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We provide a prediction of the corona virus flu propagation in selected European countries. This prediction is based on estimating sub-exponential growth rates with linear regressions (using Gompertz' law).

Milevsky and Huang (2011) investigated the optimal retirement spending policy for a utility-maximizing retiree facing a stochastic lifetime but assuming deterministic investment returns. They solved the problem using techniques from the calculus of variations and derived analytic expressions for the optimal spending rate and wealth depletion time under the Gompertz law of mortality. Of course, in the real world financial returns are stochastic as well as lifetimes, raising the question of whether their qualitative insights and approximations are generalizable or practical. We solve the retirement income problem when investment returns are indeed stochastic using numerical PDE methods, assuming the principles of stochastic control theory and dynamic programming. But then -- and this is key -- we compare the proper optimal spending rates to the analytic approach presented in Milevsky and Huang (2011) by updating the portfolio wealth inputs to current market values. Our main practical conclusion is that this simplistic approximation when calibrated properly and frequently can indeed be used as an accurate guide for rational retirement spending policy. As a by-product of our PDE-based methodology, our results indicate that even though the wealth depletion time is no longer a certainty under stochastic returns, the expected age at which liquid wealth is exhausted (i.) takes place well before the maximum lifetime and (ii.) is also well approximated by our analytical solution.

Health-care slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. This paper solves the problem of optimal dynamic consumption and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect the Gompertz' law. Optimal consumption and healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Health spending steadily increases with age, both in absolute terms and relative to total spending. Differential access to healthcare can account for observed longevity gains across cohorts.

The paper presents closed-form Delta and Gamma hedges for annuities and death assurances, in the presence of both longevity and interest-rate risk. Longevity risk is modeled through an extension of the classical Gompertz law, while interest rate risk is modeled via an Hull-and-White process. We theoretically provide natural hedging strategies, considering also contracts written on different generations. We provide a UK-population and bond-market calibrated example. We compute longevity exposures and explicitly calculate Delta-Gamma hedges. Re-insurance is needed in order to set-up portfolios which are Delta-Gamma neutral to both longevity and interest-rate risk.