## A trend in interest rates

Interest rates in Canada have declined gradually since the 1980s. Chart 1 plots the interest rate that the Government of Canada pays to borrow for a period of 10 years. This rate hovered around 10 per cent between 1987 and 1991 but has since declined steadily to reach around 2 per cent recently. Chart 1 also shows the parallel decline in the interest rate on a 5-year mortgage for a typical Canadian household. Other Canadian interest rates share this pattern.

Part of the secular decline in the 10-year interest rate is due to investors’ lower forecasts of future interest rates at a distant horizon. At this long-term horizon, the investors’ forecasts revert to an endpoint beyond which no further changes are anticipated (Kozicki and Tinsley 2001). Using econometric techniques, we estimate the shifts in the endpoint between 1987 and 2018 for Canada. Overall, the estimate exhibits three phases, in which the endpoint is increasingly lower and more stable (shown below).

### Chart 1: Nominal yields have trended down in Canada

Sources: Statistics Canada Table 34-10-0145-01 (Canada Mortgage and Housing Corporation: 5-year conventional mortgage lending rate) and Bank of Canada calculations (Bauer, Diez de los Rios and Melin, Government of Canada yield curve from 1938 to the present: 10-year yield)Last observation:

In the model, the most important components of nominal long-term interest rates are the forecasted endpoint and a term premium. This premium is the compensation for risk related to holding a series of short-term bonds. The endpoint estimates that we present are stable over long periods of time after 1995, which suggests that the influence of policy actions on long-term interest rates was largely mediated by changes in the term premium.

The interest rate endpoint is related to the idea of the neutral interest rate—the rate of interest that should prevail after the effects of business cycle shocks have dissipated. For instance, Mendes (2014) and, more recently, Chen and Dorich (2018) provide Bank of Canada estimates of the neutral rate. We provide an estimate of the endpoint over a long historical sample, but this is not necessarily the history of the neutral rate. We keep a distinction between the neutral rate and the interest rate endpoint estimates because, being the products of different methods, they do not necessarily coincide. Indeed, the current endpoint estimate appears too low relative to the neutral rate estimate. More research is in progress to understand these differences.

## Endpoint shifts in survey forecasts

Chart 2 shows forecasts of the 10-year bond yield from a series of surveys of professional forecasters (the results are available publicly from the Government of Canada website). These surveys, repeated roughly every year since 1995, collect forecasts in each year, up to 7 years ahead. The 7-year horizon is not necessarily the investors’ forecasted endpoint, but it gives a good idea of where forecasts are ultimately heading.

The endpoint of these survey-based forecasts declined gradually in our sample, but not nearly as fast as the observed bond yield (blue line). The forecasts’ endpoint seems to follow three phases. It declined early in the sample, starting from 7.3 per cent (the 2000 forecast at the end of 1994). It was then stable between 5 and 6 per cent from 1998 until the US financial crisis in 2008. Since 2008, the forecasts’ endpoint has slowly declined from 5 to 3.3 per cent (the 2022 forecast from the 2017 survey). In contrast, the observed 10-year bond yield declined more or less continuously, to around 1.8 per cent at the end of 2017. Long-term forecasts adjusted only slowly, and the differences were large compared with the 10-year bond yield.

### Chart 2: Survey forecasts of the nominal yields have also declined

Sources: Department of Finance Canada and Bank of CanadaLast observation: Dec 2017

We can say the difference is large because it generated large unexpected returns for bond investors (relative to their forecasts). Table 1 shows that annual returns from holding a 10-year bond averaged 16, 9.5 and 8 per cent in the 5-year periods 1990–95, 1995–2000 and 2000–05, respectively. These are long episodes with high returns from holding bonds, even compared with the stock market returns, which we usually expect to outperform bonds over time.

Overall, given what we know about investors’ forecasts and bond returns, our interpretation of the trend in Chart 1 is that investors continued to forecast a relatively high endpoint, close to the historical average at the time, and that they adjusted this endpoint only slowly. Implicitly, their endpoint also forecasted returns on bond portfolios that were much smaller than what we see in Table 1.

### Table 1: Returns from long-term investments in bonds and stocks

#### Average annual returns over each 5-year period, in percentage

Period | 10-year GoC bond | S&P/TSX Composite |
---|---|---|

1990–95 | 16.0 | 7.6 |

1995–2000 | 9.5 | 19.8 |

2000–05 | 8.0 | 3.7 |

2005–10 | 4.4 | 6.0 |

2010–15 | 6.2 | 7.2 |

Note: Exact periods are December 31, 1989, to May 4, 1995; May 4, 1995, to March 16, 2000; March 16, 2000, to August 16, 2005; August 16, 2005, to April 30, 2010; and April 30, 2010, to January 26, 2015.

Sources: Bloomberg and Bank of Canada calculations

## Estimating the shifting endpoint

We estimate the endpoint implicit in nominal bond yields using a classical Beveridge-Nelson model. Our approach follows in the steps of Kozicki and Tinsley (2001), who introduce the concept of a shifting endpoint (see the Appendix for details). One difference is that we do not use survey-based forecasts to inform our model-based estimates. For now, we are letting interest data govern the estimates.

Chart 3 shows the model-based shifting endpoint estimates between 1987 and 2018, as well as a constant endpoint estimate (in blue, at 4.15 per cent). The shifting endpoint exhibits three broad phases. The first phase exhibits a rapid decline from a peak in 1990 until 1996, at which point the estimated endpoint reached 4 per cent. The timing of this decline corresponds to the February 1991 Bank of Canada announcement to gradually bring down the inflation rate to around 2 per cent by 1995.

In the second phase, between 1996 and 2008, the endpoint hovered between 3 and 4 per cent. The third phase starts roughly in 2008 with another decline of the endpoint, to between 1 and 1.5 per cent, and continues until June 2018, the end of our sample.

### Chart 3: The estimated interest rate endpoint in Canada

Sources: Department of Finance Canada and Bank of CanadaLast observation:

The estimated endpoint may seem too low at the end of our sample, especially when compared with the forecasts’ endpoint in Chart 2. The period starting in 2008 features several large distortions, ignored in our model, that changed the behaviour of bond yields. Central banks around the world pushed policy rates to their lower bound, used forward guidance and launched large-scale asset purchases. More research is needed to improve our model and to estimate the endpoint as monetary policy normalizes around the world. It remains to be seen whether the endpoint will return close to the constant estimate (the blue line), which is roughly the centre of the range we saw during the second phase.

The variability of the estimated endpoint is different across the three phases. Chart 4 shows the time-varying speed of adjustment that we recover from our model. Existing models incorporate a constant speed of adjustment. The speed of adjustment was high during the transition to a new inflation target and then moderated between 1996 and 2008. Again, this decline and new plateau suggest that anchoring inflation around 2 per cent stabilized the nominal interest rate endpoint.

### Chart 4: Interest rate endpoint speed of adjustment

Sources: Reuters and Bank of Canada calculationsLast observation:

The model-based premium shows large cyclical swings that correspond mainly to the cycles between economic expansions and slowdowns. Because the estimate of the endpoint is stable, the model attributes the influence of economic cycles and monetary policy on long-term interest rates largely to changes in the term premium. The term premium movements estimated under the assumption of a constant endpoint result in a very similar picture (also shown in Chart 5).

However, the similarities between the term premium estimates in Chart 5 make it easy to overlook one important difference. The term premium based on a constant endpoint is lower for most of the sample, and the difference has been widening since 2017. The question is which is the more useful estimate. If your long-horizon forecast of interest rate is around 4 per cent—roughly the estimate in the model with a constant endpoint and close to the average level of the shifting endpoint estimate between 1995 and 2007—then your estimate of the term premium is close to zero in 2018. If your endpoint is lower than 4 per cent, then your estimate of the term premium is higher.

The contrast between these two views cannot be easily resolved. Interest rates have been low for several years. The survey-based endpoint forecast in Chart 2 has declined to almost 3 per cent. Are interest rates expected to return close to the 4 per cent historical average, which would imply expected returns that are low or negative for investors holding bonds over a long horizon? Or will interest rates remain historically low, well below 4 per cent, which would imply expected returns that are higher and positive?

### Chart 5: Estimates of the 10-year term premium

Sources: Haver Analytics (C.D. Howe Institute) Recession Indicator, Reuters and Bank of Canada calculationsLast observation:

## Conclusion

We provide a new estimate of the shifting forecast endpoint in Canadian bond yields. The shifting endpoint estimate exhibited a rapid decline between 1990 and 1995, following the announcement of the inflation target in Canada, and stabilized after 1996, consistent with the success the target had in anchoring the endpoint of inflation.

The period since 2008 shows another decline of the endpoint that persists to the end of our sample. This decline could be due to policy actions in bond markets (such as quantitative easing), other economic distortions (such as a structural increase in the demand for long-duration bonds from pension fund investors), or limitations of the model. More research is needed to improve our estimate of what the interest rate endpoint will be when monetary policy normalizes.

## Appendix

The Beveridge-Nelson model separates the evolution of interest rates into the evolution of the forecast endpoint and the evolution of cyclical components. The model estimates the cyclical component by minimizing the size of monthly interest rate forecast errors.

Let us denote by \(P_{t}\) a portfolio of nominal bond yield. In this note we have used the first three principal components of bond yield with maturities 3, 12, 24, 60, 120 and 360 months. The Beveridge-Nelson in our model is given by

$$ \begin{eqnarray*} P_{t}&=&P_{0}+\gamma i_{t}^{\ast }+\tilde{P}_{t}\\ i_{t}^{\ast } &=&i_{t-1}^{\ast }+\sigma _{i,t-1}^{\prime }u_{t}^{P} \\ \tilde{P}_{t} &=&\mathcal{E}_{t-1}+\Sigma _{t-1}^{\tilde{P}}u_{t}^{P} \\ \sigma _{i,t} &=&\bar{\sigma}_{i}v_{i,t} \\ u_{t}^{P} &\sim &i.i.d\ \ \ N\left( 0,I_{N}\right) \end{eqnarray*} $$

$$ \begin{eqnarray*} \mathcal{E}_{t} &=&\Phi \mathcal{E}_{t-1}+\left( \Phi -\Theta \right) \Sigma _{t-1}^{\tilde{P}}u_{t}^{P} \\ \ln \left( v_{it}^{2}\right) &=&\omega _{i}+\ln \left( v_{it-1}^{2}\right) +g_{i}\left( z_{t}^{i}\right) \end{eqnarray*}% $$

where

$$ z_{t}^{i}\equiv \frac{\alpha _{i}^{\prime }u_{t}^{P}}{\sqrt{\alpha _{i}^{\prime }\alpha _{i}}} $$ $$ g\left( z_{t}\right) =\sqrt{\alpha _{i}^{\prime }\alpha _{i}}z_{t}+\kappa _{i}\left( \left\vert z_{t}\right\vert -E\left[ \left\vert z_{t}\right\vert % \right] \right) , $$ $$ \omega _{i}-\kappa _{i}\sqrt{\frac{2}{\pi }}+\frac{\left( \sqrt{\alpha _{i}^{\prime }\alpha _{i}}-\kappa _{i}\right) ^{2}}{2}+\ln \left( \Phi _{N}\left( \kappa _{i}+\sqrt{\alpha _{i}^{\prime }\alpha _{i}}\right) +\Phi _{N}\left( \kappa _{i}-\sqrt{\alpha _{i}^{\prime }\alpha _{i}}\right) \right) <0, $$and \(\Phi _{N}\left( \cdot \right) \) is the CDF of a standard normal distribution.

## Acknowledgments

We thank Jason Allen, Toni Gravelle, Sharon Kozicki and Jonathan Witmer for suggestions. We also thank Liam Lindsay and Scott Woldum for research assistance.

## References

- Chen, X. S. and J. Dorich. 2018. “The Neutral Rate in Canada: 2018 Estimates.” Bank of Canada Staff Analytical Note No. 2018-22. Available at https://www.bankofcanada.ca/wp-content/uploads/2018/07/san2018-22.pdf
- Kozicki, S. and P. A. Tinsley. 2001. “Shifting Endpoints in the Term Structure of Interest Rates.”
*Journal of Monetary Economics*47 (3): 613–652. - Mendes, R. R. 2014. “The Neutral Rate of Interest in Canada.” Bank of Canada Staff Discussion Paper No. 2014-5. Available at https://www.bankofcanada.ca/wp-content/uploads/2014/09/dp2014-5.pdf

## Disclaimer

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