A financial market model for the Netherlands: A methodological refinement
The Committee Parameters (Langejan et al. (2014)) advises to use the KNW-model (after Koijen et al. (2010)) to generate a representative scenario set for feasibility studies of pension funds. The scenario set enables a stochastic analysis of such feasibility studies. The underlying KNW-model is based on an affine factor model for the term structure. Stock returns, bond returns, interest rates, and inflation depend on observed factors and two latent factors. As such, the model contains relations between key financial risk factors of pension funds. CPB’s task is to estimate the model on Dutch data and, if appropriate, to calibrate some parameters in order to fit the recommendations of the Committee Parameters. Draper (2014) describes the current methods for this estimation and calibration.
The calibration aims to adjust the Ultimate Forward Rate (UFR) and certain long-term expectations and covariances of the variables in the model. However, this calibration process introduces some arbitrariness. More specifically, the resulting parameter set may deviate substantially from the maximum likelihood set, even when taking the restrictions of the calibration into account. Instead
of calibrating the model, we show how to impose restrictions in a continuous-time affine term structure model. In this way, the parameters correspond to the optimum of a constrained maximum
likelihood estimation. The results suggest that the method in Draper (2014) provides suboptimal parameter estimates.
The main result of this paper is the derivation of closed-form expressions for the long-term (unconditional) expectations, covariances, and the term structure. The expressions are required for
the constrained likelihood optimization, and replace simulations for a long-run analysis of parameter sets. Our results apply to a wide range of continuous-time affine term strucure models with the
Markov property, including the models in Dai and Singleton (2002) and Koijen et al. (2010).
The model is outlined in Section 2. Section 3 provides expressions for the mean and covariance of possibly transformed variables in a VAR(1)-model. Section 4 presents closed-form expressions for some characteristics of the term structure in terms of the parameters. The estimation results are in Section 5. We draw conclusions in Section 6.