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.
Highlights
Two prominent approaches to model the term structure of interest rates are the classes of equilibrium and no-arbitrage models
The variants differ in the assumption about the local long run risk premium functions, which determine the change from the risk neutral to the real world measure
As the Gauss2++ model is often used for pricing purposes, the focus in the literature lies on the evolution of interest rates under the risk neutral measure Q
Summary
Two prominent approaches to model the term structure of interest rates are the classes of equilibrium and no-arbitrage models. For these applications the corresponding model needs to be regarded under the real world measure Under this measure the corresponding one factor short-rate model has the following dynamic dr(t) = (t, r) + (t, r) (t, r) dt + (t, r)dW (t), where is the market price of risk and can depend on t and r. The linear function assumes that the market price of risk in the short horizon converges linearly to a long-term level With these simplified time-dependent functions it is possible to account for the problem mentioned by Hull et al [13] and the functions can still be estimated by historical data or calibrated in a forward looking manner to interest rate forecasts. For the interest rate model they use a Gauss2++ model with a presumed constant market price of risk Following their calibration procedure allows us to compare our results to real applications in the insurance industry. To calculate performance scenarios and risk indicators the Gauss2++ model must be regarded under the real world measure P
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