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This paper explores a model of bond prices where agents have diverse prior beliefs about domestic and foreign inflation. In the long run, the foreign exchange forward premium reflects expected differences in inflation, but in the short run, it depends upon the diversity of prior beliefs. If some people have diffuse priors about a country's inflation process, then its currency commands a forward premium that is eventually dissipated. Using data on the dollar–mark premium from the 1980s, it shows that this kind of diversity really matters. Thus models with a single representative agent give an inadequate description of the data.

This paper takes the idea of heterogeneity in financial markets seriously. It develops a theory of the foreign exchange forward premium based upon the notion that people in the world economy have diverse prior beliefs about inflation. For most plausible specifications of prior beliefs, agents eventually have completely accurate knowledge about each country's inflation processes. Indeed, in the long run, yields reflect the common inflation forecasts, and the forward premium predicts expected depreciation of the spot rate accurately. Thus the asymptotic behavior of the world economy can be modeled using the artifice of a single “representative agent” having “rational expectations” about all the “fundamentals” in the world economy. But in the short run a fascinating theory of asset prices emerges. This theory has two important elements. First, the learning matters. Interest differentials depend upon the stochastic inflation history in the world economy, and the model provides a simple explanation for the “forward discount anomaly.”1 Second, the heterogeneity of beliefs matters. In this paper, I will show that it is not enough to ask traders what their inflation forecasts are; it is actually necessary to ask them how sure they are of their own forecasts. Because different classes of agents can hold more or less precise forecasts, bond yields typically have an option value inherent in them, even if everyone agrees on expected inflation. An asset is worth the sum of its expected real stream of income and the option value of reselling it at a later date. This option value can never be negative, and it is typically positive. Thus yields are lower than they would be if the world economy consisted of a single representative agent. Whether this effect is stronger for domestic or foreign assets is at the heart of the theory. One implication is that diverse precision of beliefs about a country's inflation process will raise the price of its bonds and thus lower their yields. Hence there will be a forward premium for that currency. An important insight is that all the moments characterizing agents' beliefs matter. Thus it is not appropriate to consider only each person's point forecasts of expected inflation; it matters how precise these forecasts are. How does a theory of asset prices emerge in a model where people have diverse prior beliefs? In particular, one question arises immediately: Why is this model not plagued by Milgrom and Stokey's (1982) No Trade Theorem? The answer lies in the subtle distinction between an environment in which agents have common priors but diverse ex post signals and one in which everyone has different priors but observes the same signals. Consider, for example, the 500th digit in the decimal representation of e. A speaker walks into the seminar room and offers a contract that pays $1 if that digit is 5. I may believe that it is likely to be an even number, and you may have more diffuse beliefs. We could easily announce our priors (thus establishing common knowledge), and we would both agree that you would pay more for the contract than I would. 2 Then the speaker opens a laptop and begins to read off numbers from the Taylor series expansion: 1.0, 2.0, 2.5, 2.67, and so on. After each new number, the speaker allows us to trade. It is quite possible that you and I would be willing to do so, with perhaps especially active trading once we get near the n-th term, where n! ≈ 10501. Thus the existence of a market (with limited short selling) where agents have heterogeneous priors is completely consistent with equilibrium. Now think of a similar situation, but the speaker announces that he is willing to sell for $1 a contract that pays $1000 if the 500th digit in the decimal expansion of e is not 5. He then takes out his laptop and boots up. No matter what your priors were, you would be unwilling to buy that asset precisely because the speaker has shown he has received a superior signal about its value. Now differential information destroys the very existence of equilibrium. What is the essence of the difference between the two examples? In the first case, all the priors are common knowledge and so are all the public signals. But in the second case, even if the prior beliefs are common knowledge, the signals are not. The model developed in Section 3 is akin to the first example. The analysis builds upon the work of Harrison and Kreps (1978), who showed that the asset's price typically exceeded the valuation of the most bullish trader. They stated that this was a formalization of Keynes's notion of a beauty contest. Morris (1996) extended this work to incorporate learning in a Bayesian framework, and Fisher (2003) extended his model to explain asset bubbles that arose in the foreign exchange experiments reported in Fisher and Kelly (2000). This paper makes four contributions. First, it applies Morris's (1996) work by building a model of bonds and extends it by incorporating more general stochastic processes. Second, it is a completely novel analysis of the foreign exchange forward premium. To the best of my knowledge, no one has built or calibrated a model like this in international finance. Third, the model's calibration and estimation shows that plausible priors can explain some of the forward premium for the German mark during the first half of the 1980s. That period and that currency were chosen to complement Lewis (1989) impressive empirical analysis using Bayesian techniques of a reduced form model of the exchange rate. Fourth, I actually estimate the precision with which different classes of agents hold plausible prior beliefs; then I show that there is strong evidence in favor of a model with diversity of beliefs. The typical homogeneous agent model in macroeconomics is just not supported by the data. What are this paper's main results? First, it shows that diverse prior beliefs about a country's inflation process induce a forward premium for its currency at horizons greater than one month. Second, it shows that the “peso problem” is not as simple as has been assumed; indeed, the typical interpretation of this phenomenon imposes very severe restrictions on agents' beliefs. Third, it gives a simple explanation for a strong version of the forward discount anomaly. When there is diverse prior information about a country's inflation process, its one-month forward rate will be negatively correlated with realized depreciations. Fourth, the model is calibrated and then estimated using actual data from the United States and Germany during the 1980s. The model performs well enough, although the effect of informational heterogeneity on the forward premium is not large. The calibrations of the model outperform a simple benchmark based upon covered interest parity, and they show that diversity of prior beliefs improves the model's fit. Fifth, I use a non-linear regression to test for homogeneity of beliefs in the data, and the Wald test overwhelmingly rejects the workhorse model in international finance. Diversity of beliefs really matters in these data. The rest of this paper is organized as follows. Section 2 gives a simple but extended example because the model is strikingly different from the norm in international finance. Section 3 contains a formal description of the model, and Section 4 discusses the forward premium both when there is one representative agent and when there are several agents in the world economy. Section 5 calibrates and then estimates the model for plausible specification of the agents' prior beliefs. It also shows that the models' predictions give rise to the forward premium anomaly. Section 6 gives some brief conclusions.

This paper has developed a new theory of the forward premium based upon a model that takes heterogeneity in financial markets seriously. The model has striking predictions for the forward premium; it shows that diverse beliefs about a country’s inflation process make its cur- rency trade forward at a premium in contracts whose horizons are greater than one month. The calibration of the model is perhaps plausible, but it predicts perhaps too much learning. Still, the notion that agents had diverse beliefs about monetary policy and doubted that inflation could be abated at the beginning of the 1981 is indeed intuitive. My primary empirical contribution is that I actually estimate the precision with which dif- ferent classes of agents in the world economy held plausible prior beliefs. There is evidence in these data that a model with one agent is just not an accurate description of world asset markets. This is the first time that an extension of the elegant models based upon Harrison and Kreps (1978) and Morris (1996) has been taken to the data, and my work shows that these authors were quite right to worry about diversity of beliefs in financial markets. The model has many weaknesses. First and foremost, it is ludicrous to impose that the real exchange rate is constant. My only defense is that a good model of the spot exchange rate is left to those with superior analytical powers. Another important weakness is that the model assumes that inflation process in each country is independently and identically distributed. Inflation is obviously correlated between countries and across time, but this fact is difficult to incorporate into an analytically tractable model with Bayesian learning. The calibrations and estimates are suggestive but not exhaustive. Again, my defense is that there is no other study in international finance that takes a structural model with heterogeneous priors and Bayesian learning to the data. So this empirical work is just a first step. In an important sense, my model is a better description of term structure than it is of the forward premium. 17 Most empirical analyses of the yield curve show that the expectations hy- pothesis of the term structure is not true; the spreads between certain long rates and short rates do not predict future short rates. Rudebusch (1995) argues that the monetary authority’s policy distorts bond prices at the short end of the yield curve. My model offers an alternative avenueworthy of exploration; future researchers can analyze the effect of heterogeneity of beliefs on the yield curve in a national bond market. Perhaps this paper will spur other researchers in international finance to investigate models with heterogeneous beliefs. It is remarkable that the ‘‘forward discount anomaly’’ can be ex- plained so easily in a model with risk-neutral agents. It is essential that we economists be care- ful in our interpretations of the notion of rational expectations as an equilibrium concept. Models with heterogeneous prior beliefs are more general than the usual ones with one repre- sentative agent. The equilibria described in this paper all converge to the ‘‘rational expectations equilibrium’’ if agents’ priors are well behaved. And the typical model used in international finance is a special case of the one that has been explored; after all, one can always impose that everyone’s perfectly precise priors agreed with the actual distribution of inflation. But in- formational heterogeneity and limited short selling surely characterize actual financial markets. So it would be nice to continue building models that incorporate these obvious facts.