Forward-rate unbiasedness is uncovered interest parity plus risk neutrality. It claims the forward rate is an unbiased predictor of the future spot, so that on average, across many periods, the forward's errors cancel. It does not claim the forward is a good predictor. An unbiased predictor can be wildly imprecise, and this one is.
How is forward-rate unbiasedness tested?
The standard test regresses the realised change in the spot rate on the forward premium:
%ΔS = α + β × (forward premium) + ε
Unbiasedness predicts α = 0 and β = 1. The literature, beginning with Eugene Fama's 1984 paper in the Journal of Monetary Economics and replicated across pairs and decades since, finds β well below 1 and frequently negative. A negative slope says something stronger than "the forward is a poor forecast." It says the forward gets the direction wrong: currencies at a forward discount, the high-yielders, tend to depreciate less than the discount and sometimes appreciate outright. That is the forward-premium puzzle.
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Run the arithmetic on eight illustrative annual observations, forward premium against realised depreciation, both in percent:
| Year | Forward premium | Realised depreciation | Carry excess |
|---|---|---|---|
| 1 | 2.30 | 0.40 | +1.90 |
| 2 | 2.05 | 3.60 | −1.55 |
| 3 | 3.10 | 1.90 | +1.20 |
| 4 | 1.80 | 1.20 | +0.60 |
| 5 | 2.60 | −0.80 | +3.40 |
| 6 | 3.40 | 0.30 | +3.10 |
| 7 | 1.50 | 3.10 | −1.60 |
| 8 | 2.90 | 2.40 | +0.50 |
The fitted slope is β = −0.9226 with an intercept of 3.7787 and an R-squared of 0.1630. The standard error on the slope is 0.8536, so the t-statistic against the unbiasedness hypothesis β = 1 is −2.2523, and the two-tailed 5% critical value at six degrees of freedom is 2.4469.
Read that last line carefully, because it is the honest part. Eight observations cannot reject unbiasedness at conventional significance, and they cannot reject β = 0 either, since that t-statistic is −1.0808. The point estimate is dramatic and the sample is far too small to support it. The literature's rejection rests on decades of monthly data across dozens of pairs, not on eight numbers, and anyone who runs this regression on a short series and announces a result has produced a coincidence with a p-value attached.
What does the puzzle mean for the carry trade?
What the eight numbers do show cleanly is the arithmetic of the carry trade, because the last column is the carry trade. Borrowing the low-yield currency and lending the high-yield one earns exactly the forward premium minus the realised depreciation, period by period. The mean is +0.9438 percentage points a year with a standard deviation of 1.8772, an information ratio of 0.5027, six positive years out of eight, and a worst year of −1.6000. Positive on average, negative sometimes, and the sometimes clusters.
The identity underneath is worth writing once. An unhedged rupee deposit funded in dollars returns, exactly:
(1 + i_INR) / [(1 + i_USD) × (1 + %ΔS)] − 1
At a realised rupee depreciation of 1.10%, borrowing $1,000,000 and running the round trip earns $12,385.26, which is 1.238526% of the notional borrowed and 1.186896% as a return on the grossed-up funding leg. Both figures are correct and answer different questions, so state the denominator. Practice problem P5 walks the four conversions.
Do the same trade hedged and the answer is exactly zero, because selling the ₹90,203,750 forward at 86.443460 delivers $1,043,500.00, the repayment to the cent. Covered interest parity is the reason the hedged version has no profit in it at all. The entire carry return is compensation for bearing the currency move, and the forward-premium puzzle is the observation that the compensation has, historically, been larger than the move.
What explains the forward premium puzzle?
Two explanations compete and both are partly right. One says the excess return is a risk premium: carry is short volatility, it loses in exactly the states where investors most need money, and the average gain is the price of that skew. The other says expectations are not rational in the way the test assumes, that market participants systematically under-predict the persistence of interest differentials. The distinction matters for what you do next. If it is a risk premium, the return is earned and will keep being earned at the same risk. If it is an expectational error, it can vanish the moment enough capital notices.