Variable Annuity with GMWB: surrender or not, that is the question

Under the optimal withdrawal strategy of a policyholder, the pricing of variable annuities with Guaranteed Minimum Withdrawal Benefit (GMWB) is an optimal stochastic control problem. The surrender feature available in marketed products allows termination of the contract before maturity, making it also an optimal stopping problem. Although the surrender feature is quite common in variable annuity contracts, there appears to be no published analysis and results for this feature in GMWB under optimal policyholder behaviour - results found in the literature so far are consistent with the absence of such a feature. Also, it is of practical interest to see how the much simpler bang-bang strategy, although not optimal for GMWB, compares with optimal GMWB strategy with surrender option. In this paper we extend our recently developed algorithm (Luo and Shevchenko 2015a) to include surrender option in GMWB and compare prices under different policyholder strategies: optimal, static and bang-bang. Results indicate that following a simple but sub-optimal bang-bang strategy does not lead to significant reduction in the price or equivalently in the fee, in comparison with the optimal strategy. We observed that the extra value added by the surrender option could add very significant value to the GMWB contract. We also performed calculations for static withdrawal with surrender option, which is the same as bang-bang minus the "no-withdrawal" choice. We find that the fee for such contract is only less than 1% smaller when compared to the case of bang-bang strategy, meaning that th "no-withdrawal" option adds little value to the contract.Comment: arXiv admin note: substantial text overlap with arXiv:1410.860

A Short Tale of Long Tail Integration

Integration of the form $\int_a^\infty {f(x)w(x)dx}$, where $w(x)$ is either $\sin (\omega {\kern 1pt} x)$ or $\cos (\omega {\kern 1pt} x)$, is widely encountered in many engineering and scientific applications, such as those involving Fourier or Laplace transforms. Often such integrals are approximated by a numerical integration over a finite domain $(a,\,b)$, leaving a truncation error equal to the tail integration $\int_b^\infty {f(x)w(x)dx}$ in addition to the discretization error. This paper describes a very simple, perhaps the simplest, end-point correction to approximate the tail integration, which significantly reduces the truncation error and thus increases the overall accuracy of the numerical integration, with virtually no extra computational effort. Higher order correction terms and error estimates for the end-point correction formula are also derived. The effectiveness of this one-point correction formula is demonstrated through several examples

Bayesian Model Choice of Grouped t-copula

One of the most popular copulas for modeling dependence structures is t-copula. Recently the grouped t-copula was generalized to allow each group to have one member only, so that a priori grouping is not required and the dependence modeling is more flexible. This paper describes a Markov chain Monte Carlo (MCMC) method under the Bayesian inference framework for estimating and choosing t-copula models. Using historical data of foreign exchange (FX) rates as a case study, we found that Bayesian model choice criteria overwhelmingly favor the generalized t-copula. In addition, all the criteria also agree on the second most likely model and these inferences are all consistent with classical likelihood ratio tests. Finally, we demonstrate the impact of model choice on the conditional Value-at-Risk for portfolios of six major FX rates

Computing Tails of Compound Distributions Using Direct Numerical Integration

An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly with little extra effort. High precision results of the 0.999 quantile and CVaR were obtained for compound losses with heavy tails and a very wide range of loss frequencies using the DNI, Fast Fourier Transform (FFT) and Monte Carlo (MC) methods. These results, particularly relevant to operational risk modelling, can serve as benchmarks for comparing different numerical methods. We found that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than MC and competitive with FFT in computing high quantiles and CVaR of compound distributions in the case of moderate to high frequencies and heavy tails

A unified pricing of variable annuity guarantees under the optimal stochastic control framework

In this paper, we review pricing of variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework, and review the existing numerical methods. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is very efficient when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimum accumulation benefit, optimal stochastic control, direct integration metho

Time-varying Autoregressive Modeling of Nonstationary Signals

Nonstationary signal modeling is a research topic of practical interest. In this thesis, we adopt a time-varying (TV) autoregressive (AR) model using the basis function (BF) parameter estimation method for nonstationary process identification and instantaneous frequency (IF) estimation. The current TVAR model in direct form (DF) with the blockwise least-squares and recursive weighted-least-squares BF methods perform equivalently well in signal modeling, but the large estimation error may cause temporary instabilities of the estimated model. To achieve convenient model stability monitoring and pole tracking, the TVAR model in cascade form (CF) was proposed through the parameterization in terms of TV poles (represented by second order section coefficients, Cartesian coordinates, Polar coordinates), where the time variation of each pole parameter is assumed to be the linear combination of BFs. The nonlinear system equations for the TVAR model in CF are solved iteratively using the Gauss-Newton algorithm. Using the CF, the model stability is easily controlled by constraining the estimated TV poles within the unit circle. The CF model shows similar performance trends to the DF model using the recursive BF method, and the TV pole representation in Cartesian coordinates outperforms all other representations. The individual frequency variation can be finely tracked using the CF model, when several frequency components are present in the signal. Simulations were carried on synthetic sinusoidal signals with different frequency variations for IF estimation. For the TVAR model in DF (blockwise), the basis dimension (BD) is an important factor on frequency estimation accuracy. For the TVAR model in DF (recursive) and CF (Cartesian), the influences of BD are negligible. The additive white noise in the observed signal degrades the estimation performance, and the the noise effects can be reduce by using higher model order. Experiments were carried on the real electromyography (EMG) data for frequency estimation in the analysis of muscle fatigue. The TVAR modeling methods show equivalent performance to the conventional Fourier transform method