Forward Premium Anomaly

Definition

The forward premium anomaly (also called the forward rate bias or forward premium puzzle) is the systematic failure of the covered-uncovered interest parity prediction that the forward exchange rate is an unbiased predictor of the future spot rate. In the standard Fama (1984) regression, $s_{t+1} = \alpha + \beta f_t + \varepsilon_{t+1}$, uncovered interest parity (UIP) predicts $\beta = 1$; empirically $\beta$ is typically well below 1 and often negative, implying that currencies at a forward premium subsequently depreciate less than predicted — or even appreciate.

Key Ideas

• Standard short-run form: $s_{t+1} - s_t = \alpha + \beta (f_t - s_t) + \varepsilon_{t+1}$; UIP $\to$ $\beta = 1$. Fama (1984) documented $\beta \ll 1$ across many currencies.
• Long-run cointegrating version (Evans-Lewis 1993/1995): In levels, regress $s_{t+N} = \alpha + \beta f_t + u_{t+N}$; if spot and forward rates cointegrate, $\beta = 1$ is the cointegration hypothesis. Evans-Lewis found $\beta < 1$ even at long horizons (3–5 years), and interpreted excess returns as nonstationary — a strong anomaly.
• Dynamic seemingly unrelated regression (DSUR) reexamination (Mark-Ogaki-Sul 2003): Jointly estimating cointegrating regressions for DM/USD, Yen/USD, GBP/USD (1975.1–1996.12 monthly) as a 3-equation system. Individual DSUR estimates: Germany $\beta = 0.992$, Japan $\beta = 1.000$, UK $\beta = 1.001$.
  • 3 leads+lags: joint $\chi^2(\beta=1) = 7.571$ ($p = 0.056$) — borderline rejection.
  • 2 leads+lags: joint $\chi^2(\beta=1) = 5.344$ ($p = 0.148$) — fail to reject.
  • Leads-only specification: same conclusion as 2 leads.
  • Conclusion: evidence against $\beta = 1$ is fragile and lag-length-dependent.
• Homogeneity: $\chi^2$ test of equal $\beta$ across currencies: 7.459 ($p = 0.024$) with 3 lags — some evidence of currency-specific effects, but the restricted (pooled) estimate $\beta = 0.997$ remains very close to 1.
• Mechanically: the DSUR system augments each spot–forward regression with leads/lags of all three $\Delta$forward series, exploiting the common dollar exposure across currency pairs.

How It Works

Forward rate anomaly arises because risk premia on currencies are time-varying and correlated with the forward premium itself. Under rational expectations with time-varying risk premia, $\beta$ can deviate systematically from 1. The long-run cointegrating interpretation ($\beta = 1$ as a no-arbitrage equilibrium condition) is the stronger claim — it says excess returns must be stationary for markets to be in equilibrium. DSUR tests this jointly across related currencies, reducing estimation uncertainty.

Why It Matters

The anomaly has been a central empirical challenge to rational expectations in foreign exchange (FX) markets for 40 years. If the Evans-Lewis cointegrating interpretation is correct, it represents a fundamental breakdown of no-arbitrage across currencies at long horizons. The DSUR evidence weakens this conclusion — $\beta$ is close to 1 for all three currencies individually, and joint rejection requires exactly 3 leads/lags (not 2). This lag-length sensitivity is a general caution about inference in small cointegrating systems.

Open Questions

• Is the fragility of DSUR results sensitive to the currency sample and sample period?
• Does the anomaly behave differently post-2000 in the euro era?
• How do peso problems and rare-event premia confound long-run tests?

Related

• Uncovered Interest Parity
• Dynamic Seemingly Unrelated Regression
• Nelson Mark