## The Forward Premium Puzzle is often misinterpreted

Fama (1984) found a positive covariance between interest differentials and currency appreciation against the US dollar when running pair-wise regressions with a constant. This fact, known as the forward premium puzzle (FPP), is not easily generated by standard models of exchange rate determination and, thus, spurred a large theoretical literature.

The usual (textbook) interpretation of the forward premium puzzle is as follows:

- Carry trades – a strategy of lending in high-interest-rate currencies and borrowing in low-interest-rate currencies – is profitable because of the forward premium puzzle;
- Currency risk premia are highly variable over time;
- This time variation in currency risk premia plays a role in determining changes in expected exchange rates (a fancy way of saying that ‘high interest rate currencies appreciate’).

In Hassan and Mano (2014), we argue that this interpretation is misleading on all three counts. Instead, we show that the famous carry trade has little to do with the forward premium puzzle; the carry trade exploits persistent differences in interest rate differentials across currencies, while the premium puzzle seems to be driven by interest rate movements of all currencies against the US dollar.

## Forward Premium Puzzle and Carry Trade as separate anomalies

How do we link the premium puzzle and the carry trade if the former is a result from a regression and the latter is a portfolio strategy? We link regression and portfolio approaches by noting that the covariance of currency returns with interest differentials can be expressed as the expected return of a strategy that weights each currency linearly by its interest differential. This decomposition allows us to represent regression coefficients as trading strategies and vice versa.

Figure 1 illustrates the difference between the carry trade (left panel) and the forward premium puzzle (right panel). In both panels, we plot the interest rate differential of the New Zealand dollar and the Japanese yen against the US dollar in blue. The red lines represent the average interest rate differential over the sample for each currency.

The left panel of Figure 1 shows a simple two-currency carry trade. Because the New Zealand dollar always has a higher interest rate than the Japanese yen, carry traders are always long (lending in) New Zealand and always short (borrowing in) Japan.

Contrast this with the right panel depicting the trading strategy we obtain when we re-write the forward premium puzzle regression results as a trading strategy (we call it the ‘forward premium trade’). Investors consider each of the two foreign currencies in isolation and decide to go long when the blue line is above the red line and short when it is below. In order to exploit the premium puzzle, investors go long the New Zealand dollar (lend) when it has a higher interest rate differential with the US than it usually does and short (borrow) otherwise. They thus trade in and out of the New Zealand dollar repeatedly during the sample.

**Figure 1**. Carry trade and premium puzzle, New Zealand dollar and Japanese yen against the US dollar

Although both anomalies constitute a violation of uncovered interest parity, the carry trade and the premium puzzle are thus clearly different things.^{1} But how do they relate to each other? A key contribution of this paper is that we decompose violations of uncovered interest parity into a cross-currency, a cross-time and a between-time-and-currency component. Each of the three components can be thought of as a basic trading strategy. The cross-currency component corresponds to the expected return of a one-shot (static) strategy that goes long currencies that usually have high interest rates (e.g. New Zealand) and short currencies that usually have low interest rates (Japan) and never changes that portfolio composition; the between-time-and-currency component goes long currencies that have unusually high interest rates relative to their time-and currency-specific means; and the cross-time component corresponds to a strategy of buying all foreign currencies (with equal weights) when foreign interest rates are high relative to the US.

- We show that the carry trade is driven mainly by the cross-currency component, while the premium puzzle is driven mainly by the cross-time component.

The between-time-and-currency component (the part that both anomalies have in common) is both economically and statically insignificant. The high returns on the carry trade thus result from the simple fact that some countries have permanently (or highly persistently) higher interest rates than others and that changes in exchange rates wipe out only part of this difference. Returns on the carry trade would not be significantly different if there was no forward premium puzzle at all.

## The premium puzzle is not as bad as you think

The tight link between regression coefficients and portfolio strategies forces one to think about what investors know when entering a particular strategy, and how regressions reflect that information.

Based on Fama’s regression results, generations of theorists have concluded that currency risk premia must be highly volatile. But when interpreting the fitted values from his regression as being informative about the risk premium of a representative currency pair, one implicitly assumes that the currency-specific constant is known in advance. That is equivalent to saying that traders who follow the strategy depicted in the right panel of Figure 1 know the position of the red lines with certainty. In reality, that is almost certainly not the case. In January of 1995, traders could not be certain what the average interest differential between the US and New Zealand dollar would be over the coming 15 years. If we take this uncertainty into account when running our regressions, the premium puzzle is much diminished (we get much smaller slope coefficients). As a result, currency risk premia are probably much less volatile than initially thought.

## Modelling exchange rate determination is much simpler than the premium suggested

In fact, once we make these econometric adjustments, we never reject the hypothesis that the covariance of appreciations and interest rate differentials is negative. Higher interest rate differentials are not necessarily tied to expected appreciations. The complicated view that has caused headaches for generations of theorists does not seem to have as much support in the data as previously thought.

The conclusion of our paper is that currency risk premia may be much simpler than previously thought. The first-order fact in the data is that most violations of uncovered interest parity are in the cross-section. The key to understanding these violations is thus to understand why some currencies have persistently higher interest rates than others. Next to this question, the forward premium puzzle, that has absorbed so much intellectual energy over the years, seems less central.

Some recent papers have proposed models of permanent or highly persistent interest rate differentials across countries. For example, Hassan (2013) and Martin (2013) argue that bonds issued in currencies of larger countries are safer and thus pay permanently lower risk-free interest rates.

## References

Fama, E F (1984), “Forward and spot exchange rates”, *Journal of Monetary Economics* 14, no. 3: 319-338.

Hassan, T A (2013), “Country size, currency unions, and international asset returns”, *The Journal of Finance* 68.6: 2269-2308.

Hassan, T A, and R C Mano (2014), “Forward and Spot Exchange Rates in a Multi-Currency World”, Fama-Miller Working Paper.

Martin, I (2011), “The forward premium puzzle in a two-country world”, National Bureau of Economic Research, No. w17564

## Footnote

1 The Uncovered Interest Parity condition states that the expected depreciation rate of a foreign currency is equal to the difference in interest rates between the home and the foreign currency, such that the expected return of borrowing in one currency and lending in another is always zero.