Does Incomplete Spanning in International Financial Markets Help to Explain Exchange Rates?

The paper asks whether departures from complete spanning in international securities trading can help address three exchange rate puzzles at once, and the complete-spanning benchmark is restrictive because it ties the exchange rate change to the two discount factors state by state with no slack at all. Beginning with Lucas (1982), “many models in international economics assume that the menu of contingent claims spans all states of the world,” and under complete spanning “the rate of appreciation of the real exchange rate equals the difference between the marginal utility growth rates of the foreign and domestic stand-in investor in each state of the world: ∆s = m* − m” (p. 1). That identity implies the foreign currency appreciates in bad times for foreign investors and the home currency depreciates in good times for domestic investors – an implication the authors call “sometimes viewed as undesirable and counterintuitive,” illustrating it with the Argentine peso’s 2002 depreciation on the one hand and the yen’s appreciation after the 2014 Japanese tsunami on the other (Section 1.1). Since “everyone agrees, however, that, in reality, financial markets certainly do not span all possible states of the world” (p. 1), departures from complete spanning introduce a stochastic wedge, and the question is whether that wedge is a useful degree of freedom.

The volatility puzzle of Brandt, Cochrane, and Santa-Clara (2006), the cyclicality puzzle of Kollmann (1991) and Backus and Smith (1993), and the forward premium puzzle of Fama (1984), measured on quarterly data for 15 developed countries over 1973.IV-2014.IV. The country set is Australia, Belgium, Canada, Denmark, France, Germany, Italy, Japan, Netherlands, New Zealand, Norway, Sweden, Switzerland, the U.K. and the U.S., with all exchange rates defined against the U.S. dollar (Table 1 notes). Panel A of Table 1 reports a cross-country mean annualized nominal exchange rate volatility of 11.21 percent with a bootstrapped standard error of 0.44 percent and a cross-country standard deviation of 1.57 percent; the real exchange rate figure is essentially the same at 11.12 percent. The correlation of U.S. and foreign consumption growth ranges from 0.02 to 0.35 with a cross-country mean of 0.17. Panel B reports a cross-country mean correlation between real exchange rate changes and relative consumption growth of −0.07 (standard error 0.05) and a Backus-Smith slope of −0.01, both of which the text describes as “not statistically different from zero.” Panel C reports an average carry trade excess return of 4.42 percent annualized (standard error 1.36) with a time-series standard deviation of 8.73 percent, built by sorting the countries into four portfolios on short-term nominal interest rates and going long the highest-rate portfolio against the lowest; the text summarizes this as “the average carry trade excess return is 4.4%, implying a Sharpe ratio of 0.5.” The cross-country mean UIP slope is −0.26.

The results rest on (1) the existence of a stochastic discount factor in the space of traded assets in one country and (2) the existence of a domestic and a foreign risk-free rate in which the other country’s investors can invest; the setup is an upper bound because enforcing only those two risk-free-bond Euler equations imposes the fewest possible restrictions on the wedge (Section 1, including footnote 1). The authors are explicit that these assumptions “are reasonable but not trivial” – a unique traded-asset discount factor requires the law of one price plus free portfolio formation, and both can fail under short-selling constraints, as can the existence of risk-free assets (fn. 1). On the upper-bound logic: “Naturally, if foreign investors can invest in other domestic assets, this will give rise to additional restrictions. By ignoring these additional restrictions, we are giving incomplete spanning the best shot at producing promising results that cannot be replicated by a simple reduced form model in complete markets. Our results thus provide an upper bound on the effects of incomplete spanning” (Section 1.2). Importantly, they hold the projection of the discount factor onto domestically traded assets fixed: “We do not study how market incompleteness changes the projection of the stochastic discount factor on the space of domestically traded assets itself. Such changes are interesting, and our paper therefore does not suggest that models with incomplete markets are uninteresting or not useful” (p. 5).

They imply two covariance restrictions that force the wedge to be procyclical, and Corollary 1 then shows that the wedge’s variance subtracts one-for-one from exchange rate variance. Proposition 1 states that under joint conditional lognormality of the discount factors and the wedge, covar_t(m*, eta) = −mu_eta − (1/2)var_t(eta) and covar_t(m, eta) = −mu_eta + (1/2)var_t(eta). “Since the wedges are necessarily pro-cyclical, they offset the effect of m and m*, and thus reduce the overall volatility of the exchange rate” (Section 2.1). Corollary 1 gives the exact accounting: var_t(∆s) = var_t(m) + var_t(m*) − 2cov_t(m, m*) − var_t(eta). The authors describe the economics of this as the wedges acting “as a reduction of the representative agent’s risk aversion in currency markets, thus solving the excess volatility puzzle. But these FX wedges also shrink currency risk premia and move the model towards the uncovered interest rate parity” (p. 4).

About as volatile as the maximum Sharpe ratio in the economy – 49 percent per annum under the paper’s deliberately generous parameterization, and at least 70 percent if the two countries’ discount factors are uncorrelated. Inverting Corollary 1 at a target volatility of 11 percent, “starting from a maximum Sharpe ratio of 0.50 in both countries … and a correlation across stochastic discount factors of 0.50 … the wedge must have a standard deviation of 49% per annum” (Section 2.1). The authors flag that the 0.50 discount-factor correlation is generous rather than realistic – “Such a correlation across stochastic discount factors is optimistic. In the data, the correlation of consumption growth varies between 0.02 and 0.35” – and that “if we decrease the correlation of the SDFs from 0.5 to zero, then we need an even more volatile wedge: std_t(eta) is at least equal to 70%.” They keep the 0.50 correlation throughout “to stack the deck in favor of the incomplete spanning wedge.” Figure 1 shows that only for wedge volatilities near 50 percent does exchange rate volatility fall into the plausible range, and that the wedge’s drift has no bearing on volatility. The conclusion states the point plainly: “the quantity of unspanned risk needed in currency markets is of the same size as the maximum Sharpe ratio” (Section 4).

It shrinks them sharply: the complete-markets risk premium of roughly 12 percent per year under these parameters falls below 6 percent, and to essentially zero or negative when the wedge’s drift is zero or negative. Corollary 2 gives the level risk premium as var_t(m) − covar_t(m*, m) − (1/2)var_t(eta) + mu_eta, so the wedge’s variance enters negatively. The paper’s parameters “imply a large complete markets currency risk premium of 12% on average per year, above the actual return on currency carry trades in developed countries,” which the authors note “stacks the deck in favor of incomplete markets” by leaving room for a large decline (Section 2.2). Figure 4 then shows that “the wedge introduced by the markets incompleteness reduces the risk premium to less than 6%. The risk premium is in line with its empirical value only for large drift parameters. For zero or negative drifts, the currency risk premium is essentially zero or turn negative” (Section 2.4). Three corner cases – a mean-zero wedge, a wedge-risk-neutral foreign investor, and a wedge-risk-neutral domestic investor – are each said to be “clearly rejected by the data on currency risk premia” (Section 2.2).

Only in a limited way, and Corollary 3 shows it cannot touch two of the three things the puzzle is about: the sign of the covariance is unchanged and the Backus-Smith regression slope remains exactly one, its complete-markets value. Corollary 3 establishes that covar_t(m* − m, ∆s) = var_t(∆s) ≥ 0, and that beta_Backus-Smith = covar_t(m* − m, ∆s)/var_t(∆s) = 1. The authors draw out three consequences: incomplete spanning “does not change the sign of the covariance between exchange rate changes and the difference in log stochastic discount factors spanned by asset markets”; it “does not change the slope coefficient … it is equal to one, as in complete markets”; and it “decreases the correlation between exchange rates and the stochastic discount factor … only at the cost of a lower Sharpe ratio on the currency risk premium” (Section 2.3). They add the important qualification that the relevant “good times” are defined by the marginal utility spanned by asset markets, “whereas the total marginal utility of the investor may be high or low.”

Because the drift of the wedge that would preserve a realistic risk premium is precisely the drift that makes the exchange rate’s correlation with discount factors and consumption growth worse than under complete markets. Section 2.4 fixes exchange rate volatility at 11 percent and traces out the admissible drift-volatility pairs. The upper panels of Figure 4 show the risk premium and currency Sharpe ratio rising with the drift; the lower panels show that “large values of the drifts imply exchange rate correlations with log stochastic discount factors (and consumption growth in the case of CRRA preferences) that are in absolute values even larger than their complete markets counterpart. The wedge exacerbates these features of the complete markets models that incomplete spanning is supposed to address.” The same trade-off appears in the simpler examples: at the “sweet spot” drift of mu_eta = +var_t(eta)/8, where risk premia look reasonable, “the incomplete market model is even less attractive than its complete market counterpart,” and with power utility the correlation between exchange rate changes and consumption growth “exceeds 0.7 in the relevant region of the parameter space” at a risk aversion coefficient of 10 (Section 2.3).

No – the authors relax lognormality using entropy and co-entropy, and the volatility-versus-risk-premium trade-off survives; but they are careful to say the full three-way trilemma cannot be formally proved in the general non-Gaussian preference-free case. Conditional entropy is defined as L_t(X) = log E_t(X) − E_t(log X), which equals half the variance under lognormality and otherwise “measures all higher order cumulants” (Section 3.1.1). Proposition 2 restates the wedge restrictions in these terms, and Corollaries 4 and 5 deliver the key link: the change in the currency risk premium equals the change in exchange rate entropy plus the wedge’s drift, so with a zero drift “a decrease in the entropy of the exchange rate leads to a commensurate decrease in the foreign currency risk premium.” On the cyclicality leg the authors state the limitation explicitly: “In a preference-free setting, we are not able to bound the co-entropy of exchange rates and stochastic discount factors. As a result, the trilemma that we highlight in the lognormal case cannot be formally expressed here.” Their honest summary of the remaining loophole is that it would require “a large non-stationary component in the exchange rate changes through a large drift in the stochastic FX wedge,” and that “while one cannot rule out the existence of a non-Gaussian model that would match the three exchange rate puzzles simultaneously thanks to incomplete spanning, we do not know of such a model” (fn. 2).

In each of them the wedge cannot produce a plausible exchange rate volatility and a significant currency risk premium at the same time. In a Barro-Rietz disaster version of the Merton (1976) model, calibrated following Backus, Chernov, and Martin (2011) to match the international evidence in Nakamura, Steinsson, Barro, and Ursua (2013) – risk aversion 5.19, consumption-growth mean 2.3 percent and standard deviation 1 percent, jump intensity 1.7 percent, mean jump size −38 percent, jump size volatility 25 percent, jumps common across countries but jump sizes uncorrelated – “there is no wedge that can simultaneously deliver a reasonable exchange rate volatility and a significant risk premium. When the variance of the jumps in the wedge reaches its maximum, the exchange rate volatility is still close to 20% and the currency risk premium is less than 2%” (Section 3.1.2, Figure 5). The authors report that a more conservative disaster calibration with smaller disasters can match exchange rate volatility, “but all the currency risk premia are too small,” and that “varying the coefficient of risk aversion does not resolve this tension.” A third, option-based Merton calibration with more frequent but much smaller jumps (risk aversion 8.70, jump intensity 139 percent, mean jump size −0.74 percent, jump size volatility 1.91 percent) again fails to match both moments (Figure 6). Because these models cannot even match two of the moments, the authors say they “ignore the exchange rate cyclicality puzzle” in this section.

The wedge’s drift stops being a free parameter, and the trade-off becomes strictly worse: a given proportional reduction in exchange rate volatility forces a mechanical reduction in the currency risk premium, and the UIP slope is always pushed toward one. In a discrete-time Cox, Ingersoll, and Ross specification with country-specific factors – whose real version “is isomorphic to a model in which the domestic (foreign) representative agent has power utility preferences over consumption with CRRA coefficient” and heteroskedastic consumption growth – Result 2 shows the wedge has only two free parameters, both of which pin down exchange rate volatility, after which “the law of motion of the incomplete spanning wedge is entirely determined” and “the drift term in the eta process is not a free parameter either” (Section 3.2). Result 3 then gives the level risk premium as kappa·z_t ≤ gamma·z_t, and the paper states the magnitude directly: “If incomplete spanning reduces the standard deviation of exchange rates by 50% …, then the currency risk premium is reduced by a factor of 0.25 …, implying a reduction by 75%.” Figure 7, under the parameters lambda_d = −1.07, theta = 0.004428, phi = 0.976, alpha = 0 and sigma = 0.008356 (chosen to match the mean, volatility and autocorrelation of the short rate), shows that when exchange rate volatility reaches its empirical value the currency risk premium is zero even when evaluated two standard deviations above the mean of the state variable. The authors note this holds with common as well as country-specific factors. The economic reading they offer: incomplete spanning “effectively reduces the representative agent’s risk aversion coefficient when pricing currency risk, but not for other risk sources.”

They point to two ingredients – highly correlated stochastic discount factors despite weakly correlated macro data, and segmented markets that concentrate aggregate risk among a small set of participants – while insisting each still needs direct evidence. The conclusion notes that “as suggested by complete market models, stochastic discount factors may be very highly correlated, even if macroeconomic series are not. To support this view further, researchers need to find direct evidence of such high correlations,” and that segmented-market models restricting international trade to financial intermediaries (Gabaix and Maggiori, 2015) or a subset of investors “are promising,” because “these models sever the link between aggregate quantities and real exchange rates by concentrating aggregate risk among a small pool of investors.” But the same measurement demand applies: “researchers need to show that changes in exchange rates are highly correlated with the marginal utility growth of these market participants” (Section 4). In the introduction the authors survey long-run-risk, rare-disaster and segmented-market resolutions and conclude that “while we find these assumptions plausible, we recognize that more empirical work is needed to directly validate them. There is yet no widely accepted solution to the exchange rate puzzles.”