This paper considers variable annuity (VA) contracts embedded with guaranteed minimum accumulation benefit (GMAB) riders when policyholder’s proceeds are taxed upon early surrender or maturity. These contracts promise the return of the premium paid by the policyholder, or a higher rolled-up value, at the end of the investment period. A partial differential equation valuation framework which exploits the numerical method of lines is used to determine fair fees that render the policyholder and insurer breakeven. Two taxation regimes are considered: one where capital gains are allowed to offset losses and a second where gains do not offset losses. Most insurance providers highlight the tax-deferred features of VA contracts. We show that the regime under which the insured is taxed significantly impacts prices. If losses are allowed to offset gains then this enhances the market, increasing the policyholder’s willingness to participate in the market compared to the case when losses are not allowed to offset gains. With fair fees from the policyholder’s perspective, we show that the net profit is generally positive for insurance companies offering the contract as a naked option without any hedge. We also show how investment policy, as reflected in the Sharpe ratio, impacts and interacts with policyholder persistency.

In this work, the Bishop and Love models for longitudinal vibrations are adopted to study the dynamics of isotropic rods with conical and exponential cross-sections. Exact solutions of both models are derived, using appropriate transformations. The analytical solutions of these two models are obtained in terms of generalised hypergeometric functions and Legendre spherical functions respectively. The exact solution of Love model for a rod with exponential cross-section is expressed as a sum of Gauss hypergeometric functions. The models are solved numerically by using the method of lines to reduce the original PDE to a system of ODEs. The accuracy of the numerical approximations is studied in the case of special solutions.

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

Implicit Runge–Kutta methods have a special role in the numerical solution of stiff problems, such as those found by applying the method of lines to the partial differential equations arising in physical modelling. Of particular interest in this paper are the high-order methods based on Gaussian quadrature and the efficiently implementable singly implicit methods.