We consider an extreme renewal process with no-mean heavy-tailed Pareto(II) inter-renewals and shape parameter $\alpha$ where $0\lt\alpha \leq 1$. Two steps are required to derive integral expressions for the analytic probability density functions (pdfs) of the fixed finite time $t$ excess, age, and total life, and require extensive computations. Step 1 creates and solves a Volterra integral equation of the second kind for the limiting pdf of a basic underlying regenerative process defined in the text, which is used for all three fixed finite time $t$ pdfs. Step 2 builds the aforementioned integral expressions based on the limiting pdf in the basic underlying regenerative process. The limiting pdfs of the fixed finite time $t$ pdfs as $t\rightarrow \infty$ do not exist. To reasonably observe the large $t$ pdfs in the extreme renewal process, we approximate them using the limiting pdfs having simple well-known formulas, in a companion renewal process where inter-renewals are right-truncated Pareto(II) variates with finite mean; this does not involve any computations. The distance between the approximating limiting pdfs and the analytic fixed finite time large $t$ pdfs is given by an $L_{1}$ metric taking values in $(0,1)$, where “near $0$” means “close” and “near $1$” means “far”.

In this paper, we treat a nonlinear and unbalanced $2$-color urn scheme, subjected to two different nonlinear drawing rules, depending on the color withdrawn. We prove a central limit theorem as well as a law of large numbers for the urn composition. We also give an estimate of the mean and variance of both types of balls.

This paper investigates potential biodiversity valuation tools which actuaries could use in their work. It is an initial research paper into a selection of UK-based biodiversity valuation tools identified by the Department for Environment and Rural Affairs in its publication “Enabling a Natural Capital Approach Guidance”. The “Enabling a Natural Capital Approach Guidance” publication is seen as a comprehensive practical guide to natural capital and therefore is determined a sensible starting point on which to base this research. This research paper is not intended to be an exhaustive exploration of all biodiversity tools available, but rather is intended to identify a selection of tools which may be candidates for further research into their actuarial use case. We conclude that there are tools which merit additional research, and we recommend that these tools be further investigated to understand (i) the specific actuarial use case(s), (ii) whether the tools are applicable to direct infrastructure investments only or a broad range of asset classes, and (iii) whether their scope can be extended beyond the UK.

This paper follows on from the initial position paper on “The Importance of Biodiversity Risks”, prepared by the Biodiversity and Natural Capital Working party, a volunteer group working under the Sustainability Board. This paper explores the link between zoonotic disease and biodiversity loss and aims to raise awareness and discussion within the actuarial community on why this should be an important consideration in risk management. This paper focuses on how zoonotic diseases emerge, how they are linked to biodiversity loss, the potential impacts in the future and progress within the financial sector. This paper forms part of a collection of papers prepared by volunteers under the Sustainability Board that focus on different elements of biodiversity risk considerations.

This paper highlights the urgent need for actuaries to take into account the importance, perils and impacts of global biodiversity risks. The Biodiversity and Natural Capital Working Party has been set up to take forward a series of activities including think pieces, webinars and external engagement to ensure our proactive engagement with these risks.

The COVID-19 pandemic interrupts the relatively steady trend of improving longevity observed in many countries over the last decades. We claim that this needs to be addressed explicitly in many mortality modelling applications, for example, in the life insurance industry. To support this position, we provide a descriptive analysis of the mortality development of several countries up to and including the year 2020. Furthermore, we perform an empirical and theoretical investigation of the impact a mortality jump has on the parameters, forecasts and implied present values of the popular Lee–Carter mortality model. We find that COVID-19 has resulted in substantial mortality shocks in many countries. We show that such shocks have a large impact on point and interval forecasts of death rates and, consequently, on the valuation of mortality-related insurance products. We obtain similar findings under the Cairns–Blake–Dowd mortality model, which demonstrates that the effects caused by COVID-19 show up in a variety of models. Finally, we provide an overview of approaches to handle extreme mortality events such as the COVID-19 pandemic in mortality modelling.

This paper highlights how actuarial thinking could contribute to the development of a justice perspective relating to biodiversity risks and also the implications for actuarial work. The impact of biodiversity loss as well as the use of ecosystem services is not equally distributed across society (both intra and inter generationally). The Biodiversity and Natural Capital Risk Working Party has been setup to proactively take forward a series of activities including think pieces, webinars and external engagement on these risks.

This paper introduces the concept of Natural Capital and explores the implications for actuarial work by way of case studies. It is part of a wider series of IFoA papers focussing on the risks from global biodiversity loss and how these risks can be mitigated.

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$. Also let $X_{1:n} \lt \cdots \lt X_{n:n}$ and $Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for $1\le i \lt j \le n$, the generalized spacing $X_{j:n} - X_{i:n}$ is more dispersed than $Y_{j:n} - Y_{i:n}$ according to dispersive ordering and for $2\le i \le n$, the dependence of $X_{i:n}$ on $X_{1:n}$ is less than that of $Y_{i:n}$ on $Y_{1 :n}$, in the sense of the more stochastically increasing ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for $i =n$.