in several respects. First, I assume thatthere are two classes of agents, households and intermediaries, who have different degreesof access to markets, in the spirit of Gromb and Vayanos (2002). Second, a key differencebetween this paper and the work of Farhi and Werning (2016), and also the discussion ofpecuniary externalities in Dávila and Korinek (2017), is my focus on an implementationof the constrained efficient allocation using quantity constraints, rather than agent-state- oragent-state-good-specific taxes. Studying this implementation is both realistic, in the sensethat regulation on banks takes this form, and it enables the empirical exercise that follows.

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My empirical work considers short-term arbitrages such as the fed funds/IOER spread(Bech and Klee, 2011) and CIP violations (Du et al., 2018), and hence this paper lies atthe intersection of the theoretical literature mentioned above and the empirical literatureon arbitrage opportunities. The central and most surprising result of the paper is, in effect,that this intersection exists. The techniques I use to characterize the externality-mimickingportfolio that links the theory with the data build on Hansen and Richard (1987) and Hansenand Jagannathan (1991). There is also a significant literature that studies CIP violationsin the context of particular models (as opposed to the general GEI framework). Examplesinclude Amador et al. (2017); Andersen et al. (2019); Du et al. (2020); Gabaix and Maggiori(2015); Ivashina et al. (2015). The framework I develop allows for the empirical analysisof CIP violations within a relatively minimal theoretical structure, and hence enables moregeneral conclusions about the optimality or sub-optimality of policy.

Section 2 outlines the GEI framework and presents the key result connection external-ities and arbitrage. Section 3 provides a more concrete example of the general framework,related to capital controls. Section 4 describes the externality-mimicking portfolio. Section5 describes the data I use in my empirical exercise, section 6 presents the main results, andI conclude in section 7. The internet appendix contains additional details on the empiricalanalysis, robustness exercises, and a formal discussion of the GEI framework.

2 Externalities and Arbitrage

In this section, I describe the connection between externalities and arbitrage under optimalpolicy in a GEI framework. Let S1 be the set of future states, let s0 be the initial state,and let S = S1∪{s0}. Let Js be the set of goods in each state s ∈ S. Consider the problemof a planner who cannot transfer goods between agents ex-post (in states s ∈ S1), but canmake transfers and control asset allocations in state s0. The problem of the planner is tochoose asset allocations for each agent, goods prices for each state, and initial transfersbetween agents, subject to market clearing, the constraint that transfers sum to zero, andthe constraints on feasible asset allocations (e.g. market incompleteness). For a formaldefinition of the problem, see Definition 3 in appendix section C.1

1The restriction against the planner transferring goods between agents in the future states S1 prevents theplanner from circumventing the incompleteness of markets. The ability of the planner to transfer income instate s0 ensures that the purpose of regulation is to correct externalities, and not to redistribute wealth. Thiscaptures the idea that the goal of macro-prudential regulators is not to redistribute wealth ex-ante, but ratherto influence the allocation of income across future states.

4

Generically, the solution to this planner’s problem will not coincide with a competitiveequilibrium, due to the interaction between market incompleteness and pecuniary external-ities (Geanakoplos and Polemarchakis (1986)). To better understand these externalities, Idefine, following Farhi and Werning (2016), “wedges.” Let µ j,s be the multiplier in theplanner’s problem on the goods market clearing constraint for good j in state s, scaled tothe units of prices, and let P∗j,s be the price chosen by the planner for good j in state s.

The multiplier µ j,s can be interpreted as the additional social cost of good j in state s

above the price P∗j,s (the cost agents privately perceive). In an economy in which the agents’and the planner’s preferences were perfectly aligned, the ratio of the social and private costswould be the same for all goods j ∈ Js within a given state s. Generically, this will not bethe case– competitive equilibria are constrained inefficient, and the wedges measure thisinefficiency. The wedge τr

j,s is the difference between the social/private cost ratio for thegood j ∈ Js in state s ∈ S1 and the average ratio for all goods in that state,2

πrs τ

rj,s =−

P∗j,s +µ j,s

P∗j,s+

1|Js| ∑

j′∈Js

P∗j′,s +µ j′,s

P∗j′,s.

The wedge τrj,s is scaled by a full-support “reference” measure on S1, πr

s > 0. In the applica-tions I consider, this measure is a risk-neutral measure or the physical probability measure.The wedges τr

j,s are defined in the context of this reference measure, and if defined insteadunder an alternative reference measure πr′

s would be rescaled, τr′j,s =

πrs

πr′s

τrj,s.

The wedge is positive if the social cost of a good is low relative to its price. The wedgeis also the difference between the first-order conditions of the planner and of the agents–the latter do not account for effects of their asset allocation on goods prices, and thesepecuniary externalities, due to market incompleteness, have welfare consequences.

The wedges can be compensated for by transferring income in state s between agents.Let h ∈H denote a “household,” and let i ∈ I denote an “intermediary,” two differentclasses of agents. Let Xh

I, j,s be the change in household h’s consumption of good j instate s if given a marginal unit of income, holding prices constant, evaluated at the incomeand prices that solve the planner’s problem, and let X i

I, j,s be the same income effect for

2This definition of wedges is essentially the same as the one employed by Farhi and Werning (2016), ad-justed for the difference between production and endowment economies and scaled by the reference measureπr

s . It is not necessary in what follows to define wedges for the state s0.

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intermediary i ∈I . If the wedge-weighted difference of these income effects,

∆h,i,rs = ∑

j∈Js

P∗j,sτrj,s(X

hI, j,s−X i

I, j,s),

is positive, transferring income from intermediary i to household h in state s has a benefit,from the planner’s perspective, because it alleviates pecuniary externalities.

I call the ∆h,i,rs “externalities” because they summarize the difference between the pri-

vate and social value of transferring income from intermediary i to household h in state s.3

For goods j ∈ Js with positive wedges τrj,s, the social cost of the good is lower than the

price, and it is desirable to increase demand for the good. If XhI, j,s > X i

I, j,s, then transferringincome from the intermediary to the household will indeed increase demand for the good.Summing these effects across goods determines the marginal benefit of a macro-prudentialregulation that transfers income from i to h in state s.

Under optimal policy, the marginal benefits of this macro-prudential regulation must beoffset by a marginal cost– distortions in risk-sharing. Absent regulation, agents will sharerisks by trading assets, but not account for the externalities just described. The plannerconsiders these externalities, and distorts risk-sharing to address them.

Let A be the set of assets in the economy, and let Ah be the subset tradable by householdh. Every intermediary can trade all assets a ∈ A, perhaps subject to some exogenous port-folio constraints. Households, in contrast, cannot trade directly with each other (each Ah isdisjoint). Two households might be able to trade an asset with the same payoffs, and hencecan trade indirectly via an intermediary. I will also assume that there are assets that canonly be traded by intermediaries, AI . Let Za,s({P∗j,s} j∈Js) denote the payoff of asset a ∈ A

in state s ∈ S, given the goods prices {P∗j,s} j∈Js that solve the planner’s problem.For any household h ∈H , asset a ∈ Ah, and intermediary i ∈ I , suppose the exoge-

nous portfolio constraints do not prevent the planner from reallocating the asset from i to h.In this case, under an optimal policy, the marginal externality-alleviating benefit of doingso must equal the marginal cost of further distorting risk-sharing. Let V h

I,s be the marginalutility of income for h ∈H (the derivative of the indirect utility function with respect toincome), again evaluated at the income and prices from the solution to the social planner’sproblem, and let V i

I,s be the corresponding marginal utility for i ∈I . The following propo-sition shows how the planner equates the marginal benefit of reducing externalities and themarginal cost of distorting risk-sharing.

3Farhi and Werning (2016) define an object τ iD,s, which is closely related, πr

s ∆h,i,rs = τh

D,s− τ iD,s.

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Proposition 1. In the solution to the social planner’s problem, for any h ∈H , i ∈I , and

a ∈ Ah, if the exogenous portfolio constraints do not bind with respect to asset a, then

∑s∈S

πrs ∆

h,i,rs Za,s({P∗j,s} j∈Js) = ∑

s∈S(

V iI,s

V iI,s0

−V h

I,s

V hI,s0

)Za,s({P∗j,s} j∈Js). (1)

These results holds for both the endowment economy of appendix section C and the pro-

duction economy of section 4 of Farhi and Werning (2016).

Proof. See the appendix, section D.1, or Farhi and Werning (2016).

Under optimal policies, the externalities ∆h,i,rs are equal to the difference of two stochas-

tic discount factors (SDFs), one for h ∈H and one for i ∈I . These SDFs are the ratios ofthe agents marginal utilities in state s and state s0, adjusted for any differences between thatagent’s beliefs and the reference measure πr. With incomplete markets, it is not surprisingthat these two SDFs are not identical. What is more surprising about the solution to theplanner’s problem is that the stochastic discount factors do not even price tradable assetsidentically. This is an implication of the idea that the planner, to implement the constrainedefficient allocation, must constrain or tax agents’ portfolio choices.

Let us suppose that the planner implements the constrained efficient allocation throughportfolio constraints, as opposed to taxes on agent’s portfolio holdings.4 In this implemen-tation, there will be a unique price for each asset (with taxes, each agent faces a differentprice for each asset). The planner has a great deal of latitude about the form of these con-straints– essentially all the matters is the shadow cost of the constraint, evaluated at theplanner’s desired allocation. In particular, the constraints could be a function of both assetprices and portfolio choices (like a capital requirement), but this is not required.5

What is the relationship between these portfolio constraints, efficiency, and arbitrage?6

An asset a ∈ Ah, for some h ∈ H, is “arbitrage-able” if there is a replicating portfolio inAI such that Za,s(·) = ∑a′∈AI wa′(a)Za′,s(·) for all s ∈ S, and all goods prices, and someportfolio weights wa′(a). Let A∗ denote the set of arbitrage-able securities.

The amount of arbitrage, χa, is defined as follows:4Whether the planner taxes asset holdings and rebates the proceeds lump sum or implements constraints

on portfolio choices is arbitrary, as long as the taxes or constraints implement the same asset allocation.Observationally, portfolio constraints such as capital requirements, restrictions on stock market leverage, andrestrictions on mortgage characteristics appear to be more common than taxes on asset holdings.

5For a formal definition of these portfolio constraints, see the appendix, Section §C.6I will the terms “arbitrage” and “law-of-one-price violations” synonymously; arbitrage of the “something

for nothing” variety does not occur in the economies I study.

7

Definition 1. For any arbitrage-able asset a ∈ A∗, the amount of arbitrage is

χa =−Qa + ∑a′∈AI

wa′(a)Qa′,

where Qa is the price of the asset a and Qa′ is the price of the asset a′ ∈ AI .

To simplify the analysis, I assume intermediaries share a common, homothetic utilityfunction, and hence that X i

I, j,s = X i′I, j,s for all i, i′ ∈ I , regardless of whether i and i′ have

identical incomes or not. As a result, the intermediaries aggregate, the planner is only con-cerned with total intermediary wealth, and can implement the efficient allocation withoutregulating trades between intermediaries. I summarize these key assumptions below.

Assumption 1. The intermediaries share a common, homothetic utility function, and the

planner implements the constrained efficient allocation

1. through portfolio constraints,

2. on intermediaries only,

3. and for each a′ ∈ AI , there is at least one intermediary that is not constrained with

respect to trade in that asset.

This assumption is without loss of generality in the sense that planner can implementany allocation of assets in the economy through restrictions satisfying this assumption.Because households must trade through intermediaries, by regulating the trade of interme-diaries with each household, the planner can dictate the asset allocation for all agents.7

Under this assumptions and the assumptions of Proposition 1, the household’s SDFprices the assets a ∈ Ah. That is, for any h ∈H and a ∈ Ah,

Qa = ∑s∈S

V hI,s

V hI,s0

Za,s. (2)

In contrast, because the planner will generically need to regulate asset markets, the interme-diaries’ SDF will not price the asset a ∈ Ah. The situation is different for the securities thatcan only be traded by intermediaries. Let i∗(a′) be an intermediary who is not constrained

7This point also illustrates the limits of the GEI framework I adopt. The model does not allow for eitherprivate information or hidden trade, both of which would limit the set of implementable allocations.

8

with respect to the asset a′ ∈ AI . This intermediary prices the asset,

Qa′ = ∑s∈S

V i∗(a′)I,s

V i∗(a′)I,s0

Za′,s. (3)

Now consider an arbitrage-able security a, which is in Ah for some h ∈ H. Underoptimal policies, intermediaries will not be constrained with respect to trade in the portfolioof replicating securities, and hence (3) will hold for all i ∈I and a′ ∈ AI . Combining (2)and (3) illustrates the relationship between the arbitrage on asset a and the externalities.

Proposition 2. In the solution to the social planner’s problem, under assumption 1, for any

arbitrage-able asset a ∈ A∗ that is tradable by household h, and any intermediary i ∈ I, if

the exogenous portfolio constraints do not bind for a or its replicating portfolio,

χa = ∑s∈S

πrs ∆

h,i,rs Za,s. (4)

Proof. See the appendix, section D.2.

This equation is essentially a transformation of the planner’s first-order condition inthe planning problem, combined with the implementation assumptions. It shows that anarbitrage-able asset will be cheap relative to its replicating portfolio if its payoffs occurmainly in states in which the planner would like to transfer wealth from intermediariesto households. One immediate implication is that constrained efficient allocations willgenerically feature arbitrage.8 The absence of arbitrage is not a sign of efficiency, but

rather a sign of inefficiency in the presence of incomplete markets.

I next provide a concrete example to further illustrate the connection between externali-ties and arbitrage. The remainder of the paper then conducts a revealed preference exerciseusing (4). Given an observed pattern of arbitrage (χa for various assets), what can wesay about the ∆

h,i,rs that would rationalize current policies as optimal? Are these revealed

externalities consistent with the expected pattern of externalities across states?

8Generically, externalities are non-zero and will lie in the span of the arbitrage-able asset’s payoff space.

9

3 An Example of Externalities as Arbitrage

This section describes a modified version of Fanelli and Straub (2018) (and see also exam-ple 5.4 of Farhi and Werning (2016)) to further illustrate the meaning of Proposition 2. Inthis example, a planner will optimally limit foreign-currency lending by intermediaries (i.e.use capital controls) to stabilize the real exchange rate (the pecuniary externality). This willcreate a CIP violation, and the size and direction of this CIP violation will be determinedby the externalities as in (4).

There are two types of domestic households, Ricardians and non-participants (H =

{r,n}). In each state, there are two goods, tradable and non-tradable, Js = {T,NT}. Bothhouseholds have log utility preferences over a Cobb-Douglas aggregate of tradables andnon-tradables, with share parameter α on tradables.

The future state can be either good (g) or bad (b), S1 = {g,b}. Non-participants areendowed with non-tradables YNT and tradables Y n

T,s, with Y nT,g >Y n

T,b. Ricardian householdsare endowed only with tradables Y r

T . Only Y nT,s varies across states; otherwise, all states are

identical. Foreign intermediaries are risk-neutral, consume tradables only, and have a largeendowment of tradables in all states. The discount factor for both households is β < 1, andthe discount factor for intermediaries is one.

The tradables price is stable in the foreign currency, and the domestic monetary author-ity stabilizes the domestic price index. The exchange rate is therefore es = (

PNT,sPT,s

)1−α , and Iwill use the tradable good as the numeraire (PT,s = 1). Ricardians can trade both a foreign-currency risk-free bond ar, with Zar,s(PNT,s) = 1, and a domestic risk-free bond ad , withZad ,s(PNT,s) = (PNT,s)

1−α . Intermediaries can trade both of these bonds, an intermediary-only foreign currency risk-free bond aI , and a currency forward a f at exchange rate f ,Za f ,s(PNT,s) = (PNT,s)

1−α − f . The bonds ar and ad are both arbitrage-able: ZaI ,s(·) =Zar,s(·) and f ZaI ,s(·)+Za f ,s(·) = Zad ,s(·). Non-participants cannot trade any assets.

The planner’s problem is to maximize a weighted sum of household utility, subjectto a participation constraint for the foreign intermediaries. Because there is a completemarket traded between intermediaries and Ricardians, the participation constraint of theintermediaries creates a unified budget constraint for the Ricardians.

Let πps be the physical probability measure,9 let Ts0 be the transfer in state s0 to non-

participants, and let λ p and λ r be the strictly positive Pareto weights on the households.

9Because intermediaries are risk-neutral, this is also the intermediaries’ risk-neutral measure.

10

The planner’s problem solves

max{Ir

s≥0,Ins≥0}s∈S,{PNT,s≥0}s∈S,Ts0

∑h∈{r,n}

λh∑s∈S

πps V h

s (Ihs ,PNT,s),

subject to non-tradable market clearing, YNT = ∑h∈{r,p}

XhNT,s(I

hs ,PNT,s),∀s ∈ S,

the non-participants budget constraints, Ins = PNT,sYNT +Y n

T,s +1{s = s0}Ts0,∀s ∈ S,

and the Ricardian budget constraint, Irs0+π

pg Ir

g +πpb Ir

b ≤ 2Y rT −Ts0.

The functional forms in this example yield simple expressions for the indirect utilityand demand functions. The demand function is Xh

NT,s(Ihs ,PNT,s) = (1−α)

Ihs

PNT,s, and

V hs (I

hs ,PNT,s) =

β [ln(Ihs )− ln(P1−α

NT,s )+(1−α) ln(1−α)] s ∈ {g,b},

ln(Ihs )− ln(P1−α

NT,s )+(1−α) ln(1−α) s = s0.

The market clearing condition highlights the pecuniary externality present in the model.If the Ricardian households sell a bond, reallocating income from the states in S1 to the states0, this will increase the price of the non-tradable good in s0 and reducing the price in thestates {g,b}. These price changes have an effect on welfare because the poor householdsface incomplete (in this case, non-existent) markets. Specifically, the additional social costof the non-tradable good, µNT,s, is determined by the planner’s first-order condition withrespect to PNT,s, and can be written for s ∈ S1 as

µNT,s

P∗NT,s= β

Ins0

λ n πps (

λ n +λ r

Ins + Ir

s− λ n

Ins),

where λ n

Ins0

µNT,s is the multiplier on the goods market clearing constraint. In states in which

the income share of non-participants, Ins

Ins +Ir

s, is lower than the relative welfare weight λ n

λ n+λ r ,the planner would like to increase non-participant incomes. Because non-participants arenet sellers of non-tradables, it is desirable in this case to increase the non-tradable price,and as a result the social cost of non-tradables is less than the private cost (µNT,s < 0). Notethat I have ignored goods market clearing for tradables (Walras’ law), and assume withoutloss of generality that µT,s = 0.

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The wedges are πps τ

pNT,s =−π

ps τ

pT,s =−

12

µNT,sP∗NT,s

, and the externalities simplify to

∆r,i,ps =−(1−α)

µNT,s

P∗NT,sπps= (1−α)β

Ins0

Ins(1− λ n +λ r

λ nIns

Ins + Ir

s).

In states in which the non-participant income share is low, the externalities are positive,meaning that it is desirable to transfer income from foreign intermediaries to Ricardianhouseholds in order to increase demand for non-tradables.

Let us now consider the first-order conditions of the planner’s problem with respect toRicardian households’ income and with respect to the transfer Ts0 . We have, for s ∈ {g,b},

βπps

λ r

Irs− (1−α)

µNT,s

P∗NT,s

λ n

Ins0

=λ r

Irs0

− (1−α)µNT,s0

P∗NT,s0

λ n

Ins0

=λ n

Ins0

− (1−α)µNT,s0

P∗NT,s0

λ n

Ins0

.

The transfer ensures that the goods market at date zero is efficient (a little algebra showsthat µNT,s0 = 0). This leads to a Ricardian complete markets Euler equation,

1−βIrs0

Irs= π

ps ∆

r,i,ps ,

and summing this equation weighted by the payoffs Za,s(P∗NT,s) gives a version of (1) forlog-utility households and risk-neutral intermediaries.

We can see from these equations that the externalities must be non-zero in the solutionto the planner’s problem (and hence that competitive equilibria are inefficient). If the ex-ternalities were zero, the income shares

Ing

Ing+Ir

gand In

bInb+Ir

bwould both equal λ n

λ n+λ r , and theRicardian Euler would require that incomes were equal, Ir

g = Irb, and therefore that the non-

participant incomes are equal, Ing = In

b . Market clearing in non-tradables requires that ifincomes are identical across {g,b}, so are prices. But if the non-tradable price is the samein g and b, non-participant income cannot be equal in those states by the assumption thatY n

T,g > Y nT,b, and therefore the externalities must be non-zero.

The externalities will manifest as arbitrages under Assumption 1. The households willprice the foreign-currency and domestic-currency bonds ar and ad , as in (2). The foreignintermediaries will price the intermediary-only (i.e. offshore) foreign-currency risk-free

12

bond aI and the currency forward a f , as in (3). The resulting arbitrages are

χar = πpg ∆

r,i,pg +π

pb ∆

r,i,pb ,

χad = πpg ∆

r,i,pg (P∗NT,g)

1−α +πpb ∆

r,i,pb (P∗NT,g)

1−α .

The first of these arbitrages is a difference between two risk-free rates– the rate intermedi-aries use when borrowing or lending with each other and the rate they use when borrowingor lending to the Ricardian households. The second is a CIP violation that involves the do-mestic currency bonds, a currency forward, and the intermediary-only interest rate. Thesetwo arbitrages closely resemble the arbitrages I study in the empirical exercise that follows.

The non-tradable price will be lower in b than in g. Consequently, the domestic bondhas a lower return in b than in g. In the absence of regulation, in the competitive equilibriumof this example the Ricardians will borrow from intermediaries using the foreign currencybond (because β < 1 and the Ricardians have no income risk). The planner, to increasethe price of non-tradables in state b relative to state g, would instead prefer that Ricardiansborrow from intermediaries using the domestic bond, thereby increasing Ricardian incomein b relative to g. A macro-prudential regulation limiting the quantity of foreign-currencylending by intermediaries is one way of implementing this outcome. Depending on param-eters, the planner might also limit the total amount of lending by intermediaries.

Because the foreign-currency and domestic bonds have different returns, if we observedthose returns, and also observed the arbitrages χar and χad and the probabilities π

pb and π

pg ,

we could recover the externalities ∆r,i,pg and ∆

r,i,pb . Perfect recovery is possible because the

set of arbitrage-able securities A∗ = {ar,ad} forms a complete market. In the empiricalexercise that follows, I will not be able to rely on market completeness, and therefore turnto the question of how to estimate externalities when faced with incomplete markets.10

4 The Externality-Mimicking Portfolio

Suppose a financial economist observes prices for (a subset of) the arbitrage-able assetsA∗ tradable by some household h, along with the prices of the corresponding replicatingportfolios of intermediary-only assets (e.g. derivatives). From these prices, the financial

10Another difference to note is that my empirical exercise considers arbitrage in developed market curren-cies, and the relevant macro-prudential regulation is not capital controls but rather leverage restrictions onintermediaries. I chose an example based on capital controls for its relative simplicity (and of course such anexample would be relevant for currencies with capital controls).

13

economist can calculate the arbitrage χa. Further suppose that the financial economistbelieves that these arbitrages are caused by regulation, and not by other frictions that areexogenous from the perspective of regulators. Examples include the post-GFC coveredinterest parity violations documented by Du et al. (2018) and the difference between thefederal funds rate and the IOER rate documented by Bech and Klee (2011).

If regulatory policy is optimal, equation (4) will hold. If in addition the set A∗ formeda complete market, there would be a one-to-one mapping between the arbitrages and theexternalities ∆

h,i,ss , given the reference measure πr

s (as in the preceding example). Althoughthere are multiple examples of assets with arbitrage, it would be a stretch to say that thefinancial economist observes a complete market in arbitrage-able securities. As a result, itwill not be possible, empirically, to recover ∆

h,i,rs for all states s ∈ S.

We can, however, project the externalities ∆h,i,rs on to a lower-dimensional space, and

require that this projection be consistent with the amount of arbitrage we observe. Thisprocedure, which I describe next, builds on Hansen and Richard (1987), who study theprojection of a stochastic discount factor onto the space of returns. Those authors showthat this projection is equivalent to minimizing the variance of a stochastic discount factor,subject to the constraint that the SDF price a set of assets.11 Hansen and Jagannathan (1991)then show that the portfolio whose returns are the projection of the stochastic discountfactor is also the portfolio with the maximum available Sharpe ratio.

I provide analogous results for the externalities ∆h,i,rs . Consider the projection of the ex-

ternalities onto the space of returns of arbitrage-able assets, under the reference probabilitymeasure. Define the “externality-mimicking portfolio” as the portfolio whose returns trackthe projected externalities. This portfolio is also the difference between a household’s SDFprojected onto the space of returns and an intermediary’s SDF projected onto the space ofreturns.12 Its returns have a lower variance than the externalities, and the portfolio has ahigher “Sharpe ratio due to arbitrage” than any other portfolio, where the “Sharpe ratio dueto arbitrage” is the difference between the Sharpe ratio on the portfolio of arbitrage-ableassets and the Sharpe ratio on its replicating portfolio.

I use the space of returns Ra,s =Za,sQa

as opposed to the space of payoffs, and thereforeassume that every arbitrage-able asset and its replicating portfolio have strictly positive

11This equivalence holds if the constraint that the SDF be positive does not bind.12This property follows from the linearity of L2 projections, which I employ (following Hansen and

Richard (1987)) for simplicity. Other authors such as Sandulescu et al. (2018) have explored projectionsusing Lp norms.

14

prices.13 I assume for simplicity that A∗ includes a risk-free asset, whose return is R f .For each arbitrage-able asset a∈A∗, the return on its replicating portfolio can be written

as RIa,s = (1− χa)Ra,s, where

χa =χa

Qa +χa(5)

is the arbitrage scaled by the price of the replicating portfolio. Intuitively, when the asset ischeaper than its replicating portfolio (χa > 0), its returns are higher. Let RI

f = (1− χ f )R f

be the return on the replicating portfolio of the risk-free asset.Let µA∗,r and ΣA∗,r be the mean vector and variance-covariance matrix of the returns

Ra,s for each a ∈A∗, under the measure πrs , and let µA∗,I,r and ΣA∗,I,r be the same for the

returns RIa,s. I assume there are no redundant risky arbitrage-able assets (ΣA∗,r and ΣA∗,I,r

have full rank). Let χA∗ be the vector of scaled arbitrages χa for the risky assets in A∗.Given a portfolio θ of risky assets in A∗, consider both the Sharpe ratio and the Sharpe

ratio of the replicating portfolio,14

SA∗,I,r(θ) =

θ T µA∗,I,r

RIf−∑a∈A∗ θa

(θ T ΣA∗,I,rθ)12

.

The portfolio weights θ do not need sum to one (the units of θ are “dollars”, not percent-ages), but the Sharpe ratio is homogenous of degree zero. Define the Sharpe ratio SA∗,r(θ)

along similar lines, with (µA∗,r,ΣA∗,r,R f ) in the place of (µA∗,I,r,ΣA∗,I,r,RIf ).

Because the prices of the replicating portfolios are not the same as the prices of thearbitrage-able assets, an allocation in dollars to arbitrage-able assets and the same dollarasset allocation to the replicating portfolios are in fact claims to different cashflows.15 Iwould like instead to compare portfolios that are claims to the same cashflows, but perhapshave different prices. To that end, define the portfolio transformation θ(θ) by

θa(θ) =Qa

χa +Qaθa.

13This is without loss of generality, given that there is a risk-free arbitrage-able security, as one couldalways add some amount of the risk-free security to another other security to ensure that its price is positive,while still ensuring that a replicating portfolio exists.

14The definition of the Sharpe ratio given here is signed (i.e. no absolute value), and might be scaled bythe inverse of RI

f when compared to other definitions of the Sharpe ratio.15For example, if both intermediaries and households can buy stocks at $1/share but intermediaries pay

$2/bond whereas households pay $3/bond, an allocation of $4 split equally between stocks and bonds meanstwo shares and one bond for the intermediaries, but one share and one bond for the households.

15

This transformation converts an allocation in dollars at the replicating portfolio prices to anallocation in dollars at the arbitrage-able asset prices.

I define the “Sharpe ratio due to arbitrage” is the difference between the Sharpe ratioon a set of claims and the replicating Sharpe ratio of those same claims,

SA∗,I,r(θ) = SA∗,r(θ(θ))−SA∗,I,r(θ).

A little algebra shows that the Sharpe ratio due to arbitrage is the ratio of the excess arbi-trage to the volatility of the portfolio,

SA∗,I,r(θ) =θ T · (χA∗− χ f

µA∗,I,r

RIf)

(θ T ΣA∗,I,rθ)12

.

I next define the externality-mimicking portfolio and describe its properties. The externality-mimicking portfolio has weights on the risky assets equal to

θA∗,r = (ΣA∗,I,r)−1(χA∗− χ f

µA∗,I,r

RIf

), (6)

and a weight on the risk-free asset equal to

θA∗,rf =−(θ A∗,r)T µA∗,I,r

RIf

+1

(RIf )

2 χ f . (7)

The following proposition summarizes the properties of this portfolio.

Proposition 3. The externality-mimicking portfolio with portfolio weights θ A∗,r has the

following properties:

1. The externalities can be expressed as the return on the externality mimicking portfolio

plus a zero-mean residual, uncorrelated with the returns of all arbitrage-able assets

a ∈ A∗: for all i ∈ I,

∆h,i,rs = ∑

a∈A∗RI

a,sθA∗,ra + ε

A∗,rs ,

∑s∈S1

πrs RI

a,sεA∗,rs = 0 ∀a ∈ A∗.

2. The variance of the externalities under the reference measure, ∑s∈S1 πrs (∆

h,i,rs − χ f

RIf)2,

16

is weakly greater than the variance of the externality-mimicking portfolio’s return,

(θ A∗,r)T ΣA∗,I,rθ A∗,r.

3. Let mI,rs be any SDF that prices the replicating portfolios under the measure πr

s . Then

mrs = mI,r

s +∑a∈A∗ RIa,sθ

A∗,ra is the solution to the problem:

minm∈R|S1|

∑s∈S1

πrs (ms−mI,r

s )2 subject to

∑s∈S1

πrs msRa,s = 1 ∀a ∈ A∗.

4. The Sharpe ratio due to arbitrage of the externality-mimicking portfolio, SA∗,I,r(θ A∗,r),

is weakly greater than the Sharpe ratio due to arbitrage of any other portfolio of as-

sets in A∗.

Proof. See the appendix, section D.3.

The first two claims follow from the least-squares projection. The third is the analogof Hansen and Richard (1987), and shows that the estimated household SDF mr

s = mI,rs +

∑a∈A∗ RIa,sθ

A∗,ra is the one that makes the household’s and intermediary’s SDFs as close as

possible, subject to the constraint that it price the arbitrage-able assets A∗ (and thereforediffer sufficiently from mI,r

s , which prices the replicating portfolios).16 The fourth claim isthe analog of Hansen and Jagannathan (1991), and shows that the externality-mimickingportfolio is also the one the maximizes the Sharpe ratio due to arbitrage.

The externality-mimicking portfolio is a reflection of what regulation is actually ac-complishing. Consider a state s in which the externality mimicking portfolio has a negative10% return. If policy is optimal, the best linear prediction of the externalities in this stateis negative 10%. That is, the planner would be indifferent between being able to trans-fer ex-post one extra dollar from households to intermediaries in state s and receiving anadditional 10%×πr

s dollars in the initial state s0.The externality-mimicking portfolio is defined in the context of the reference mea-

sure πr. In my empirical exercises, I focus on the intermediaries’ risk-neutral measure,

π is = Ri

fV i

I,s

V iI,s0

, and consider the physical (or actual) probability measure, π p, in robustness

exercises. The two corresponding externalities are the “risk-neutral externalities” ∆h,i,is and

16That is, mrs maximizes a “market integration” measure along the lines of Chen and Knez (1995).

17

the “physical externalities” ∆h,i,ps , which are linked by the relationship

πps ∆

h,i,ps = π

is∆

h,i,is .

This connection reflects the usual equivalence in asset pricing between state-dependentpreferences and beliefs. Using the risk-neutral externality mimicking portfolio has a par-ticular advantage, which is that all expected returns are equal to the risk-free rate, and henceobservable. Moreover, options and quanto prices (which I presume are traded only by inter-mediaries) can reveal risk-neutral variances and covariances (Martin (2017); Kremens andMartin (2019)). If we consider only arbitrage-able assets for which options and quantosare available (currencies and the S&P 500), no estimation is required when constructingthe risk-neutral externality mimicking portfolio. Using the physical measure, in contrast,requires estimating both expected returns and a variance-covariance matrix.

The externality-mimicking portfolio reveals what externalities would justify observedpatterns of arbitrage under an optimal policy. The next step in our “revealed preference”exercise is to ask whether the recovered externalities make sense. We generally expect thatthe externalities (and hence the mimicking portfolio returns) are negative in “bad” statesof the world. That is, empirically, governments seem tempted to bailout intermediaries inbad times, not in good times. To test whether regulations are consistent with this intuition,we need to define what we mean by “bad” states. I consider two definitions, which resultin two different empirically feasible tests. The first definition is to define bad times asbeing bad for the intermediaries, which is to say that the intermediaries’ stochastic discountfactor (SDF) is high. The second definition involves studying a particular situation– the“stress tests” conducted by the Federal Reserve– in which the regulator is concerned aboutexternalities, and would like intermediaries to have more wealth. Presumably, the ideabehind the stress tests is to ensure that intermediaries have sufficient wealth in the stressscenario so as to avoid a bailout ex-post.

The first approach yields a simple test. The covariance of the intermediary SDF and therisk-neutral externality-mimicking portfolio, under the physical measure, is

Covp(π i

s

RIf π

ps,∆h,i,i

s ) =1

RIf(

χ f

RIf− ∑

s∈S1

πps ∆

h,i,is )

=− 1RI

f(θ A∗,i)T · (µA∗,I,p−RI

f )+Covp(π i

s

RIf π

ps,εA∗,i

s ).

18

In other words, if the externalities are negatively correlated with the intermediaries’ SDF,the expected excess return of the risk-neutral externality mimicking portfolio under thephysical measure should be positive. The covariance term between the projection error andthe intermediary SDF shows that this test would be biased if there are components of theSDF that are not spanned by the space of returns, and which are correlated with componentsof the externalities that are also unspanned.

The second approach uses stress tests to identify a particular state (the stress test sce-nario) in which externalities should be negative. The purpose of the stress test is to ensurethat intermediaries have sufficient wealth in the stress scenario. To the extent that the stresstests are effective, they must operate by inducing the intermediaries to hold different assetsand issue different liabilities than they otherwise would have. Consequently, the interme-diaries’ counterparties (the households) must also hold different assets and issue differentliabilities than they otherwise would have. In other words, if the regulations act to raiseintermediaries’ wealth in certain scenarios, they must lower the wealth of households inthose scenarios (at least in an endowment economy). That is, stress test scenarios are astatement when the regulator perceives negative externalities associated with transferringwealth from intermediaries to households (negative ∆

h,i,rs ). Consequently, if the regulations

are having the desired effect, returns on the externality mimicking portfolio in the stresstest scenario should be negative. This test is biased if the unspanned component of theexternalities is large in absolute value in the stress scenario.

To summarize, I will estimate the risk-neutral externality mimicking portfolios, andthen conduct two tests. First, I will estimate the portfolio’s expected return under the phys-ical measure, and check if this is positive. Second, I will examine the portfolio’s return inthe stress scenario, and check if it is negative. That is, we expect

1RI

f(θ A∗,i)T · (µA∗,I,p−RI

f )> 0, (8)

and, if s∗ ∈ S1 is the stress scenario state,

RIf θ

A∗,i,if + ∑

a∈A∗RI

a,s∗θA∗,ia < 0. (9)

As a robustness check, in appendix section B I also study the stress scenario returns for thephysical-measure externality-mimicking portfolio.

19

5 Data

In this section, I describe the arbitrages, data sources, and additional assumptions I use toconduct the tests described in the previous section.

The set of arbitrage-able securities A∗ should be limited to arbitrages induced by regu-lation. I therefore focus on short maturity arbitrages, to avoid issues like the classic limitsto arbitrage argument (Shleifer and Vishny (1997)) and debt overhang (Andersen et al.(2019)). This precludes many of the arbitrages documented in the literature. I focus onarbitrages appeared after but not before the global financial crisis, reasoning that these ar-bitrages are likely to be induced by post-crisis regulatory changes. Two classes of arbitragethat fit these criteria are the difference between the federal funds rate and interest on excessreserves and CIP violations (Bech and Klee (2011); Du et al. (2018)).

Constructing the externality-mimicking portfolio requires determining what is the “as-set” a ∈ A∗ and what is the “replicating portfolio.” My framework offers two ways of mak-ing this distinction: assets a∈A∗ are tradable by households, whereas replicating portfoliosare not, and intermediaries’ trades in assets a∈A∗ are regulated, whereas trades in the repli-cating portfolio are not. To a first approximation, the difference between “cash assets” and“derivatives” lines up with both of these distinctions: derivatives are both less accessible tohouseholds and less regulated under various leverage and capital requirements.

Consider first the fed funds/IOER arbitrage, which is the difference of two risk-freerates. A bank can earn interest on excess reserves held at the Fed, whereas a householdcannot. If there is no meeting of the FOMC within the next month, the bank is essentiallyguaranteed to earn one month’s worth of interest at the current overnight rate.17 A house-hold could instead purchase treasury bills, highly-rated commercial paper, repo agreements(via money market funds), bank deposits, or the like. There is a significant literature argu-ing that treasury bonds are special, relative to other bonds that appear to be close substitutes(e.g. Fleckenstein et al. (2014)), perhaps due to their additional liquidity or some other kindof benefits. For this reason, I use 1-month OIS swap rates, which closely track the yieldsof one-month maturity highly-rated commercial paper in the US, as a proxy for a risk-free rate available to households that provides no liquidity benefits. These rates tend tobe higher than the rates on one-month constant maturity treasuries, but lower than LIBORrates (which may include credit risk).18 In the notation of the model, R f is the one-month

17In rare circumstances, the Fed might change the interest on reserves between meetings, but such changeshave low ex-ante likelihood and are unlikely to materially alter the expected interest rate.

18For example, on August 19th, 2016 (a little over one month before the next FOMC meeting), the one-

20

OIS swap rate, and RIf is the interest rate on excess reserves.

Next, consider covered interest parity violations. Here, guided by the “cash vs. deriva-tives” heuristic, I assume that households can purchase foreign-currency bonds, but cannottrade derivatives easily.19 The asset a ∈ A∗ is therefore a claim to one euro in one month,and households can purchase this asset by spot exchanging dollars for euros and then pur-chasing a risk-free euro-denominated bond. Following Du et al. (2020), I use OIS rates asproxies for the risk-free rates available to households in various currencies. The replicatingportfolio involves an intermediary earning the dollar IOER rate for one month and using aone-month FX forward to lock in the dollar/euro exchange rate.20

The third arbitrage I study, in robustness exercises in Appendix Section B.3, is an arbi-trage between the SPDR S&P 500 ETF and options on that ETF, which trade on the CBOEunder the ticker SPY. This arbitrage is closely related to the classic index-future arbitrageinvolving S&P 500 futures (e.g. Chung (1991); MacKinlay and Ramaswamy (1988); Milleret al. (1994)). I describe this arbitrage in more detail in the Appendix, Section A.3.

My data sample begins on Jan 4th, 2011, and runs through March 12th, 2018. Mysample is restricted to include only days on which the settlement of the one-month currencyforward occurs before the next FOMC meeting, excluding days with an FOMC meeting.Because the FOMC holds eight scheduled meetings each year, roughly one quarter of allnon-weekend days are included in the dataset. My data on spot and forward exchange rates,FX options, and OIS rates are from Bloomberg. I use the London closing time for all ofthese instruments, following Du et al. (2018). I focus on the euro, yen, and pound, becausethese currencies are major currencies that are modeled explicitly in the Federal Reserve’sstress test scenarios, and the Australian dollar, which plays a role in the “carry trade.” Fordetails of the data construction, see appendix section §A.

Information on the “stress test” scenarios comes from the Federal Reserve’s website.21

The “severely adverse” scenario described in the tests shows, among other variables, thelevel of euro, yen, and pound, as well as the Dow Jones Industrial Average, at a quarterly

month constant-maturity treasury rate was 27bps, the AA non-financial one-month commercial paper rateswas 37bps, the one-month OIS rate was 40bps, the interest on excess reserve rate was 50bps, and one-monthLIBOR was 52bps.

19That is, either the household literally cannot trade derivatives, or the transactions costs on derivativestrades for households are high enough that the households cannot profitably execute the arbitrage.

20Implicitly, I am assuming that the default risk on the forward contract is negligible (or, to be moreprecise, that the pricing data reflects forward rates available to a risk-free counterparty). Du et al. (2018)argue, persuasively in my view, that this risk is negligible.

21https://www.federalreserve.gov/supervisionreg/dfa-stress-tests.htm

21

frequency. I collect both the one and four-quarter percentage changes for each of the assetsI study, and in my analysis will pretend that these are returns that occur over a one-monthhorizon. For AUD, which is not explicitly modeled in the stress test scenarios, I impute thereturns in each stress scenario by running a daily regression predicting AUD returns usingthe contemporaneous GBP, EUR, JPY, and stock returns over the preceding 720 days, andthen using these regression coefficients along with the one or four-quarter stress returns.

To conduct the tests described in the previous section, several additional assumptionsare required. To construct the risk-neutral externality mimicking portfolio, I require a fullvariance-covariance matrix under the risk-neutral measure. I construct such a matrix fromcurrency options on each currency possible currency pair. For details, see appendix sec-tion §A. To estimate that portfolio’s expected excess returns under the physical measure, anestimate of expected excess returns is required. Motivated by Meese and Rogoff (1983) andthe related literature, I assume that exchange rates are random walks over my one-monthhorizon. As a result, the expected excess return of using the IOER rate and a forward topurchase, say, one yen one month from now, is the determined by the difference betweenthe forward and spot exchange rate.

Table 1 presents the sample means and standard deviations of the arbitrage associatedwith each currency and the risk-free arbitrage. Conceptually, these statistics correspond tothe term χa defined in (5). For example, for euros, it represents the percentage differencein price, in dollars today, of purchasing a single euro one month in the future by buyingthe euro at spot today and saving at OIS (the asset a ∈ A∗), and obtaining the same euroone month in the future by savings at the IOER rate and using a currency forward (thereplicating portfolio). I also present the difference between the dollar OIS rate and IOER(R f vs. RI

f ). The arbitrages have a one month horizon, but are scaled to annualized values.This table also shows the option-implied volatility and correlations of each currency

(with respect to the US dollar).22 Finally, the table reports the empirical correlations be-tween the currencies and both the SPDR ETF and the daily He et al. (2017) (HKM) inter-mediary capital factor, as well as the quanto-implied correlations between the currenciesand the S&P 500 (as in Kremens and Martin (2019), see appendix section A for details).Here, a positive correlation means appreciation relative to the dollar when the S&P 500 haspositive returns.

From table 1, we can observe several notable features of the data. First, observe that22A version of the table with physical measure estimated volatilities and correlations is in the appendix,

section §B. The average volatilities and correlations are strikingly similar to their risk-neutral counterparts.

22

intermediaries are able to earn a higher rate of interest than households (IOER vs. OIS).However, the positive sign on the euro, pound, and yen arbitrages implies that it is moreexpensive for intermediaries to use derivatives to purchase e.g. a euro one month in thefuture in exchange for a dollar today than it is for intermediaries to use products also avail-able to households. The opposite pattern holds for the Australian dollar. Note also that theeuro and Australian dollar are positively correlated with the S&P 500 and the HKM factor,while the yen is negatively correlated with the S&P 500. Immediately, by proposition 2, wecan observe that the AUD and JPY CIP violations do not have the expected sign, if eitherthe S&P 500 or the HKM factor is a reasonable proxy for the externalities.

Figure 1 shows the time series of the “risk-neutral” excess arbitrages, χa − χ f , forAustralian dollar, euro, and yen in my sample. Using the risk-neutral excess arbitrage, asopposed to the physical measure excess arbitrage, eliminates the dependence on an estimateof expected returns. We can observe that the magnitude of arbitrage is quite volatile, andthere is significant positive co-movement between the euro and yen arbitrage (these factsare essentially the same as those documented in Du et al. (2018)), while the Australiandollar arbitrage behaves quite differently.

6 Results

I begin by constructing the risk-neutral externality-mimicking portfolio, using the euro,Australian dollar, and yen assets, and a risk-free asset. This portfolio can be constructed atdaily frequency using (6) and date on the arbitrages and the risk-neutral variance-covariancematrix implied by FX options prices. Figure 2 displays the time series of the portfolioweights on the risky assets (EUR, AUD, JPY).

A few patterns in the data are apparent. First, the portfolio is generally long yen andeuro and short AUD, and long currencies overall. That is, the portfolio is short US dol-lars and short the carry trade.23 The “short US dollars” part is likely to generate positiveexpected returns, whereas the “short the carry trade” generates negative expected returns,and this latter effect will dominate (see table 2 below). Interpreted through the lens of themodel, this portfolio implies that a strong desire to transfer wealth from households to inter-mediaries (negative externalities) coincides with an appreciation of the US dollar and highreturns for the carry trade. If we assume that the planner would like to transfer wealth to

23In the terminology of Lustig et al. (2011), the portfolio is long the “level” factor and short the “slope”factor (the slope is with respect to interest rates).

23

intermediaries in “bad times,” the first part seems sensible, in light of the safe haven role ofthe US dollar (see, e.g., Maggiori (2017)), but the second is surprising. Lustig and Verdel-han (2007) show that negative carry trade returns are associated with falls in consumption,and we would generally presume that these times are times when the planner would likeintermediaries to have relatively more wealth. Second, the noticeable spikes in the euroand yen CIP deviations that occur around quarter and year ends result in large changes tothe portfolio weight. This is not surprising, as there is no corresponding large change inimplied volatilities that would offset the effect. Interpreted through the lens of the model,suddenly binding constraints could only be justified by large changes in externalities, andhence in the externality mimicking portfolio.

I next consider the predictions that this portfolio has about other arbitrages. I delib-erately excluded GBP from the set of currencies used to form the externality-mimickingportfolio. This allows me to test whether the arbitrage predicted using the externality-mimicking portfolio is consistent with the arbitrage actually observed for the dollar-poundcurrency pair. Formally, I compute

χGBP− χ f = ΣA∗,I,iGBP θ

A∗,i, (10)

where θ A∗,i is the externality-mimicking portfolio in (6) and ΣA∗,I,iGBP is the covariance, under

the intermediaries’ risk-neutral measure, between the dollar-pound exchange rate and therisky assets used to form the externality-mimicking portfolio (EUR, AUD, JPY). This isequivalent to computing the excess arbitrage under the projected externalities, which willcoincide with the arbitrage under the true externalities if there is no covariance between thepound and the error term in the projection.

Figure 3 displays the results graphically. The actual excess arbitrage in pounds is con-structed from OIS rates in dollars and pounds, and the spot and forward dollar-pound ex-change rates (using excess arbitrage eliminates the dependence on the IOER rate). Thepredicted excess arbitrage is constructed entirely from those same variables in euros andyen, along with options prices on all six possible currency pairs, which are used to bothconstruct the externality mimicking portfolio (in the matrix ΣA∗,I,i) and to construct the co-variances Σ

A∗,I,iGBP . That is, the set of financial instruments used to construct the actual and

predicted excess arbitrages do not overlap. Nevertheless, the predicted and actual excessarbitrages track each other, except near the end of 2011. The R2 of a regression of the actualarbitrage on the predicted arbitrage, with no constant, is 83%.

24

I next consider the expected return of this portfolio (the first test described in the pre-vious section). Intuitively, because the portfolio is generally short the carry trade, the ex-pected return on the portfolio is negative. This contradicts the intuition that the externalitiesshould be negatively correlated with the stochastic discount factor.

Figure 4 presents the time series of expected excess returns on the portfolio, and Table 2formally tests whether the average expected return over my sample is greater than or equalto zero (a one-sided test). I show results for the full sample, only for dates for which thetrade crosses a quarter-end, and only for dates for which the trade crosses a year-end.24 Ialso formally test whether the quarter-end dates are different from other dates, and whetherthe year-end dates are different from other quarter-end dates. Both Bech and Klee (2011),for fed funds vs. IOER, and Du et al. (2018), for CIP, have documented that the arbitragespikes near quarter-ends. As these results demonstrate, the problem of negative expectedreturns documented above is particularly acute at quarter and year-ends.

I now turn to the second test, using the stress tests. Once per year, the Federal Reservedescribes a “severely adverse” scenario and requires banks to maintain various leverageand capital ratios in this scenario. In Table 3, I present the returns of the yen, euro, andstocks in the stress test scenarios, at both the one quarter and four quarter horizons, foreach stress test conducted. A general pattern emerges: recent stress tests have involvedsizable euro depreciations relative to the dollar, and sizable yen appreciations. This patternis consistent with the observation that, during my sample, stock market declines tend tocoincide with euro depreciation and yen appreciation relative to the dollar, and that thesesorts of correlations might influence how the Federal Reserve constructs the stress testscenarios. The stock return itself very negative in all of these scenarios.

The scenarios do not specify a return for the Australian dollar, presumably because itwould be virtually impossible to the specify the returns of every asset a bank might hold.I impute the return of the AUD using the stress tests returns on euro, yen, and pounds(not shown in Table 3) and the stock market. Because of the Australian dollar’s positivecorrelation with the stock market and negative correlation with the yen, the imputed returnsare quite negative. When banks calculate their stress scenario returns, they likely performsome kind of imputation along these lines.

Each of the stress test scenarios is associated with a particular date (listed in table 3)

24The trade “crosses quarter end” if the settlement dates of the spot and forward FX trades are before andafter the end of some quarter, respectively. Crossing year end is defined the same way, but only for the fourthquarter.

25

which is the date at which the scenario starts. For each date in my sample that is alsowithin 180 calendar days of the stress test date, I report the returns of the risk-neutralexternality-mimicking portfolio under the associated stress test scenario. Requiring thatthe relevant financial market data come from a day that is within 180 days of the stress testdate effectively assigns almost all of the days in my sample to a single stress test per date,dropping only a handful of days that are far from any stress test date.

If regulation is optimal, we should expect that the returns of the externality mimickingportfolio are negative. What I find in the data, however, is that this is not the case. At almostall points in time, the portfolio is long low-interest-rate currencies (EUR and/or JPY) andshort high-interest-rate currencies (AUD), and as a result has positive returns in the stressscenario, because the carry trade performs poorly in the stress scenario.

Figure ?? presents the stress scenario returns graphically, and Table 4 formally testswhether returns are negative, averaging across dates near a particular stress test. The p-values correspond to a one-sided test that the mean is less than or equal to zero. I am ableto reject the hypothesis that returns are negative on average for all four stress test yearsbeginning in 2014.

In summary, the risk-neutral externality mimicking portfolio has negative expected re-turns and positive returns in the stress scenario, which is the exact opposite of what wewould expect under the presumption that the planner would like intermediaries to havemore wealth relative to households in bad states as opposed to good states.

On possible explanation, of course, is that the goal of regulation is to get intermediariesto take more, not less, macro-prudential risk, and regulatory policy is in fact accomplishingits goals. A more likely explanation, in my view, is that the current regulatory apparatus isnot accomplishing its macro-prudential objectives.

Du et al. (2018) shows that the direction of the CIP arbitrage across currencies is pre-dicted by the direction of the carry trade. A simple interpretation of this fact is that house-holds or their proxies want to do the carry trade, and intermediaries are induced by thearbitrage to take the other side. Leverage constraints, such as the “supplementary lever-age ratio” described in D’Hulster (2009), prevent the intermediaries from fully satisfyinghouseholds’ demands. If this story is correct, households are trying to take macro-economicrisk and insure intermediaries from those risks, but regulation (the leverage ratio) is limitingthis risk transfer, which is the exact opposite of what an optimal policy would do.

For robustness, in appendix section B, I present three sets of additional results. Thefirst set of these results uses the physical measure externality-mimicking portfolio instead

26

of the risk-neutral externality-mimicking portfolio in the stress test exercise. These resultsshow that the estimated physical and risk-neutral measure covariance matrices are similar,and that the stress test results do not depend on the choice of reference probability measure.Appendix section A.2 discusses the construction of the portfolio and appendix section B.2presents the results.

The second set of results incorporates an equity-based arbitrage between the SPDRETF and SPY options into the risk-neutral externality-mimicking portfolio. The purposeof this robustness exercise is to demonstrate that the puzzling results of the main analysisare not driven by the choice of arbitrages to include in the portfolio. Including the SPDR-SPY arbitrage in the externality-mimicking portfolio greatly increases both the complexityand data requirements of the exercise, and also increases the noisiness of the results dueto the imprecision in the measurement of the arbitrage. For these reasons, I do not includeit in the main analysis. With this arbitrage included in the portfolio, the results of theexpected return test are quite similar– expected returns are robustly negative, contrary toexpectations. The results for the stress test are “better,” in the sense that more but notall of the stress returns are sharply negative.25 This effect is driven by the combination ofvery negative equity returns in the stress scenario and a small positive SPDR-SPY arbitrage(on average). The data is described in appendix sections A.3 and A.4 and the results arepresented in appendix section B.3.

The third and final set of results uses “carry” and “dollar” portfolios of currencies in-stead of individual currencies. These results support the interpretation that the externality-mimicking portfolio is short USD (which was expected) and short the carry trade (whichwas not expected). The negative expected returns and positive returns in the stress sce-nario that I document are due to the short carry aspect of the portfolio. The portfolios aredescribed in appendix section A.5 and the results are presented in appendix section B.4.

7 Conclusion

Under optimal policy, there is a close connection between the externalities a planner isattempting to correct with regulation and the arbitrage that regulation creates. Using thisconnection, we can assess whether macro-prudential policies are achieving their objectives.

25Taken together, these results suggest that the stress returns might not be negative enough. Just becausethe returns in the stress scenario are negative and not positive does not mean macro-prudential regulation isworking optimally.

27

This paper develops a method of backing out a set of externalities that would rational-ize a particular pattern of arbitrage across assets. This method constructs an externality-mimicking portfolio, whose returns are a projection of the externalities that would ratio-nalize existing policy onto the space of returns. I argue that these externalities shouldnegatively covary with the SDF, and be particularly negative in “stress” scenarios. Usingthese intuitions, I develop two simple tests: does the externality-mimicking portfolio havepositive expected returns, and does it have negative returns in the Federal Reserve’s stresstests? I show, in current data, using currencies, the answer to both these questions is no,implying an inconsistency in current regulatory policy.

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29

Tables and Figures

Figure 1: Time Series of Excess Arbitrage-1

000

100

200

300

Exce

ss A

rb. (

bps)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

JPY EURAUD

Notes: This figure plots the excess arbitrage χa− χ f , where χa and χ f are defined as in (5), for the yen, euro, and Australian dollar. Thesample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting.

30

Figure 2: Externality-Mimicking Portfolio Weights

-50

510

15

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

JPY EURAUD

Notes: This figure plots excess the risk neutral measure portfolio weight θA∗,ra , defined in (6), where A∗ contains the yen, euro, and

Australian dollar, as well as a risk-free asset, and the reference measure is the intermediaries’ risk-neutral measure. The sample is all UStrading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting. The variance-covariance matrixused in the computation is inferred from currency options.

Figure 3: Actual vs. Predicted Excess Arbitrage in Pounds

050

100

150

200

Exce

ss A

rb. (

bps)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

GBP Pred. GBP

Notes: This figure plots excess the actual pound excess arbitrage χGBP− χ f , as defined in (5), along with the predicted value defined asin (10). The risk-neutral externality-mimicking portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, aswell as a risk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are inferred fromcurrency options. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMCmeeting.

31

Figure 4: Risk-Neutral EMP Expected Returns

-200

-100

010

020

0Ex

cess

Ret

urn

(bps

)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

Notes: This figure plots excess expected return under the physical measure of the risk-neutral externality mimicking portfolio (5) definedin (6), under the assumption that currencies follow a random walk. The excess return is censored at +/- 200bps to enhance readability.The risk-neutral externality-mimicking portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, as well as arisk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are inferred from currencyoptions. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting.

Table 1: Summary Statistics for ArbitragePounds Euros Yen Aus.

DollarOIS-IOER

Arbitrage Mean (bps/year) 6.7 22.4 28.3 -15.4 -12.5Arbitrage SD (bps/year) 28.2 37.7 37.2 18.5 2.8

OI Vol. (bps/year) 859 950 977 1073 -OI Corr. with Pound/USD 1.00 0.56 0.22 0.47 -OI Corr. with Euro/USD 0.56 1.00 0.31 0.51 -OI Corr. with Yen/USD 0.22 0.31 1.00 0.26 -

Empirical Corr. with SPDR 0.23 0.10 -0.34 0.37 -Empirical Corr. with HKM 0.26 0.17 -0.31 0.31 -Implied Corr. with S&P 500 0.28 0.11 -0.29 0.50 -

N 444 444 444 444 444Notes: This table presents summary statistics for the sample of all US trading days from Jan 4, 2011 to March 12, 2018 at least one monthbefore an FOMC meeting. Arbitrage mean χa is defined using (5) for a claim to e..g. one euro in one month, priced in dollars today. TheOIS-IOER arbitrage is the risk-free arbitrage, based on a claim to one dollar in one month. Arbitrage SD is the daily standard deviation ofχa. OI Vol. and OI Corr. variables for currencies are the time-series mean of a daily series extracted from variance-covariance matricesimplied by currency options. Empirical Corr. with SPDR and Empirical Corr. with HKM are the time-series means of the correlationsbetween the currency returns and the SPDR ETF (which tracks the S&P 500) and with the He et al. (2017) daily intermediary capitalfactor, as estimated on a rolling basis by the methodology described in appendix section A.2. Implied Corr. with S&P 500 is based onthe time-series mean of the currency correlation with the S&P 500 extracted from quanto options and described in appendix section A.4.

32

Table 2: Risk-Neutral EMP Expected ReturnsN Mean (bps) Standard Deviation (bps) Test P-Value

Full Sample 444 -222 25.9 ≥ 0 0.0000Quarter-Ends 155 -431 69.7 ≥ 0 0.0000

Year-Ends 46 -1005 209.2 ≥ 0 0.0000QE - NQE -321 = 0 0.0000

YE - NYE QE -816 = 0 0.0000Notes: This table reports the excess expected return under the physical measure of the risk-neutral externality mimicking portfolio (5)defined in (6), under the assumption that currencies follow a random walk. The set A∗ is the JPY, EUR, and AUD arbitrages, and therisk-free rate arbitrage. The full sample is all US trading days from Jan 4, 2011 to March 12, 2018 at least one month before an FOMCmeeting. The quarter-end and year-end sub-samples are restricted to days on which a quarter- or year-end occurs between the spot FXsettlement date and the one-month FX settlement date. The QE-NQE and YE-NYE QE report the mean difference between quarter-endvs. non-quarter-end dates and year-end vs. non-year-end quarter-end. Test indicates the hypothesis about the mean being tested, andP-Value reports the associated p-value.

Table 3: Stress Test “Severely Adverse” ScenariosEuro Stocks Yen AUD* *Imputed

Stress TestDate

One-QuarterReturn

Four-QuarterReturn

One-QuarterReturn

Four-QuarterReturn

One-QuarterReturn

Four-QuarterReturn

One-QuarterReturn

Four-QuarterReturn

9/30/12 -7.7 -15.4 -19.3 -51.5 2.8 -1.0 -8.8 -24.59/30/13 -14.3 -21.4 -26.5 -49.5 3.1 -1.1 -17.2 -28.79/30/14 -12.0 -13.4 -16.3 -57.1 7.6 6.5 -5.1 -17.2

12/31/15 -7.7 -13.9 -20.2 -50.7 2.7 5.1 -7.7 -18.312/31/16 -9.1 -11.9 -34.0 -49.7 3.3 7.5 -17.0 -24.112/31/17 -6.6 -10.9 -51.3 -62.8 11.7 4.6 -23.7 -32.9

Notes: This table reports the percentage changes in the level of the euro, yen, and the Dow Jones Total Stock Market Index (“Stocks”)during the first one or four quarters of the associated “Severely Adverse Scenario” from that year’s stress test. These percentage changesare treated as returns in my analysis. Stress Test Date lists the date on which that year’s scenario begins. AUD shows the imputed returnfor the Australian dollar, using the imputation method described in the text.

33

Table 4: Risk-Neutral Returns in Stress ScenarioStressTestDate

N Mean(1Q,%)

S.D.(1Q,%)

P-value(1Q)

Mean(4Q,%)

S.D.(4Q,%)

P-value(4Q)

9/30/12 63 -0.8 0.3 0.9923 -1.8 1.0 0.95939/30/13 59 -0.9 0.4 0.9838 -2.1 0.7 0.99879/30/14 62 4.8 0.6 0.0000 9.9 1.0 0.0000

12/31/15 60 1.9 0.4 0.0000 4.5 0.8 0.000012/31/16 61 2.8 0.8 0.0000 7.0 1.2 0.000012/31/17 45 31.4 5.0 0.0000 28.6 4.3 0.0000

Notes: This table reports the mean and standard deviation of stress test scenario returns for the risk-neutral externality-mimickingportfolio portfolio. The risk-neutral EMP portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, as wellas a risk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are computed fromcurrency options prices. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before anFOMC meeting and within 180 days of a stress test date. Each of these dates is assigned to the nearest stress test date. N reports thenumber of dates assigned to each stress test, and P-Value reports the p-value associated with a one-sided hypothesis test that the meanreturn is negative. Results are reported for both one-quarter (1Q) and four-return (4Q) returns from the stress test scenarios.

34

Internet Appendix

A Details on Data Construction

A.1 The Risk-Neutral Variance-Covariance Matrix

The risk-neutral variance-covariance matrix ΣA∗,I,i is defined as variance-covariance matrix,under the intermediaries’ risk-neutral measure, of the returns RI

a,s. In the case of currencies,these are the returns of investing at the IOER rate for one month and using a currencyforward to purchase, say, yen, and then immediately exchanging back to dollars the spotrate. Let S j,t be the spot exchange rate of currency j per dollar (e.g. euros per dollar, yenper dollar), and let Fj,t be the one-month forward rate. The empirical analog of the returnRI

a,s is

R j,t = RIf ,t

Fj,t

S j,t+1,

where RIf ,t is the gross IOER rate accumulated over the one-month time horizon. Note that,

because RIf ,t is the risk-free rate available to intermediaries, we must have E∗t [

Fj,tS j,t+1

] = 1where E∗t denotes expectations taken under the intermediaries’ risk-neutral measure.

To construct the risk-neutral variance-covariance matrix of currency returns, I use daily,London-closing at-the-month 1-month implied volatilities from Bloomberg for each cur-rency pair. The volatilities are “percentage” volatilities from a log-normal Garman andKohlhagen (1983) model. If S j,t and S j′,t are two exchange rates vs. the US dollar at time t

(e.g. euros per dollar and yen per dollar), then

Cov∗t [∆s j,t+1,∆s j′,t+1] =12(V ∗t [∆s j,t+1]+V ∗t [∆s j′,t+1]−V ∗t [∆s j,t+1−∆s j′,t+1])

where ∆s j,t = ln(S j,t+1S j,t

) and Cov∗ and V ∗ denote the risk-neutral variance and covariance,respectively. Under the assumption of log-normality,

Cov∗t [S j,t

S j,t+1,

S j′,t

S j′,t+1] = E∗t [exp(−∆s j,t+1−∆s j′,t+1)]−E∗t [exp(−∆s j,t+1)]E∗t [exp(−∆s j′,t+1)]

=S j,tS j′,t

Fj,tFj′,t(exp(Cov∗t [∆s j,t+1,∆s j′,t+1])−1),

35

where F1,t and F2,t are the forward rates. It follows that

ΣA∗,I,ij, j′,t = (RI

f ,t)2(exp(

12(V ∗t [∆s j,t+1]+V ∗t [∆s j′,t+1]−V ∗t [∆s j,t+1−∆s j′,t+1]))−1).

In theory, I should I make an adjustment to Bloomberg implied volatilities to use a dis-count rate associated with the IOER rate, instead of the more standard OIS rate. However,the difference amounts to about a one basis point in the option price, and hence is negligible.I have also experimented with more sophisticated methods of computing the risk-neutralvariance-covariance matrix (the “SVIX” method of Martin (2017)). Such methods avoidlog-normality assumptions at the expense of additional data requirements and complexity,and have little impact on my results.

A.2 The Physical Expected Returns and Variance-Covariance Matrix

Expected returns under the physical measure are required both to conducted the expectedreturns test described in the text and to construct the physical measure externality-mimickingportfolio. The latter also requires an estimate of the physical measure variance-covariancematrix of returns.

As described in the text, I assume that currencies are random walks. Specifically, Iassume log-normal exchange rates and that the log-exchange rate is a martingale. Theexpected excess return of using the IOER rate and a forward to purchase, say, one yen onemonth from now, is the determined by the difference between the forward and expectedexchange rate. That is,

µA∗,I,pj,t = RI

f ,tEt [Fj,t

S j,t+1],

where St+1 is the exchange rate in foreign currency per dollar, Ft is the one-month forwardrate, RI

f ,t is the IOER rate accumulated over the next month, and expectations are takenunder the physical measure. Under the stated assumptions,

µA∗,I,pj,t = RI

f ,tFj,t

S j,texp(

12

Vt [∆s j,t+1])

where Vt [∆st+1] is the conditional variance of the log change in the exchange rate. Conse-quently, armed with an estimate for Vt [∆st+1], we can construct expected returns.

I estimate a daily physical-measure variance-covariance matrix using an exponentiallyweighted moving average of the daily series, with a decay factor of 0.97 (a procedure

36

known as the “RiskMetrics” methodology, see for example Alexander (2008)). This avoidsusing future information to estimate the variance-covariance matrix. I initialize my varianceand covariance estimates at the beginning of 2011 with the realized variance/covariance for2010. I then scale my daily estimated variance-covariance matrix to a one-month horizon.26

I use this estimated variance-covariance matrix of log returns both construct my esti-mates of mean returns, as above, and to construct a variance-covariance matrix for arith-metic returns, as described in appendix section A.1, under the assumption of log-normality.In this case,

Covt [S j,t

S j,t+1,

S j′,t

S j′,t+1] = Et [exp(−∆s j,t+1−∆s j′,t+1)]−Et [exp(−∆s j,t+1)]Et [exp(−∆s j′,t+1)]

=µ

A∗,I,pj,t µ

A∗,I,pj′,t

(RIf ,t)

2

S j,tS j′,t

Fj,tFj′,t(exp(Covt [∆s j,t+1,∆s j′,t+1])−1)

and

ΣA∗,I,pj, j′,t = (µA∗,I,p

j,t µA∗,I,pj′,t )(exp(

12(Vt [∆s j,t+1]+Vt [∆s j′,t+1]−Vt [∆s j,t+1−∆s j′,t+1]))−1).

A.3 The SPDR ETF/SPY Option Arbitrage

In this section, I describe an equity-related arbitrage that I include in a robustness exercise.The arbitrage I consider is an arbitrage between the SPDR S&P 500 ETF and optionson that ETF, which trade on the CBOE under the ticker SPY. This arbitrage is closelyrelated to the classic index-future arbitrage involving S&P 500 futures (e.g. Chung (1991);MacKinlay and Ramaswamy (1988); Miller et al. (1994)).

The arbitrage I study considers the cost of purchasing a share of the SPDR ETF andholding it for one month as compared to the cost of purchasing that ETF via put-call parity(by buying a call and selling a put with a one month horizon). The ETF share itself isthe arbitrage-able asset (both because it is easily purchased by households and becauseregulations affect intermediaries’ trade in equity shares). The intermediary, to replicatethe ETF, can buy a call on the ETF, sell a put on the ETF at the same strike, and investenough cash at the IOER rate over the next month to cover the exercise price of the put/call.Regardless of whether the ETF ends up above or below the strike price, the intermediary

26More sophisticated approaches that incorporate higher-frequency data might yield better results. See, forexample, Ghysels et al. (2006).

37

will end up owning the ETF in one month.The particular details of this arbitrage are complicated by the fact that the SPY options

are “American” and not “European” options, meaning they can be exercised at any time.I deal with this issue by employing the Margrabe (1978) bound on American put prices,which in my setting (short time horizons and low interest rates) is reasonably tight. I discussthis issue in more detail below.

The ETF and the replicating portfolio will generate identical payoffs as long as there areno dividends over the course of the month (more precisely, that an ex-dividend date doesnot occur within the month). The ETF has ex-dividend dates quarterly, usually on the thirdFriday of March, June, September, and December.27 I therefore limit my sample to avoidthese dates. This illustrates one of the two main advantages the ETF-based arbitrage hasover the traditional S&P 500 cash-futures arbitrage. The stocks of the S&P 500 index paydividends often, and hence most studies of index arbitrage assume either perfect foresightof dividends or use a dividend forecast, whereas no such assumptions are required for theETF arbitrage. The second advantage relates to transactions costs and stale prices. Thetraditional index arbitrage involves buying and selling 500 stocks, generating substantialtransactions costs and exacerbating the issue that prices might not be synchronized. Usingthe ETF, which is one of the most actively traded securities in the equity market and has avery small bid-offer, mitigates many of these issues. Of course, synchronizing the optionsprices and the ETF price is still critically important, as in Van Binsbergen et al. (2012).

For this arbitrage, I am assuming that the costs associated with posting margin on theoptions are negligible. That is, the margin is sufficiently small, and the interest rate the in-termediary receives on the posted margin sufficiently close to the IOER rate, that these costsare negligible. This assumption is also, implicitly, being applied to the margin required bycounterparties in the OTC market for FX swaps when studying CIP violations.

Because the SPY options are American, not European, I construct arbitrage bounds asopposed to a single arbitrage measure. It is straightforward to observe that an Americancall or put must be weakly more valuable than its European counterpart, but the possibilityof early exercise implies that this weak inequality might be strict.

Let pa(K) ∈ A and ca(K) ∈ A denote the American put and American call of strikeK. Following the argument of Margrabe (1978), let pa(K) be an American put option toexchange the ETF for an amount that grows at the IOER rate, K exp(t · ln(RI

f )), where t is

27The prospectus, available at https://us.spdrs.com/library-content/public/SPDR_500%20TRUST_PROSPECTUS.pdf,describes the details of how the ex-dividend dates are determined.

38

the time of the exchange. Because the option matures prior to the next FOMC meeting,the IOER rate is assumed to be constant. Because the intermediary is always indifferentbetween buying the ETF and the risk-free bond, there is never any advantage to early ex-ercise. Consequently, the put option pa has the same value as a European style put optionwith the same expiry and a strike KRI

f . Moreover, because RIf ≥ 1, the option pa is more

valuable than the American put pa. Hence, by no-arbitrage, the following inequalities holdfor the American put:

0≤ Qpe(K) ≤ Qpa(K) ≤ Qpa(K) = Qpe(KRIf ),

where pe(K) ∈ A is the European put of strike K. Using essentially the same argument,let ca be an American call option to buy the ETF in exchange for K

RIfexp(t · ln(RI

f )) dollars

at time t. Early exercise is again never optimal, and hence this call’s value is equal tothe European call with strike K and the same expiry. Moreover, this call dominates theAmerican call with strike K and the same expiry, and hence early exercise is never optimaland the American and European calls have the same value. That is,

Qca(K) = Qce(K),

where ce(K) ∈ A is the European call of strike K.Let us now consider how to replicate the ETF. If I observed European options prices, I

would calculate the arbitrage, for any strike K, as

χe = Qce(K)−Qpe(K)+KRI

f−Qe,

where Qe is the ETF price. Using the put inequalities derived above,

χe ≥ χe,min(K) = Qca(K)−Qpa(K)+KRI

f−Qe,

and, for K′ = KRI

f,

χe ≤ Qca(K′RIf )−Qpa(K′)+K′−Qe. (11)

If RIf is sufficiently close to one (and one month’s worth of interest is indeed quite small),

these bounds will be tight.One implementation issue that arises from these inequalities is that the strike KRi

f is un-

39

likely to be traded. However, by the convexity of call prices (another cashflow dominanceargument),

Qce(KRJf )≤ αQce(K1)+(1−α)Qce(K2)

for any K1 ≤ KRIf ≤ K2 such that αK1 +(1−α)K2 = KRI

f . Choosing K1 and K2 to be asclose as possible to KRI

f generates the tightest bound. If KRIf is greater than the maximum

traded strike Kmax, then Qce(KRIf )≤Qce(Kmax). Using these bounds along with (11) generates

an implementable upper-bound, which I will call χe,max(K).The scaled arbitrage χe defined in (5) is monotonically increasing in χa (under the

assumption of positive prices), and therefore

χe ≥χe,min(K)

Qe +χe,min(K)

andχe ≤

χe,max(K)

Qe +χe,max(K).

Because these bounds must hold for all K, we are free to choose the greatest lower boundand least upper bound from the set of available strikes.

Another empirical issue to consider in the implementation of this trading strategy iswhether to use bids and offers or mid-prices. Bid-offers are wide in options markets, andlikely substantially overstate the bid-offer associated with “delta one” trades. That is, buy-ing a call and selling a put together likely has a much smaller bid-offer than doing thosetrades separately. For this reason, authors such as van Binsbergen et al. (2019) use mid-prices. However, mid-prices can exhibit strange behavior when bid-offers are particularlywide (which is why those authors use outlier-robust methods of analysis). To deal with thisissue, I consider only strikes K with sufficiently small bid-offers. In particular, I restrictattention to values of χe,min(K) which the difference between the mid-price and the lowestbound that can be constructed from the various bids and offers is less than 0.05% of thespot price Qe. Under this restriction, the lower bound χe,min(K)

Qe+χe,min(K) constructed from midprices is at most ~5bps too high in the worse-case scenario. Similarly, I require that thedifference between χe,max(K) and the highest bound constructed from bids and offers beless than 0.05% of Qe. From these two sets of valid strikes (one for χe,min and one forχe,max), I choose the strikes that generate the tightest possible bounds on χe.

After finding the maximum and minimum bounds, χe,min and χe,max, I define the esti-

40

mated arbitrage as

χe =

χe,min max(χe,min, χe,max)> χRF

χe,max min(χe,max, χe,max)< χRF

χRF otherwise.

In other words, I will assume that there is zero risk-neutral excess arbitrage (χe− χRF) ifthis is possible, and assume the minimum amount, in absolute value terms, if it is not possi-ble.28 In practice, my final dataset never has χe = χRF , because the bounds are sufficientlytight.

The dataset is a high-frequency (minute-level) dataset of options quotes purchased fromthe CBOE DataShop. From this dataset (which contains quotes for all minutes the ex-change is open), I have extracted the five minutes on each day immediately preceding theBloomberg London closing time. On most days, this is 12:55pm-12:59pm EST, althoughthe EST hour moves around due the asynchronous use of daylight savings time in the USand UK.

This dataset contains SPY options of many different expiries. Because I am interested inone-month options (where one-month is defined based on FX trading conventions) that donot cross an SPY dividend date, I restrict attention to expiries between 21 and 58 days in thefuture. These cutoffs ensure expiries are roughly one month and using these specific cutoffssimplifies the logic of determining whether an expiry occurs after the next ex-dividend dateon the SPY. I also require that each expiry have at least eleven different strikes quoted tobe included in the dataset.

The result of these restrictions and calculations is a dataset containing many estimatesof χe on each day (five minutes times the number of valid expiries). From this set, for eachday I search for minute/expiry pairs with non-missing data, expiries that cross neither thenext SPY ex-dividend date nor the next FOMC meeting, and that have no arbitrage viola-tions based on the bids and offers of options prices.29 Among the surviving minute/expirypairs, I choose first the expiries that are closest to the FX market definition of one month,and then among those choose the minute-expiry pair with the narrowest bid-offer for therelevant arbitrage bounds. This procedures results in a unique value for χe on each day.

28Note that, because I am using mid prices, it is possible to have χe,min > χe,max.29This last filter eliminates a few days with bad options quotes.

41

A.4 Expectations, Variance, and Covariance with the SPDR Arbitrage

To use the SPY arbitrage described in the preceding sub-section in my exercise, I requiresestimates of its variance and covariance with currency returns under both the physical andrisk-neutral measures, as well as an estimate of its expected return under the physical mea-sure.

I use, as an empirical analog of the one-month return RIa,s = (1− χa)Ra,s,

Re,t = (1− χe,t−1)Qe,t

Qe,t−1,

where Qe,t is the spot SPY price. Note that this definition does not include dividends,because I have restricted attention to dates on which dividends will not occur over the nextmonth.

To compute expected returns under the physical measure, I assume an equity premiumof 5% (roughly the average value of Martin (2017) in recent years). Although many pre-dictors of time-varying equity returns have been documented in the literature, over a one-month horizon most of these predictors are quite weak, and it seems reasonable to use anestimate of the unconditional equity premium. Under this assumption,

µA∗,I,pt = (1− χe,t)(R f ,t−1+1.05∆t),

where ∆t is the time (in years) to the next month under the FX market convention and R f ,t

is the US OIS rate accumulated over that month.To compute the physical-measure variance-covariance matrix, I use a daily series of

surprise log-returns,

re,t = ln(Qe,t)− ln(Qe,t−1)− ln(R f ,t−1+1.05∆t),

consistent with how I construct surprise currency returns. I then use the same “Risk-Metrics” methodology described in appendix section A.2.

I compute the risk-neutral variance-covariance matrix using the SVIX method of Martin(2017) and data on quanto options from Markit (as in Kremens and Martin (2019)). Apply-ing the SVIX methodology of Martin (2017) (in particular, equation (11) of that paper) tothe SPY options data used to construct the arbitrage series χe, I compute V ∗t−1[

Qe,tQe,t−1

], andthen scale by (1− χa,t−1)

2 to compute the variance.

42

I extract covariances from data on quanto options on the S&P 500. A quanto call optionis, for example, the right to buy the S&P 500 for a fixed amount of euros at a certain date.Such options are traded in OTC markets, and Markit provides a pricing service to helpthat dealers that trade these options mark their books. The prices represent the (trimmed,cleaned) averages of prices submitted by participating dealers. My data set include pricesfor call and put options for all of currencies used in this paper. Unfortunately, these optionshave a twenty-four month expiry (this is essentially the only traded expiry), and the data ismonthly rather than daily. I discuss how I deal with both of these issues below. My use ofthe quanto options is also complicated by the presence of arbitrage (CIP violations). I dealwith this issue by pricing the options under the assumptions of the framework developed inthis paper, and then extracting a risk-neutral covariance from those pricing formulas.

Let S j,t be the spot exchange rate (e.g. euros per dollar). The dollar price of a quantocall (qc) option, as a percentage of the spot price, with a strike equal to the current spotprice is

qc j,t =

1RI

f ,t,t+24E∗t [

X j,tS j,t+24

max{Qe,t+24−Qe,t ,0}]

Qe,t,

where X j,t is the agreed-upon fixed exchange rate for the quanto option and RIf ,t,t+24 is the

intermediaries’ cumulative discount factor over the next two years. The Markit data use theconvention X j,t = S j,t .

Quanto-put (qp) prices follow an analogous formula, and by put-call parity, for thestrikes in my data,

RIf ,t,t+24(qc j,t−qp j,t) = E∗t [

S j,tQe,t+24

S j,t+24Qe,t]−E∗t [

S j,t

S j,t+24].

My data also includes hypothetical prices for quanto call and put options under the assump-tion of zero correlation between the foreign exchange rate and the S&P 500. Under thisassumption, the price of the quanto call with X j,t = S j,t is

zc j,t = E∗t [S j,t

S j,t+24]×

1RI

f ,t,t+24E∗t [max{Qe,t+24−Qe,t ,0}]

Qe,t

and hence is equal to the (inverse) forward premium multiplier by the price of the vanilla(standard) call option on the S&P 500. That is, by asking for “zero-correlation” quanto callprices, Markit is not asking dealers to price a new exotic instrument but rather to report the

43

levels of two standard contracts along with the the price of the quanto call option.Again by put-call parity,

RIf ,t,t+24(zc j,t− zp j,t) = E∗t [

S j,t

S j,t+24]E∗t [

Qe,t+24

Qe,t]−E∗t [

S j,t

S j,t+24],

where zp j,t is the zero-correlation quanto put price.It follows that

E∗t [Qe,t+24

E∗t [Qe,t+24]

Fj,t,t+24

S j,t+24] =

1+ Fj,t,t+24S j,t

RIf ,t,t+24(qc j,t−qp j,t)

1+ Fj,t,t+24S j,t

RIf ,t,t+24(zc j,t− zp j,t)

, (12)

where Fj,t,t+24 is the two-year forward price.As mentioned previously, the quanto dataset is a monthly dataset (with a few missing

observations) of 24-month expiry options. The goal of this exercise is to extract one-monthhorizon covariances. To that end, I assume that correlations are constant over horizon, sothat I can extract a 24-month risk-neutral correlation and then pretend it is a one-monthcorrelation. I will use the most recent non-missing observation for each quanto currency.In the data, the correlations I extract move slowly over time.

Ignoring these issues for a moment, the quantity of interest is

ΣA∗,I,ij,e,t = E∗t [Re,t+1R j,t+1]−E∗t [Re,t+1]E∗t [R j,t+1],

where R j,t is the intermediary currency return defined in A.1. This is

ΣA∗,I,ij,e,t = (1− χe,t)RI

f ,tE∗t [Qe,t+1]

QtCov∗t [

Qe,t+1

E∗t [Qe,t+1],

Fj,t

S j,t+1]

= (RIf ,t)

2Corr∗t [Qe,t+1

E∗t [Qe,t+1],

Fj,t

S j,t+1]V ∗t [

Qe,t+1

E∗t [Qe,t+1]]

12V ∗t [

Fj,t

S j,t+1]

12 . (13)

Under the assumption that correlations are constant across horizon,

Corr∗t [Qe,t+1

E∗t [Qe,t+1],

Fj,t

S j,t+1] =

Cov∗t [Qe,t+24

E∗t [Qe,t+24],

Fj,t,t+24S j,t+24

]

V ∗t [Qe,t+24

E∗t [Qe,t+24]]

12V ∗t [

Fj,t,t+24S j,t+24

]12

=E∗t [

Qe,t+24E∗t [Qe,t+24]

Fj,t,t+24S j,t+24

]−1

V ∗t [Qe,t+24

E∗t [Qe,t+24]]

12V ∗t [

Fj,t,t+24S j,t+24

]12. (14)

44

I compute risk-neutral variances V ∗t [Qe,t+24

E∗t [Qe,t+24]] and V ∗t [

Fj,t,t+24S j,t+24

] from Bloomberg at-the-money 2-year at-the-money SPX implied volatilities and 2-year FX volatilities (implicitlyassuming log-normality). As discussed earlier, using an SVIX-based calculation wouldavoid log-normality assumptions at the expense of increased data requirements, computa-tional complexity, and uncertainty related to bid-offers and illiquidity of out-of-the-moneyoptions. Under an assumption of log-normality,

V ∗t [Qe,t+24

E∗t [Qe,t+24]] = exp(V ∗t [ln(Qe,t+24)])−1,

and likewiseV ∗t [

Fj,t,t+24

S j,t+24] = exp(V ∗t [ln(S jt+24)])−1.

One last implementation concerns the intermediaries’ two-year discount factor, RIf ,t,t+24.

In the main text, I study only dates at least one month before an FOMC meeting and use theIOER rate as the one-month rate. This approach does not allow me to construct a two-yearrate. For simplicity, I use instead the two-year OIS rate and then add the sample meanspread between IOER and fed funds (see Table 1). This adjustment makes almost no dif-ference to the estimated correlations. van Binsbergen et al. (2019) offer a better approach:extracting a two-year intermediary discount factor using “box” trades (put-call parity fordifferent strikes). As with the SVIX methodology, this approach is theoretically superiorbut increases data requirements and concerns about issues related to illiquidity, bid-offerspreads, and the like.

A.5 Construction of the Dollar and Carry Portfolios

As an additional robustness exercise, I construct an externality mimicking portfolio underthe assumption that A∗ includes a risk-free asset and two portfolios of currency trades,which I will refer to as dollar and carry.

These portfolios are portfolios of five developed-market currencies vs. the US dollar.The five currencies (plus the US dollar) in the portfolio are: euro, yen, pound, Australiandollar, and Canadian dollar. To select these currencies, I started with the nine non-US-dollar G10 currencies. I removed the New Zealand dollar, Swedish krona, and NorwegianKrone due to limited data availability for FX options and OIS swaps. I removed the Swissfranc both because of problems with its OIS rate (discussed in Du et al. (2020)) and becauseof the pegging and de-pegging events that occur during the sample period.

45

From these five non-USD currencies, I define the dollar and carry portfolios, in thespirit of the factor approach of Lustig et al. (2011). Dollar is an equal-weighted basket ofthe five currencies vs. USD. Note that this portfolio is short USD vs. these other curren-cies, not long. This sign convention helps make the portfolio definition consistent with theexercise in the main text. Carry is a long-short portfolio that is long the two currencieswith the smallest 1m forward premium and short the two countries with the largest forwardpremium. For almost all of the sample, this means long AUD and CAD and short JPY andEUR. To ensure a positive price, I add some of the risk-free USD investment to the Carryportfolio. This has no effect on the excess arbitrage χcarry− χRF , and hence no effect onresulting externality-mimicking portfolio.

Expected returns and variance-covariance matrices for these portfolios can be con-structed from the assumed expected returns and variance-covariance matrices of the in-dividual currencies (as described in the previous parts of this appendix section). Becausesome of the currencies in these portfolios are not explicitly modeled in the stress tests, Iimpute returns using the same procedure used in the main text for AUD.

B Additional Results

B.1 Predicted Arbitrage in Other Currencies

This sub-section presents the predicted vs. actual arbitrage using the risk-neutral externality-mimicking portfolio for three additional currencies: CAD, CHF, and SEK. These currencieshave enough OIS swap and options data available to make these predictions, although forboth CHF and SEK the sample size is reduced.

46

Figure 1: Actual vs. Predicted Excess Arbitrage in Canadian Dollar

-50

050

100

Exce

ss A

rb. (

bps)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

CAD Pred. CAD

Notes: This figure plots excess the actual CAD excess arbitrage χCAD− χ f , as defined in (5), along with the predicted value defined asin (10). The risk-neutral externality-mimicking portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, aswell as a risk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are inferred fromcurrency options. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMCmeeting.

Figure 2: Actual vs. Predicted Excess Arbitrage in Swiss Franc

010

020

030

0Ex

cess

Arb

. (bp

s)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

CHF Pred. CHF

Notes: This figure plots excess the actual CHF excess arbitrage χCHF − χ f , as in (5), along with the predicted value defined as in (10).The risk-neutral externality-mimicking portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, as well as arisk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are inferred from currencyoptions. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting.The two vertical lines indicate the beginning and end of a period during which CHF was pegged to EUR.

47

Figure 3: Actual vs. Predicted Excess Arbitrage in Swedish Krona

-100

010

020

0Ex

cess

Arb

. (bp

s)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

SEK Pred. SEK

Notes: This figure plots excess the actual SEK excess arbitrage χSEK − χ f , as in (5), along with the predicted value defined as in (10).The risk-neutral externality-mimicking portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, as well as arisk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are inferred from currencyoptions. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting.

B.2 Results using the Physical Measure

This sub-section presents results using the physical-measure externality mimicking portfo-lio, as described in appendix section A.2. Table 1 presents a version of the summary statis-tics table, with an estimated variance-covariance matrix in the place of an option-impliedvariance-covariance matrix. I then present the portfolio weights, predicted vs. actual GBParbitrage, and stress test returns for the physical measure. The results are similar to theircounterparts from the main text.

48

Table 1: Summary Statistics for Physical-Measure ArbitragePounds Euros Yen Aus.

DollarOIS-IOER

Arbitrage Mean (bps/year) 6.7 22.4 28.3 -15.4 -12.5Arbitrage SD (bps/year) 28.2 37.7 37.2 18.5 2.8

Empirical Vol. (bps/year) 812 852 922 1046 -Empirical Corr. with

Pound/USD1.00 0.58 0.18 0.46 -

Empirical Corr. with Euro/USD 0.58 1.00 0.32 0.46 -Empirical Corr. with Yen/USD 0.18 0.32 1.00 0.24 -

Empirical Corr. with SPDR 0.23 0.10 -0.34 0.37 -Empirical Corr. with HKM 0.26 0.17 -0.31 0.31 -

N 444 444 444 444 444Notes: This table presents summary statistics for the sample of days from Jan 4, 2011 to March 12, 2018 at least one month before anFOMC meeting. Arbitrage mean χa is defined using (5) for a claim to e..g. one euro in one month, priced in dollars today. The OIS-IOER arbitrage is the risk-free arbitrage, based on a claim to one dollar in one month. Arbitrage SD is the daily standard deviation of χa.Empirical Vol. and Empirical Corr. variables for currencies are the time-series means of the correlations between the currency returns,as estimated on a rolling basis by the methodology described in appendix section A.2.. Empirical Corr. with SPDR and Empirical Corr.with HKM are the time-series means of the correlations between the currency returns and the SPDR ETF (which tracks the S&P 500)and with the He et al. (2017) daily intermediary capital factor, as estimated on a rolling basis by the methodology described in appendixsection A.2.

Figure 4: Externality-Mimicking Portfolio Weights, Physical Measure

-4-2

02

46

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

JPY EURAUD

Notes: This figure plots excess the physical measure portfolio weight θA∗ ,ra , defined in (6), where A∗ contains the yen, euro, and

Australian dollar, as well as a risk-free asset, and the reference measure is the physical probability measure. The sample is all US tradingdays from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting. The variance-covariance matrix used inthe computation is estimated using the RiskMetrics methodology.

49

Table 2: Physical Returns in Stress ScenarioStressTestDate

N Mean(1Q,%)

S.D.(1Q,%)

P-value(1Q)

Mean(4Q,%)

S.D.(4Q,%)

P-value(4Q)

9/30/12 63 -1.0 0.3 0.9998 -2.8 0.8 0.99959/30/13 59 -1.2 0.4 0.9963 -2.7 0.7 0.99999/30/14 62 6.2 0.9 0.0000 11.7 1.4 0.0000

12/31/15 60 1.9 0.5 0.0002 4.0 0.9 0.00012/31/16 61 1.4 0.8 0.0384 4.9 1.1 0.000012/31/17 45 35.7 4.3 0.0000 31.8 3.3 0.0000

Notes: This table reports the mean and standard deviation of stress test scenario returns for the physical measure externality-mimickingportfolio portfolio. The physical EMP portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, as well asa risk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are computed using theRiskMetrics methodology. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before anFOMC meeting and within 180 days of a stress test date. Each of these dates is assigned to the nearest stress test date. N reports thenumber of dates assigned to each stress test, and P-Value reports the p-value associated with a one-sided hypothesis test that the meanreturn is negative. Results are reported for both one-quarter (1Q) and four-return (4Q) returns from the stress test scenarios.

Figure 5: Actual vs. Predicted Excess Arbitrage in Pounds, Physical Measure

050

100

150

200

Exce

ss A

rb. (

bps)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

GBP Pred. GBP

Notes: This figure plots excess the actual pound excess arbitrage χGBP− χ f , as defined in (5), along with the predicted value defined asin (10). The physical externality-mimicking portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, as wellas a risk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are estimated using theRiskMetrics methodolgy. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before anFOMC meeting.

50

B.3 Results including Equity Arbitrage

This sub-section presents results for a risk-neutral externality mimicking portfolio that in-corporates JPY, EUR, and AUD arbitrages as well as the SPY-based arbitrage described inappendix section A.3 and the risk-free rate arbitrage. The covariances and expected returnsunder the physical measure used in this section are described in appendix section A.4.

Table 3: Summary Statistics for Arbitrage including SPYPounds Euros Yen Aus.

DollarSPY OIS-

IOER

Arbitrage Mean (bps/year) 6.7 22.4 28.3 -15.4 5.4 -12.5Arbitrage SD (bps/year) 28.2 37.7 37.2 18.5 47.5 2.8

OI Vol. (bps/year) 859 950 977 1073 1566 -OI Corr. with Pound/USD 1.00 0.56 0.22 0.47 0.28 -OI Corr. with Euro/USD 0.56 1.00 0.31 0.51 0.11 -OI Corr. with Yen/USD 0.22 0.31 1.00 0.26 -0.29 -

Empirical Corr. with SPDR 0.23 0.10 -0.34 0.37 1.00 -Empirical Corr. with HKM 0.26 0.17 -0.31 0.31 0.66 -Implied Corr. with S&P 500 0.28 0.11 -0.29 0.50 1.00 -

N 444 444 444 444 312 444Notes: This table presents summary statistics for the sample of all US trading days from Jan 4, 2011 to March 12, 2018 at least onemonth before an FOMC meeting. The SPY statistics are restricted to dates at least one month before a SPDR ex-dividend date. Arbitragemean χa is defined using (5) for a claim to e..g. one euro in one month, priced in dollars today. The OIS-IOER arbitrage is the risk-freearbitrage, based on a claim to one dollar in one month. Arbitrage SD is the daily standard deviation of χa. OI Vol. and OI Corr. variablesfor currencies are the time-series mean of a daily series extracted from variance-covariance matrices implied by currency options, SPYoptions, and quanto options. Empirical Corr. with SPDR and Empirical Corr. with HKM are the time-series means of the correlationsbetween the currency returns and the SPDR ETF (which tracks the S&P 500) and with the He et al. (2017) daily intermediary capitalfactor, as estimated on a rolling basis by the methodology described in appendix section A.2. Implied Corr. with S&P 500 is based onthe time-series mean of the currency correlation with the S&P 500 extracted from quanto options and described in appendix section A.4.

51

Figure 6: Risk-Neutral Externality-Mimicking Portfolio Weights with SPY

-40

-20

020

40

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

JPY EURAUD SPDR

Notes: This figure plots excess the risk neutral measure portfolio weight θA∗,ra , defined in (6), where A∗ contains the yen, euro, and

Australian dollar, as well as a risk-free asset and the SPDR ETF, and the reference measure is the intermediaries’ risk-neutral measure.The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting and onemonth before a SPDR ex-dividend date. The variance-covariance matrix used in the computation is inferred from currency options, SPYoptions, and quanto options.

52

Table 4: Risk-Neutral EMP Expected Returns with SPYN Mean (bps) Standard Deviation (bps) Test P-Value

Full Sample 312 -111 43.3 ≥ 0 0.0053Quarter-Ends 93 -425 125.9 ≥ 0 0.0005

Year-Ends 23 -1495 439.9 ≥ 0 0.0013QE - Full -447.0 = 0 0.0000YE - QE -1421 = 0 0.0000

Notes: This table reports the excess expected return under the physical measure of the risk-neutral externality mimicking portfolio (5)defined in (6), under the assumption that currencies follow a random walk. The set A∗ is JPY, EUR, and AUD, a risk-free rate asset, andthe SPDR ETF. The full sample is all US trading days from Jan 4, 2011 to March 12, 2018 at least one month before an FOMC meetingand at least one month before a SPDR ex-dividend date. The quarter-end and year-end sub-samples are restricted to days on which aquarter- or year-end occurs between the spot FX settlement date and the one-month FX settlement date. The QE-NQE and YE-NYE QEreport the mean difference between quarter-end vs. non-quarter-end dates and year-end vs. non-year-end quarter-end. Test indicates thehypothesis about the mean being tested, and P-Value reports the associated p-value.

Figure 7: Actual vs. Predicted Excess Arbitrage in Pounds, Risk-Neutral Measure withSPY

010

020

030

0Ex

cess

Arb

. (bp

s)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

GBP Pred. GBP

Notes: This figure plots excess the actual pound excess arbitrage χGBP− χ f , as defined in (5), along with the predicted value defined asin (10). The risk-neutral externality-mimicking portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, aswell as a risk-free asset and the SPDR ETF. The variance-covariance matrix used in the computation and the covariances with the poundare inferred from currency options, SPY options, and quanto options. The sample is all US trading days from Jan 4, 2011 to March 12,2018 that are at least one month before an FOMC meeting and one month before a SPDR ex-dividend date.

B.4 Results with Dollar and Carry Portfolios

This sub-section presents for a risk-neutral externality mimicking portfolio that incorpo-rates “Carry” and “Dollar” arbitrages, as well as a risk-free rate arbitrage. Carry and Dollar(defined in appendix section A.5) are portfolios of currency trades. Carry is long two

53

Table 5: Risk-Neutral Portfolio with SPY, Returns in Stress ScenarioStressTestDate

N Mean(1Q,%)

S.D.(1Q,%)

P-value(1Q)

Mean(4Q,%)

S.D.(4Q,%)

P-value(4Q)

9/30/12 28 -1.9 0.6 0.9989 -6.0 2.1 0.99569/30/13 52 -0.7 1.5 0.6786 -17.7 4.3 0.99999/30/14 54 2.5 1.3 0.0304 -14.3 2.0 1.0000

12/31/15 53 0.1 0.7 0.4302 0.4 1.6 0.414212/31/16 55 -23.4 1.6 1.0000 -30.2 2.1 1.000012/31/17 41 0.0 5.4 0.5013 -16.7 4.8 0.9994

Notes: This table reports the mean and standard deviation of stress test scenario returns for the risk-neutral externality-mimickingportfolio portfolio. The risk-neutral EMP portfolio is constructed with an A∗ that contains the yen, euro, and Australian dollar, and theSPDR ETF, as well as a risk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound arecomputed from currency options, SPY options, and quanto options. The sample is all US trading days from Jan 4, 2011 to March 12,2018 that are at least one month before an FOMC meeting, one month before a SPDR ex-dividend date, and within 180 days of a stresstest date. Each of these dates is assigned to the nearest stress test date. N reports the number of dates assigned to each stress test, andP-Value reports the p-value associated with a one-sided hypothesis test that the mean return is negative. Results are reported for bothone-quarter (1Q) and four-return (4Q) returns from the stress test scenarios.

low-forward-premium currencies (AUD and CAD for most of the sample) and short thearbitrage in two high-forward-premium currencies (JPY and EUR for most of the sample).Dollar is an equally weighted portfolio of currencies vs. USD. Note when interpreting theresults below that Dollar is short USD, not long USD.

Table 6: Summary Statistics for Carry and Dollar ArbitrageCarry Dollar OIS-

IOER

Arbitrage Mean (bps/year) -44.1 8.6 -12.5Arbitrage SD (bps/year) 31.4 23.3 2.8

OI Vol. (bps/year) 835 680 -OI Corr. with Carry 1.00 0.15 -OI Corr. with Dollar 0.15 1.00 -

Empirical Corr. with SPDR 0.56 0.21 -Empirical Corr. with HKM 0.46 0.21 -Implied Corr. with S&P 500 0.68 0.30 -

N 444 444 444Notes: This table presents summary statistics for the sample of all US trading days from Jan 4, 2011 to March 12, 2018 at least onemonth before an FOMC meeting. Arbitrage mean χa is defined using (5) for a claim to e..g. one euro in one month, priced in dollarstoday. The OIS-IOER arbitrage is the risk-free arbitrage, based on a claim to one dollar in one month. Arbitrage SD is the daily standarddeviation of χa. The OI Vol. and OI Corr. variables are the time-series mean of a daily series extracted from variance-covariance matricesimplied by currency options. Empirical Corr. with SPDR and Empirical Corr. with HKM are the time-series means of the correlationsbetween the currency returns and the SPDR ETF (which tracks the S&P 500) and with the He et al. (2017) daily intermediary capitalfactor, as estimated on a rolling basis by the methodology described in appendix section A.2. Implied Corr. with S&P 500 is based onthe time-series mean of the currency correlation with the S&P 500 extracted from quanto options and described in appendix section A.4.

54

Figure 8: Risk-Neutral Externality-Mimicking Portfolio Weights with Carry and Dollar

-10

-50

510

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

Dollar Carry

Notes: This figure plots excess the risk neutral measure portfolio weight θA∗,ra , defined in (6), where A∗ contains the Carry and Dollar

portfolios, as well as a risk-free asset, and the reference measure is the intermediaries’ risk-neutral measure. The sample is all US tradingdays from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meeting. The variance-covariance matrix used inthe computation is inferred from currency options.

Figure 9: Actual vs. Predicted Excess Arbitrage in Pounds, Risk-Neutral Measure withCarry and Dollar

050

100

150

200

Exce

ss A

rb. (

bps)

1-Jan-11 1-Jan-12 1-Jan-13 1-Jan-14 1-Jan-15 1-Jan-16 1-Jan-17 1-Jan-18Date

GBP Pred. GBP

Notes: This figure plots excess the actual pound excess arbitrage χGBP− χ f , as defined in (5), along with the predicted value definedas in (10). The risk-neutral externality-mimicking portfolio is constructed with an A∗ that contains the Carry and Dollar portfolios, aswell as a risk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are inferred fromcurrency options. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMCmeeting.

55

Table 7: Risk-Neutral EMP Expected Returns with Carry and DollarN Mean (bps) Standard Deviation (bps) Test P-Value

Full Sample 444 -151 20.7 ≥ 0 0.0000Quarter-Ends 155 -315 56.1 ≥ 0 0.0000

Year-Ends 46 -763 169.9 ≥ 0 0.0000QE - Full -252 = 0 0.0000YE - QE -638 = 0 0.0000

Notes: This table reports the excess expected return under the physical measure of the risk-neutral externality mimicking portfolio (5)defined in (6), under the assumption that currencies follow a random walk. The set A∗ is the Carry and Dollar portfolios and a risk-freerate asset. The full sample is all US trading days from Jan 4, 2011 to March 12, 2018 at least one month before an FOMC meeting. Thequarter-end and year-end sub-samples are restricted to days on which a quarter- or year-end occurs between the spot FX settlement dateand the one-month FX settlement date. The QE-NQE and YE-NYE QE report the mean difference between quarter-end vs. non-quarter-end dates and year-end vs. non-year-end quarter-end. Test indicates the hypothesis about the mean being tested, and P-Value reports theassociated p-value.

Table 8: Stress Test “Severely Adverse” ScenariosCarry* *Imputed Dollar* *Imputed

Stress TestDate

One-QuarterReturn

Four-QuarterReturn

One-QuarterReturn

Four-QuarterReturn

9/30/12 -7.4 -16.5 -4.1 -11.69/30/13 -8.9 -13.1 -10.5 -16.59/30/14 -2.1 -10.5 -3.2 -7.7

12/31/15 -3.8 -10.5 -4.1 -8.812/31/16 -11.3 -18.2 -7.9 -11.012/31/17 -23.6 -25.9 -8.0 -14.2

Notes: This table reports the imputed returns of the Carry and Dollar portfolios during the first one or four quarters of the associated“Severely Adverse Scenario” from that year’s stress test, using the imputation method described in the text. Stress Test Date lists thedate on which that year’s scenario begins.

C General Equilibrium with Intermediaries

This appendix section describes a general equilibrium endowment economy with incom-plete markets (GEI) that features a distinction between households and intermediaries. It isan endowment economy30 version of the economy studied by Farhi and Werning (2016),with a specific asset structure that I will introduce below.

The economy has a set of future states, S1, and an initial state, s0. Let S = S1 ∪{s0}denote the set of all states, and let Js be the set of goods in each state s∈ S. The government

30For simplicity, I also assume flexible prices and a simpler form of constraints on government transfers.

56

Table 9: Risk-Neutral Portfolio with SPY, Returns in Stress ScenarioStressTestDate

N Mean(1Q,%)

S.D.(1Q,%)

P-value(1Q)

Mean(4Q,%)

S.D.(4Q,%)

P-value(4Q)

9/30/12 63 0.9 0.4 0.0144 0.4 0.9 0.32349/30/13 59 -1.8 0.4 1.0000 -2.9 0.6 1.00009/30/14 62 0.1 0.1 0.1707 3.9 0.6 0.0000

12/31/15 60 -0.5 0.1 0.9992 0.1 0.3 0.396512/31/16 61 3.7 0.6 0.0000 7.9 1.0 0.000012/31/17 45 27.5 4.4 0.0000 20.6 3.0 0.0000

Notes: This table reports the mean and standard deviation of stress test scenario returns for the risk-neutral externality-mimickingportfolio portfolio. The risk-neutral EMP portfolio is constructed with an A∗ that contains the Carry and Dollar portfolios, as well as arisk-free asset. The variance-covariance matrix used in the computation and the covariances with the pound are computed from currencyoptions. The sample is all US trading days from Jan 4, 2011 to March 12, 2018 that are at least one month before an FOMC meetingand within 180 days of a stress test date. Each of these dates is assigned to the nearest stress test date. N reports the number of datesassigned to each stress test, and P-Value reports the p-value associated with a one-sided hypothesis test that the mean return is negative.Results are reported for both one-quarter (1Q) and four-return (4Q) returns from the stress test scenarios.

can transfer income between agents in the initial state, s0, but not any other state, and thesetransfers must sum to zero. The goods available in each state are denoted by the set Js.31

Households h ∈H maximize expected utility,

∑s∈S

Uh({Xhj,s} j∈Js;s),

where Uh({Xhj,s} j∈Js;s) is the utility of household h in state s, inclusive of the household’s

rate of time preference and the probability the household places on state s. I will assumenon-satiation for at least one good in each state, implying that each household places non-zero probability on each state in S1, and that the utility functions are differentiable. Notethat I will refer to each h as “a household,” and assume price-taking; nothing would changeif we thought of each h as representing a mass of identical households. In each state s ∈ S,household h ∈ H has an endowment of good j ∈ Js equal to Y h

j,s. In state s0, the householdmight also receive a transfer T h.

The set of securities available in the economy, A, has securities which offer payoffsZa,s({Pj,s} j∈Js) for security a ∈ A in state s ∈ S. Note that the payoff may be a function ofgoods prices, which are endogenous, and I will assume that the payoffs are homogenousof degree one in prices, so that it is without loss of generality to fix the price for one good(the numeraire) in each state. Let Dh

a denote the quantity of security a purchased or sold by

31Separating contingent commodities into states and goods available in each state will give meaning to thefinancial structure described below.

57

household h, and let Qa be the “ex-dividend” price at time zero (i.e. under the conventionthat Za,s0 = 0).

In state s, the household’s income used for consumption (i.e. consumption expenditure)is

Ihs =

∑ j∈Js Pj,sY hj,s +∑a∈A Dh

aZa,s({Pj,s} j∈Js) if s 6= s0,

T h +∑ j∈Js Pj,sY hj,s−∑a∈A Dh

aQa if s = s0.(15)

That is, for all states except the initial state, consumption expenditure is equal to income,and is the value of the household endowment plus the payoffs of the household’s assetholdings. In the initial state, income used for consumption is the value of the endowmentplus any transfers, less the purchase price of the household’s asset holdings. In all states,there is a budget constraint for consumption in the state,

∑j∈Js

Pj,sXhj,s ≤ Ih

s .

Household asset allocations are constrained by limited participation constraints or otherkinds of limits. The constraints on households’ asset positions are summarized by

Φh({Dh

a}a∈A)≤~0, (16)

where Φh is a vector-valued function, convex in Dh. Limited participation is the key formof constraint I am attempting to capture with these Φ functions, but it is not the only kindof constraint that fits into this framework.

Having defined expenditure and prices, I define the standard indirect utility function ineach state,

V h(Ihs ,{Pj,s} j∈Js;s) = max

{Xhj,s∈R+} j∈Js

Uh({Xhj,s} j∈Js;s)

subject to

∑j∈Js

Pj,sXhj,s ≤ Ih

s .

Using these indirect utility functions, we can write the portfolio choice problem as

max{Dh

a∈R}a∈A∑s∈S

V h(Ihs ,{Pj,s} j∈Js;s)

subject to the budget constraints that define income (equation (15)) and the constraints onasset allocation (equation (16)). Note that I fold the household’s discounting and subjective

58

probability assessments into the state-dependent direct and indirect utility functions. In acompetitive equilibrium (that is, taking asset prices Q and goods prices P as given, definedbelow), this is the problem households solve when choosing their asset allocation.

I will call the other type of agents in the economy intermediaries, and use i ∈ I todenote a particular intermediary. Intermediaries are like households (in the sense that allof the notation above applies, with some i ∈ I in the place of an h ∈H ), except that theyface different constraints on their portfolio choices, and they share a common set of util-ity functions. In particular, households are constrained to trade only with intermediaries,but intermediaries can trade with both households and other intermediaries. Intermediariesalso share a common set of homothetic utility functions; that is, U i(·;s) = U i′(·;s) for alls ∈ S and i, i′ ∈ I, and U i(·;s) is homothetic for all s ∈ S. The assumption of a common,homothetic (but state-dependent) utility function ensures that redistributing wealth acrossintermediaries does not influence relative prices in goods markets, and hence is not partic-ularly useful from a planner’s perspective.

The constraint that households can trade only with intermediaries, but not each other,can be implemented using this notation in the following way. The set of assets, A, is asuperset of the union of disjoint sets {Ah}h∈H , denoting trades with household h. Fora given household h, the function Φh implements the requirement that, for all a ∈ A \Ah, Dh

a = 0. To be precise, if a ∈ A \ Ah and Dha 6= 0, then there exists an element of

Φh(Dh) strictly greater than zero. The set of assets also includes assets that cannot betraded by any household. Define AI = A \ (∪h∈HAh) as the set of securities tradable onlyby intermediaries.

For arbitrage between the asset market AI and the asset market Ah to exist, intermedi-aries must face financial constraints. The approach of this paper, in contrast to the much ofthe existing literature on arbitrages, is to assume that the constraints faced by intermediariesare induced by government policy. That is, I will assume that some of the Φi functions arethe government’s policy instrument; in contrast, the Φh functions are assumed to be exoge-nous. The assumption that the Φh cannot be augmented by regulation does not constrainthe social planner. Because all trades are intermediated, and the government can constrainintermediaries, the government can effectively control all of the trades in the economy, andtherefore implement any allocation that could be implemented with agent-specific taxes (asin Farhi and Werning (2016)).

The notion of equilibrium is standard:

Definition 2. An equilibrium is a collection of consumptions Xhj,s and X i

j,s, goods prices

59

Pj,s, asset positions Dha and Di

a, transfers T h and T i, and asset prices Qa such that:

1. Households and intermediaries maximize their utility over consumption and assetpositions, given goods prices and asset prices, respecting the constraints that con-sumption be weakly positive and the constraints on their asset positions,

2. Goods markets clear: for all s ∈ S and j ∈ Js,

∑h∈H

(Xhj,s−Y h

j,s) = ∑i∈I

(X ij,s−Y i

j,s),

3. Asset markets clear: for all a ∈ A,

∑h∈H

Dha + ∑

i∈IDi

a = 0, (17)

4. The government’s budget constraint balances,

∑h∈H

T h + ∑i∈I

T i = 0. (18)

The definition of equilibrium presumes price-taking by households and intermediaries.Absent government constraints, each household h can trade with every intermediary, andthe price of asset a ∈ Ah will be pinned down by competition between intermediaries. Theequilibrium definition supposes that this will continue to be the case, even if the govern-ment places asymmetric constraints on intermediaries– for example, by granting a singleintermediary a monopoly over trades with a particular household. In this case, it is as if thehousehold had all of the bargaining power. Such a policy is unlikely to optimal, and willnever be the unique optimum.

I next describe a planner’s problem for this economy. I assume that the planner is unableto redistribute resources ex-post (doing so would allow the planner to circumvent limitedparticipation). Instead, in the spirit of Geanakoplos and Polemarchakis (1986), I will allowthe planner to trade in asset markets on behalf of agents, trading for each agent only inmarkets she can participate in, to maximize a weighted sum of the household’s indirectutility functions, subject to an ex-ante participation constraint for intermediaries.

60

Definition 3. The constrained planner’s problem is

max{Dh

a∈R}a∈A,h∈H ,{Dia∈R}a∈A,i∈I ,{Pj,s∈R+}s∈S, j∈Js ,{T i∈R}i∈I ,{T h∈R}h∈H

∑h∈H

λh∑s∈S

V h(Ihs ,{Pj,s} j∈Js;s),

subject to the intermediaries’ ex-ante participation constraint,

∑s∈S

V i(Iis,{Pj,s} j∈Js;s)≥ V i, ∀i ∈I ,

household’s limited participation constraints,

Φh({Dh

a}a∈A)≤~0, ∀h ∈H ,

intermediaries exogenous portfolios constraints,

Φi,exog({Di

a}a∈A)≤~0, ∀i ∈I ,

the definition of incomes Ihs and Ii

s (15), market clearing in assets (17), the government’sbudget constraint (18), and goods market clearing for each state s ∈ S and good j ∈ Js,

∑h∈H

(Xhj,s(I

hs ,{Pj′,s} j′∈Js)−Y h

j,s) = ∑i∈I

(X ij,s(I

is,{Pj′,s} j′∈Js)−Y i

j,s).

Here, Xhj,s(I

hs ,{Pj′,s} j′∈Js) denotes the demand function for good j by agent h in state s.

Note that the definition of the social planner’s problem includes only exogenous constraintson intermediaries’ trades (Φi,exog). Note also that I have chosen to write the planner’sproblem as maximizing household utility subject to an ex-ante participation constraint forintermediaries, because this fits best into the example from the text; nothing would changeif instead the planner maximized a weighted combination of all agents’ utilities. Next, Idefine constrained (in)efficiency:

Definition 4. A competitive equilibrium is constrained efficient if there exists Pareto weightsλ h and outside options V i such that the allocation of assets and goods in the competitiveequilibrium coincides with the solution to the planner’s problem. Otherwise, the competi-

61

tive equilibrium is constrained inefficient.

Lastly, I will define macro-prudential regulation. Regulation, in this framework, are ad-ditional convex functions Φi,reg({Di

a}a∈A) such that the intermediaries face the constraintΦi(·) =

[Φi,reg Φi,exog

]. As discussed in the text, for notational simplicity I assume that

these functions depend only on asset quantities and not on asset prices. Because the equi-librium gradient on these constraints is the only quantity that matters for equilibrium, thisassumption is without loss of generality.

D Proofs

D.1 Proof of Proposition 1

Consider a perturbation to the solution of the planner’s problem in which the planner re-allocations an asset a ∈ A from the intermediary i ∈I to the household h ∈H . If such aperturbation is feasible, we must have (by differentiability)

−λh0V h0

I,s0 ∑s∈S

∑j∈Js

µ j,s[XhI, j,s−X i

I, j,s]Za,s({P∗j,s} j∈Js) = ∑s∈S

(λ iV iI,s−λ

hV hI,s)Za,s({P∗j,s} j∈Js),

where λ i is the multiplier on intermediary i’s participation constraint and µ j,sλh0V h0

I,s0is

the multiplier on the goods-market clearing constraint. Note that I have normalized themultiplier to units of dollars, rather than social utility, using the marginal utility of anarbitrary household h0 ∈H .

By the definitions of the wedges, this is

λh0V h0

I,s0 ∑s∈S

∑j∈Js

(πrs τ

rj,s−

1|Js| ∑

j′∈Js

µ j′,s

P∗j′,s)P∗j,s[X

hI, j,s−X i

I, j,s]Za,s({P∗j,s} j∈Js) =

∑s∈S

(λ iV iI,s−λ

hV hI,s)Za,s({P∗j,s} j∈Js).

By the identity

∑j∈Js

XhI, j,sP

∗j,s = 1,

62

this simplifies using the definition of the externalities to

λh0V h0

I,s0 ∑s∈S

πrs ∆

h,i,rs Za,s({P∗j,s} j∈Js) = ∑

s∈S(λ iV i

I,s−λhV h

I,s)Za,s({P∗j,s} j∈Js).

By the first-order condition for the transfer between h0 and h,

λh0V h0

I,s0= λ

hV hI,s0

and likewise λ h0V h0I,s0

= λ iV iI,s0

, and therefore

∑s∈S

πrs ∆

h,i,rs Za,s({P∗j,s} j∈Js) = ∑

s∈S(V i

I,s

V is0

−V h

I,s

V hI,s0

)Za,s({P∗j,s} j∈Js).

D.2 Proof of Proposition 2

If the asset a∈ Ah is tradable by household h in the solution to the social planner’s problem,and the exogenous portfolio constraints do not bind, then under assumption 1, (2) musthold. If the asset a′ ∈ AI is tradable by i∗(a′), then (3) holds.

Applying proposition 1 to the asset a′ and the intermediaries i and i∗, and differencingthe equations, it follows that (3) holds for any intermediary i ∈ I . The result followsimmediately.

D.3 Proof of Proposition 3

The first two of these claims are simply the definition of the least-squares projection. Thatis

(∑s∈S

πrs RI

s(RIs)

T )−1(∑s∈S

πrs RI

s∆h,i,rs ) = (∑

s∈Sπ

rs RI

s(RIs)

T )−1

[χa

χ f

]

=

[(ΣA∗,I,r +µA∗,I,r(µA∗,I,r)T ) RI

f µA∗,I,r

RIf (µ

A∗,I,r)T (RIf )

2

]−1[χa

χ f

]

=

[(ΣA∗,I,r)−1 −(RI

f )−1(ΣA∗,I,r)−1µA∗,I,r

−(RIf )−1(µA∗,I,r)T (ΣA∗,I,r)−1 (RI

f )−2(1+(µA∗,I,r)T (ΣA∗,I,r)−1µA∗,I,r)

][χa

χ f

]

=

[θ A∗,r

θA∗,rf

].

63

where RIs is the vector form of {RI

a,s}a∈A∗ . The mean externalities, by construction, are

∑s∈S

πrs ∆

h,i,rs =

χ f

RIf,

and the claim about the variance follows.I prove the third claim below. Define the Lagrangian, using Ra,s(1− χa) = RI

a,s, as

minm∈R|S1|

maxθ∈R|A∗|

12 ∑

s∈S1

πrs (ms−mI,r

s )2−

∑a∈A∗

θa[(1− χa)− ∑s∈S1

πrs msRI

a,s].

The FOC for m isπ

rs (ms−mI,r

s )+πrs ∑

a∈A∗θaRI

a,s = 0.

Plugging this back into the problem,

maxθ∈R|A∗|

12 ∑

s∈S1

πrs ( ∑

a∈A∗θaRI

a,s)2−

∑a∈A∗

θa[ ∑s∈S1

πrs ((1− χa)−mI,r

s RIa,s +RI

a,s ∑a′∈A∗

θa′RIa′,s)],

which simplifies, using the assumption that mI,rs prices the replicating portfolios (∑s∈S1 πr

s mI,rs RI

a′,s =

1), to

maxθ∈R|A∗|

∑a∈A∗

θaχa−12 ∑

s∈S1

πrs ( ∑

a∈A∗θaRI

a,s)2. (19)

It follows immediately that θ ∗, the solution to this problem, is the projection of χa ontothe space of returns, and hence is the externality-mimicking portfolio. Observe, by con-struction, that the mean return of the externality-mimicking portfolio is χ f

RIf= ( 1

RIf− 1

R f).

Therefore,mr

s = mI,rs − ∑

a∈A∗θ

A∗,ra RI

a,s

is the household SDF that minimizes the variance of the difference between SDFs subjectto the constraint that it price the arbitrage.

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Lastly, consider the fourth claim: the externality mimicking portfolio maximizes theSharpe ratio due to arbitrage,

SA∗,I,r(θ) = SA∗,r(θ(θ))−SA∗,I,r(θ)

=

θ(θ)T µA∗,r

R f−∑a∈A∗ θa(θ)

(θ(θ)T ΣA∗,I,rθ(θ))12−

θ T µA∗,I,r

RIf−∑a∈A∗ θa

(θ T ΣA∗,I,rθ)12

=∑a∈A∗ θa(χa− χ f

µA∗,I,r

RIf)

(θ T ΣA∗,I,rθ)12

.

Suppose not; let θ be some portfolio with a higher ratio. Note that the Sharpe ratio due toarbitrage is homogenous of degree zero. Moreover, mixing in some amount of the risk-freerate does not change this ratio,

∑a∈A∗ θa(χa− µA∗,I,r

RIf

χ f )

(θ T ΣA∗,I,rθ)12

=∑a∈A∗(θa +1(a = f ))(χa− µA∗,I,r

RIf

χ f )

((θa +1(a = f ))T ΣA∗,I,r(θa +1(a = f ))T )12.

It is therefore without loss of generality to suppose that

∑a∈A∗

θaµA∗,I,r = ∑

a∈A∗θaµ

A∗,I,r

and

∑a∈A∗

θaχa = ∑a∈A∗

θaχa.

But in this case, θ must achieve a higher payoff than the externality-mimicking portfolioin (19), a contradiction.

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