## Abstract

Interest rates are very persistent. Modeling the persistent component of interest rates has important consequence for forecasting. Factor models of the term structure are restricted VAR models that project over a long-horizon the one-period risk-free rate to obtain yields at longer horizon as the sum of the expected future monetary policy and the term premia. The included factors are typically mean reverting and the equilibrium real rates are considered constant (think, for example, of the standard Taylor rule), partial adjustments to equilibrium yields are then used to rationalize the persistence in the observed data. As a result, the empirical models feature a very high level of persistence that makes long-horizon predictions inherently inaccurate. This paper relates the common persistent component of the U.S. term structure of interest rates to the age composition of population. The composition of age structure determines the equilibrium rate in the monetary policy rule and therefore the persistent component in one-period yields. Fluctuations in demographics are then transmitted to the whole-term structure via the expected policy rate components. We build an affine term structure model (ATSM) which exploits demographic information to capture the dynamics of yields and produce useful forecasts of bond yields and excess returns that provide economic value for long-term investors.

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## Notes

“...

*adequate preparation for the coming demographic transition may well involve significant adjustments in our patterns of consumption, work effort, and saving*...” Chairman Ben S. Bernanke, Before The Washington Economic Club, Washington, D.C., October 4, 2006. Also, research agenda questions (Theme 5) stated on Bank of England website stresses the importance of secular trends, in particular demographics, in determining equilibrium interest rates.When young adults, who are net borrowers, and the retired, who are dissavers, dominate the economy, savings decline and interest rates rise. The idea is certainly not a new one as it can be traced in the work of Wicksell (1936), Keynes and Maynard (1936), Modigliani and Brumberg (1954), but it has received relatively little attention in the recent literature.

In principle, there are many alternative choices for the demographic variable, MY. However, using a proxy derived from a model is consistent with economic theory (Giacomini and Ragusa, 2014) and reduces the risk of a choice driven by data mining. Importantly, MY is meant to capture the relative weights of active savers investing in financial markets. Our results are robust to an alternative specification of the demographic variable, middle-aged to old (MO) ratio, another variable consistent with the GMQ model.

The Bureau of Census currently publishes, on its website, the projections for the age structure of the population with a forecasting horizon up to fifty years ahead and historical Census reports back to 1950 are available to avoid look-ahead bias.

While the nominal short rate is directly observable, the real rate depends on how the expected inflation is modeled. Here, we predict the inflation from an autoregressive model using growth rate of GDP deflator. Our main analysis will be based on observable nominal rates, since the demographic channel affects the nominal rates via the (unobserved) real rates regardless of different inflation specifications (Gozluklu

*et al*, 2015).There are other alternative views in the literature which argue for a unit root in the spot rates (Dewachter and Lyrio, 2006; Christensen

*et al*, 2011; Hamilton*et al*, 2015) or suggest a near unit root process to model the persistent component (Cochrane and Piazzesi, 2005; Jardet, Monfort and Pegoraro, 2013; Osterrieder*et al*, 2012).Theoretical models suggest a link between equilibrium rate and growth in the economy, measured either via output or consumption growth. However, empirical evidence on economic fundamentals driving equilibrium rate is at best weak (Hamilton

*et al*, 2015).In a nutshell, high policy inertia should determine high predictability of the short-term interest rates, even after controlling for macroeconomic uncertainty related to the determinants of the central bank reaction function. This is not in line with the empirical evidence based on forward rates, future rates (in particular federal funds futures), and VAR models.

Life-cycle investment hypothesis suggests that agents should borrow when young, invest for retirement when middle-aged, and live off their investment once they are retired.

Recent literature also shows that consumption smoothing across time rather than the risk management across states is the primary concern of the households (Rampini and Viswanathan, 2014).

We adopt Cochrane and Piazzesi (2005) notation for log bond prices: \(p_{t}^{(n)}=\log\) price of n-year discount bond at time t. The continuously compounded spot rate is then \(y_{t}^{(n)}\equiv -\frac{1}{n} p_{t}^{(n)}\)

Note that under the assumption of perfectly stationary demographic structure, the relative cohort size, middle-aged over young population is the same as the middle-aged over retired population.

The implications of the evidence for stock market predictability are further investigated by Favero

*et al*(2011).Ang

*et al*(2008) have solved the identification problem of estimating two unobservables, real rates, and inflation risk premia, from only nominal yields by using a no-arbitrage term structure model that imposes restrictions on the nominal yields. These pricing restrictions identify the dynamics of real rates (and the inflation risk premia).Over a longer sample period 1900Q1–2013Q4, the demographic variable MY explains about 15 percent of the variation in real short rates once we use a AR(1) model with stochastic volatility to extract the expected inflation.

The Bureau of Census website provides projections for demographics variable up to 2050 and the current 5-year yield depends on the values of MY over the next five years.

One-year Treasury bond yields are taken from Gurkaynak

*et al*,'s dataset. Middle-young ratio data are available at annual frequencies from Bureau of Census (BoC) and it has been interpolated to obtain quarterly series.Improvement in mortality rates that have generated over the last forty years difference between actual population and projected population are mostly concentrated in older ages, after 65.

The results are robust when we construct a smaller panel with balanced data. The demographic data are collected from World Bank database.

Bond yield are collected from Global Financial data. Long-term bond yields are 10-year yields for most of the countries, except Japan (7-year), Finland, South Korea, Singapore (5-year), Mexico (3-year), and Hong Kong (2-year).

We thank an anonymous referee for pointing out that for some countries the domestic bond markets may be less important for life-cycle investment behavior, e.g., households from the emerging markets may instead demand foreign assets, in particular, U.S. securities (dollarization).

The report provides annual forecasts from 1975 to 2000 and five-year forecasts from 2000 to 2050.

We do not have age structure data for Cayman Islands, Middle East countries, and rest of the world. So, we account for 60 percent of foreign bond holdings as of June 2012. Source: Demographic data 1960–2000 from World Bank Population Statistics, Data 2011–2050 from U.S. Census International Database.

The t-statistics MY\(_{t}\) coefficient in the OLS regression of the 3-month rate on its own lag and the demographic variable.

The probability of observing a t-statistics of −3.02 on the coefficient on MY\(_{t}\) is 0.032 for the nominal rate and a t-statistics of −3.12 on the coefficient on MY\(_{t}\) is 0.018 for the real rate.

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## Author information

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## Additional information

* Carlo Favero holds a D.Phil. from Oxford University. He is a full professor of Deptment of Finance at Bocconi University and become Head of department in 2013. He is a research fellow of CEPR in the International Macroeconomics programme. He is president of IGIER at Bocconi University and a member of the scientific committee of CIDE. He has been advisor to the Italian Ministry of Treasury and consulting the European Commission, the World Bank and the European Central Bank; his email address is: carlo.favero@unibocconi.it. Arie Gozluklu is an Associate Professor of Finance at the University of Warwick. His research focuses on empirical asset pricing, international finance and market microstructure. His work has been published in Journal of Financial and Quantitative Analysis, Journal of Financial Markets and Financial Management. He has a master’s degree in economics from Pompeu Fabra and a Ph.D. from Bocconi University. His research has been presented in international conferences including European Finance Association, China International Conference in Finance, NBER Summer Institute and Banque de France; his email address is: arie.gozluklu@wbs.ac.uk. Haoxi Yang is an Assistant Professor of School of Finance at Nankai University. Her primary research interests are asset pricing, financial econometrics and macro-finance. Her research has been presented at conferences organized by the Society for Financial Econometrics (SoFiE), the Financial Management Association (FMA), European Economic Association (EEA), China International Conference in Finance (CICF). She holds a Ph.D. in Finance from Bocconi University; her email address is: haoxi.yang@outlook.com.

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## Appendices

### Appendix A: Derivation of Demographic ATSM

We consider the following model specification for pricing bonds with macro and demographic factors:

Bond prices can be recursively computed as follows:

where \(E_{t}^{Q}\) denotes the expectation under the risk-neutral probability measure, under which the dynamics of the state vector \(X_{t}\) are characterized by the risk-neutral vector of constants \(\mu ^{Q}\) and by the autoregressive matrix \(\Phi ^{Q}\)

To derive the coefficients of the model, let us start with \(n=1\):

\(A_{1}=-\delta _{0}\), \(B_{1}=-\delta _{1}\) and \(\Gamma _{1}=\gamma _{0} ^{1}=-\delta _{2}\), Then for \(n+1,\)we have \(P_{t}^{(n+1)}=E_{t}[m_{t+1} P_{t+1}^{(n)}]\)

To simplify the notation, we define \([-\delta _{2},\Gamma _{n} ]\equiv \left[ -\delta _{2},\gamma _{0}^{n}\text {, }\gamma _{1}^{n}\cdots ,\gamma _{n-1}^{n}\right]\)

Then, we can find the coefficients following the difference equations:

### Appendix B: Data Description

### Demographic Variables

The U.S. annual population estimates series are collected from U.S. Census Bureau and the sample covers estimates from 1900–2050. Middle-aged to young ratio, MY\(_{t}\) is calculated as the ratio of the age group 40–49 to age group 20–29. Past MY\(_{t}\) projections for the period 1950–2013 are hand-collected from various past Census reports available at http://www.census.gov/prod/www/abs/p25.html. MY projections under different fertility rates are based on BoC’s 1975 population estimation and projections report.

### Spot Rate

3-Month Treasury Bill rate is taken from Goyal and Welch (2008) extended collecting data from St. Louis FRED database.

### Bond Yields

Bond yields are collected from Gurkaynak* et al* (2007) dataset, end of month data.

### Core Inflation

Time series of core inflation are collected from St. Louis FRED database.

### International Data

International bond yields are collected from Global Financial Data up to 2011. Benchmark bond yield is the 10-year constant maturity government bond yields. For Finland and Japan, shorter maturity bonds, 5-year and 7-year, respectively, are used, since a longer time series is available. International MY\(_{t}\) estimates for the period 1960–2008 are from World Bank Population estimates and projections from 2009 to 2050 are collected from International database (U.S. Census Bureau).

### Macro Factors

Stationary output and inflation factors are constructed following the data appendix of Ludvigson and Ng (2009). Data series of Group 1 (output) and Group 7 (prices) are extended up to 2013Q4 using data from Bureau of Economic Analysis (BEA) and St. Louis FRED databases.

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Favero, C.A., Gozluklu, A.E. & Yang, H. Demographics and the Behavior of Interest Rates.
*IMF Econ Rev* **64**, 732–776 (2016). https://doi.org/10.1057/s41308-016-0020-2

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DOI: https://doi.org/10.1057/s41308-016-0020-2

### JEL

- E43
- E44
- J11