# Uncovering uncovered interest parity: exchange rates, yield curves and business cycles

Simon Lloyd and Emile Marin

The textbook uncovered interest parity (UIP) condition states that the expected change in the exchange rate between two countries over time should be equal to the interest rate differential at that horizon. While UIP appears to hold at longer horizons (around 5-10 years), it is regularly rejected at shorter ones (0-4 years). In a recent paper, we argue that interest rates at other maturities — captured in the slope of the yield curve — reflect information about the pricing of ‘business cycle risks’, which can help explain departures from UIP. A country with a relatively steep yield curve slope will tend to experience a depreciation in excess of the UIP benchmark, at business cycle frequencies especially.

Uncovered interest parity

Underpinned by a number of assumptions (including risk neutrality and rational expectations), UIP predicts that the currencies of countries with relatively high interest rates today should depreciate over time, and vice versa for low interest rate economies. This adjustment acts to equalise returns on domestic and foreign assets measured in a common currency.

However, an overwhelming share of empirical evidence rejects the UIP condition, typically using short-horizon interest rates and exchange rate moves of around 4 years or less, giving rise to the UIP puzzle (a.k.a the Fama puzzle). At these tenors, the currencies of high-yield countries tend to excessively appreciate (or insufficiently depreciate), complicating the forecasting of exchange rates. On the other hand, a body of evidence has failed to reject the hypothesis at longer horizons, out to around 10 years.

A role for the yield curve?

The failure of UIP at short-horizons is especially puzzling in light of its long-horizon success. But the stark textbook prediction — that the expected $k$-period exchange rate change should be proportional to the differential between $k$-period yields — is the result of strong assumptions, including that of risk neutrality. In reality, investors can buy a range of assets in a variety of currencies and across a spectrum of maturities. When investors are risk averse (ie concerned about uncertainty in the future), interest rate differentials and exchange rates will reflect investors’ perceptions of risks, as well as expected returns. From a cross-country perspective, exchange rate risk premia, which arise to compensate investors for risk, should therefore be predictable by information contained in the entire yield curve. In other words: $k$-period exchange rate changes are likely to be related to a range of interest differentials, not only the $k$-period tenor, along the yield curve that influence risk-averse investors’ portfolio choice.

In a recent Working Paper, we test whether information in the entire yield curve, over and above interest rate differentials, can improve our understanding of exchange rate dynamics. We extend an otherwise standard regression test for UIP — a regression of $k$-period exchange rate changes $(e_{t+k} - e_t)$ on $k$-period interest rate differentials $(i_{t,k}-i_{t,k}^*)$ — by adding the relative slope and curvature of yield curves as explanatory variables:

$\underbrace{e_{t+k} - e_t = \alpha_k + \beta_{1,k}\left( i_{t,k} - i_{t,k}^* \right)}_{\text{Standard UIP Regression}} + \underbrace{\beta_{2,k} S_t^R + \beta_{3,k} C_t^R}_{\text{Additional Terms}} + u_{t,t+k}$

where $e_t$ is the exchange rate of the domestic economy versus the foreign economy, defined as the home price of one unit of foreign currency such that an increase in $e_t$ corresponds to a domestic depreciation. $S_t^R$ represents the difference between countries’ yield curve slopes (defined as the difference between 10-year and 6-month yields) and $C_t^R$ measures differences in how curved countries’ yield curves are (measured as a combination of short, medium and long-term rates).

We estimate both the standard UIP and augmented regressions using data on six major currencies (Australia dollar, Canadian dollar, Swiss franc, euro, Japanese yen and Great British pound) vs US dollar, from 1980:01 to 2017:12, at a range of horizons $k$, from 6 months to 10 years.

Consistent with the existing empirical evidence, the UIP coefficient $\beta _1,_k$ estimates from the standard UIP regressions (Figure 1) demonstrate the rejection of UIP at short horizons and the failure to reject it at longer horizons. At 6 to 36-month horizons, the estimate is negative, indicating that high short-term interest rate currencies tend to appreciate, instead of depreciate as UIP predicts. In contrast, longer-horizon estimates are positive and close to 1 — ie the $k$-period exchange rate change is proportional to the $k$-maturity interest rate differential. At these horizons, in line with UIP, high interest rate currencies tend to depreciate proportionally to interest rate differentials.

Figure 1: Coefficient depicting relationship between interest rate differentials and exchange rate changes at different horizons from standard UIP regression

Notes: UIP coefficient estimates from standard UIP regression measuring relationship between exchange rate changes and interest rate differentials at different horizons, from 6 months to 10 years. Red bars denote error bands.

Mirroring the UIP puzzle, interest rate differentials only explain a small proportion of exchange rate variation at short-to-medium horizons (Figure 2). This has contributed to the idea that exchange rates do not mirror macroeconomic fundamentals, a phenomenon dubbed the ‘exchange rate disconnect’. But augmenting the UIP regression with measures of the relative yield curve slope and curvature helps to increase the fit for exchange rates. The improvement is particularly pronounced at 3 to 5-year horizons, where interest rate differentials alone capture less than 3pct of variation in yields. The addition of relative slope and curvature almost treble the share of explained variation.

Figure 2: Proportion of exchange rate variation explained by interest rate differentials, and the relative yield curve slope and curvature

Notes: Fraction of exchange rate variation explained by a UIP condition (red) and the yield curve-augmented specification (black) at different horizons, from 6 months to 10 years.

Focusing on the contribution of the yield curve, Figure 3 plots how a 1pp increase in a country’s yield curve slope relative to the US affects exchange rates, in pct. The estimates reveal a tent-shaped relationship between exchange rates and the relative slope across horizons. At short horizons (6-months and 1-year) and longer-horizons (6 to 10-years), estimates suggest no relationship between the two. But at medium horizons (1.5 to 5.5-years), the relative slope and exchange rates tend to be positively related, with the relationship strongest at the 3.5-year horizon.

Figure 3: Coefficient depicting relationship between relative yield curve slope and exchange rate changes at different horizons

Notes: Relative slope coefficient estimates from yield curve-augmented specification (black) at different horizons, from 6 months to 10 years.

Exchange rates and the business cycle

The fact the positive relationship between exchange rates and the relative yield curve slopes is strongest at business cycle frequencies suggests that yield curve slopes reflects information about countries’ relative future economic prospects which are relevant for exchange rates. This is unsurprising when considering that the yield curve is often cited as a good predictor of future GDP growth.

To understand why a steep yield curve slope is associated with a subsequent exchange rate depreciation, consider two things. First, an exchange rate depreciation increases a domestic investors’ return on foreign assets when evaluated in domestic currency terms. Second, a steep yield curve today indicates that future economic prospects and returns are expected to be strong. Knowing better times lie ahead in the distant future, a risk-averse investor will value nearer-term returns relatively highly — wanting to reallocate returns from the distant future to nearer-term. When this desire is stronger for an investor in one country relative to another — reflected by a relatively steep yield curve — their valuation of nearer-term returns will be comparatively high. Reflecting the data, a transitory exchange rate depreciation is required in the nearer-term (relative to UIP) for currencies with a relatively steep yield curve, in order to reallocate returns in response to ‘business cycle risk’.

Yield curve inversions and the return of UIP

Although this relationship between yield curve slopes and exchange rates persists over time, we also show that yield curve inversions are associated with a change in exchange rate dynamics, consistent with evidence that the yield curve is a harbinger of downturns.

Figure 4 presents estimates of UIP coefficients $\beta _1,_k$ over two different periods: periods in which yield curves are upward sloping — as they are on average — and periods in which they are inverted. Consistent with the UIP puzzle, the coefficient estimates are below 1 at short horizons. High interest rate currencies appreciate relative to the UIP. But during inversions, the coefficient reverses significantly in sign at short horizons (6 to 18-months). At these frequencies, high-yield currencies depreciate excessively relative to UIP when yield curves are inverted.

Figure 4: Relationship between interest rate differentials and exchange rate changes at different horizons when yield curves are upward sloping (black) and inverted (red)

Notes: UIP coefficient estimates at different horizons (from 6 months to 10 years) from UIP condition when domestic yield curves slope upwards (black) and when they invert (red). Grey shaded area and red bars denote error bands.

These novel results are difficult to reconcile in standard models, but in our working paper we interpret them through the lens of a literature studying ‘rare events’. Consider again the role of yield curve inversions as a harbinger of downturns. In such a ‘rare event’, investors’ return valuations can evolve in a non-linear fashion. With a domestic yield curve inversion signalling an increase in the likelihood of a domestic downturn, domestic risk-averse investors will value returns disproportionately highly relative to foreign investors. By the same reasoning as before, the exchange rate will depreciate in the near-term to reallocate returns.

These findings speak to a growing literature on the ‘New Fama Puzzle’, which highlight changes in exchange rate dynamics following the global financial crisis. In particular, our results suggest that these ‘flips’ in UIP coefficients are a more pervasive feature of exchange rate dynamics, arising around economic downturns more generally, rather than the global financial crisis specifically.

Conclusion

Overall, our results suggest that information in the entire yield curve, over and above spot interest rate differentials, can help to explain exchange rates. A country with a relatively steep yield curve slope will tend to experience a depreciation in excess of the UIP benchmark at business cycles frequencies. We argue that these dynamics arise because the yield curve slope captures information about risk-averse investors’ expectations of future economic prospects, which exchange rates respond to. Although the relationship between the relative yield curve slope and exchange rates is persistent, we also demonstrate that yield curve inversions signal changes in exchange rate dynamics, adding further nuance to the UIP puzzle.

Simon Lloyd works in the Bank’s Global Analysis Division and Emile Marin works at the University of Cambridge.