## Introduction

A flexible exchange rate is an essential component of the Bank of Canada’s monetary policy framework. This is because it can act as a shock absorber for the Canadian economy. The Canadian dollar is typically influenced by systematic risk factors—ongoing day-to-day risks in the overall economy. In particular, three systematic risk factors can affect Canada’s dollar:

- the average exposure of the Canadian dollar to fluctuations in the US dollar
- cross-country differences in interest rates
- oil price movements

In fact, as shown by Nolin, Kyeong and Fontaine (2018), the importance of these systematic factors increased during and after the 2008–09 global financial crisis.

In this paper, we build on these findings in two ways. First, we show that the role of systematic currency risk factors has remained significant since the global financial crisis, as they continue to account for up to 80% of exchange rate variation. Second, following Feunou, Fontaine and Krohn (2022), we decompose the exchange rate into yield differential and risk premium components. We then analyze the extent to which the three systematic risk factors explain the dynamics of these components. This allows us to show that the co-movement of risk premiums in currency markets with global risk factors is, on average, lower than their co-movements with the exchange rate itself. In addition, we find that the explanatory power of systematic risk factors has declined recently, in particular since the beginning of 2022. This suggests that the impacts of idiosyncratic or country-specific factors have increased in recent months and movements in foreign exchange risk premiums have at least partially de-coupled from movements in systematic currency risk factors.

## Decomposing currency dynamics

To understand the different drivers of exchange rate dynamics, we employ the present-value decomposition introduced by Feunou, Fontaine and Krohn (2022). Those authors note that differences in the term structures of two countries’ bond markets impact movements in the exchange rate between those two countries. With this methodology, the daily exchange rate is decomposed into parts that are explained by bond market factors and by an expected carry risk premium component. The model can be summarized by the following present-value decomposition:

\(\displaystyle\, s_t\) \(\displaystyle=\, E_t\ [ s_{t+m}\ ]\) \(\displaystyle+\,m(y_t^*-y_t )\) \(\displaystyle-\,m({TP}_t^*-{TP}_t )\ -{∑_{j=1}^T} E_t \ [r_{t+j} \ ] ,\) \(\displaystyle\, (1)\)

where movements in the daily nominal spot exchange rate can be written as the sum of the expected future spot rate *m* periods ahead (\(\displaystyle E_t\ [ s_{t+m}\ ]\)), the yield differential between a foreign and a domestic country (\(\displaystyle m(y_t^*-y_t )\)), a term premium differential between these countries (\(\displaystyle m({TP}_t^*-{TP}_t ))\), and an expected carry risk premium component \(\displaystyle ({∑_{j=1}^T} E_t \ [r_{t+j} \ ])\). We refer to the sum of the last two components as the foreign exchange risk premium (FXRP), since it measures additional sources of risk that investors are exposed to after they have bought or sold the foreign currency (\(\displaystyle s_t\)).

The risk premium components help rationalize the apparent disconnect between the yield differential and currency returns (Meese and Rogoff 1983). While one might expect an increase in the yield differential to be accompanied by a future appreciation of the foreign currency, that is not always the case. This is because a simultaneous increase in the risk premium components can alleviate or offset the positive correlation between yield differentials and currency returns. For instance, an increase in the compensation required by investors to hold long-term nominal bonds in the foreign country relative to the domestic country (i.e., a widening of the term premium differential) could lead to a decline in the exchange rate. In contrast, a shrinking term premium differential or lower carry risk premium can amplify an increase in the exchange rate. Therefore, assessing risk premium dynamics and their drivers is crucial to understanding currency movements.

## Measuring systematic drivers of exchange rates

Following Fontaine and Nolin (2017), we construct three currency risk factors to capture systematic market-wide movements. We use daily data from Bloomberg Finance L.P. for the G9 currencies between January 2010 and May 2023. Exchange rates are expressed in US dollars per unit of foreign currency—i.e., a higher price (or positive returns) is associated with a depreciation of the US dollar.

- First, we calculate the dollar portfolio as an equally weighted portfolio of long positions in foreign currencies (Lustig, Roussanov and Verdelhan 2011) to capture systematic movements of the US dollar against a cross-section of major currencies. An increase in the dollar portfolio means that the broad basket of major currencies appreciates, on average, versus the US dollar.
- Second, we construct the cross-sectional carry factor as the difference between long positions in currencies with high interest rate differentials and short positions in currencies with low interest rate differentials (Lustig and Verdelhan 2007). It captures the fact that currencies associated with high interest rates tend to appreciate against the US dollar, while those associated with low interest rates tend to depreciate (Menkhoff et al. 2012).
- Third, we construct an oil factor by allocating currencies to different portfolios based on their time-varying sensitivity to changes in oil prices. The factor moves with the systematic exposure to oil and commodity prices that impacts export- and import-oriented economies by a different degree.

We provide more details about how the global factors are constructed in the Appendix.

## The importance of systematic risk factors since the global financial crisis

To understand the importance of these three global currency risk factors in the dynamics of the Canadian dollar over time, we estimate the following regression using a 252-day rolling window and document the variation in explanatory power:1

\(\displaystyle\, y_t\) \(\displaystyle=\,\beta Dollar_t\) \(\displaystyle+\, θCarry_t\) \(\displaystyle+\, \omega Oil_t\) \(\displaystyle+\, \varepsilon_t .\) \(\displaystyle\, (2)\)

The dependent variable \(\displaystyle\, y_t\) measures either log spot returns (\(\displaystyle \Delta S_{t}\)), the daily log changes in the FXRP, or the FXRP’s two components—the term premium differential and the expected carry risk component—which we obtain from the present-value decomposition in equation (1). Chart 1 shows the time-varying adjusted \(\displaystyle\, \bar{R^{2}}\) from these regressions.

### Chart 1: Time-varying explanatory power of global currency factors

Chart 1 suggests that the explanatory power of global currency risk factors is large for the log spot returns of the Canadian dollar (blue line). The adjusted\(\displaystyle\, \bar{R^{2}}\) ranging from 25% to 80% corroborates the findings in Verdelhan (2018) and Fontaine and Nolin (2017). It suggests that systematic currency risk factors have continued to be important drivers of the Canadian dollar since the global financial crisis.

The novelty of our analysis is that we also assess the role of these global risk factors for the FXRP and its components. The explanatory power of the global risk factors for the FXRP (Chart 1, green line) varies significantly over time, with an average of 24% over the 12-year sample period, significantly lower than their average explanatory power for spot returns. Moreover, it decreases significantly between 2013 and 2015, between 2017 and 2018, for a short period in early 2020, and from the beginning of 2022 onward. These occasional but frequent declines in the adjusted\(\displaystyle\, \bar{R^{2}}\)suggest that country-specific risk factors—which are not captured by movements in systematic factors—played an increasingly important role during these episodes. For instance, despite flight-to-safety dynamics and heightened dollar demand during the onset of the COVID‑19 pandemic, the decline in early 2020 can most likely be attributed to a sudden rise in global and country-specific uncertainty caused by the pandemic. The decline in 2017 and the one since 2022 seem to coincide with periods of increasing policy rates, suggesting a growing importance of monetary policy decisions for currency risk premiums.

The explanatory power of the global risk factors differs for the two components of the FXRP. The average explanatory power for the carry risk premium (Chart 1, yellow line) is about 35%, and its evolution over time largely resembles the dynamics of the FXRP described earlier. In contrast, the explanatory power for the term premium differential (Chart 1, red line) is considerably lower and close to zero for most of the sample period. This suggests that the expected carry component drives most of the explanatory power for the FXRP and that factors outside the currency space primarily affect time series dynamics of the term premium differential.

The previous section focused on the explanatory power of the three global risk factors combined; here we estimate the effect of each factor on the foreign exchange risk premium and its two components. Table 1 shows the regression outcomes from equation (2). All global risk factors are statistically significant at the 1% level. The dollar and carry factors are negatively related to the FXRP as well as to the expected carry risk premium component, suggesting that a market-wide average depreciation of the dollar is associated with increases in risk premiums. Similarly, larger divergences of interest rates across countries are associated with higher risk premiums, which is in line with the observation that carry trade returns tend to be higher during times of high volatility.

In contrast, the sign of the oil factor is positive, indicating that risk premiums decrease in periods when the dollar appreciates. While the term premium differential also increases with higher values of the dollar factor, it has a positive exposure to oil and declines in response to increasing carry factor returns. Table 1 suggests that an increase of 1 standard deviation in the dollar factor is associated with a decrease in the FXRP and the expected carry risk premium by 0.28 and 0.4 standard deviations, respectively. The impact of the oil factor is smaller and approximately only half the size of the other systematic risk factors. This supports the findings in Nolin, Kyeong and Fontaine (2018) that oil price changes have become relatively less important for the dynamics in the Canadian dollar in recent years.

### Table 1: Regression results—exchange rate components and global currency factors

FXRP | FXRP | FXRP | FXRP | Expected carry risk premium component | Term premium differential component | |
---|---|---|---|---|---|---|

Dollar factor | -0.37 (-67.22) |
-0.28 (-52.02) |
-0.4 (-57.06) |
0.11 (34.66) |
||

Carry factor | -0.36 (-32.33) |
-0.25 (-19.24) |
-0.26 (-13.16) |
-0.04 (-3.78) |
||

Oil factor | 0.2 (11.02) |
0.16 (22.88) |
0.16 (25.80) |
0.03 (3.46) |
||

Adjusted R^{2} |
0.14 | 0.13 | 0.04 | 0.23 | 0.34 | 0.01 |

N | 3,486 | 3,486 | 3,486 | 3,486 | 3,486 | 3,486 |

Note: FXRP is foreign exchange risk premium.

Lastly, we show how exposure to global currency factors is changing over time. Chart 2 plots the time-varying coefficients from equation (1) based on 252-day rolling window regressions. For brevity, we present here only the results for the FXRP.

### Chart 2: Time-varying exposure to global currency factors

The FXRP’s co-movement with the oil factor is consistently positive. However, the exposures to the carry and dollar factors are negative and co-move negatively with each other (correlation coefficient: -0.49). This suggests that the dollar and carry factors contribute more substantially to the total level of explanatory power than the oil factor does. We assess this observation systematically in the next section.

## A recent increase in country-specific risk factors

Finally, we assess the individual marginal explanatory power of the dollar, carry and oil factors. To this end, we conduct a sequential regression approach where we account explicitly for the common co-movements across factors.

- First, we orthogonalize all factors, estimate univariate regressions with each orthogonalized factor as the single explanatory variable on the right-hand side, and then document the explanatory power of each regression.
- Second, since the explanatory power depends on the order in which a factor has been orthogonalized, we repeat the estimation for all possible permutations. For each iteration, we document the adjusted\(\displaystyle\, \bar{R^{2}}\), yielding a distribution of explanatory power for each factor.

This approach confirms the previous findings. The dollar factor is the most relevant across all permutations, explaining about 11% of the variability of the FXRP. The marginal explanatory power of the carry and oil factors is substantially lower, at 3% and 1%, respectively.

However, the individual contributions have changed noticeably since 2021. Although most of the explanatory power continues to originate from the dollar factor (ranging from 9% to 23%), Chart 3 shows that the role of the carry factor grew substantially in 2021 and in the first months of 2022 (ranging from 2% to 22%). This increase might be explained by an environment of heightened uncertainty and a larger divergence in policy rates across countries, compared with the low interest rate environment that existed after the global financial crisis and until the COVID‑19 pandemic began. The oil factor contribution remains small, which is consistent with the previous exercise, but it also increased temporarily between mid-2021 and mid-2022. This suggests a transitory, stronger co-movement between currency markets and commodity prices following the start of the pandemic.

Furthermore, the combined explanatory power of the factors has notably declined since early 2022, from over 40% (at January 26, 2022) to 12% at the end of the sample (April 12, 2023). This points to a rising importance of country-specific risk for foreign exchange risk premiums, even though the explanatory power for currency spot returns during the same period remains stable and high (recall Chart 1). This diverging pattern highlights the importance of the decomposition in equation (1), because it identifies that the relevant factor structure of the exchange rate varies across subcomponents and over time. In fact, the results suggest that systematic factors pick up common movements across the yield differential, while country-specific factors seem to matter more for currency risk premiums during the recent period of heightened uncertainty after the pandemic began.

### Chart 3: Average marginal contribution of global currency factors and fixed announcement dates

## Conclusion

We assess the role of global currency factors for risk premiums of the bilateral exchange rate between the Canadian and US dollars. We do this by breaking down exchange rate dynamics into a yield differential and a risk premium term that consists of the term premium differential and an expected carry risk premium. The results show that the dollar portfolio explains, on average, 40% of the variation in the foreign exchange risk premium. While the role of the carry factor has increased over the past three years as global policy rates have begun to diverge, we also find that the overall role played by global currency factors has declined. This suggests that country-specific factors may have begun to play a larger role for foreign exchange risk premiums, particularly in the period after the start of the pandemic, which has also coincided with rapidly rising policy rates.

## Appendix

The dollar portfolio is defined as:

\(\displaystyle\, Dollar_t\) \(\displaystyle=\, \frac{1} {N} {∑_{i=1}^N} \Delta S_{i,t}\) \(\displaystyle,\)

where \(\displaystyle \Delta S_{i,t} = S_{i,t} - S_{i,t-1}\) refers to the daily change in the log spot rate \(\displaystyle s_{i}\) of country *i* between day *t* and *t-1*.

The carry factor is defined as:

\(\displaystyle\, Carry_t\) \(\displaystyle=\, \frac{1} {N_{H_{FD}}} {∑_{i \varepsilon H_{FD}}} \Delta S_{i,t}\) \(\displaystyle-\, \frac{1} {N_{L_{FD}}} {∑_{i \varepsilon L_{FD}}} \Delta S_{i,t}\) \(\displaystyle,\)

where \(\displaystyle N_{{H}_{FD}} (N_{{L}_{FD}})\) refers to the number of currencies with a high (low) forward discount.

To construct the carry factor, we first sort currencies into portfolios based on the one-month forward discount on the previous day, and then allocate currencies with high (low) forward discount to portfolio 3 (portfolio 1). The carry factor is then constructed as the return difference from taking a long position in portfolio 3, while shorting currencies that were allocated to portfolio 1.

To construct the oil factor, we take a two-step approach. First, we estimate a 252-day rolling window regression, whereby log spot returns of currency i serve as the dependent variable and log changes in West Texas Intermediate (WTI) is the independent variable. In addition, we add the dollar portfolio and the carry factor as control variables. The estimated regression is of the form:

\(\displaystyle\, \Delta S_{i,t}\) \(\displaystyle=\, \alpha_{i,t}\) \(\displaystyle+\, \beta_{i,t} \Delta WTI_{t}\) \(\displaystyle+\, \omega_{i,t} Carry_t\) \(\displaystyle+\, θ_{i,t} Dollar_t\) \(\displaystyle+\, \varepsilon_{i,t}\) \(\displaystyle.\)

Second, we sort currencies into portfolios based on their exposure to oil price dynamics (i.e., \(\displaystyle \beta_{i,t}\)). We allocate currencies with high (low) exposure to portfolio 3 (portfolio 1) and then construct the oil factor by taking long positions in currencies with high exposure, and short positions in currencies with low exposure. Therefore, the oil factor is:

\(\displaystyle\, Oil_t\) \(\displaystyle=\, \frac{1} {N_{H_{\beta}}} {∑_{i \varepsilon H_{\beta}}} \Delta S_{i,t}\) \(\displaystyle-\, \frac{1} {N_{L_{\beta}}} {∑_{i \varepsilon L_{\beta}}} \Delta S_{i,t}\) \(\displaystyle.\)

We orthogonalize each factor by considering all possible permutations of the three global factors, i.e., we consider six possible orderings. For each set of orthogonalized factors, we then follow these steps:

- Regress the independent variable onto the first factor. The resulting R-squared\(\displaystyle\, R_{1}\) is the marginal contribution of the first factor to explain the independent variable.
- Regress the residuals \(\displaystyle\, E_{t,1}\) onto the second factor.
- The resulting R-squared\(\displaystyle\, R_{2}\) must be adjusted to obtain the marginal contribution of the second factor. So, we perform the operation\(\displaystyle\, (1-R_{1})*R_{2}\), which gives us the individual contribution of the second factor. We perform a similar analysis for the third factor.

\(\displaystyle\, Y_t\) \(\displaystyle=\,\beta_{t,1} F_{t,1}^\bot\) \(\displaystyle+\, \varepsilon_{t,1}\)

\(\displaystyle\, E_{t,1}\) \(\displaystyle=\,\beta_{t,2} F_{t,2}^\bot\) \(\displaystyle+\, \varepsilon_{t,2}\)

Once we have completed these steps for each of the possible permutations, we average the individual contribution of each factor. The resulting value corresponds to the average marginal contribution of a given factor and is the number we refer to throughout the text.

## Endnotes

- 1. The findings are robust to different lengths of the window used for the rolling regressions.[←]

## References

Feunou, B., J.-S. Fontaine and I. Krohn. 2022. “Real Exchange Rate Decompositions.” Bank of Canada Staff Discussion Paper No. 2022-6.

Fontaine, J.-S., and G. Nolin 2016. “The Share of Systematic Variations in the Canadian Dollar—Part I.” Bank of Canada Staff Analytical Note No. 2016-15.

Fontaine, J.-S. and G. Nolin 2017. “The Share of Systematic Variations in the Canadian Dollar—Part II.” Bank of Canada Staff Analytical Note No. 2017-1.

Lustig H. and A. Verdelhan. 2007. “The Cross Section of Foreign Currency Risk Premia and Consumption Growth Risk.” *American Economic Review* 97 (1): 89–117.

Lustig H., N. Roussanov and A. Verdelhan. 2011. “Common Risk Factors in Currency Markets.” *Review of Financial Studies* 24 (11): 3731–3777.

Meese, R. A. and K. Rogoff. 1983. “Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?” *Journal of International Economics* 14 (1-2): 3–24.

Menkhoff, L., L. Sarno, M. Schmeling and A. Schrimpf. 2012. “Carry Trades and Global Foreign Exchange Volatility.” *Journal of Finance* 67 (2): 681–718.

Nolin, G., J. Kyeong and J.-S. Fontaine. 2018. “The Share of Systematic Variations in the Canadian Dollar—Part III.” Bank of Canada Staff Analytical Note No. 2018-13.

Verdelhan A. 2018. “The Share of Systematic Variation in Bilateral Exchange Rates.” *Journal of Finance* 73 (1): 375–418.

## Disclaimer

Bank of Canada staff analytical notes are short articles that focus on topical issues relevant to the current economic and financial context, produced independently from the Bank’s Governing Council. This work may support or challenge prevailing policy orthodoxy. Therefore, the views expressed in this note are solely those of the authors and may differ from official Bank of Canada views. No responsibility for them should be attributed to the Bank.

DOI: https://doi.org/10.34989/san-2024-20