Fama-Bliss Forward Rate Regressions

The pure expectations hypothesis says a forward rate is just the market's unbiased forecast of the future spot rate — if the one-year-forward-in-one-year rate is 5%, the theory says spot rates a year from now should average 5%, with no risk premium baked in. Fama and Bliss tested this directly with a simple regression, and the result is one of the most cited pieces of evidence against pure expectations in fixed income.

The regression is:

$$r_{t+1} - y_t = \alpha + \beta (f_t - y_t) + \varepsilon_{t+1},$$

where $y_t$ is today's short-term spot yield, $f_t$ is today's forward rate for the future period, and $r_{t+1}$ is the future spot rate that actually materializes. $f_t - y_t$ is the "forward spread" — how much steeper the forward rate is than today's spot rate. Pure expectations predicts $\beta = 1$: the forward spread should be a one-for-one unbiased forecast of the coming change in rates.

What Fama and Bliss actually found is $\beta$ reliably greater than 1 (often 1.5-3 depending on maturity) at longer horizons — the forward spread overpredicts the direction correctly but underpredicts its magnitude relative to what pure unbiasedness implies, and it also has real forecasting power for excess bond returns, not just rate changes. This is now read as evidence that the term premium (compensation for bearing interest-rate risk over the holding period) varies over time and is partly forecastable from the current shape of the yield curve, rather than being the constant the simplest theories assume.