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.

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# 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 *...*

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- Properties of the term structure
- $y(\infty)$ and $y(0)$
- Slope of the short end -Instantaneous covariances
- Unconditional finite-time yield dynamics
- Conditional finite-time yield dynamics ###

- The forward curve
- Expected excess returns and risk premier
- Bond portfolios
- Local efficient frontier
- Finite time portfolio evolution

- Sensitivity of prices and yields to volatilities
- Partial yield curve inversion using linearity in some parameters
- Sensitivity of yields to constant risk premia

## EFM(2)

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- Introduction
- Model specification
- Diagonalisation of ${A^\star}^\prime$ and $A$
- Special results for EFM(2)
- u(t)
- $y(\infty)$
- $\mu_\infty$

## OU-processes

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- Momenta
- OU-process in VAR form
- Maximum likelihood estimation
- Autocorrelation function and power spectrum
- Short observation times
- Estimation of OU processes
- Unrestricted estimation
- Restricted estimation
- Non-stationary OU-processes

## Diagonalisation of outer product matrices

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## Model specification - EFM(1)

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