Accounting for Changes in Long-Term Interest Rates: Evidence from Canada

Abstract

We provide a novel perspective on changes in global long-term interest rates using a dynamic term structure model of Canadian nominal and real yields with adjustments for term, liquidity, and inflation risk premiums. For the period from 1996 to 2021, we find that the steady-state or equilibrium short-term real interest rate fell by more than 4 percentage points, long-term inflation expectations edged down modestly, and real bond and inflation risk premiums varied with little longer-run trend. In contrast, our yield decomposition attributes the post-pandemic rise in interest rates largely to a sharp reversal and increase in the equilibrium real rate.

The secular decline in global long-term nominal interest rates in the decades before the coronavirus disease 2019 (COVID-19) pandemic has been the subject of much discussion and research. Indeed, the possibility of a lower new normal for interest rates was at the center of key economic, financial, and policy debates about the transformation of the economy and bond market dynamics. Still, the source and permanence of that gradual decline in interest rates remain unresolved—especially given a general post-pandemic rise in global interest rates. Therefore, in this article, we estimate a comprehensive dynamic term structure model on an under-utilized data set to provide a new perspective on the underlying nature of the changes in long-term interest rates.

Accounting for movements in long-term interest rates during the past few decades requires assessing the relative importance of nominal inflationary forces, real economic factors, and risk premiums. In the 1980s and 1990s, falling inflation expectations clearly played a substantial role in lowering long-term yields, but in the 2000s and 2010s, actual inflation as well as survey-based measures of longer-run inflation expectations remained relatively stable. Instead, the source of the general decline in interest rates in the first two decades of the century appears to reflect a variety of longer-run real-side factors—such as slower productivity growth and an aging population (e.g. Rachel and Smith 2015; Christensen and Rudebusch 2019; Bauer and Rudebusch 2020). More recently, the post-pandemic rise in interest rates may reflect various influences including elevated inflation, higher government debt levels, increased investment for the transition to a greener economy, and greater defense spending (e.g. Rogoff 2023). These shifts in economic fundamentals can persistently affect nominal and real yield curves by changing the steady-state level of the safe real short-term interest rate—the so-called equilibrium or natural or neutral rate of interest. Many researchers have used macroeconomic models and data to try to pin down the equilibrium real rate. Inspired by Christensen and Rudebusch (2019, henceforth CR), we use a financial model that accounts for nominal and real factors.¹ Varying term, inflation risk, and liquidity risk premiums can also play a role in changing long-term yields, and one advantage of a financial modeling approach is that it allows explicitly for risk premiums to account for changes in long-term interest rates (e.g. Bauer and Rudebusch 2020). In particular, as price inflation fell and became better anchored at low levels in many countries before the pandemic, the inflation risk premium may have declined (e.g. Wright 2011; Bauer, Rudebusch, and Wu 2014). In contrast, during the recent spell of more rapid price increases, the inflation risk premium may have reversed some of that decline. Our analysis assesses the relative contribution to bond yields of each of these various components.

For our analysis, we employ Canadian government bond prices, which provides a new—or at least, a relatively under-studied—perspective on persistent changes in long-term interest rates. Besides its novelty, a Canadian case study has several other advantages. First, Canada has deep and liquid markets for government debt. Also, the Bank of Canada has engaged much less than most other major central banks in unconventional monetary policies such as large-scale asset purchases (also known as quantitative easing or QE).² As a consequence, Canadian bond yields are less directly affected by such policies (although we explicitly explore the possibility of indirect spillover effects from QE in other countries).³ Arguably then, the Canadian government bond market offers a cleaner, less-managed setting for analyzing these questions than the euro area, Japan, the U.K., or the United States, where such policies have been implemented for extended periods. Also, as the underlying factors affecting long-term interest rates are likely global in nature—such as worldwide demographic shifts or disinflationary and inflationary pressures—the Canadian government bond market may well be about as informative as any other major sovereign bond market. Furthermore, the Canadian government holds a AAA credit rating with a stable outlook from all major rating agencies, which also contrasts with some of its G7 peers. Therefore, there is no credit risk to account for in our Canadian bond price data, which is an additional advantage. Finally, the Canadian government has been issuing inflation-indexed debt since 1991, and Canada therefore offers a relatively long history of both nominal and real yields, only rivaled by U.K. and U.S. samples.⁴

We examine a sample of Canadian nominal government bond yields along with prices of Canadian government Real Return Bonds (RRBs), which have coupon and principal payments indexed to the Canadian Consumer Price Index (CPI). The indexed debt provides compensation to investors for the erosion of purchasing power due to price inflation with prices that can be expressed directly in terms of real yields. We assume that the embedded longer-term expectations in these asset prices reflect financial market participants’ views about the steady state of the Canadian economy, including the natural rate of interest and the long-run level of inflation (i.e. the perceived Bank of Canada inflation target). Still, the use of RRBs for measuring the steady-state short-term real interest rate and inflation level does pose empirical challenges. One problem is that despite the long history and fairly large notional amount of outstanding RRBs, these securities potentially face appreciable liquidity risk as they tend to have smaller trading volumes and wider bid-ask spreads than Canadian nominal government securities. Presumably, investors require a premium for bearing the liquidity risk associated with holding RRBs, but the extent and time variation of this liquidity premium is unknown and apparently unresearched. This contrasts with numerous studies of the liquidity risk associated with U.S. Treasury Inflation-Protected Securities (TIPS) (e.g., Campbell et al. 2009; Pflueger and Viceira 2016; Christensen and Rudebusch 2019).

To estimate the natural rate of interest and long-term inflation expectations in the presence of liquidity and real and nominal term premiums, we use an arbitrage-free dynamic term structure model of nominal and real yields augmented with a liquidity risk factor. The identification of the liquidity risk factor comes from its unique loading for each individual RRB security as in Andreasen et al. (2021, henceforth ACR). Similar to ACR, our analysis combines a standard sample of nominal yields with the prices of individual RRBs. The underlying mechanism for identifying liquidity risk assumes that, over time, an increasing proportion of the outstanding inventory of each RRB is locked up in buy-and-hold investors’ portfolios. Given forward-looking investor behavior, this lock-up effect means that a particular bond’s sensitivity to the market-wide liquidity factor will vary depending on how seasoned the bond is and how close to maturity it is. In a careful study of nominal U.S. Treasuries, Fontaine and Garcia (2012) find a pervasive liquidity factor that affects all bond prices with loadings that vary with the maturity and age of each bond. By observing a cross section of RRB prices over time—each with a different time-since-issuance and time-to-maturity—we can identify the overall RRB liquidity factor and each bond’s loading on that factor. This technique is particularly useful for analyzing inflation-indexed debt when only a limited sample of bonds may be available as in our case.⁵

As in ACR, the frictionless Canadian nominal and real yields follow the model of nominal and real yields introduced in Christensen, Lopez, and Rudebusch (2010, henceforth CLR), referred to throughout as the CLR model. We estimate CLR models and their liquidity-augmented extension, denoted CLR-L models. The theoretical arbitrage-free formulation of the CLR-L model also provides identification of a time-varying real term premium in the pricing of RRBs. Identifying the liquidity and term premiums allows us to estimate the underlying frictionless real rate term structure and the natural rate of interest, which we measure as the average expected real short rate over a 5-year period starting five years ahead as in CR. Finally, to account for the zero lower bound (ZLB) that may have constrained the behavior of the nominal yields in our Canadian data periodically since the Global Financial Crisis (GFC), we also estimate a shadow-rate version of the CLR-L model that enforces the ZLB for the nominal yields.⁶

Our preferred estimate of the natural rate of interest, $rt*$⁠, is shown in Figure 1 along with measures of the 10-year nominal and real Canadian government bond yields. Both nominal and real long-term yields trended down together from the mid-1990s until early 2021 and sharply increased from late 2021 until the end of our sample, and this concurrence suggests little net change in inflation expectations or the inflation risk premium, which is supported by our results as evidenced by the estimated 10-year expected inflation also shown in Figure 1. Instead, the estimated equilibrium real rate is identified as the primary driver of the long-term yield declines since it fell from above 4% to below −1% in the 1996–2020 period. Accordingly, our results show that much of the 6-percentage-point decline in longer-term Canadian bond yields represents a reduction in the natural rate of interest, while 10-year inflation expectations are estimated to have declined only about 0.7 percentage point during that 25-year period before reversing about half of that modest decline during the last 3 years of our sample. Importantly, a sharp reversal in our $rt*$ estimate accounts for the significant joint increase in nominal and real yields since 2021.

Furthermore, we find that the average liquidity premiums embedded in the yields of RRBs exhibit notable time variation reaching highs of close to 35 basis points around the peak of the GFC in the fall of 2008, and another sharp spike of almost similar magnitude around the peak of the COVID-19 pandemic in spring 2020. However, outside of those two periods of market stress, the average liquidity premium has been in a fairly narrow, slightly negative range. For the entire sample, the estimated RRB liquidity premiums average −4.8 basis points. This can be compared to the results of ACR, who report that the average liquidity premium in U.S. TIPS is estimated at 34 basis points for the 1997–2013 period. The difference in liquidity premium levels across the TIPS and the Canadian RRB markets is likely to be due to the much greater relative liquidity of nominal versus indexed U.S. Treasury securities compared to the more modest liquidity advantage of Canadian fixed-coupon government securities over Canadian RRBs.

In line with the existing literature on TIPS, we rely on a joint modeling of the nominal and real yield curves (e.g. Christensen, Lopez, and Rudebusch 2010; Abrahams et al. 2016; D’Amico, Kim, and Wei 2018). Although these joint specifications can also be used to estimate the steady-state real rate similar to our analysis, this earlier work has emphasized only the measurement of inflation expectations and risk. Importantly, our methodology requires a stable dynamic relationship between the pricing factors of both nominal and real bonds. This assumption is particularly demanding during the period from 2009 to 2015 and again in the 2020–2021 period when the U.S. Federal Reserve kept the overnight federal funds rate at its effective ZLB, which drove Canadian short-term yields to historical lows as well. As a consequence, the dynamic interactions of Canadian short- and medium-term nominal yields were likely affected (see Swanson and Williams 2014; Christensen and Rudebusch 2015, for discussions). However, our results for the shadow-rate version of the CLR-L model that respects the ZLB for nominal yields are very similar to the results from the unconstrained CLR-L model. This suggests that the dynamic properties of Canadian nominal yields are at most mildly affected by the nonnegative constraint on the overnight policy rate set by the Bank of Canada.

To demonstrate the applicability of the CLR-L model for economic analysis, we estimate it at daily frequency and use it to examine the market reaction to the decision by the Canadian Finance Department to terminate RRB issuance. This sudden policy change was announced on November 3, 2022, and came as a surprise to most investors and market observers. Our results indicate that the increase in RRB yields in response to the announcement does not reflect an increase in the RRB liquidity premium, rather this premium exhibited a declining pattern in the months following the announcement. Thus, it does not appear as if investors positioned themselves for somewhat slower RRB market trading going forward. Overall, though, the market reaction was tempered, and the RRB market has continued to function on par with the past through the end of our sample. Hence, for now, RRB trading remains active despite no new issuance has come to market since before November 2022. These findings also support our usage of the RRB data through the end of our sample, and presumably will support their usage well into the future given that the longest dated outstanding RRB can be expected to continue to trade until 2054.

As another policy-relevant application of our model, we examine international spillover effects from foreign unconventional monetary policies. Here, we follow Christensen and Rudebusch (2012) and analyze the responses of Canadian government bond yields to announcements by the U.S. Federal Reserve and the Bank of England of plans to buy longer-term government debt, commonly referred to as QE, in response to the GFC. Using our preferred shadow-rate B-CLR-L model estimated at daily frequency to account for the near-zero short-term nominal yields at the time, we decompose the nominal yield responses into changes in expectations about future monetary policy and changes in term premiums. We also decompose the responses of the difference between nominal and real yields of comparable maturity, known as breakeven inflation (BEI), into changes in expected inflation and changes in the inflation risk premium. We find that U.S. QE announcements mainly lowered Canadian bond yields through their impact on risk premiums with much smaller effects on Canadian policy and inflation expectations. These findings appear consistent with strong portfolio rebalancing spillover effects, while signaling effects play at best a secondary role. Although U.K. QE announcements produce much smaller reactions in the Canadian bond markets, it remains the case that the spillover effects seem to operate through the risk premiums. Thus, the portfolio balance channel seems to be relatively more important than the signaling channel when it comes to international spillover effects, at least for Canada during the specific period we consider. We stress that the GFC period was characterized by major market dislocations and associated elevated liquidity risk premiums. As a consequence, spillover effects may be different in more normal times. However, we leave it for future research to examine that conjecture further.

The analysis in this article relates to several important literatures. Most directly, it speaks to the burgeoning literature on measurement of both the natural rate of interest and long-term inflation expectations. Second, our estimates of the real yield curve that would prevail without trading frictions have implications for asset pricing analysis on the true slope of the real yield curve. Third, our results relate to research on financial market liquidity. Indeed, the RRB liquidity premiums we estimate may serve as a benchmark for assessing liquidity premiums in other fixed-income markets in Canada and elsewhere. Furthermore, the paper is among the first to document what happens to trading and market dynamics when a government decides to terminate issuance of inflation-indexed debt. Finally, the paper contributes to the rapidly growing literature on the economic consequences of the COVID-19 pandemic.

The remainder of the article is structured as follows. Section 1 contains the data description, while Section 2 provides a description of the no-arbitrage term structure models we use and presents the empirical results, including an analysis of the RRB liquidity premium and its empirical determinants. Section 3 examines the Canadian bond market response to the surprise cancellation of all future RRB bond issuance. Section 4 analyzes our market-based estimates of long-term inflation expectations and the natural rate, while Section 5 is dedicated to an analysis of spillover effects on Canadian interest rates from U.S. and U.K. unconventional monetary policies. Finally, Section 6 concludes the article and offers directions for future research. Supplementary Appendix contains additional technical model details, estimation results, and robustness checks.

1 Canadian Government Bond Data

This section describes the Canadian government bond data used in our model estimation. As for the size of the Canadian government bond market, at the end of June 2024, the total outstanding notional amount of marketable bonds issued by the government of Canada was CAD1,076.5 billion of which CAD70.9 billion, or 6.6%, represented RRBs.⁷ The Canadian government bond market is equivalent to about 37% of Canadian nominal GDP, and the Canadian government holds a AAA rating with a stable outlook by all major rating agencies.

1.1 Nominal bonds

The Bank of Canada produces daily zero-coupon yield curves from a subset of the available universe of Canadian government fixed-coupon bonds using an “exponential spline model.”⁸ The database starts in January 1986 and is updated every Thursday with a two-week reporting lag. Note that, while the database contains the time series of maturities in quarterly increments from three months to thirty years, we limit our focus to a representative sample with the following eleven fixed maturities: 0.25, 0.5, 1, 2, 3, 5, 7, 10, 15, 20, and 30 years. Furthermore, we limit the sample to monthly data covering the period from January 31, 1991, to June 28, 2024, where the start date matches the launch of the RRB market in 1991 and also marks the initial introduction of the Bank of Canada’s current inflation targeting regime.⁹

Figure 2 shows time series of the zero-coupon yields with maturities of 3 months, 2 years, 5 years, 10 years, and 30 years. First, we note the downward trend of the general yield level between 1991 and 2021. The 10-year yield dropped from above 10% to below 1% during this 30-year period. Also notable is the sharp, partial reversal since then that has brought interest rates in Canada back up close to 4%. Second, as in U.S. Treasury yield data, there is clear business cycle variation in the shape of the yield curve around these persistent trends. It is these characteristics that are the practical motivation behind our choice of using a three-factor model for the Canadian nominal government bond yield curve, adopting an approach similar to what is standard for U.K. and U.S. nominal yield data; see Christensen and Rudebusch (2012). This choice is also supported by a principal component analysis of our nominal bond yield sample, as described in Supplementary Appendix B.

Finally, regarding the important question of a lower bound, the Bank of Canada only temporarily lowered its conventional policy rate to zero in response to the COVID-19 pandemic. As a consequence, the bond yields in the data have remained well above zero outside of this brief period. Thus, our benchmark model is an unconstrained Gaussian model. However, as a robustness check, we consider a shadow-rate specification of our benchmark model that respects the ZLB for the nominal yields. This allows us to assess the impact on our results from this constraint.

1.2 Real bonds

Since 1991, the Canadian government has issued RRBs, which have semi-annual interest payments that are adjusted for inflation using the changes in the all-items Canadian CPI without seasonal adjustment.¹⁰ These bonds have all been 30-year bonds and are issued only once every 3–5 years. As a consequence, there is a limited universe of RRBs with a total of nine bonds having been issued by the end of our sample. Despite the limited number of RRBs in the early years and their correspondingly narrow maturity range, we start our sample of RRB prices in December 1995, when the second RRB was issued and started trading. Coincidentally, December 1995 also happens to be the time when the Bank of Canada’s inflation target was lowered to its current 2% level; see Kozicki (2024). Crucially, this choice provides us with the longest possible perspective on persistent changes in Canadian real yields. Table 1 contains the contractual details of all nine RRBs as well as the number of monthly observations of each in our sample, while the time-varying maturity distribution of the nine RRBs in our sample is illustrated in Figure 3, where each security is represented by a downward-sloping line showing its remaining years to maturity at each date. Moreover, as noted by Gürkaynak et al. (2010) and ACR, prices of inflation-indexed bonds near their maturity tend to be somewhat erratic because of the indexation lag in their payouts. Therefore, to facilitate model estimation, we censor the prices of RRBs from our sample when they have less than one year to maturity.

Figure 4 shows the yields to maturity for all nine RRBs. The significant persistent fluctuations in real yields over this 28-year period are clearly visible. Canadian long-term real yields were close to 4% in the late 1990s and had dropped below zero by the time of the pandemic in spring 2020. Since then Canadian real yields have bounced back more than 2 percentage points. One empirical question is to what extent this persistent decline between 1996 and 2020 and the subsequent partial reversal represent changes in the natural real rate or are driven by other factors such as liquidity or other risk premiums.

1.2.1 The liquidity risk of RRBs

Before turning to our models and their estimation, we examine RRB bid-ask spreads to provide support for the ACR approach to identify liquidity premiums. The spreads are constructed by converting the bid and ask prices into the corresponding yield to maturity and calculating the difference with all data downloaded from Bloomberg. Figure 5 shows two series of bid-ask spreads for Canadian RRBs, one represents the bid-ask spread of the first ever RRB issued in 1991, the other tracks the bid-ask spread of the most recently issued (on-the-run) RRB—a sequence of different underlying RRBs. Both series are smoothed four-week averages and measured in basis points.¹¹ Similar to what ACR document for U.S. TIPS, the RRB bid-ask spreads are wider for more seasoned securities than for recently issued securities. Rational, forward-looking investors are aware of these dynamics and the fact that future market liquidity of a given security is likely to be below its current market liquidity. This gives rise to liquidity premiums in the security price that serve as compensation for assuming the risk that it may be difficult to sell the security back to the market in the future at a satisfactory price and without incurring higher transaction costs. This pattern in observed measures of current market liquidity of RRBs is consistent with the factor loading of the liquidity risk factor in our approach that is intended to model the effects on current RRB prices of expected future market liquidity conditions. Although we mainly think of these premiums as liquidity discounts, we note that, given their high credit quality, these bonds may be viewed by investors as very safe assets and hence trade at a safety premium; see Christensen and Mirkov (2022). We stress that the model we use is flexible enough to accommodate such outcomes.

2 Model Estimation and Results

In this section, we first detail the CLR-L model that serves as the benchmark in our analysis before we describe the restrictions imposed to achieve econometric identification of the model. We then compare its estimates to those from the CLR model without a liquidity adjustment and the shadow-rate B-CLR-L model that respects the ZLB for nominal yields. Finally, we describe the RRB liquidity premium implied by the estimated CLR-L model and examine its empirical determinants.

2.1 The CLR-L model

To begin, let $Xt=(LtN,St,Ct,LtR,Xtliq)$ denote the state vector of the five-factor CLR-L model. Here, $LtN$ and $LtR$ denote the level factor unique to the nominal and real yield curve, respectively, while S$t$ and C$t$ represent slope and curvature factors common to both yield curves. Finally, $Xtliq$ represents the added liquidity risk factor unique to the RRB prices.

The instantaneous nominal and real risk-free rates are defined as

Note that the differential scaling of the real rates to the common slope factor is captured by the parameter $αR$⁠.¹²

The risk-neutral $Q$-dynamics of the state variables used for pricing are given by

where $Σ$ is assumed to be a diagonal matrix as per Christensen et al. (2011).

Based on the $Q$-dynamics above, nominal zero-coupon bond yields preserve a Nelson and Siegel (1987) factor loading structure

where $AN(τ)$ is a convexity term that adjusts the functional form in Nelson and Siegel (1987) to ensure absence of arbitrage (see Christensen et al. 2011). The assumed Nelson and Siegel (1987) factor structure for the nominal yields is supported by a principal component analysis of our nominal yield data; see Supplementary Appendix B.

On the other hand, due to the lower liquidity in the market for real bonds, real yields are sensitive to liquidity pressures. As a consequence, the pricing of RRBs is not performed with the standard real discount function, but rather with a discount function that accounts for the liquidity risk:

where $t0i$ denotes the date of issuance of the specific real bond and $βi$ is its sensitivity to the variation in the liquidity factor. Furthermore, the decay parameter $λL,i$ is assumed to vary across securities as well.

ACR show that the net present value of one consumption unit paid by real bond i at time $t+τ$ has the following exponential-affine form

which implies that the model belongs to the class of Gaussian affine term structure models. Note also that, by fixing $βi=0$ for all i, we recover the CLR model.

Now, consider the whole value of the real bond i issued at time $t0i$ with maturity at $t+τi$ that pays an annual coupon C^R semi-annually. Its price is given by¹³

Unlike U.S. TIPS, Canadian RRBs have no embedded deflation protection option, which makes their pricing straightforward. The only minor omission in the bond price formula above is that we do not account for the lag in the inflation indexation of the real bond payoff. Grishchenko and Huang (2013) and D’Amico, Kim, and Wei (2018) find that this adjustment normally is within a few basis points for the implied yield on U.S. TIPS. It is likely to be very small for our Canadian data as well as our sample is dominated by very long-term bonds where this adjustment tends to be tiny.

Finally, within the CLR-L model described above, nominal bonds are assumed to have a minimum of liquidity risk, which is consistent with the empirical findings of Andreasen et al. (2019). Therefore, the nominal bonds are valued using the standard nominal zero-coupon yield described in Equation (3).

So far, the description of the CLR-L model has relied solely on the dynamics of the state variables under the $Q$-measure used for pricing. However, to complete the description of the model and to implement it empirically, we will need to specify the risk premiums that connect the factor dynamics under the $Q$-measure to the dynamics under the real-world (or historical) $P$-measure. It is important to note that there are no restrictions on the dynamic drift components under the empirical $P$-measure beyond the requirement of constant volatility. To facilitate empirical implementation, we use the essentially affine risk premium specification introduced in Duffee (2002). In the Gaussian framework, this specification implies that the risk premiums $Γt$ depend on the state variables; that is,

where $γ0∈R5$ and $γ1∈R5×5$ contain unrestricted parameters; see Supplementary Appendix A.

Thus, the resulting unrestricted five-factor CLR-L model has $P$-dynamics given by

This is the transition equation in the extended Kalman filter estimation of the CLR-L model.

2.1.1 The shadow-rate B-CLR-L model

The CLR-L model relies on the standard affine specification for the nominal short rate, which does not enforce the ZLB. But, the policy rate in Canada was constrained by the ZLB in the first years following the COVID-19 pandemic, and possibly even occasionally in the earlier period from January 2009 to December 2015 when the U.S. monetary policy rate was constrained by the ZLB and Canadian nominal yields were low for that reason. To enforce this lower bound within our benchmark CLR-L model, we follow Black (1995) and replace $rtN$ in Equation (1) by $rtN=max{LtN+St,0}$ and solve for nominal yields using the approximation in Christensen and Rudebusch (2015), but the model is otherwise identical to the CLR-L model described above and denoted the B-CLR-L model.

2.2 Model estimation and econometric identification

While nominal yields in the CLR-L model have a standard affine formulation given by Equation (3), RRB prices are nonlinear functions of the state variables.¹⁴ Due to this nonlinearity of the RRB pricing formulas, the model cannot be estimated with the standard Kalman filter. Instead, we use the extended Kalman filter as in Kim and Singleton (2012); see ACR for details. To make the fitted errors comparable across RRBs of various maturities, we follow ACR and scale each bond price by its duration. Thus, the measurement equation for the real bond prices takes the following form:

where $P^tR(t0i,τi)$ is the model-implied price of real bond i and $DtR(τi)$ is its duration, which is fixed and calculated before estimation. See Andreasen et al. (2019) for evidence supporting this formulation of the measurement equation.

Since the liquidity risk factor is a latent factor that we do not observe, its level is not identified without additional restrictions. As a consequence, we let the first Canadian government RRB issued, that is, the thirty-year RRB with 4.25% coupon issued on December 10, 1991, with maturity on December 1, 2021, have a unit loading on the liquidity factor, that is, $βi=1$ for this security. This choice implies that the $βi$ sensitivity parameters measure liquidity sensitivity relative to that of the thirty-year 2021 RRB.

Furthermore, we note that the $λL,i$ parameters can be hard to identify if their values are too large or too small. As a consequence, we impose the restriction that they fall within the range from 0.0001 to 10, which is without practical consequences as demonstrated by ACR. Also, for numerical stability during the model optimization, we impose the restriction that the $βi$ parameters fall within the range from 0 to 250, which turns out not to be a binding constraint at the optimum.

Finally, we assume that all nominal yield measurement equations have i.i.d. fitted errors with zero mean and standard deviation $σεN$⁠. Similarly, all RRB price measurement equations have fitted errors that are assumed to be i.i.d. with zero mean and standard deviation $σεR$⁠.

2.3 Estimation results

This section presents our benchmark estimation results. In the interest of simplicity, focus is in this section devoted to a version of the CLR-L model where $KP$ and $Σ$ are diagonal matrices. As shown in ACR, these restrictions have hardly any effects on the estimated liquidity premium for each RRB, because it is identified from the model’s $Q$-dynamics, which is independent of $KP$ and only display a weak link to $Σ$ through the small convexity adjustment in yields. Furthermore, we stress that we relax this assumption in Section 4 when we analyze estimates of long-term inflation expectations and $rt*$⁠, which are indeed sensitive to the specification of the models’ $P$-dynamics.

Given that the CLR-L model includes standard nominal Canadian government zero-coupon bond yields, it seems natural to first explore how well it fits nominal yields. Table 2 documents that it provides a very satisfying fit to all nominal yields, where the overall root mean-squared error (RMSE) is just 12.95 basis points. The corresponding CLR model without a liquidity factor gives broadly a similar fit with an overall RMSE of 13.50 basis points, while the B-CLR-L model that respects the ZLB of nominal yields provides a slightly better fit with an overall RMSE of 12.66 basis points.¹⁵ Thus, accounting for the liquidity risk of RRBs does not affect the ability of the CLR-L model to match nominal yields, which is consistent with the results in ACR. Still, we note some signs of poorer fit in the very long end of the nominal yield curve across the models.

The impact of accounting for liquidity risk is, however, more apparent in the RRB market. The first two columns in Table 3 show that the RRB pricing errors produced by the CLR model indicate a reasonable fit, with an overall RMSE of 7.61 basis points. The following two columns reveal a substantial improvement in the pricing errors when correcting for liquidity risk, as the CLR-L model has a very low overall RMSE of just 2.53 basis points, while the next two columns show that the B-CLR-L model provides an equally accurate fit with an overall RMSE of 2.57 basis points. Hence, accounting for liquidity risk leads to a notable improvement in the ability of our benchmark model to explain RRB market prices.

The remaining columns of Table 3 report the estimates of the specific parameters attached to each RRB within the CLR-L and B-CLR-L models. We note that the 4 most recently issued RRBs in our sample are not exposed to liquidity risk, as their $βi$s are insignificantly different from zero at the conventional 5% level. In contrast, in the B-CLR-L model, it is only the most recently issued RRB that is insensitive to liquidity risk. Although the estimated values of $λL,i$ vary notably across securities, this does not imply major differences in the sensitivity of the RRBs to the liquidity risk factor as shown in Figure 6. The liquidity adjustment for all RRBs is increasing in t due to the strong mean-reversion in $Xtliq$ under the $Q$-measure (⁠$κliqQ=12.05$ according to Table 4). Thus, liquidity risk operates as a traditional slope factor within the CLR-L model, although its steepness varies across the universe of RRBs.

The remaining estimated model parameters are provided in Table 4, which shows that the dynamics of the four frictionless factors, $(LtN,St,Ct,LtR)$⁠, are very similar across the CLR, CLR-L, and B-ClR-L models, both under the $P$- and the $Q$-measure. We draw the same conclusion from Figure 7, which plots the estimated factors in all three models. The differences tend to be short-lived and barely noticeable. For that reason it is also the case that the frictionless instantaneous real rates $rtR=LtR+αRSt$ are almost indistinguishable across the three models as documented in Figure 7f. Finally, Figure 7e shows the estimated liquidity factors, $Xtliq$⁠, which are unique to the CLR-L and B-CLR-L models and also very similar.

2.4 The estimated RRB liquidity premium

We now use the estimated CLR-L model to extract the liquidity premium in the RRB market. To compute this premium, we first use the estimated parameters and the filtered states ${Xt|t}t=1T$ to calculate the fitted RRB prices ${P^tRRB,i}t=1T$ for all outstanding securities in our sample. These bond prices are then converted into yields to maturity ${y^tc,i}t=1T$ by solving the fixed-point problem

for $i=1,2,…,nRRB$⁠, meaning that ${y^tc,i}t=1T$ is approximately the real rate of return on the ith RRB if held until maturity (see Sack and Elsasser 2004). To obtain the corresponding yields with correction for liquidity risk, a new set of model-implied bond prices are computed from the estimated CLR-L and B-CLR-L models but using only their frictionless part, i.e., using the constraints that $Xt|tliq=0$ for all t as well as $σ55=0$ and $θliqQ=0$⁠. These prices are denoted ${P˜tRRB,i}t=1T$ and converted into yields to maturity $y˜tc,i$ using Equation (4). They represent estimates of the prices that would prevail in a world without any financial frictions. The liquidity premium for the ith RRB is then defined as

Figure 8 shows the average RRB liquidity premium $Ψ¯t$ across the outstanding RRB at a given point in time. The average estimated RRB liquidity premium clearly varies notably over time with a maximum of 34 basis points achieved at the peak of the financial crisis and a low of −31 basis points in the spring of 2000. Note also the sharp short-lived spike at the onset of the COVID-19 pandemic in spring 2020. Importantly for our analysis, they are stationary near zero with an average for the entire period equal to −4.84 basis points and a standard deviation of 10.79 basis points.

In Figure 8, we also show the average RRB liquidity premium implied by the B-CLR-L model that accounts for the asymmetric behavior of nominal yields near the ZLB. We note that accounting for this asymmetry matters little for the estimated RRB liquidity premiums, although the estimated liquidity premiums implied by the shadow-rate B-CLR-L model tend to be slightly below those implied by the unconstrained Gaussian CLR-L model.¹⁶

In summary, these results allow us to conclude with great confidence that changes in RRB liquidity premiums are not a factor behind the persistent decline in Canadian long-term interest rates observed in Figure 1.¹⁷ Still, thanks to their significant time variation, it is important to account for the liquidity premiums in our analysis as we demonstrate later on.

2.4.1 Determinants of the RRB liquidity premium

In this section, we aim to show that the RRB liquidity premium is related to observable proxies for liquidity risk as well as factors that are fundamental determinants of the prices of RRBs. For ease of exposition, we limit our focus to the estimated RRB liquidity risk premium series implied by our benchmark CLR-L model.

The first variable we consider is the VIX options-implied volatility index, which represents near-term uncertainty in the Standard & Poor’s 500 stock market index. This is a widely used gauge of investor fear and risk aversion. The second variable is the yield difference between the seasoned (off-the-run) 10-year U.S. Treasury as provided by Gürkaynak, Sack, and Wright (2007) and the most recently issued (on-the-run) U.S. Treasury of the same maturity from the H.15 series at the Federal Reserve Board of Governors. The on-the-run security is typically the most traded security and therefore penalized the least in terms of liquidity premiums, which explains the mostly positive spread. For our analysis, the important thing to note is that, if there is a wide yield spread between liquid on-the-run and comparable seasoned U.S. Treasuries, we would expect liquidity premiums in the Canadian bond market to also be elevated. The third variable is the U.S. TED spread, which is calculated as the difference between the three-month U.S. LIBOR and the three-month U.S. T-bill interest rate. This spread represents a measure of the perceived general credit risk in global financial markets. To control for factors related to the uncertainty about the interest rate environment, we include the MOVE index. The next variable is the 10-year U.S. Treasury yield from the Federal Reserve’s H.15 database, which is included to control for reach-for-yield effects. This may be particularly relevant for our sample during the period between December 2008 and December 2015 and again in the 2020–2021 period when U.S. short-term interest rates were constrained by the ZLB. Our final two variables represent domestic factors that are fundamental to the prices of RRBs, namely Canadian CPI inflation and the Canadian government debt-to-GDP ratio.¹⁸

We now run standard linear regressions to more formally assess the relative importance of each of these seven variables. First, we run regressions with each explanatory variable in isolation with the results reported in columns (1)–(7) of Table 5. The VIX, the U.S. Treasury on-the-run premium, the TED spread, and the MOVE index all have estimated positive coefficients on a stand-alone basis, but none of them are statistically significant at the conventional 5% level. In contrast, the 10-year U.S. Treasury yield, the Canadian CPI inflation, and the Canadian government debt-to-GDP ratio all have highly statistically significant negative coefficients in these stand-alone regressions.

A 1 percentage point increase in the 10-year U.S. Treasury yield is associated with a decline in the average RRB liquidity premium of slightly more than 3 basis points. During most of our sample, U.S. long-term interest rates were declining thanks to slowing growth and low inflation. In that environment, inflation-indexed bonds like RRBs do not represent particularly attractive investment opportunities, which may explain the tendency for the RRB liquidity premiums to be slightly higher under those circumstances.

A 1 percentage point increase in the Canadian CPI inflation tends to coincide with RRB liquidity premiums that are almost 2 basis points lower. Given that higher domestic CPI inflation directly increases the cash flows of RRBs, we interpret the decline in the RRB liquidity premium as a sign that RRBs are more convenient assets to hold under those conditions.

Lastly, a 1 percentage point increase in the Canadian government debt-to-GDP ratio is associated with a decline in the average RRB liquidity premium of 0.6 basis point. Since greater government debt likely requires increased issuance of RRBs, we interpret the negative coefficient as a positive supply effect whereby increased RRB issuance leads to more trading and lower financial frictions in the market for these bonds.

These encouraging preliminary results made us look for a preferred regression model the results for which are reported in column (8) of Table 5. In addition to the three significant variables discussed above, the preferred model includes the MOVE index that now has a highly statistically significant positive coefficient. This implies that increased uncertainty about the near-term interest rate environment in the U.S. tends to coincide with higher liquidity premiums in the RRB market. Moreover, this preferred regression produces an adjusted R² of 0.57.

For completeness, the final column of Table 5 reports the results of regressing the average RRB liquidity premium from the CLR-L model on all seven explanatory variables jointly. We note that this only produces a slightly higher adjusted R² of 0.59. Moreover, the four explanatory variables from our preferred regression model preserve the sign, size, and high statistical significance of their coefficients in this joint regression. We interpret this as a sign of the robustness of our preferred regression model, even though we note that the model also leaves a notable part of the variation in the RRB liquidity premium series unexplained.

3 The Termination of the RRB Program

In this section, we examine the market response to the Canadian government’s decision to terminate the RRB program announced on November 3, 2022.

3.1 The announcement

The decision to cease issuance of RRBs was made by the Canadian Department of Finance and communicated to the public through its 2022 Fall Economic Statement released on November 3, 2022. In Annex 2, page 69, a brief section on the RRB program contains the following notable paragraph:¹⁹

“The government has decided to cease issuance of Real Return Bonds (RRBs) effective immediately. This decision reflects low demand for this product and will allow the government to promote liquidity by consolidating funding within our core funding sectors.”

The announcement caught investors and market observers by surprise as there had been no prior indication of any changes to the RRB program. Moreover, conversations with staff at the Bank of Canada, who act as the debt manager for the Department of Finance, indicate that the Bank was not consulted in this matter.

In response, one asset manager, SLC Management, issued a note on November 4, 2022, stating:²⁰

“Real return bond markets froze after the announcement. We expect that over the next few days institutional investors, investment managers and index providers will have discussions on a path forward for their existing products and allocations. We suspect that total return investors may no longer wish to hold on to their RRBs as the asset class becomes increasingly illiquid, and this could result in some additional supply in the short to medium term. However, investors that hold real return bonds to back inflation-linked liabilities, such as pension plans and insurance companies, will likely hold their existing positions to maturity and may look to absorb any additional supply to satisfy any future planned RRB purchases.”

According to the same note, the missing issuance in the fiscal year that would end on March 31, 2023, would equal about 0.5% of all government bonds, or CAD1.4 billion. Given that RRBs are only issued with long maturities, this would represent about 6% of the 30-year government bond issuance in that fiscal year.

Regarding any market implications of this announcement, we note that, on one hand, it seems investors feared selling pressures and associated market illiquidity. However, at the same time, there is reason for cautious optimism that some current long-term holders of RRBs, such as pension funds, would remain willing buyers knowing that no further supply would come to market. The purpose of this section is to assess both the immediate market reaction to this surprise announcement as well as any longer-term impact on the functioning of the RRB market.

3.2 The market reaction

In the following, we assess the impact of the termination of all future RRB issuance on both RRB yields directly and on our estimated RRB liquidity risk premiums.

To begin, we examine the immediate market reaction to the surprise announcement. Table 6 reports the one- and two-day yield changes for the RRBs outstanding at the time of the announcement. The results indicate a notable market reaction with a two-day change in the observed RRB yields of about 10 basis points in the 4- to 28-year maturity range.

Since the responses in Table 6 reflect changes in yields to maturity, they are sensitive to both the bond coupon sizes and the shape of the underlying real yield curve and therefore hard to interpret and compare across bonds. For a cleaner read, Table 7 reports the response of fitted real zero-coupon yields. We focus on the important 5- to 10-year maturity range that is commonly used in the construction of BEI measures. Note the initial one-day reaction between 6 and 8 basis points, which increases to a material response between 10 and 23 basis points with a two-day event window.

Although this appears like a strong market reaction, this assessment is tempered by the fact that nominal yields increased by a similar magnitude as documented in Table 8. This is suggestive of Canadian bond markets reacting for other reasons, including the U.S. Federal Open Market Committee (FOMC) meeting on November 2, 2022, when the FOMC decided to raise the target for the overnight federal funds rate by 0.75%.

Still, at face value, the market reaction suggests that investors wanted to get out of holding RRBs following the announcement. Alternatively, they may have become concerned about the trading conditions in the RRB market going forward and demanded a higher liquidity premium as a result.

3.3 Impact on RRB market conditions

In this section, we explore the latter conjecture above by examining data on trading conditions for evidence of any dislocations to the functioning of the RRB market.

First, we examine bid-ask spreads of the outstanding set of RRBs. We think of these spreads as representative measures of the current trading conditions in the market for these bonds. Figure 9 shows four-week moving averages of the bid-ask spreads for each RRB from January 2020 through the end of our sample. We first note the general upward trend in the bid-ask spread series that is also evident in Figure 5 and caused by the fact that the bonds become more seasoned and less liquid as time passes. In addition, we do see some tendency for greater volatility of the bid-ask spreads in 2023, but this change seems to have started already in late 2021, well before the November 2022 announcement. We speculate that this increase in volatility may reflect the monetary policy tightening cycle on-going at the time as dealers and intermediaries manage their bond holdings more actively to avoid losing money on long-term bond inventories in a rising interest rate environment. Importantly, though, there is no major change in the general bid-ask spread levels in the period following the November 3, 2022, announcement. Thus, the RRB trading conditions do not seem to have fundamentally changed.

Second, we assess whether there seems to have been any impact on the performance of the CLR-L model and its ability to fit the RRB bond prices. To that end, we estimate the CLR-L model using daily data instead of the monthly frequency considered so far. Figure 10 shows the resulting average estimated RRB liquidity premium series since the start of 2020 through the end of June 2024. First, there is hardly any reaction in the estimated RRB liquidity premium in the days following the announcement to cease all future RRB issuance. This contrasts with the market movements at the peak of the pandemic in spring 2020 when the RRB liquidity premium series spiked up more than 30 basis points. Thus, based on our model results, investors did not seem to worry much about the future liquidity in the RRB market despite no new supply being issued.

Second, in the time since the announcement, the estimated RRB liquidity premium has declined and turned even more negative. This suggests that RRBs have become scarce and now trade at a notable convenience premium. Crucially, this decline underscores the advantage of our model approach that allows us to account for the variation in these bond-specific premiums in individual RRB prices.

As for the model fit specifically, Figure 11 shows the daily fitted error series for all 9 RRBs in our sample going back to December 1995. We note some tendency for larger and more volatile fitted errors since late 2021, but it is not clear that it is tied to the November 3, 2022, announcement.

To examine this in greater detail, Table 9 reports the summary statistics of the fitted errors for each RRB broken down into the pre- and post-announcement period with the full sample statistics reported for completeness. We note somewhat higher fitted errors for five of the RRBs in the period since the termination of RRB issuance, but this is partially offset by improved fit for the remaining three RRBs. As a result, the overall fit has only deteriorated modestly. Hence, the CLR-L model clearly remains able to fit and explain the variation in the entire RRB yield curve.

3.4 Summary

To summarize, despite a negative initial reaction with RRB prices declining, or equivalently, RRB yields rising, the decision by the Canadian Finance Department to cease all future RRB issuance does not seem to have impacted RRB market conditions much, if at all.

With a mildly negative RRB liquidity premium from November 2022 through the end of our sample, there is little that points to disadvantageous trading conditions of RRBs in the secondary market. Data for bid-ask spreads of individual RRBs support this interpretation. Moreover, the CLR-L model remains able to provide a very close fit to the RRB bond price data. As a consequence, it remains valid to continue to use the RRB bond price data despite the decision by the Canadian government to terminate all future RRB issuance.

Furthermore, based on our results, RRB liquidity premiums actually declined in the year after the announcement. Thus, the liquidity discount has become a scarcity convenience premium. This suggests that, rather than terminating the RRB program, the Canadian government could benefit from resuming and increasing RRB issuance.

With remaining maturities up to 30 years, it is feasible for the existing set of RRBs to continue to trade for several decades, even without any new supply coming to market. Thus, it seems recommendable to continue to use the RRB data for both market monitoring and policy analysis.

Overall, this decision appears to be short sighted. The Canadian government spent thirty years on establishing this market and incurred the high real rates early on visible in Figure 4, only to shut down the program at the first sight of a meaningful spike in inflation. This comes across as a less than well contemplated decision. Furthermore, both fiscal and monetary policy benefit from access to directly observable measures of real yields. As we document in this paper, we can measure and decompose BEI and track the underlying inflation expectations among investors in the Canadian government bond market. Moreover, the RRB market offers direct measures of real yields that can be decomposed to obtain estimates of the natural real rate that can be used to assess the stance of monetary policy—as we demonstrate later on—and the sustainability of public finances. Finally, this analysis provides a concrete example of how a model like ours can leverage the RRB pricing data to examine the implications of important policy decisions like the one made by the Finance Department when it ceased all future RRB issuance.

4 A New Normal for Canadian Interest Rates?

After first defining BEI and the natural real rate, we select a preferred specification of the CLR-L model’s objective $P$-dynamics consistent with the data. We use this preferred CLR-L model to account for liquidity and term premiums in RRB prices and obtain estimates of investors’ long-term inflation expectations and an associated measure of the equilibrium real rate, which we compare to other market- and macro-based estimates from the literature. We also leverage our natural real rate estimate to measure the stance of monetary policy in Canada. Finally, we end the section using our preferred model to make 10-year projections for both long-term expected inflation and the equilibrium real rate as of June 2024.

4.1 Definition of BEI and the natural rate

To begin the analysis in this section, we note that BEI is defined as

that is, the difference between nominal and real yields of the same maturity, which can then be decomposed into the sum of expected inflation $πte(τ)$ and the inflation risk premium $ϕt(τ)$⁠.

CLR show that the market-implied average rate of inflation expected at time t for the period from t to $t+τ$ is

and the associated inflation risk premium for the same time period is

where $Πt$ is the price level and $MtR$ is the real stochastic discount factor.

This last equation highlights that the inflation risk premium can be positive or negative. It is positive if and only if

That is, the riskiness of nominal bonds relative to real bonds depends on the covariance between the real stochastic discount factor and inflation, and is ultimately determined by investor preferences.

Furthermore, following CR, our definition of the equilibrium real rate of interest $rt*$ is

that is, the average expected real short rate over a 5-year period starting five years ahead where the expectation is with respect to the objective $P$-probability measure. As noted in the introduction, this 5yr5yr forward average expected real short rate should be little affected by short-term transitory shocks.²¹ Alternatively, $rt*$ could be defined as the expected real short rate at an infinite horizon as discussed in Bauer and Rudebusch (2020). However, this quantity will depend crucially on whether the factor dynamics exhibit a unit root. As is well known, the typical spans of time series data that are available do not distinguish strongly between highly persistent stationary processes and non-stationary ones. Our model follows the finance literature and adopts the former structure, so strictly speaking, our infinite-horizon steady state expected real rate is constant. However, our data sample likely has insufficient information in the 10-year to infinite horizon to definitively pin down that steady state.

4.2 Model selection

For decompositions of BEI and estimation of the natural real rate and associated inflation, nominal, and real risk premiums, the specification of the mean-reversion matrix $KP$ is critical as noted earlier. To select the best fitting specification of the CLR-L model’s real-world dynamics, we use a general-to-specific modeling strategy in which the least significant off-diagonal parameter of $KP$ is restricted to zero and the model is re-estimated. This strategy of eliminating the least significant coefficient is carried out down to the most parsimonious specification, which has a diagonal $KP$ matrix. The final specification choice is based on the value of the Bayesian information criterion (BIC) as in Christensen, Lopez, and Rudebusch (2014).²²

The summary statistics of the model selection process are reported in Table 10. The BIC is minimized by specification (17), which has a $KP$ matrix given by

The estimated parameters of the preferred specification are reported in Table 11. The estimated $Q$-dynamics used for pricing and determined by (⁠$Σ$⁠, $λ$⁠, $αR$⁠, $κliqQ, θliqQ$⁠) are close to those reported in Table 4 for the CLR-L model with diagonal $KP$⁠. This implies that both model fit and the estimated RRB liquidity premiums from the preferred CLR-L model are similar to those already reported and therefore not shown here, but can be found in Supplementary Appendix C. Furthermore, the estimated objective $P$-dynamics in terms of $θP$ and $Σ$ are also qualitatively similar to those reported in Table 4. Finally, we note that the liquidity factor matters for the expected excess return of nominal bonds through $κ25P$ in addition to its effect on RRB pricing, while the slope factor is important for the expected return of both nominal bonds and RRBs. This comes about both through its own direct effect on their pricing and through its dynamic interactions with the other state variables. In contrast, the third factor, i.e. the curvature factor, is the only independent factor without any dynamic interactions with any of the other state variables.

4.3 Empirical BEI decomposition

In this section, we describe the decomposition of the 10-year BEI implied by our estimation results.

The starting point for the decomposition is the fitted 10-year BEI rate from the CLR model, which offers the cleanest and most straightforward fit of the raw bond data without any adjustments. This measure of 10-year BEI is shown in Figure 12. Now, the estimated 10-year frictionless BEI from the CLR-L model, which does not contain any RRB liquidity risk premiums, is also shown in Figure 12. It fluctuates above and below the 10-year fitted BEI as the estimated RRB liquidity premiums switch sign. Hence, the difference between these two series represent an alternative measure of the RRB liquidity premiums, which is different from the estimate in Figure 8 as the former has a constant maturity.

As explained in Section 4.1, the CLR-L model also provides a decomposition of the estimated 10-year frictionless BEI into an expected CPI inflation component and the associated inflation risk premium both shown in Figure 12.

The 10-year inflation risk premium is variable and was mostly positive until mid-2015. Since then the Canadian inflation risk premium seems to have switched sign and has remained mostly mildly negative.²³ Notably, it dropped back into negative territory the last year of our sample despite the recent spell of high inflation also on vivid display in Figure 12. This premium was also significantly negative in the late 1990s when Canadian CPI inflation barely averaged 1%. Moreover, the inflation risk premium experienced a temporary softening around the peak of the financial crisis in late 2008, when CPI inflation in Canada and elsewhere started to fall; see Christensen, Lopez, and Rudebusch (2012) for a U.S. analysis of this episode. It also turned negative briefly in late 2015 and early 2016 when global energy and commodity prices fell sharply. In addition, it fell further into negative territory around the peak of the COVID-19 pandemic in spring 2020 when price inflation also feel sharply. Finally, we note that it did temporarily turn positive during the post-pandemic economic reopening when inflation in Canada and elsewhere spiked up to levels not seen in more than 40 years. Based on these observations we consider the estimated inflation risk premium from the CLR-L model to be reasonable in terms of both its level and time-series variation.

Most existing studies have found inflation risk premiums to be positive on average and relatively stable; see Ang, Bekaert, and Wei (2008), Buraschi and Jiltsov (2005), and Hördahl and Tristani (2014), among many others. Thus, the tilt toward a negative inflation risk premium the past decade seems to be a more recent and novel development that merits monitoring going forward. At face value, it entails that investors in the Canadian bond markets fear future recessions happening in the context of disinflationary price dynamics similar to the GFC and the pandemic. In that regard, the recent spell of high inflation is not very informative in shedding light on the crucial question of the sign of the inflation risk premium because it happened in the context of a booming economy in both Canada and the United States.

In comparison with the inflation risk premium, the estimated 10-year inflation expectations are much less variable, although still characterized by a very persistent and gradual decline since the mid-1990s when bond investors’ long-term inflation expectations were about a quarter point above the Bank of Canada’s 2% inflation target. By the time of the pandemic, these expectations had fallen to a low of 1.52%, or about a 0.7 percentage point drop, before partially reversing that decline during the last three years of our sample to a level of 1.82% by June 2024. This contrasts with the responses to the Consensus Forecasts survey of professional forecasters, who twice a year are asked about their expectations for inflation over the following 10 years. The mean responses in each survey since October 1995 are shown with blue crosses in Figure 12 and have remained very close to 2% since the late 1990s. As a consequence, the forecasters view the gyrations in Canadian long-term BEI rates to be entirely driven by changes in the inflation risk premiums.

In support of the decline in our preferred CLR-L model’s estimated 10-year expected inflation, we note that Canadian 12-month CPI inflation averaged 2.04% in the 12 and a half years from January 1996 through the end of June 2008. In contrast, it only averaged 1.58% in the 12 and a half years from July 2008 through the end of December 2020. Thus, the decline in the model-implied 10-year expected inflation series is of a magnitude similar to the decline in actual inflation outcomes during this 25-year period.

We take the modest increase in the 10-year expected inflation in the 2021-2024 period against the backdrop of the outsized increase in realized inflation as a sign that investors’ inflation expectations in Canada were, and continue to remain, well anchored close to the Bank of Canada’s 2% inflation target.

4.4 Estimates of the natural rate

Our market-based measure of the natural rate is the average expected real short rate over a 5-year period starting five years ahead. This 5yr5yr forward average expected real short rate should capture the persistent trends in the natural real rate.

Figure 13 shows the preferred CLR-L model decomposition of the 5yr5yr forward frictionless real yield based on the standard definition of the real term premium

where $ytR(τ)$ is the fitted frictionless real zero-coupon yield with maturity in $τ$ years; see Supplementary Appendix A for details. The solid grey line is the 5yr5yr forward real term premium, which has fluctuated around a fairly stable level since 1996 with spikes in September 2001, during the GFC in the 2008–2009 period, and also during the peak of the COVID-19 pandemic in spring 2020. Hence, as suggested by theory, the term premium is countercyclical and elevated during economic recessions. In contrast, the estimate of the natural real rate of interest implied by the CLR-L model—the black line in Figure 13—shows a very gradual and persistent decline from above 4% in late 1995 to below −1% by spring 2020 followed by a sharp partial reversal during the last three years of the sample that leaves it at 1.83% by June 2024. Thus, much of the downward trend in the 5yr5yr forward real yield in the 1995–2020 period is driven by declines in this measure of $rt*$⁠. Similarly, the increase in the 5yr5yr forward real yield since 2021 is fully accounted for by a matching increase in $rt*$⁠, while the real term premium has declined slightly on net during this most recent three-year period.

To examine the sensitivity of our $rt*$ estimate to our choice to define $rt*$ as the average expected real short rate over a 5-year period starting 5 years ahead, we consider three alternative definitions that all embed a longer view about the time it takes for the economy to reach steady state. The first assumes that this takes 7 years and then measures the neutral real rate as the average expected real short over the following 3 years. It is referred to as the 7yr3yr $rt*$ estimate. The second alternative takes an even longer view and assumes that it takes nine years to reach steady state and then measures the neutral real rate as the average expected real short rate over a short 1-year period. It is referred to as the 9yr1yr $rt*$ estimate. Finally, the third alternative takes the longest view and assumes that it takes 10 years for the economy to reach steady state and then measures $rt*$ as the average expected real short rate over a 5-year horizon as in our benchmark definition. It is referred to as the 10yr5yr $rt*$ estimate. Figure 14 shows all four $rt*$ estimates, which are very close to each other thanks to the high estimated persistence of the state variables within our preferred CLR-L model. Hence, our $rt*$ estimate of the neutral real rate has very little sensitivity to the specific definition of $rt*$ used. Thus, our reported results are very robust from that perspective.

4.4.1 Comparison of estimates of the natural rate

In this section, we compare other existing estimates of the equilibrium or natural interest rate in the literature to our Canadian estimate. To start, we compare the Canadian $rt*$ estimate from our preferred CLR-L model to the U.S. finance-based estimate generated by CR using solely the prices of U.S. TIPS. These two market-based estimates of the natural rate are shown in Figure 15. Their high positive correlation and similar downward trend for the overlapping period are both evident. Moreover, they share the feature that much of their common declines happened outside of the GFC. This suggests that the factors depressing both Canadian and U.S. interest rates during the overlapping period are likely to be global in nature and do not just reflect the immediate financial market dislocations during the crisis.

Now, we turn to the crucial comparison of our finance-based estimate of $rt*$ with estimates based on macroeconomic data. Figure 16 shows the $rt*$ estimate from our preferred CLR-L model, along with the macro-based estimate of $rt*$ from Holston, Laubach, and Williams (2017, henceforth HLW), which is the filtered estimate generated by applying the approach described in Laubach and Williams (2003) to Canadian macroeconomic series. The $rt*$ estimate from HLW starts in 1961 at a level above 5%. Since then it has followed a steady declining trend that leaves it close to 1.5% by the end of 2023 and hence close to the finance-based estimate.

The second macro-based estimate of $rt*$ is taken from Del Negro et al. (2019, henceforth DGGT). They estimate a flexible vector autoregression model with common trends for a sample of data from seven advanced economies, including Canada, covering the period from 1870 to 2016 and here extended through the end of 2020.²⁴ They use annual data on short- and long-term government bond yields, consumer prices, and real consumption per capita in addition to Moody’s Baa corporate bond yields. In their analysis, it is an assumption of no arbitrage in the long run that implies a factor structure for the trend of real interest rates across countries. They find that real interest rates across these seven countries share a global common trend that has been particularly pronounced since the 1970s. Moreover, using regression analysis, they find that declining consumption growth and increasing convenience yields from the safety and liquidity offered by government bonds from these countries are the main drivers of declining real rates the past 40 years. Their $rt*$ estimate for Canada is shown with a solid blue line in Figure 16. Note that their $rt*$ estimate was increasing back in the 1960s and 1970s before starting a pronounced secular decline in the early 1980s. The trend lower continues through the end of the shown sample and leaves it close to zero by 2020 and very close to our finance-based $rt*$ estimate at the time.

The third macro-based estimate of $rt*$ is taken from Ferreira and Shousha (2023, henceforth FS) and shown with a solid green line in Figure 16. They consider a panel of 11 advanced economies and estimate the longer-run neutral real interest rates while accounting for changes in the global supply of safe assets and their convenience yields in addition to productivity and demographics and global spillovers from their developments. Their $rt*$ estimate for Canada starts in 1960 and fell steadily until the mid-1970s. It reversed some of the decline in the early 1980s and remained fairly stable until the late 1990s. It then steadily declined until 2008 when it reached a low of 0.85%. Since then it has gradually trended higher and stood at 1.76% by the end of 2023. This upward trend the past 15 years with a net increase of about 1 percentage point sets it apart from the other macro-based estimates, which mostly trend lower during this period.

The fourth and final macro-based estimate of $rt*$ comes from Davis et al. (2024, henceforth DFHMT). They introduce a unified no-arbitrage macro-finance model with two trend factors used to estimate the natural rate $rt*$ for 10 advanced economies, including Canada. Using a multitude of data sources on trend growth and inflation in addition to risk premium series, DFHMT also underscore the need for a coherent model approach like ours. Importantly, the interpretation of their natural rate $rt*$ is consistent with Laubach and Williams (2003) of representing a medium-run real rate anchor for monetary policy. Our finance-based definition taken from CR is intended to capture the same concept. Hence, the DFHMT $rt*$ estimate should be comparable to ours. One notable difference, though, is that, by relying on a single average yield equation, their estimation is not fully exploiting all the information in the yield curve unlike our approach. Their $rt*$ series is shown with a solid yellow line in Figure 16. While increasing from the 1960s and into the 1990s, the DFHMT $rt*$ estimate peaked in the mid-1990s. In the subsequent more than 25 years, $rt*$ fell more than 4 percentage points according to their estimate and ends the sample slightly above zero. For the overlapping period this entails a close similarity between their $rt*$ estimate and our finance-based $rt*$ estimate. Overall, this pattern aligns well with the observed RRB yields shown in Figure 4. Moreover, using panel regressions, DFHMT relate their $rt*$ estimates to economic growth and demographic variables and find that slowing growth and population aging have been significant factors in driving down the natural rate $rt*$ globally, and in Canada as well. Given the similarity of our finance-based $rt*$ to their estimate, we speculate that our estimate is likely influenced by those same factors.

Finally, we note that the Bank of Canada publishes an annual analysis of the level of the neutral rate in Canada. The most recent release of this analysis described in Adjalala et al. (2024) indicates an estimate between 2.25 and 3.25% for the neutral rate using five different approaches. Assuming inflation at 2% in the long run consistent with the Bank of Canada’s inflation target, this would translate into a range between 0.25 and 1.25% for the natural real rate. This puts our $rt*$ estimate of 1.85% at the end of our sample a notch above the range considered relevant by staff at the Bank of Canada. However, in light of the involved estimation uncertainty and the disagreement across macro-based models on full display in Figure 16, we consider our $rt*$ estimate to be reasonable and broadly consistent with those implied by standard macroeconomic models.

4.4.2 The stance of Canadian monetary policy

In this section, as an additional application of our market-based estimate of $rt*$⁠, we follow Christensen and Mouabbi (2024) and use it to construct a measure of the stance of the Bank of Canada’s monetary policy.

As described in Christensen and Mouabbi (2024), the stance of monetary policy would in theory be given by the difference between the current real instantaneous short rate and its neutral level as reflected in $rt*$⁠, that is, it would be defined as

The intuition behind this definition is straightforward. When the current real short rate is above its neutral level, interest rates of all kinds are likely to be above their steady-state level and will provide some headwind for new economic activity through higher borrowing costs and help slowdown the economy. And vice versa, when the current real short rate is below its neutral level, the general interest rate level is likely to be below what is needed to maintain trend growth, and businesses and households may take advantage of that by making investments in new projects or housing at cheap financing rates, which will help boost economic activity.

Unfortunately, the instantaneous real short rate is not directly observable because we do not have a continuous measure of the very short end of the RRB yield curve, given that individual RRBs reach maturity infrequently as noted in Figure 3. Furthermore, as explained earlier, RRBs, like other inflation-indexed bonds, tend to have rather erratic prices close to maturity thanks to both low liquidity and the unpredictability of the final inflation adjustments to be earned—the sudden and very sharp spike in Canadian CPI inflation in 2021–2022 period is very illustrative in this regard.²⁵ Thus, to make the definition above operational, we consider instead two proxies that we think of as reasonable substitutes for r$t$⁠. The first proxy is the one-year fitted real RRB yield from an estimation of the CLR model without censoring any bond price information, that is, RRB prices remain in the sample until they mature. This provides the best possible coverage around the one-year maturity point but comes at the cost of adding noise from the prices of RRBs close to maturity. Still, one can argue that this yield measures the full actual real yields observed in financial markets—including noise and frictions—and hence represents the most realistic real-world equivalent to the textbook real short-term interest rate embedded in the definition of $ζt$⁠. The second proxy is the 1-year frictionless real yield implied by our preferred CLR-L model. This is a cleaner and more stable measure of the one-year real yield as it adjusts for the noise from the bond-specific risk premiums in the RRB prices. However, in doing so, it may be different from the textbook concept of the real short rate r$t$ appearing in the original definition of $ζt$⁠. Moreover, as RRB prices with less than one year to maturity are censored in the estimation of our preferred CLR-L model, it may capture the short end of the RRB real yield curve less accurately.

The resulting two empirical measures of the stance of Canadian monetary policy are shown in Figure 17. In general, the two measures are very similar and highly positively correlated (99%), although we note greater volatility in the measure based on the fitted real yield in 2021. Hence, they paint a very similar picture for the stance of monetary policy in Canada the past 28 years. First, monetary policy in Canada reached a restrictive stance in the years ahead of the mild recession in 2000 caused by the burst of the IT stock price bubble at the time. Second, monetary policy in Canada was tight going into the GFC in 2007 and remained above neutral into 2009 before finally reaching an accommodative level that remained in place until 2018 when the U.S. Federal Reserve had also raised its overnight policy rate for some time. Third, at the peak of the COVID-19 pandemic in spring 2020, Canadian monetary policy briefly reached a very restrictive stance before becoming very accommodative relatively quickly. Finally and similar to the United States, the response of the Bank of Canada to the spike in inflation following the global economic reopening after the pandemic was delayed, which had the implication that monetary policy remained very accommodative for an extended period of time and did not reach a tightening posture until spring 2022 according to our measures. This may have contributed to prolonging the spell of high inflation in Canada during this period, but it falls outside the focus of this paper to make any determinations to that effect, so we leave it for future research to explore that question further.

Based on these observations we think of our empirical market-based measures of the stance of Canadian monetary policy as realistic. Moreover, they can be estimated at daily frequency and used for truly real-time policy analysis. This represents a major advantage relative to existing macro-based estimates, which are only available with a lag and may be subject to significant data revisions.

4.5 Projections of expected inflation and the natural rate

Given the debate among researchers, investors, and policymakers about the persistence of the recent rise in interest rates, we analyze the outlook for long-term inflation expectations and the natural rate based on our preferred CLR-L model. We follow the approach of Christensen, Lopez, and Rudebusch (2015) and simulate 10,000 factor paths over a 10-year horizon conditioned on the shapes of the nominal and real yield curves and investors’ embedded forward-looking expectations as of the end of our sample (i.e. using estimated state variables and factor dynamics as of June 28, 2024). The simulated factor paths are then converted into forecasts of 10-year expected inflation and $rt*$⁠. Figure 18 shows the median projection and the 5th and 95th percentile values for the simulated 10-year expected inflation and natural rate over a 10-year forecast horizon.²⁶

The median projections of both 10-year expected inflation and $rt*$ show essentially no change in either measure over the coming decade. We note that for the projected 10-year expected inflation the 90% confidence band is fairly narrow, even at the 10-year projection horizon, consistent with the stable in-sample 10-year expected inflation series. For the $rt*$ estimate the projected values are notably more uncertain. The upper 95th percentile continues to rise rapidly reaching the high levels last seen in the mid-1990s towards the end of the projection period, while the lower 5th percentile represents outcomes with the natural rate returning to the near-zero values prevailing in the decade before the pandemic.

We take the stable projected long-term expected inflation as additional strong evidence that long-term inflation expectations in Canada continue to remain anchored close to the central bank target despite the recent spell of highly elevated inflation. Furthermore, the fact that the 90% confidence band of both 10-year expected inflation and $rt*$ are on par with the historical variation in each respective series since 1996 suggests that the model’s estimated factor dynamics imply a realistic amount of uncertainty about investors’ long-term inflation expectations and the natural real rate.

This exercise also demonstrates that our preferred CLR-L model is able to simultaneously imply modest uncertainty about long-term inflation expectations, while allowing for quite significant uncertainty about the medium-term steady state real rate as reflected in our definition $rt*$⁠. Mapping to a Taylor-based rule for monetary policy, these findings imply little uncertainty about long-run inflation, or $π*$⁠, in Canada, but much uncertainty about the neutral level of the overnight policy rate thanks to the persistent and significant trends in $rt*$⁠.

Finally, our market-based estimate of $rt*$ remains relevant to the debate about the source of the persistent trends in the equilibrium real rate. In particular, although our measure of the real rate fluctuated a bit at the start of the GFC, our average $rt*$ estimate appears to be dominated by deeper and more structural persistent trends. This suggests that flight-to-safety and safety premium explanations of the trends in the equilibrium real rate, in particular following the GFC, are unlikely to be key drivers of the downtrend in global interest rates in the decades prior to the pandemic (as proposed by Hall 2016, among others). Instead, our estimate appears more broadly consistent with many of the explanations that attribute the decline in the natural rate to real-side fundamentals such as changing demographics (e.g. Carvalho, Ferrero, and Nechio 2016; Favero, Gozluklu, and Yang 2016; Gagnon, Johannsen, and Lopez-Salido 2016). Lastly, the notable uptick in 2022 along with the sharp increases in global long-term interest rates suggests that, even with inflation brought back under control, real interest rates in Canada may not return to their pre-pandemic lows, which contrasts with the view of Blanchard (2023).

5 International Spillover of Unconventional Monetary Policy

To further demonstrate the potential usefulness of our model for policymakers and investors, we examine the spillover to Canadian bond markets from foreign unconventional monetary policies enacted in response to the GFC of 2008–2009.²⁷ Given the absence of any domestic Canadian unconventional monetary policies during that period, Canada is a useful laboratory for studying such international spillover effects.²⁸

To the best of our knowledge, we are the first to examine the impact of foreign QE announcements on domestic BEI and underlying inflation expectations and risk premiums in a major advanced economy. In addition, we quantify their effects on Canadian nominal yields and their embedded risk premiums as others have done; see Bauer and Neely (2014, henceforth BN) for an example.

As this exercise is intended to be more illustrative than comprehensive, we limit our focus to the QE events considered in Christensen and Rudebusch (2012, henceforth CR). Their study represents a useful benchmark as it offers an international perspective by studying the domestic bond market reactions to U.S. Federal Reserve and Bank of England announcements regarding their respective QE programs launched as part of the early response to the financial crisis.²⁹ Moreover, that study relies on dynamic term structure models of nominal yields estimated at a daily frequency for its assessment. Hence, we provide an extension by estimating our preferred shadow-rate B-CLR-L model of nominal and real yields using daily data. We note that, although the policy rate in Canada never went below 0.5% during the period examined here, we rely on the shadow-rate specification of our preferred CLR-L model for this exercise to account for the possibility that investors may have expected Canadian short-term nominal interest rates to be constrained by the ZLB similar to their U.S counterparts.³⁰

We begin by analyzing spillover effects from the Fed’s first QE program followed by a similar analysis of spillovers from QE announcements made by the Bank of England.

5.1 U.S. QE

In late 2008, the Federal Reserve lowered its target policy rate—the overnight federal funds rate—effectively to its ZLB. Given a deteriorating outlook for economic growth and a perceived threat of price deflation, the Fed began to purchase longer-term securities to push down bond yields and provide additional monetary policy stimulus to the economy.

To shed some light on the various channels through which these foreign bond purchases may have affected Canadian bond prices, we examine the responses of Canadian bond yields using an event study methodology as in CR. Specifically, we quantify the effects as the changes in Canadian interest rates over two-day intervals around announcements of future bond purchases.³¹ Although widely used, this is an imperfect technique. We use a 2-day window to allow investors in the Canadian bond market to fully digest the news reflected in each announcement and determine its implications for the Canadian economy and monetary policy. The drawback of this choice is that other news may have been released that significantly affected Canadian interest rates and obscured the effects we are trying to assess.³²

5.1.1 Nominal yield decompositions

We estimate the effect of the Fed’s QE announcements on expected short-term interest rates and inflation as well as associated risk premiums with our preferred B-CLR-L model estimated using daily data through the end of June 2024. We focus on the model decomposition of the reaction of the 10-year nominal yield and BEI rate over two-day periods.³³

Recall that nominal term premiums are defined as the difference in expected nominal return between a buy-and-hold strategy for a $τ$-year nominal bond and an instantaneous rollover strategy at the risk-free nominal short rate $rtN$

Based on this formula, our preferred B-CLR-L model is used to decompose the Canadian 10-year nominal zero-coupon yield into three components:

1. the estimated average expected nominal short rate until maturity;

2. the nominal term premium defined as the difference between the model fitted nominal yield and the average expected nominal short rate; and

3. a residual that reflects variation not accounted for by the model.

This breakdown of the 10-year nominal yield for the full sample is shown in Figure 19a.

Table 12 reports the result of these decompositions for our eight U.S. QE announcements. Overall, these indicate small changes in the nominal short-rate expectations component—accounting, at most, for about a third of the total reduction. Consequently, signaling effects appear secondary in the declines in Canadian nominal bond yields around these announcements. This contrasts with the findings of Christensen and Rudebusch (2019) and Bauer and Rudebusch (2014), both of which find significant signaling effects in the U.S. Treasury bond market reaction to these events. More importantly, these results also contrast with those reported by BN, who consider a slightly different set of event windows and use term structure models of Canadian nominal yields to decompose the changes in the 10-year nominal yield into changes in the expected nominal short rate and the nominal term premium. They interpret their results to suggest significant signaling spillover effects onto the Canadian government bond market from the U.S. QE events. Instead, our results suggest declines in the nominal term premiums to be the primary source behind the observed strong market reaction. Finally, the unexplained residuals account for another 20% of the market reaction, underscoring that the outcomes of the decompositions are associated with some uncertainty. We take this evidence to imply that Canadian bond markets were affected by these U.S. policy actions mainly through a portfolio re-balance channel; see Christensen and Krogstrup (2019) for a discussion. This is also consistent with the results of Dahlhaus, Kristina, and Abeer (2018), who use a FAVAR model to examine the international spillover effects on the broader Canadian economy from the U.S. QE programs and find that they operated mainly through a financing channel. Furthermore, our results are in line with the analysis in Neely (2015), which suggests that simple portfolio balance effects may have played an important role as an international spillover transmission mechanism of unconventional monetary policy to Canada and other advanced economies.

Another approach in the literature uses vector autoregressions (VAR) to estimate the yield impulse responses to changes in the quantity of asset purchases by the Federal Reserve. Kabaca and Tuzcuoglu (2022) use a VAR to show that a 1% asset purchase announcement by the U.S. Federal Reserve (as a share of U.S. GDP) is followed by a (median) 10 basis point decline in the 10-year Canadian yield and a two basis point decline in the 1-year Canadian yield. Similarly, Fratzscher, Lo Duca, and Straub (2018) include Canada in a basket of advanced economies and show that QE in the U.S. significantly lowered yields in this set of advanced economies.

5.1.2 BEI decompositions

Building on Equation (6), the B-CLR-L model allows us to make a decomposition of the $τ$-year BEI as follows:

where the $τ$-year constant maturity RRB liquidity premium, $Ψt(τ)$⁠, is given by the difference between the fitted BEI and the corresponding frictionless BEI implied by our preferred B-CLR-L model.

While the result of this decomposition of BEI at the 10-year maturity for the full sample is shown in Figure 19b, the results of the 10-year BEI decomposition on the eight U.S. announcement dates are reported in Table 13. Here, we see only very small reactions in the estimated inflation expectations. Instead, the market reactions are driven by a mix of the inflation and RRB liquidity risk premiums. On net, the inflation risk premium increases a bit, while the negative of the RRB liquidity risk premium declines, that is, the liquidity risk premiums increase as well. This suggests that investors grew somewhat more fearful about upside risks to inflation, but clearly not to the point that they felt compelled to raise their expectations for inflation. Furthermore, the results indicate that this happened in the context of somewhat deteriorating financial market conditions as reflected in the uptick in the liquidity premiums they demand for holding RRBs.

5.2 U.K. QE

The Bank of England first announced its asset purchases in February 2009 to provide additional stimulus during the GFC. By March 2009, the facilities were operational, and the Bank of England had set the policy rate to 0.5%, close to the effective ZLB. Despite substantial stimulus coming from fiscal policy and the main interest rate being cut close to zero, the Bank of England decided that QE would help stimulate nominal spending and stabilize inflation relative to its target.³⁴ For the U.K. analysis, we choose to go with a 1-day event window for each announcement given that their release would happen before or early in each day’s Canadian trading session. This timing contrasts with the U.S. announcements, which mostly came toward the end of the trading session on those days.

5.2.1 Nominal yield decompositions

To illuminate the effects of U.K. QE announcements on Canadian yields, we again use our preferred B-CLR-L model to decompose the 10-year nominal yield changes into the estimated average expected nominal short rate until maturity, the nominal term premium, and a residual.

Table 14 presents the results for these decompositions for our seven U.K. announcement days. We find that for all days besides the May 2009 announcement, the change in the average expected nominal short rate is close to zero. This result is similar to what we find for the U.S., indicating that the signaling effect plays a minor role in the changes in Canadian nominal bond yields around these announcement days. While the net effect on the nominal term premium is smaller than the expectations component, the change within each event tends to be larger for the term premium than the expectations component. Taken together with the evidence from the U.S. data, it appears that foreign announcements regarding QE impact domestic yields through a term premium channel rather than a signaling channel, a result that contrasts with findings of U.S. Federal Reserve announcements on the U.S. Treasury bond market, but is consistent with the U.K. findings reported by CR. Notably, the residual component has the most considerable net change indicating that the results for the U.K. come with more uncertainty than observed for the U.S. case.

5.2.2 BEI decompositions

Finally, we examine the B-CLR-L model-implied decompositions of the response of the Canadian 10-year BEI to the U.K. QE announcements. These results are reported in Table 15. On net, the change in the 10-year BEI is four basis points indicating that U.K. QE announcements had a minimal effect on inflation expectations in Canada. Of these four basis points, most are ascribed to changes in the 10-year inflation risk premium, and only one comes from changes in the 10-year expected inflation. Across all announcements, changes in inflation expectations and inflation risk components for the 10-year BEI tend to be zero or close to it, except for an increase in the inflation risk premium of three basis points around the March and May 2009 Bank of England QE announcements. Overall, these small responses indicate little, if any, spillover effects from the QE announcements in the U.K. to Canadian inflation expectations. Together with the U.S. Federal Reserve QE announcement effects on Canadian yields, these results are the first to document that asset purchase announcements appear to have limited effects on foreign inflation expectations and inflation risk premiums despite the significant spillover effects on nominal yields.

Our results are thus comparable to other findings in the literature on the effects of U.S. QE announcements on Canadian yields. However, relative to the earlier literature, we consider movements in BEI and inflation risk premiums on QE announcement days. We find that QE spillovers to these variables are smaller than for the nominal yields. Additionally, we provide the first analysis of the effects of U.K. QE announcements on Canadian yields and find U.S. announcements tend to have a relatively larger effect on Canadian yields.

6 Conclusion

Given the historic persistent trends in yields in recent decades, many researchers have investigated the factors affecting interest rates. Much of this work has focused on the steady-state level of the safe real short-term interest rate based on macroeconomic models and U.S. data. Instead, we consider a finance-based decomposition of interest rates from empirical dynamic term structure models estimated on a sample of standard Canadian nominal bond yields combined with the prices of Canadian inflation-indexed bonds known as RRBs. By adjusting for both RRB liquidity premiums and nominal and real term premiums, we uncover investors’ expectations for the underlying frictionless real short rate for the 5-year period starting five years ahead. This measure of the natural rate of interest exhibits a persistent gradual decline over the 1996–2021 period that accounts for about two thirds of the general decline in Canadian yields. It also accounts for much of the sharp spike in Canadian interest rates following the economic reopening after the COVID-19 pandemic. Furthermore, model projections that exploit the estimated factor dynamics suggest that this measure of the natural rate is likely to remain near its current higher level for the foreseeable future.

Equally important, our joint model of nominal and real yields also produces estimates of investors’ inflation expectations and associated inflation risk premiums. Here, we find that a modest decline in investors’ long-term inflation expectations of about 0.7 percentage point in the 1996–2021 period contributed further to the decline in Canadian long-term nominal bond yields prior to the pandemic. Moreover, about half of that decline has been retraced in the years since then in response to the transitory spell of high inflation in the 2022–2023 period. Crucially, our model-based projections as of the end of June 2024 indicate that investors’ long-term inflation expectations are likely to experience a further modest increase in the coming years, which will make them approach the Bank of Canada’s 2% inflation target. We interpret this as a sign that Canadian inflation expectations are very well anchored near the central bank’s desired level.

For monetary policy analysis, the proposed finance-based approach to estimate both inflation expectations and the natural real rate also offers notable advantages as they are available in real-time and not subject to data revision unlike estimates based on macroeconomic data. Since our measures are based on the forward-looking information priced into the active RRB market and can be updated at daily frequency, they could serve as an important input for real-time monetary policy analysis. As one envisioned application, we follow the literature and calculate a measure of the stance of monetary policy that suggests that monetary policy in Canada remains restrictive at the end of our sample. We demonstrate two further applications by first examining the market reaction to the surprise announcement by the Canadian Finance Department to permanently halt RRB issuance made public on November 3, 2022. We document that the RRB market has continued to function on par with the historical experience prior to the announcement. This supports our continued usage of the RRB data. We then proceed to an examination of the international spillovers to Canadian interest rates from the first round of U.S. and U.K. QE programs launched around the peak of the GFC. Using high-frequency daily analysis, we document significant effects from these programs on bond risk premiums, which points to a portfolio rebalancing effect as an international QE transmission mechanism.

For future research, our methods can be further expanded along an international dimension. With a significant degree of capital mobility, the natural rate will depend on global saving and investment, so the joint modeling of inflation-indexed bonds in several countries could be informative similar to HLW. Finally, our measure could be incorporated into an expanded joint macroeconomic and finance analysis—particularly with an eye towards further understanding the determinants of the persistent trends in interest rates. In this regard, Bauer and Rudebusch (2020) show that accounting for fluctuations in the natural rate substantially improves long-range interest rate forecasts and helps predict excess bond returns.

Supplemental Material

Supplemental material is available at Journal of Financial Econometrics online.

We thank conference participants at the 2nd LTI/Bank of Italy Workshop on Long-Term Investors’ Trends: Theory and Practice, the 37th International Conference of the French Finance Association (AFFI), the IFABS 2021 Oxford Conference, the INQUIRE Residential Seminar 2021, the RCEA Conference on Recent Developments in Economics, Econometrics, and Finance, the Midwest Finance Association 2022 Annual Meeting, and the IFABS 2022 Naples Conference for helpful comments and suggestions, including our discussants Guillaume Roussellet, Fan Dora Xia, and Juan Arismendi Zambrano. We also thank seminar participants at the Bank of Canada, the Federal Reserve Board, and the Norges Bank for helpful comments and suggestions. Furthermore, we thank Antonio Diez de los Rios for helpful comments on an early draft of the article. Finally, we thank Gus Kmetz, Caroline Paulson, Dori Wilson, and Simon Zhu for outstanding research assistance. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Federal Reserve System.

Footnotes

References