R. Roll, S. Yan An Explanation of the Forward Premium 'Puzzle' Content: 1. The puzzle

2. Sample data

3. The tests

4. Plausibility of the results

5. The shortcomings

6. The results

7. Conclusion 2. Data Foreign exchange data:

US dollar in 8 foreign currencies Puzzle in detail... The non-stationary of spot and forward exchange rates of the previous model, suggested: The Unit Root Problem Is the OLS well-specified? Conclusion Changes in spot exchange rates are noisy Interest Rate Parity: The negative relation to the subsequent change in spot exchange rates Pretty straightforward: buy it where its higher ...but this correlation is close not perfect risk premium nominal interest = real + inflation ... and then... they found it: The puzzle in the forward exchange premium by Corneliu Ciobu the forward exchange rate is an unbiased predictor of the future spot rates. Two time series samples:

From July 1974 - December 1994 &

March 1985 - July 1997 Calculations are done using log of forward and spot exchange rates Using monthly data only Simple explanatory model of the forward rate forecast: F test fail to reject: Although... for distinctive countries, for the sub-periods we reject the null hypothesis The reason: long-term swings in the level of both spot and forward exchange rates Unit Root was tested;

The tests (ADF and Phillips/Parron) support the suspicion of non-stationary

The critical levels are: -4 (1%) and -3.5(5%) Failing to reject unit roots for both exchange rates it suggests that the OLS might produce spurious results. OLS coefficients have no asymptotic distribution, therefore unreliable even in large samples. Table 2 provides the results of the Unit Roots test These provide very divergent results for different time-series. The explanatory variable is the forward premium The puzzle itself Goodhart, McMahon and Ngama that these negative estimates in b are due to structural breaks and outliers. The regressions have an uniformly negative slope coefficients. The explanatory power is very low, less than 5 % The unit root hypothesis is soundly rejected However...

The forward premium appears to non-stationary Data was corrected for possible errors The paradox of Forward Premium non-stationary: - always stationary - sometimes not Therefore, explanations:

1. forward premium is indeed non-stationary;

but the noise of spot exchange rate differential is overwhelming Table 5 analyzing this issue. 2. An alternative explanation: Neither variable has unit root, although forward premium is nearly non-stationary Reason:

weak power;

as in previous example, vary large sample size required. 3. b=0 Possibility Forward exchange rate - no correlation to future spot rates. It explain why the its estimate is negative in some of the cases Therefore, if the spot exchange rate is a random walk, then regression should produce b=0 The Plausibility of unit root Since forward premium is possibly non-stationary, then nominal rates imply non-stationary, as well. Since: Therefore: 1. unit root of the inflation differential 2. Time-varying risk premium differential. Inflation: There are no international forces to control the expectations Countries pursue divergent monetary policies Hyperinflation, as example of non-stationary process But the hyperinflation come to an end

In infinite horizon it cannot be non-stationary Risk premium as the source of non-stationary: Risk premium suggests that, since the nominal payoffs are certain than the real rates should be attributable to inflation risk. Therefore inflation process the driving force behind the unit foot of the forward premium The actual inflation difference appears to be stationary; Near non-stationary, but this is masked be large unexpected inflation. For now: is close to being ill-specified. 5. Further Tests: 5.1 Standard methods: The results do not shed much light on the puzzle Therefore we reconsider: Combining the assumptions: Dissatisfaction with OLS Models Their solution is to fit an cointegrating vector Therefore, they pursued a different technique for forecasting functions. A new procedure: Not a spurious OLS regression The example, The most elementary function: This data reveal the most unbiased estimates of future spot rates. For delta =1 reveal the forecast errors that decidedly stationary. The observed negative b's are spurious, while the reason that so many countries experienced them is the cross-country dependency The correlation is sample dependent The levels of the exchange rates are nearly non-stationary 7. Conclusion The newly devised test of forecasting for a financial variable is dependent on market expectations. According to it, the forward rates are virtually unbiased estimates of future spot rates. The End Puzzle Solved.

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# An explanation of the Forward premium Puzzle

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