A diffusion approximation to a risk process under dynamic proportional reinsurance is considered. The goal is to minimise the discounted time in drawdown; that is, the time where the distance of the present surplus to the running maximum is larger than a given level $d > 0$ . We calculate the value function and determine the optimal reinsurance strategy. We conclude that the drawdown measure stabilises process paths but has a drawback as it also prevents surpassing the initial maximum. That is, the insurer is, under the optimal strategy, not interested in any more profits. We therefore suggest using optimisation criteria that do not avoid future profits.

Drawdown (respectively, drawup) of a stochastic process, also referred as the reflected process at its supremum (respectively, infimum), has wide applications in many areas including financial risk management, actuarial mathematics, and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.

Subterranean turions (tubers) of hydrilla lose viability when desiccated. Experimental data showed that freshly collected tubers had a moisture content between 50 and 60% and more than 90% viability. When desiccated, there was an approximate 2% increase in tuber mortality with each percent decline in moisture content. However under field conditions, the tuber bank within the exposed sediments of a northern Texas reservoir showed no decline in number or tuber viability throughout a 12-mo continuous drawdown. Apparently, the buried tubers were never subject to sufficient dessication to damage them. Finally, an experimental pond with an extensive hydrilla tuber bank was manipulated through six flood/drawdown cycles to determine the effects of short-term drawdowns on tuber survival and quiescence. Initially, the pond had a tuber bank of about 676 and 305 tubers m−2 in the shallow and deep zones, respectively. Although the tuber number was reduced to fewer than 15 to 30 tubers m−2 by these repetitive drawdowns, hydrilla tubers were not eradicated from the pond.

This paper investigates the effect of recent regulatory changes to the compulsory annuitisation of tax-privileged pension savings, on the demand for annuities and other retirement products. We find that the demand for annuities has fallen by almost 75 per cent from its peak in 2012, and the demand for income drawdown products has increased. There is some evidence that people at younger ages and with smaller pension pots are choosing not to annuitise, and hence the average size of an annuity purchase has increased.

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Drawdowns measuring the decline in value from the historical running maxima over a given period of time are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focused on the side of severity by studying the first drawdown over a certain prespecified size. In this paper we extend the discussion by investigating the frequency of drawdowns and some of their inherent characteristics. We consider two types of drawdown time sequences depending on whether a historical running maximum is reset or not. For each type we study the frequency rate of drawdowns, the Laplace transform of the nth drawdown time, the distribution of the running maximum, and the value process at the nth drawdown time, as well as some other quantities of interest. Interesting relationships between these two drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent drawdowns are also proposed and priced.

This paper provides a detailed overview of variable annuities. Consideration is given first to the definition of the term variable annuity. Common terminology used in the variable annuity market is introduced. The current state of the United Kingdom and other international markets is described. Then, by reference to a simplified product, an analysis of customer outcomes, pricing, reserving, risk management and hedging is carried out. The paper ends with a description of current U.K. pensions legislation and how it potentially constrains product development.