A parent who supports a permanently dependent child faces a concrete problem: guaranteeing the child an income if the parent dies first. This paper studies the two-life contract that answers this need for a child with Down syndrome, with parent and child governed by distinct mortality laws, and computes the full distribution of its net present value by path enumeration, not merely its mean. The provision is small and identical for the two benefit designs, a lump sum or a life annuity, because they are reversionary; but the risk borne by the insurer lies in the rest of the distribution. The choice of design barely alters the capital except when both lives are old, and never rescues the product. The dominant risk is hard to diversify for two reasons: the low prevalence of the condition prevents assembling the portfolio that would dilute its specific component, and the beneficiary’s longevity is systematic: it shifts the provision of the whole portfolio at once, so it does not average out with volume and sets a capital floor that the Solvency II standard formula understates by a factor of about five. Nor does product redesign help: a three-life variant reduces the absolute variance of the liability but worsens the capital per unit of premium. The three levers of private insurance, diversifying within a portfolio, mutualising the systematic risk, and redesigning the product, fail simultaneously, placing the cover beyond the reach of the private market at an affordable premium.
The Pareto distribution is widely used for modelling large losses, and its popularity is well justified. In particular, practitioners can develop good intuition for reasonable Pareto alphas. Every sufficiently regular severity distribution behaves locally like a Pareto distribution, which motivates the concept of local Pareto alphas. These provide a powerful tool for comparing tail heaviness across severity distributions. In this study, we use local Pareto alphas as the defining input for constructing tailor-made severity distributions. This is a useful reparameterization of the standard construction of a survival function from its hazard, allowing for greater interpretability in large-loss modeling. We propose an algorithm that approximates the resulting local Pareto distributions with piecewise Pareto distributions at a prescribed accuracy. This approximation scheme is practically valuable, as piecewise Pareto models allow efficient computation of key quantities such as the CDF, PDF, quantile function, layer means, and layer variances. A computational implementation is freely available in the R package LocalPareto, which is used for illustration in various numerical examples.
We examine the key challenges associated with insuring climate-related risks in the context of a changing climate. Adopting a balance-sheet perspective on insurance company management, we analyze the concept of insurability and how it may be reshaped by climate change. Focussing on non-life perils, we further explore the roles of regulation, risk prevention and mitigation in addressing these issues, and discuss a range of potential solutions – and limitations – that emerge. These challenges also open up interesting opportunities for both academic research and practical innovation within the insurance industry.
In this bicentenary year of Gompertz (1825) [23] we advance a conjecture; that the split between Newtonian and Leibnizian forms of calculus occurring in the eighteenth century had a long-term influence on actuarial science in the nineteenth and twentieth centuries. When the growing life insurance industry in Britain needed mathematical expertise, a profession with Newtonian roots was created to supply it, while elsewhere in Europe it was found in universities with Leibnizian roots. We consider the consequences up to the present day, when analysis in the Leibnizian branch has led via Kolmogorov [49] to modern financial mathematics.
