Bayesian Value-at-Risk Backtesting: The Case of Annuity Pricing

Leung, Melvern and Li, Youwei and Pantelous, Athanasios and Vigne, Samuel (2019): Bayesian Value-at-Risk Backtesting: The Case of Annuity Pricing.

Abstract

We propose a new Unconditional Coverage backtest for VaR-forecasts under a Bayesian framework that significantly minimise the direct and indirect effects of p-hacking or other biased outcomes in decision-making, in general. Especially, after the global financial crisis of 2007-09, regulatory demands from Basel III and Solvency II have required a more strict assessment setting for the internal financial risk models. Here, we employ linear and nonlinear Bayesianised variants of two renowned mortality models to put the proposed backtesting technique into the context of annuity pricing. In this regard, we explore whether the stressed longevity scenarios are enough to capture the experienced liability over the foretasted time horizon. Most importantly, we conclude that our Bayesian decision theoretic framework quantitatively produce a strength of evidence favouring one decision over the other.

• Item Type: MPRA Paper
• Language: English
• Keywords: Bayesian decision theory; Value-at-Risk; Backtesting; Annuity pricing; Longevity risk
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C44 - Operations Research ; Statistical Decision Theory; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing; G - Financial Economics > G1 - General Financial Markets > G17 - Financial Forecasting and Simulation; G - Financial Economics > G2 - Financial Institutions and Services > G22 - Insurance ; Insurance Companies ; Actuarial Studies; G - Financial Economics > G2 - Financial Institutions and Services > G23 - Non-bank Financial Institutions ; Financial Instruments ; Institutional Investors

References

Berger, J. O. and Sellke, T. (1987). Testing a point null hypothesis: The irreconcilability of p values and evidence. Journal of the American Statistical Association, 82(397):112-122.

Berkowitz, J. and O'Brien, J. (2002). How accurate are value-at-risk models at commercial banks? The Journal of Finance, 57(3):1093-1111.

Cairns, A., Blake, D., and Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73(4):687-718.

Cairns, A. J., Blake, D., Dowd, K., Coughlan, G. D., Epstein, D., Ong, A., and Balevich, I. (2009). A quantitative comparison of stochastic mortality models using data from england and wales and the united states. North American Actuarial Journal, 13(1):1-35.

Campbell, S. D. (2007). A review of backtesting and backtesting procedures. Journal of Risk, 9(2):1-18.

Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika, 81(3):541-553.

Christoffersen, P. (2009). Value{at{risk models. In Handbook of Financial Time Series, pages 753-766. Springer.

Christoffersen, P. and Pelletier, D. (2004). Backtesting value-at-risk: A duration-based approach. Journal of Financial Econometrics, 2(1):84-108.

Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 4(39):841-862.

Debon, A., Montes, F., and Puig, F. (2008). Modelling and forecasting mortality in Spain. European Journal of Operational Research, 189(3):624 - 637.

Drenovak, M., Rankovi�c, V., Ivanovi�c, M., Uro�sevi�c, B., and Jelic, R. (2017). Market risk management in a post-Basel II regulatory environment. European Journal of Operational Research, 257(3):1030-1044.

Eckert, J. and Gatzert, N. (2018). Risk-and value-based management for non-life insurers under solvency constraints. European Journal of Operational Research, 266(2):761-774.

Fung, M. C., Peters, G. W., and Shevchenko, P. V. (2017). A uni�ed approach to mortality modelling using state-space framework: characterisation, identi�cation, estimation and forecasting. Annals of Actuarial Science, 11(2):343{389.

Glasserman, P., Heidelberger, P., and Shahabuddin, P. (2002). Portfolio value-at-risk with heavytailed risk factors. Mathematical Finance, 12(3):239{269.

Goodman, S. N. (2001). Of p-values and bayes: a modest proposal. Epidemiology, 12(3):295{297.

Hari, N., De Waegenaere, A., Melenberg, B., and Nijman, T. E. (2008). Longevity risk in portfolios of pension annuities. Insurance: Mathematics and Economics, 42(2):505{519.

Harvey, C. R. (2017). Presidential address: the scienti�c outlook in �nancial economics. The Journal of Finance, 72(4):1399{1440.

Konicz, A. K. and Mulvey, J. M. (2015). Optimal savings management for individuals with de�ned contribution pension plans. European Journal of Operational Research, 243(1):233{247.

Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 3(2):73{84.

Lee, R. D. and Carter, L. R. (1992). Modeling and forecasting us mortality. Journal of the American Statistical Association, 87(419):659{671.

Leung, M., Fung, M. C., and O`Hare, C. (2018). A comparative study of pricing approaches for longevity instruments. Insurance: Mathematics and Economics, 82:95 { 116.

Li, Y., Zeng, T., and Yu, J. (2014). A new approach to bayesian hypothesis testing. Journal of Econometrics, 178:602{612.

Nieto, M. R. and Ruiz, E. (2016). Frontiers in VaR forecasting and backtesting. International Journal of Forecasting, 32(2):475{501.

Plat, R. (2011). One-year value-at-risk for longevity and mortality. Insurance: Mathematics and Economics, 49(3):462{470.

Wied, D., Wei�, G. N., and Ziggel, D. (2016). Evaluating value-at-risk forecasts: A new set of multivariate backtests. Journal of Banking & Finance, 72:121{132.

Ziggel, D., Berens, T., Wei�, G. N., and Wied, D. (2014). A new set of improved value-at-risk backtests. Journal of Banking & Finance, 48:29-41.