# The Economic Factor Model and a class of affine term structure models

The economic factor model is an affine term structure model where the factors have a particular economic interpretation. The general affine n-factor term structure model is presented and the EFM as a special case is developed in detail. The results include general expressions for dynamics bond prices, forward curves and

The economic factor model is an affine term structure model where the factors have a particular economic interpretation. The general affine n-factor term structure model is presented and the EFM as a special case is developed in detail. The results include general expressions for dynamics bond prices, forward curves and expected excess returns and risk premia.
For the special case EFM(2) the numerical calculations are presented in detail. A section on the Ornstein-Uhlenbeck process collects all the necessary results.

# The general n-factor Gaussian model

## Notation

Matrix notation is used throughout. Primes$^\prime$ denote transpose. The vector $e_i\in\mathbb{R}^n$ is Cartesian basis vector $i$. Objective measure is $P$ and risk-neutral measure is $P^\star$. In order to avoid clutter the stars$^\star$ are occasionally dropped. Although this is always indicated: be careful.

## Bond prices

Consider the general Gaussian real-world n-factor model under objective
measure
$$P:\quad d\xi = (A\xi+a),d t + \Sigma,d W, , A\in\mathbb{R}^{n\times n}, a\in\mathbb{R}^n \quad .$$ Via Girsanov transform to risk-neutral measure $P^\star$ with $d W = \lambda(\xi) d t+ d W^\star$ and $\lambda(\xi)$ affine we get:
$$\lambda(\xi)=\Sigma^-(\Lambda\xi+\lambda^1), , \Lambda\in\mathbb{R}^{n\times n}\quad .$$ Introducing $A^\star=A+\Lambda$ and $a^\star=a+\lambda^1$ leads to risk neutral dynamics
$$P^\star:\quad d\xi = (A^\star\xi+a^\star),d t + \Sigma,d W^\star \quad ,$$ The price $p(\tau,T)$ of a T-Bond (i.e. a zero-bond of maturity $T$) at time $\tau$ then satisfies the PDE
$$-r p + \frac{\partial p}{\partial\tau} + \left(\frac{\partial p}{\partial\xi}\right)^\prime (a^\star+A^\star\xi)+ {1\over 2}\:\textsf{tr}\:\Sigma^\prime \left(\frac{\partial^2 p}{\partial \xi\partial \xi}\right) \Sigma = 0.$$ Setting $t=T-\tau$ the ansatz $p(\tau,T) = p(t) = \exp\left(-u(t)^\prime\xi + v(t)\right)$ leads to the following two ODEs
$$\frac{\partial u}{\partial t} - {A^\star}^\prime u = (1,0,0)^\prime$$ $$\frac{\partial v}{\partial t} = -u^\prime a^\star + {1\over 2} u^\prime V u \qquad ,$$ with initial conditions $u(0) = 0$ and $v(0)=0$. The function $u(t)$ has a natural interpretation as duration with respect to the rates $\xi$, since $\partial\ln(P)/\partial\xi = u$. The solutions to the two ODEs are ...