RDP 2018-02: Affine Endeavour: Estimating a Joint Model of the Nominal and Real Term Structures of Interest Rates in Australia Appendix A: The Affine Term Structure Model

This section documents the mathematics of the ATSM that we use, and follows the expository style and terminology of Ang and Piazzesi (2003) closely.

First we derive pricing equations for the general class of ATSM that we use. Let Inline Equation be the price of a nominal bond at time t that pays one dollar at time t + n. The no-arbitrage assumption implies that an equivalent martingale (or risk-neutral) measure, denoted by Q, exists such that

where rt is the nominal short rate (Harrison and Kreps 1979). Assume that the nominal short rate is given by Inline Equation, where Xt is a stochastic process that describes all economic and financial factors relevant to bond pricing and which evolves according to

for μ a vector, θ and ∑ matrices, and εt+1 ~N(0,IN). Now denote the Raydon-Nikodym derivative, which converts the risk-neutral measure to the real-world measure, by ξt + 1; for any random variable Zt + 1,

where Inline Equation without a superscript is understood to be under the real-world measure. In our case, we assume that the so-called market price of risk is given by

for λ0 a vector and λ1 a matrix, and that the Raydon-Nikodym derivative linking the real-world and risk-neutral measures is given by

Note that Inline Equation is often referred to as the pricing kernel or stochastic discount factor, which summarises how agents discount pay-offs under different states of the world.

Given the above, we now show by induction that bond prices are exponentially affine in Xt. Assume that Inline Equation; we will show that this implies that Inline Equation. Starting from Equation (A1),

where line 2 follows from Equations (A3) and (A5) and the assumption that Inline Equation, line 3 follows from Equation (A2), line 5 follows from the moment-generating function of a multivariate N(0,IN) random variable, and line 6 follows from Equation (A4) and the assumed functional form of rt. Line 6 has the desired functional form of expInline Equation, and so by equating coefficients that do and do not depend on Xt we can deduce that

To start the induction and also provide starting values for the above recursion, note that Inline Equation, which implies that A0 = 0, B0 = 0. One can also directly calculate Inline Equation so that A1 = −ρ0 and B1 = −ρ1.

Next consider an inflation-indexed bond Inline Equation that pays one unit of consumption good, or Qt + n/Qt units of nominal value, at time t + n. Here Qt is the price level, we define inflation πt by Qt/Qt – 1 = exp(πt), and we assume that

for π0 a scalar and π1 a vector. As with the nominal case, start by assuming that Inline Equation. We will show by induction that this implies that Inline Equation. The no-arbitrage condition for an inflation-indexed bond is given by

where, using the fact that Inline Equation, we define Inline Equation and Inline Equation. But, given these re-definitions, the last line now matches line 2 from the nominal bond case, so that

To provide starting values for the above recursion, note that Inline Equation, which implies that Inline Equation.

Two things are worth noting. First, in the pricing equations above, the price of risk parameters λ0 and λ1 are confounded with the factors describing the evolution of Xt, being μ, θ and ∑. As such, a cross-section of bond prices is not enough to fully identify the model. Second, and related to the first point, one would obtain the same pricing equations if one assumed that investors were risk neutral (so that λ0 = 0, λ1 = 0) and Inline Equation for Inline Equation and Inline Equation.