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

Abstract

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.

Capital requirement modeling for market and non-life premium risk in a dynamic insurance portfolio

“…According to Berninger and Pfeiffer (2021), the real-world processes for the state variables satisfy the following stochastic differential equations: where and are the constant speeds of mean reversion, and are the constant diffusion coefficients, while and are the Standard Brownian Motions under the real measure with instantaneous correlation . Furthermore, and are the deterministic mean reversion levels, and we here assume that they are given by step functions: where , , , and are real-valued constants, is a constant time parameter, and is the indicator function.…”

“…The zero-coupon bond price formula is also found to be exactly the same as in the real-world case, even though the state variables correspond now to the values of the processes under the risk-neutral measure. As described by Berninger and Pfeiffer (2021), using the explicit solutions of the differential equations both in the real and risk-neutral world, and using equation (11) and (12), the relation between the expected zero-coupon interest rate (continuously compounded) at time for a term of under the real measure and the risk-neutral measure is given by: …”

“…According to Berninger and Pfeiffer (2021), the real-world processes for the state variables satisfy the following stochastic differential equations:…”

“…We point out that, by specifying the models under real-world probabilities and risk-neutral probabilities, it is possible to carry out either risk management analysis, or market-consistent valuations, as proposed by Gambaro et al. (2018) or Berninger and Pfeiffer (2021). Such model specification can also be used for planning and management purposes.…”

“…On the other hand, Solvency II requires that capital requirements according to an IM are calculated using real-world probabilities to predict the risk drivers' behavior for the first year, even though risk-neutral probabilities must be used again to determine the market-consistent value of the basic Own Funds of the insurer after the first year. We point out that, by specifying the models under real-world probabilities and risk-neutral probabilities, it is possible to carry out either risk management analysis, or marketconsistent valuations, as proposed by Gambaro et al (2018) or Berninger and Pfeiffer (2021). Such model specification can also be used for planning and management purposes.…”