This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Lévy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. [Scand. Actuar. J., 2014, 1–20], and Luo and Shevchenko [Int. J. Financ. Eng., 2014, 2, 1–24], to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. [op. cit.]. We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging intra-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Lévy framework. While the second moment and variance have been considered in existing literature, we show that the Value-at-Risk (VaR) may also be of interest as a risk measure to minimise risk in variable annuities portfolios.

There are three main challenges facing pay-as-you-go public pension systems. First, pension systems need to provide an adequate income for pensioners in the retirement phase. Second, participants wish a fair level of benefits in relation to the contributions paid. Last but not least, the pension system needs to be financially sustainable in the long run. In this paper, we jointly analyse the adequacy, fairness and sustainability of both defined benefit and defined contribution schemes. Finally, risk sharing mechanisms, that involve changes in the key variables of the system, are designed to restore the financial sustainability while we study their consequences on the adequacy and fairness of the system.

We analyse the effect of post-financial crisis unemployment dynamics on the Spanish pension system’s financial health using Aggregate Accounting. We compare the basic scenario where the current labour market dynamics persist with a full employment (best-case) scenario. We find that economic risk is the main driver of unsustainability in the short run. However, in the long run, the main driver of expenditures lies in the ageing demographic structure. Our results suggest that future reforms should increase labour market participation but confirm that recent pension reforms do not attain sustainability in the long run, indicating the need of further pension reforms.

The notional defined contribution model combines pay-as-you-go financing and a defined contribution pension formula. This paper aims to demonstrate the extent to which liquidity and solvency indicators are affected by fluctuations in economic and demographic conditions and to explore the introduction of an automatic balancing mechanism (ABM) into the pension scheme. We demonstrate that the introduction of an ABM reduces the volatility of the buffer fund and that, in most cases, the automatic mechanism that re-establishes solvency produces the highest value of the risk-adjusted notional factor.