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The new Fama puzzle

The ‘Fama puzzle’ is the finding that ex post depreciation and interest differentials are negatively correlated, contrary to what theory suggests. This column re-examines the puzzle for eight advanced country exchange rates against the US dollar, over the period up to February 2016. The rejection of the joint hypothesis of uncovered interest parity and rational expectations still occurs, but with much less frequency. In contrast to earlier findings, the Fama regression coefficient is positive and large in the period after the Global Crisis, but survey-based measures of exchange rate expectations reveal greater evidence in favour of uncovered interest parity.

Uncovered interest parity (UIP) - the proposition that anticipated exchange rate changes should offset interest rate differentials - is one of the most central concepts in international finance, defining whether financial capital is freely mobile and substitutable across borders (Frankel 1982). This is a critical question for the allocation of assets in different currencies; it is also a factor in considering the spillover effects of macroeconomic policy. In a new paper (Bussière, et al., 2018), we re-examine UIP.

At the same time, empirical validation of this concept has proven elusive. In fact, the failure of the joint hypothesis of UIP and rational expectations – sometimes termed the unbiasedness hypothesis – is one of the most robust empirical regularities in the literature. The most commonplace explanations – such as the existence of an exchange risk premium, which drives a wedge between forward rates and expected future spot rates – have little direct empirical verification.

Why revisit?

Several developments have prompted this revisit. 

  • First and foremost, the last decade includes a period in which short rates have effectively hit the interest rate zero lower bound. This point is clearly illustrated in Figure 1, where we plot one-year interest rates for a set of eight selected countries. This development affords us the opportunity to examine whether the Fama puzzle is a general phenomenon or one that is regime-dependent.

Figure 1 One-year yields on Eurocurrency deposits

  • Second, we now have more indicators for risk aversion for extended period of time. This potentially allows us to distinguish between competing explanations for the failure of the unbiasedness hypothesis. Specifically, we can examine whether the inclusion of these risk proxies alters the Fama puzzle. 
  • Third, we now have long spans of survey data on expectations spanning the post-crisis period.

The new puzzle

We obtain the following findings. First, Fama’s (1984) finding that interest rate differentials point in the wrong direction for subsequent ex-post changes in exchange rates is by and large replicated in regressions for the full sample, ranging from 1999 to February 2016. However, the results change if the sample is truncated to apply to only the most recent decade, the period for which interest rates are essentially at zero. For that period, interest differentials correctly signal the right direction of subsequent exchange rate changes, but with a magnitude that is altogether not reconcilable with the arbitrage interpretation of UIP. In other words, we obtain positive coefficients at exactly a time of high risk when it would seem less likely that UIP would hold. This finding is illustrated for eight exchange rates (against the US dollar) in Figure 2, at the one-year horizon.

Figure 2 OLS regression coefficient of ex post depreciation on interest differential, at one-year horizon, for interest rates over 1999M01-2006M08 period (blue bar), and over 2006M09-2015M02 period (red bar)

Note: Gey line at value of unity, consistent with uncovered interest parity and rational expectations.

One might be tempted to assert that this is a result special to dollar currency pairs. We’ve checked this possibility, and we find that for the most part, non-dollar cross pairs also exhibit this switch in coefficients. 

For greater insight into how the correlations look before and after the Global Crisis, Figure 3 presents a scatterplot for the UK-US relationship, again at the one-year horizon.

Figure 3 One-year ex-post depreciation of the US dollar against the pound versus corresponding one-year US-UK yield differential: Pre-crisis versus post-crisis

The slope of the red line is -2.1 in the pre-crisis period, and +10 in the post-crisis period. In both instances, the coefficients are statistically significantly different from the posited value of one.

Is it (mostly) risk? Or expectations?

The failure of the unbiasedness hypothesis could arise either because risk drives a wedge between interest rates and expected depreciation (Engel 1996, Engel 2014), or because expectations of exchange rate depreciation are not unbiased. On the first point, we find that the inclusion of a proxy variable for risk – namely, the VIX – results in Fama regression coefficients that are only slightly more in line with anticipated values of unity. This finding suggests that changes in the elevation of risk as measured by the VIX do not explain the forward rate bias, at least not in a direct linear fashion.

The use of survey-based measures of expectations data (from Consensus Forecasts, which start in 2003) provides the following insights. First, interest differentials and anticipated exchange rate changes as measured by survey data are positively correlated, consistent with the proposition that investors tend to equalise at least partially expected returns expressed in common currency terms (see also Chinn and Frankel 2016 for results for 1986-2009). Second, for the five major currency dollar pairs (euro, yen, Swiss franc, pound, Canadian dollar) where the Fama coefficient switches sign going from the 2003-2006 period to post-crisis period, the result arises in large part because the correlation of expectations errors (defined as expected minus actual) and interest differentials switches sign. (The pound is one example where the coefficient does not switch).

This can be seen from the fact that the regression coefficient in large samples equals a value of unity if three components (shown in Figure 4) are zero: (1) the comovement of covered interest differential and the interest differential (red bar); (2) the comovement of the risk premium and interest differential (blue bar); and (3) the comovement of forecast error and interest differential (green bar).

Figure 4 Theoretical beta, estimated beta, and components

Note: Sample periods pertain to interest rate data.

It might be surprising that the covered interest differential does not show up as being important in these calculations, given their appearance post-crisis (Du et al. 2017). They do at the three-month horizon, but even then, only to a very small degree. That is because expectations errors as measured by the survey data are so much larger, and change their covariation with interest differentials so much more profoundly.

Concluding thoughts

Despite the elevated risk in the global economy over the past decade, the change in the comovement of exchange rate risk with the interest differential does not appear to be the primary reason why the Fama coefficient has been so large in recent years (although the altered behaviour of exchange risk does play a role). Rather, how expectations errors comove with the interest differential appears of central importance.


Bussière, M, M D Chinn, L Ferrara, and J Heipertz (2018), “The New Fama Puzzle,” NBER Working Paper 24342.

Chinn, M and J Frankel (2016), “A Quarter Century of Currency Expectations Data: The Carry Trade and the Risk Premium,” mimeo (August).

Du, W, A Tepper and A Verdelhan (2017), “Deviations from Covered Interest Rate Parity,” NBER Working Paper 23170.

Engel, C (1996), “The Forward Discount Anomaly and the Risk Premium: A Survey of Recent Evidence,” Journal of Empirical Finance 3(June): 123-92.

Engel, C (2014), “Exchange Rates and Interest Parity.” Handbook of International Economics, vol. 4, pp. 453-522.

Fama, E (1984), "Forward and Spot Exchange Rates." Journal of Monetary Economics 14: 319-38.

Frankel, J (1982), "A test of perfect substitutability in the foreign exchange market." Southern Economic Journal 49(2): 406-416.

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