11.9 Uncovered interest parity, the forward premium puzzle and PPP

11.9.1UIP and the Fama regression#

Uncovered interest parity (UIP) says that, with risk-neutral investors, expected depreciation equals the rate differential:

with $s=\ln S$. Combined with CIP, $f_t - s_t = i_t-i_t^*$, so UIP says the forward is an unbiased predictor of the future spot. Fama (1984, J. Monetary Econ. 14(3):319–338 [V]) tested this with

where UIP predicts $a=0$, $b=1$. The robust finding across major currency pairs and decades is $b<1$, often $b<0$: high-interest currencies tend, if anything, to appreciate. This is the forward premium puzzle.

Proof

$\Delta s_{t+1} = q_t + \eta_{t+1}$ with $\eta$ a forecast error uncorrelated with time-$t$ information. Therefore $\operatorname{Cov}(\Delta s, f-s) = \operatorname{Cov}(q, q+p) = \operatorname{Var}(q)+\operatorname{Cov}(p,q)$. Dividing by $\operatorname{Var}(q+p)$ gives (11.21). The denominator is positive, so $b<0$ iff the numerator is negative. Next, $b<\tfrac12 \iff 2\operatorname{Var}(q)+2\operatorname{Cov} < \operatorname{Var}(q)+\operatorname{Var}(p)+2\operatorname{Cov} \iff \operatorname{Var}(q)<\operatorname{Var}(p)$.∎

So a negative $b$ means the risk premium is more volatile than expected depreciation and negatively correlated with it. The forward premium is mostly a time-varying risk premium. Section 11.9.2 asks what risk that premium pays for.

Small-sample caveat $f-s$ is highly persistent, so (11.20) suffers the Stambaugh-type bias of predictive regressions. That bias does not overturn the finding, but standard errors from OLS are optimistic. m11_parity.py checks the OLS test's size under the null on simulated data: rejection about 5% at $n=240$.

11.9.2The carry trade#

A carry trade borrows in a low-rate (funding) currency and invests in a high-rate (target) currency. In logs, its excess return is

Under UIP, $\mathbb E[xr]=0$. If (11.20) holds with slope $b$, then $\mathbb E_t[xr_{t+1}] = (1-b)(i_t-i_t^*) - a$, positive whenever $b<1$ and the differential is positive. Carry has earned positive average returns historically. Brunnermeier, Nagel and Pedersen (2008, NBER Macroeconomics Annual 23:313–348 [V]) show the returns come with negative skewness. High-rate currencies "go up the stairs and come down the elevator" when funding liquidity dries up and carry positions are unwound together, as in 2008 and the yen-carry unwind of August 2024. The premium is compensation for crash risk.

INR as a carry currency The rupee offers a positive rate gap against the dollar, so foreign investors in Indian bonds earn carry. Through 2026 the INR–USD policy-rate gap narrowed to about 1.6 pp (5.50% vs a 3.875% midpoint). At the same time FX volatility rose sharply, with the rupee reaching a record low in March (11.11). The carry-to-volatility ratio, the trade's Sharpe proxy, fell, which is one reason FPI bond and equity flows turned negative.

11.9.3Purchasing power parity#

Absolute PPP: $S = P/P^*$, where $P$ and $P^*$ are price levels. Relative PPP: $\Delta s = \pi - \pi^*$. The real exchange rate is $q = s + p^* - p$. PPP says $q$ is constant. Empirically:

1. PPP fails badly at horizons under two years, because nominal exchange rates are far more volatile than price ratios.
2. Real exchange rates mean-revert slowly. For an AR(1) $q_t=\rho q_{t-1}+e_t$ at annual frequency, the half-life is $\ln 0.5/\ln\rho$. The consensus range of 3–5 years surveyed by Rogoff (1996, "The Purchasing Power Parity Puzzle", J. Econ. Lit. 34(2):647–668) implies $\rho\approx0.79$–$0.87$.
3. Balassa–Samuelson: fast productivity growth in a country's traded-goods sector raises its non-traded prices, so fast-growing economies show real appreciation over decades.

For INR, relative PPP is a useful long-run anchor. India's inflation has typically run 2–4 pp above US inflation, and the rupee's long-run nominal depreciation is of that order, while year-to-year moves are dominated by flows and oil. The RBI publishes 40-currency nominal and real effective exchange rate indices (NEER/REER) in its Bulletin. A REER well above 100 (overvaluation) is a standard argument for tolerating depreciation.