Demographics and the Behavior of Interest Rates

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.

Appendices

Appendix A: Derivation of Demographic ATSM

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

$$\begin{gathered} y_{t}^{{(n)}} = - \frac{1}{n}\left( {A_{n} + B_{n}^{\prime } X_{t} + \Gamma _{n} {\text{MY}}_{t}^{n} } \right) + \varepsilon _{{t,t + 1}}\quad \varepsilon _{{t,t + n}} \sim N(0,\sigma _{n}^{2} )\hfill\\X_{t} = \mu + \Phi X_{{t - 1}} + \nu _{t}\quad \nu _{t} \sim i.i.d.N(0,\Omega ) \hfill \\ y_{t}^{{(1/4)}} = \delta _{0} + {\mathbf{\delta }}_{1}^{\prime } X_{t} + \delta _{2} {\text{MY}}_{t} \hfill \\ \Lambda _{t} = \lambda _{0} + \lambda _{1} X_{t} \hfill \\ m_{{t + 1}} = \exp ( - y_{t}^{{(1)}} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} - \Lambda _{t}^{\prime } v_{{t + 1}} ) \hfill \\ P_{t}^{{(n)}} \equiv \left[ {\frac{1}{{1 + Y_{{t,t + n}} }}} \right]^{n} ,{\mkern 1mu} y_{t}^{{(n)}} \equiv \ln \left( {1 + Y_{t}^{{(n)}} } \right) \hfill \\ \Gamma _{n} {\text{MY}}_{t}^{n} \equiv \left[ {\gamma _{0}^{n} ,\gamma _{1}^{n} \cdots ,\gamma _{{n - 1}}^{n} } \right] \left[ {\begin{array}{*{20}c} {{\text{MY}}_{t} } \\ {{\text{MY}}_{{t + 1}} } \\ \vdots \\ {{\text{MY}}_{{t + n - 1}} } \\ \end{array} } \right] \quad X_{t} = \left[ {\begin{array}{*{20}l} f_{t}^{\pi }\\ f_{t}^{x} \\ {f_{t}^{{u,1}} } \hfill \\ {f_{t}^{{u,2}} } \hfill \\ {f_{t}^{{u,3}} } \hfill \\ \end{array} } \right] \hfill \\ \end{gathered}$$

Bond prices can be recursively computed as follows:

$$\begin{aligned} P_{t}^{{(n)}} & = E_{t} [m_{{t + 1}} P_{{t + 1}}^{{(n - 1)}} ] = E_{t} [m_{{t + 1}} m_{{t + 2}} P_{{t + 2}}^{{(n - 2)}} ] \\ & = E_{t} [m_{{t + 1}} m_{{t + 2}} \cdots m_{{t + n}} P_{{t + n}}^{{(0)}} ] = E_{t} [m_{{t + 1}} m_{{t + 2}} \cdots m_{{t + n}} 1] \\ & = E_{t} \left[ {\exp \left( {\mathop {\mathop \sum \limits^{{n - 1}} }\limits_{{i = 0}} \left( { - y_{{t + i,t + i + 1}} - \frac{1}{2}\Lambda _{{t + i}}^{\prime } \Omega \Lambda _{{t + i}} - \Lambda _{{t + i}}^{\prime } \nu _{{t + i + 1}} } \right)} \right)} \right] \\ & = E_{t} [\exp (A_{n} + B_{n}^{\prime } X_{t} + \Gamma _{n}^{\prime } {\text{MY}}_{t}^{n} )] = E_{t} \left[ {\exp \left( { - ny_{t}^{{(n)}} } \right)} \right] \\ & = E_{t}^{Q} \left[ {\exp \left( { - \mathop {\mathop \sum \limits^{{n - 1}} }\limits_{{i = 0}} y_{{t + i}}^{{(1)}} } \right)} \right], \\ \end{aligned}$$

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 $\Phi ^{Q}$

$$\mu ^{Q}=\mu -\Omega \lambda _{0}\quad\text { and }\quad\Phi ^{Q}=\Phi -\Omega \lambda _{1}$$

To derive the coefficients of the model, let us start with $n=1$:

$$P_{t}^{(1)}=\exp (-y_{t}^{(1/4)})=\exp (-\delta _{0}-\delta _{1}^{\prime } X_{t}-\delta _{2}\text {MY}_{t})$$

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

$$\begin{gathered} = E_{t} [\exp ( - y_{t}^{{(1/4)}} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} - \Lambda _{t}^{\prime } \nu _{{t + 1}} )\exp (A_{n} + B_{n}^{\prime } X_{{t + 1}} + \Gamma _{n} {\text{MY}}_{{t + 1}}^{n} )] \hfill \\ = \exp ( - y_{t}^{{(1/4)}} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} + A_{n} )E_{t} [\exp ( - \Lambda _{t}^{\prime } \nu _{{t + 1}} + B_{n}^{\prime } X_{{t + 1}} + \Gamma _{n} {\text{MY}}_{{t + 1}}^{n} )] \hfill \\ = \exp ( - y_{t}^{{(1/4)}} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} + A_{n} + \Gamma _{n} {\text{MY}}_{{t + 1}}^{n} )E_{t} [\exp ( - \Lambda _{t}^{\prime } \nu _{{t + 1}} + B_{n}^{\prime } (\mu + \Phi X_{t} + \nu _{{t + 1}} ))] \hfill \\ = \exp [ - \delta _{0} - \delta _{1}^{\prime } X_{t} - \delta _{2} MY_{t} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} + A_{n} + \Gamma _{n} {\text{MY}}_{{t + 1}}^{n} + B_{n}^{\prime } (\mu + \Phi X_{t} )]E_{t} [\exp ( - \Lambda _{t}^{\prime } \nu _{{t + 1}} + B_{n}^{\prime } \nu _{{t + 1}} )] \hfill \\ = \exp [ - \delta _{0} - \delta _{1}^{\prime } X_{t} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} + A_{n} - \delta _{2} {\text{MY}}_{t} + B_{n}^{\prime } (\mu + \Phi X_{t} ) + \Gamma _{n} {\text{MY}}_{{t + 1}}^{n} ]\exp \{ E_{t} [( - \Lambda _{t}^{\prime } + B_{n}^{\prime } )\nu _{{t + 1}} ] + \frac{1}{2}var[( - \Lambda _{t}^{\prime } + B_{n}^{\prime } )\nu _{{t + 1}} ]\} \hfill \\ = \exp [ - \delta _{0} - \delta _{1}^{\prime } X_{t} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} + A_{n} + B_{n}^{\prime } (\mu + \Phi X_{t} ) + \left[ { - \delta _{2} ,\gamma _{0}^{n} {\text{, }}\gamma _{1}^{n} \cdots ,\gamma _{{n - 1}}^{n} } \right]{\text{MY}}_{t}^{{n + 1}} ]\exp \{ \frac{1}{2}var[( - \Lambda _{t}^{\prime } + B_{n}^{\prime } )\nu _{{t + 1}} ]\} \hfill \\ \end{gathered}$$

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

$$\begin{aligned} = & \exp \left\{ { - \delta _{0} - \delta _{1}^{\prime } X_{t} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} + A_{n} + B_{n}^{\prime } (\mu + \Phi X_{t} ) + [ - \delta _{2} ,\Gamma _{n} ]{\text{MY}}_{t}^{{n + 1}} } \right\} \\ & \times \exp \left\{ {\frac{1}{2}E_{t} [( - \Lambda _{t}^{\prime } + B_{n}^{\prime } )\nu _{{t + 1}} \nu _{{t + 1}} \prime ( - \Lambda _{t} + B_{n} )]} \right\} \\ = & \exp \left\{ { - \delta _{0} - \delta _{1}^{\prime } X_{t} - \frac{1}{2}\Lambda _{t}^{\prime } \Omega \Lambda _{t} + A_{n} + B_{n}^{\prime } (\mu + \Phi X_{t} ) + [ - \delta _{2} ,\Gamma _{n} ]{\text{MY}}_{t}^{{n + 1}} } \right\} \\ & \times \exp \left\{ {\frac{1}{2}[\Lambda _{t}^{\prime } \Omega \Lambda _{t} - 2B_{n}^{\prime } \Omega \Lambda _{t} + B_{n}^{\prime } \Omega B_{n} )]} \right\} \\ = & \exp \left\{ { - \delta _{0} + A_{n} + B_{n}^{\prime } \mu + (B_{n}^{\prime } \Phi - \delta _{1}^{\prime } )X_{t} - B_{n}^{\prime } \Omega \Lambda _{t} + \frac{1}{2}B_{n}^{\prime } \Omega B_{n} + [ - \delta _{2} ,C_{n} ]MY_{t}^{{n + 1}} } \right\} \\ = & \exp \left\{ { - \delta _{0} + A_{n} + B_{n}^{\prime } \mu + (B_{n}^{\prime } \Phi - \delta _{1}^{\prime } )X_{t} - B_{n}^{\prime } \Omega (\lambda _{0} + \lambda _{1} X_{t} ) + \frac{1}{2}B_{n}^{\prime } \Omega B_{n} + [ - \delta _{2} ,\Gamma _{n} ]{\text{MY}}_{t}^{{n + 1}} } \right\} \\ = & \exp \left\{ {A_{1} + A_{n} + B_{n}^{\prime } (\mu - \Omega \lambda _{0} ) + \frac{1}{2}B_{n}^{\prime } \Omega B_{n} + (B_{n}^{\prime } \Phi - B_{n}^{\prime } \Omega \lambda _{1} + B_{1}^{\prime } )X_{t} + [ - \delta _{2} ,\Gamma _{n} ]{\text{MY}}_{t}^{{n + 1}} } \right\} \\ \end{aligned}$$

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

$$\begin{gathered} A_{{n + 1}} = A_{1} + A_{n} + B_{n}^{\prime } (\mu - \Omega \lambda _{0} ) + \frac{1}{2}B_{n}^{\prime } \Omega B_{n} \hfill \\ B_{{n + 1}}^{\prime } = B_{n}^{\prime } \Phi - B_{n}^{\prime } \Omega \lambda _{1} + B_{1}^{\prime } \Gamma _{{n + 1}} = [ - \delta _{2} ,\Gamma _{n} ]. \hfill \\ \end{gathered}$$

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.