The Gauss2++ model: a comparison of different measure change specifications for a consistent risk neutral and real world calibration

Especially in the insurance industry interest rate models play a crucial role, e.g. to calculate the insurance company's liabilities, performance scenarios or risk measures. A prominant candidate is the 2-Additive-Factor Gaussian Model (Gauss2++ model)—in a different representation also known as the 2-Factor Hull-White model. In this paper, we propose a framework to estimate the model such that it can be applied under the risk neutral and the real world measure in a consistent manner. We first show that any time-dependent function can be used to specify the change of measure without loosing the analytic tractability of, e.g. zero-coupon bond prices in both worlds. We further propose two candidates, which are easy to calibrate: a step and a linear function. They represent two variants of our framework and distinguish between a short and a long term risk premium, which allows to regularize the interest rates in the long horizon. We apply both variants to historical data and show that they indeed produce realistic and much more stable long term interest rate forecast than the usage of a constant function, which is a popular choice in the industry. This stability over time would translate to performance scenarios of, e.g. interest rate sensitive fonds and risk measures.