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The New Fama Puzzle

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Abstract

We re-examine the historically common finding that ex post depreciation and the forward premium are negatively correlated, usually termed the forward premium puzzle. When covered interest differentials are zero, this finding is equivalent to the rejection of the joint hypothesis of uncovered interest parity (UIP) and full information rational expectations. We term this result the Fama puzzle (1984), given the difficulty in identifying a time-varying risk premium that could rationalize this result. In our analysis, the rejection occurs for eight exchange rates against the US dollar, but does not survive into the period during and in the decade after the financial crisis. Strikingly, in contrast to earlier findings, the Fama coefficient—the coefficient on the interest differential—then becomes large and positive; this is what we term the New Fama Puzzle. Using survey based measures of exchange rate expectations, we find much more consistant evidence in favor of UIP. Hence, the explanation for the switch in the Fama coefficient in the wake of the global financial crisis is mostly a change in how expectations errors and interest differentials co-move.

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Fig. 1

Source Thomson Reuters Datastream

Fig. 2

Source Thomson Reuters Datastream, and authors’ calculations

Fig. 3

Source International Financial Statistics and authors’ calculations

Fig. 4

Source policyuncertainty.com and CBOE

Fig. 5

Source Thomson Reuters Datastream, International Financial Statistics, and authors’ calculations

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Notes

  1. If there are no covered interest differentials (as should be the case in the absence of capital controls and capital requirements), then the forward premium equals the interest differential. A regression of depreciation on the forward premium is equivalent to a regression of depreciation on interest differentials. We re-examine this point in the theoretical section.

  2. In fact, Fama did not interpret the negative coefficient as a puzzle, as he attributed the result to the presence of a time varying risk premium. Engel (1996) surveys the failure of the portfolio balance models and consumption capital asset pricing models to provide a risk premium basis for the Fama result. See also Chinn (2006) and more recently Engel (2014). Most recently, Corsetti and Marin (2020) argue no puzzle exists given the role of “Peso events’.

  3. The question of exchange rate developments in light of interest rate differentials is obviously important for policy makers in general (and central bankers in particular, see for instance Coeuré 2017).

  4. For ease of exposition, log approximations are used. In the empirical implementation, exact formulas are used. We have examined data at three month and one year horizons (\(\mathrm{h}\in \left[\mathrm{3,12}\right]\)), using monthly data. This means the regression residuals are serially correlated under the null hypothesis of rational expectations and uncovered interest parity. We account for this issue by using robust standard errors. We report results for h=12, in order to conserve space; h=3 results are reported in the Appendix Tables 2-4.

  5. See Dooley and Isard (1980) for discussion and Popper (1993) for a review of the pre-2008 experience, in which the covered interest differential is attributed to political risk.

  6. Note that we use offshore yields rather than sovereign bond yields, thereby mitigating the convenience yield channel emphasized by Engel (2016).

  7. See Engel (1996) for a discussion of how the forward rate and the expected spot rate might deviate even under rational expectations and risk neutrality.

  8. Note that the definition of the expectation or forecast error is the negative of the convention, i.e., actual minus forecast.

  9. Similar results are cited in surveys by MacDonald and Taylor (1992) and Isard (1995). Meese and Rogoff (1983) show that the forward rate is outpredicted by a random walk, which is consistent with the failure of the unbiasedness hypothesis.

  10. Flood and Rose (19962002) note that including currency crises and devaluations, one finds more evidence for the unbiasedness hypothesis.

  11. See Hassan and Mano (2017) for a different perspective on how the Fama puzzle relates to the carry trade.

  12. Chinn and Meredith (2004) tested the UIP hypothesis at five year and ten year horizons for the Group of Seven (G7) countries, and found greater support for the UIP hypothesis holding at these long horizons than at shorter horizons of three to twelve months. The estimated coefficient on the interest rate differentials were positive and were closer to the value of unity than to zero in general.

  13. We adopt the standard assumption of no default risk. In general, this is believed to hold, although during the height of the global financial crisis, counterparty risk was perceived as high (along with liquidity issues), so that covered interest parity did not hold (Coffey et al. 2009; Baba and Packer 2009).

  14. Since we are examining one year horizons, the interest rate sample is truncated at 2020M09. Results for three month horizon, reported in Tables 5, 6, and 7, are truncated at 2021M06.

  15. Engel et al. (2021) finds weaker rejection of unbiasedness using a longer sample for the early period, and an alternative estimator for standard errors. They find in a 2007–2020 period, positive coefficients but a general failure to reject unity for the slope coefficient.

  16. We have also conducted the analysis with a first breakpoint at 2006M08, and a second at 2018M01. That breakpoint incorporates exchange rate changes up to 2007M08, which could be considered as the beginning of the Global Financial Crisis, with the turmoil on the US housing market. Using this setup, we again obtain the same pattern of coefficient sign reversals.

  17. As indicated by the Survey of Professional Forecasters forecasts of the three month Treasury yield; this is discussed further in Sect. 5.

  18. Figure 11 shows the corresponding graphs for all the currencies.

  19. The absolute size of the coefficients is larger after the first period; mechanically, this arises because the regression coefficient is a covariance divided by the variance of the interest differential, and the variance of interest differentials are much smaller post-Crisis, as illustrated in Fig. 2.

  20. If the exchange risk premium is a mean zero random error term, there is no need to include a proxy variable. If, however, there is a central bank reaction function that essentially makes the error term correlated with the interest differential (as in a Taylor rule), then the estimates obtained from a simple Fama regression will be biased. Variants of this approach include McCallum (1994), in which the central bank responds to exchange rate depreciation, and Chinn and Meredith (2004), in which exchange rate depreciation feeds into output and inflation gaps that determine central bank policy rates. See also Mark and Wu (1998) and Engel (2014).

  21. Note that we also evaluate inflation differentials (and industrial production growth differentials) as proxies for a premium, in this case a liquidity premium, in line with Engel et al.’s (2019) model of forward rate bias (and high interest-high value currencies). However, we do not obtain empirical evidence for the usefulness of those variables in explaining the Fama puzzle.

  22. See Berg and Mark (2018) for discussion of uncertainty and the risk premium.

  23. Kalemli-Ozcan and Varela (2021) investigate how the deviation from survey-implied UIP moves with the VIX, as opposed to how ex post depreciation moves.

  24. In other words, we are assuming Classical measurement error, in line with most other analyses. Constant bias would be impounded in the constant. Time varying bias would be much more problematic.

  25. An additional complication is that the interest rates and exchange rates do not align precisely in this data set. Interest rates are sampled at end-of-month, while exchange rates forecasts are sampled usually at the second Monday of the month by Consensus Forecasts. We have cross checked the results for the euro using Currency Forecasters Digest/FX Forecasts data (as used in Chinn and Frankel 2020). The results are the same when the expected, futures and spot rates are exactly aligned.

  26. Skeptics of survey based measures argue that reported forecasts are read off of interest differentials. Chinn and Frankel (1993) note the pattern of relationship between expected spot rates and forwards was consistent with the idea that survey respondents use other information in judging future exchange rate movements. In addition, Cheung and Chinn (2001) survey foreign exchange traders, and find that interest differentials are only one of the inputs forecasters use.

  27. At the three-month horizon, the A component is slightly more important, but remains less significant than the B and C components.

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Acknowledgements

We would like to thank Agnès Bénassy-Quéré, Yin-Wong Cheung, Alexander Chudik, Jeffrey Frankel, Jim Hamilton, Jean Imbs, Ben Johannsen, Joe Joyce, Steve Kamin, Evgenia Passari, Arnaud Mehl, Lucio Sarno, and conference participants at the Banque de France-Sciences Po. “Workshop on Recent Developments in Exchange Rate Economics,” the “Jean Monnet Workshop on Financial Globalization and its Spillovers,” and seminars at the Banque de France, ECB, Dallas Fed, Brandeis, University of Adelaide, UC Riverside, and James Madison University. Jonas Heipertz gratefully acknowledges financial support by INET. The views expressed do not necessarily reflect those of the Banque de France, the Eurosystem, or NBER.

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Correspondence to Menzie Chinn.

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Appendix

Appendix

See Tables 4, 5, 6, 7, 8 and Fig. 11.

Table 4 Estimated fama coefficients for the various sub-samples for selected base currencies (12 month horizon)
Table 5 Fama regression results (3 month horizon)
Table 6 Fama regression augmented with VIX results (3 month horizon)
Table 7 UIP regressions results using survey data on exchange rate expectations (3 month horizon)
Table 8 Data sources
Fig. 11
figure 11figure 11figure 11figure 11

Scatterplot of the 1 Year Ex-Post Depreciation Rate (1 Year Ahead) on 1 Year Eurodeposit Rate Differential (decimal format). Note: Top graph is Early Period, middle graph is Middle Period, and bottom graph is Late Period. CA denotes Canadian dollar, CH denotes Swiss franc, DK denotes Danish krone, EA denote Euro, JP denotes Japanese yen, NO denote Norwegian krone, SW denotes Swedish krona, and UK denotes British pound. Regression line in red. Authors’ calculations based on International Financial Statistics and Thomson Reuters Datastream data

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Bussière, M., Chinn, M., Ferrara, L. et al. The New Fama Puzzle. IMF Econ Rev 70, 451–486 (2022). https://doi.org/10.1057/s41308-022-00161-z

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