Scientific Monographs, Bank of Finland
No E:44/2012:
Stochastic modeling of financing longevity risk in pension insurance
Vesa Ronkainen ()
Abstract: This work studies and develops tools to quantify and
manage the risks and uncertainty relating to the pricing of annuities in
the long run. To this end, an idealized Monte-Carlo simulation model is
formulated, estimated and implemented, which enables one to investigate
some typical pension and life insurance products. The main risks in pension
insurance relate to investment performance and mortality/longevity
development. We first develop stochastic models for equity and bond
returns. The S&P 500 yearly total return is modeled by an uncorrelated and
Normally distributed process to which exogenous Gamma distributed negative
shocks arrive with Geometrically distributed interarrival times. This
regime switching jump model takes into account the empirical observations
of infrequent exceptionally large losses. The 5-year US government bond
yearly total return is modeled as an ARMA(1,1) process after suitably
log-transforming the returns. This model is able to generate long term
interest rate cycles and allows rapid year-to-year corrections in the
returns. We also address the parameter uncertainty in these models.
We
then develop a stochastic model for mortality. The chosen mortality
forecasting model is the well-known model of Lee and Carter (1992), in
which we use the Bayesian MCMC methods in the inference concerning the time
index. Our analysis with a local version of the model showed that the
assumptions of the Lee-Carter model are not fully compatible with Finnish
mortality data. In particular we found that mortality has been lower than
average for the cohort born in wartime. However, because the forecasts of
these two models were not significantly different, we chose the more
parsimonious Lee-Carter model. Although our main focus is on the total
population data, we also analysed the data for males and females
separately. Finally we build a flexible model for the dependence structure
that allows us to generate stochastic scenarios in which mortality and
economic processes are either uncorrelated, correlated or shock-correlated.
By using the simulation model to generate stochastic pension
cash-flows, we are then able to analyse the financing of longevity risk in
pension insurance and the resulting risk management issues. This is
accomplished via three case studies. Two of these concentrate on the
pricing and solvency questions of a pension portfolio. The first study
covers a single cohort of different sizes, and the second allows for
multiple cohorts of annuitants. The final case study discusses individual
pension insurance from the customer and long-term points of view.
Realistic statistical long-term risk measurement is the key theme of this
work, and so we compare our simulation results with the Value-at-Risk or
VaR approach. The results show that the limitations of basic VaR approach
must be carefully accounted for in applications. The VaR approach is the
most commonly used risk measurement methodology in insurance and finance
applications. For instance, it underlies the solvency capital requirement
in Solvency II, which we also discuss in this work.
Keywords: equities; stocks; jump model; bond; longevity; Lee-Carter model; stochastic mortality; cohort mortality; dependence model; asymmetric dependence; parameter uncertainty; stochastic annuity; pension; cohort size; solvency; internal model; (follow links to similar papers)
JEL-Codes: G12; J11; (follow links to similar papers)
124 pages, May 25, 2012
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