Dynamic pricing of general insurance in a competitive market

A model for general insurance pricing is developed which represents a stochastic generalisation of the discrete model proposed by Taylor (1986). This model determines the insurance premium based both on the breakeven premium and the competing premiums offered by the rest of the insurance market. The optimal premium is determined using stochastic optimal control theory for two objective functions in order to examine how the optimal premium strategy changes with the insurer’s objective. Each of these problems can be formulated in terms of a multi-dimensional Bellman equation. In the first problem the optimal insurance premium is calculated when the insurer maximises its expected terminal wealth. In the second, the premium is found if the insurer maximises the expected total discounted utility of wealth where the utility function is nonlinear in the wealth. The solution to both these problems is built-up from simpler optimisation problems. For the terminal wealth problem with constant loss-ratio the optimal premium strategy can be found analytically. For the total wealth problem the optimal relative premium is found to increase with the insurer’s risk aversion which leads to reduced market exposure and lower overall wealth generation

Optimal strategies for pricing general insurance

Optimal premium pricing policies in a competitive insurance environment are investigated using approximation methods and simulation of sample paths. The market average premium is modelled as a diffusion process, with the premium as the control function and the maximization of the expected total utility of wealth, over a finite time horizon, as the objective. In order to simplify the optimisation problem, a linear utility function is considered and two particular premium strategies are adopted. The first premium strategy is a linear function of the market average premium, while the second is a linear combination of the break-even premium and the market average premium. The optimal strategy is determined over the free parameters of each functional form. It is found that for both forms the optimal strategy is either to set a premium close to the break-even or not to sell insurance depending on the model parameters. If conditions are suitable for selling insurance then for the first premium strategy, in the case of no market average premium drift, the optimal premium rate is approximately ¯p(0)/aT above break-even where ¯p(0) is the initial market average premium, a is a constant related to the elasticity of demand and T is the time horizon. The optimal strategy for the second form of premium depends on the volatility of the market average premium. This leads to optimal strategies which generate substantial wealth since then the market average premium can be much larger than break-even leading to significant market exposure whilst simultaneously making a profit. Monte-Carlo simulation is used in order to study the parameter space in this case

Pricing general insurance with constraints

Deterministic control theory is used to find the optimal premium strategy for an insurer in order to maximise a given objective. The optimal strategy can be loss-leading depending on the model parameters, which may result in negative premium values. In such circumstances, it is optimal to capture as much of the market as possible before making a profit towards the end of the time horizon. In reality, the amount by which an insurer can lower premiums is constrained by borrowing restrictions and the risk inherent in building up a large exposure. Consequently, the effect of constraining the pricing problem is analysed with two forms of constraint: a bounded premium and a solvency requirement. If a lower bound is placed on the premium then an analytical solution can be found, which is not necessarily a smooth function of time. The optimal premium strategy is described in qualitative terms, without recourse to specifying particular parameter values, by considering the value of the terminal optimal premium. Solvency constraints lead to an optimisation problem which is coupled to the state equations and so there is no analytical solution. Numerical results are presented for a subset of the parameter space using control parameterisation, which turns the optimisation problem into a nonlinear programming problem

A streamtube model of rotating turbidity currents

A rotating turbidity current is modeled as a steady tube of sediment-laden water traversing along and down a uniform slope representing the continental shelf. The model includes parameterizations of erosion of sediment from the seabed into the plume, deposition onto the seafloor, bottom friction, and turbulent entrainment of the overlying ambient seawater. The purpose of the work is to study how these parameterizations affect the path the current takes on the slope and to determine criteria which affect the qualitative behavior of the model solutions. It is found that there are three possible final states to the model. The first is near-geostrophic flow along the slope which is similar to the uniform state found in streamtube models. The second is the catastrophic state which describes an erosive current entraining and depositing a large amount of sediment and traveling downslope much like an avalanche. In the third state the current decelerates and flows along slope whilst depositing its load. The current is extinguished on a length scale 2(aQ*/vsKfr0)1/2 from the source where a is the aspect ratio, Q* is the initial volume flux, vs is the settling velocity, Kf is a frictional coefficient and r0 is the deposition rate. The state the current adopts depends on the initial conditions and model parameters. A linear stability analysis finds that the geostrophic state is always unstable while the catastrophic state is always stable. Thus the model predicts either catastrophic flow down slope or depositional flow along-slope. Given a sufficiently large initial disturbance the current will always ignite. This behavior is illustrated with parameters appropriate to the Kveitehola outflow. However the catastrophic state is unrealistic in that the turbulent energy required to maintain the sediment in suspension can be greater than that available to the current. Consequently a five-equation model is introduced which contains a turbulent kinetic energy equation and a parameterization of interfacial shear stress based on turbulence. With this model the Coriolis force has a much greater impact on the plume\u27s path and the sediment erosion is not so unreasonably large

Optimal management of an insurer's exposure in a competitive general insurance market

The qualitative behavior of the optimal premium strategy is determined for an insurer in a finite and an infinite market using a deterministic general insurance model. The optimization problem leads to a system of forward-backward differential equations obtained from Pontryagin’s Maximum Principle. The focus of the modelling is on how this optimization problem can be simplified by the choice of demand function and the insurer’s objective. Phase diagrams are used to characterize the optimal control. When the demand is linear in the relative premium, the structure of the phase diagram can be determined analytically. Two types of premium strategy are identified for an insurer in an infinite market, and which is optimal depends on the existence of equilibrium points in the phase diagram. In a finite market there are four more types of premium strategy, and optimality depends on the initial exposure of the insurer and the position of a saddle point in the phase diagram. The effect of a nonlinear demand function is examined by perturbing the linear price function. An analytical optimal premium strategy is also found using inverse methods when the price function is nonlinear

Exploring the anthelmintic properties of Australian native shrubs with respect to their potential role in livestock grazing systems

We measured in vitro anthelmintic activity in extracts from 85 species of Australian native shrub, with a view to identifying species able to provide a degree of worm control in grazing systems. Approximately 40% of the species showed significant activity in inhibiting development of Haemonchus contortus larvae. The most active extracts showed IC50 values of 60–300 mg/ml. Pre-incubation with polyvinylpolypyrrolidine removed the activity from some extracts, implicating tannins as the bioactive agent, while in other cases the pre-incubation had no effect, indicating the presence of other anthelmintic compounds. Plant reproductive maturity (onset of flowering or fruiting) was associated with increasing anthelmintic activity in some species. Variability was observed between plants of the same species growing in different environments, while variation between individual plants of the same species within a single field suggests the existence of distinct chemotypes. Significant activity against adult H. contortus worms in vitro was also demonstrated in a limited number of extracts tested against this life stage. Our study indicates that there is potential for Australian native shrubs to play an anthelmintic role in grazing systems, and highlights some plant biology factors which will need to be considered in order to maximize any anthelmintic effects.A. C. Kotze, J. O’Grady, J. Emms, A. F. Toovey, S. Hughes, P. Jessop, M. Bennell P. E. Vercoe and D. K. Revel

Patient journey following lumbar spinal fusion surgery (LSFS): protocol for a multicentre qualitative analysis of the patient rehabilitation experience (FuJourn)

INTRODUCTION: There has been a 65% increase in lumbar spinal fusion surgery (LSFS) worldwide over the last 13 years, with costs of £26 million to the UK National Health Service annually. Patient dissatisfaction with outcome and persistent pain and disability incurs further costs. Three trials provide low-quality evidence for the role of physiotherapy. Our UK surveys investigating physiotherapy/surgeon practice concluded rehabilitation should be tailored to the individual patient owing to considerable clinical heterogeneity. This study will explore the perceptions of patients who undergo LSFS to inform precision rehabilitation. METHODS AND ANALYSIS: A qualitative study, using interpretive phenomenological analysis, will recruit a purposive sample (n=40) to ensure patterns of similarity and difference in their journeys can be explored. In-depth semistructured interviews will be undertaken following discharge from hospital and at 12 months postsurgery. Patients' preoperative and postoperative experiences, underlying attitudes and beliefs towards the surgical intervention, facilitators and barriers to recovery, adherence to advice and physiotherapy, experiences of rehabilitation and return to normal function/activity/work will be explored. A 12-month patient diary will provide real time access to patient data, capturing a weekly record of life as lived, including symptoms, medication, experiences of stages of recovery, rehabilitation adherence, healthcare professional appointments, attitudes, their feelings and experiences throughout their journey. Data will be analysed in a number of stages in accordance with interpretive phenomenological analysis, supported using NVivo software. Analysis of the first interviews and patient diaries will afford a rich density of data to build an overall understanding of the patients' lived experiences, informing the 12-month interview. Strategies (eg, reflexivity) will ensure trustworthiness. ETHICS AND DISSEMINATION: The study has ethical approval (IRAS 223283). Findings will ensure that patient-driven data inform precision rehabilitation by understanding the patient journey. Findings will be disseminated through peer-reviewed journals and conferences

Mitigation of carbon using Atriplex nummularia revegetation

The use of abandoned or marginally productive land to mitigate greenhouse gas emissions may avoid competition with food and water production. Atriplex nummularia Lindl. is a perennial shrub commonly established for livestock forage on saline land, however, its potential for carbon mitigation has not been systematically evaluated. Similarly, although revegetation is an allowable activity to mitigate carbon within Article 3.4 of the United Nations Framework Convention on Climate Change's Kyoto Protocol, there is a paucity of information on rates of carbon mitigation in soils and biomass through this mechanism. For six sites where A. nummularia had been established across southern Australia four were used to assess changes in soil carbon storage and four were used to develop biomass carbon sequestration estimates. A generalised allometric equation for above and below ground biomass was developed, with a simple crown volume index explaining 81% of the variation in total biomass. There were no significant differences in soil organic carbon storage to 0.3 m or 2 m depth compared to existing agricultural land-use. Between 2.2 and 8.3 Mg C ha−1 or 0.2–0.6 Mg C ha−1 yr−1 was sequestered in above and below ground biomass and this translates to potential total sequestration of 1.1–3.6 Tg C yr−1 on saline land across Australia. Carbon income and forage grazing may thus provide a means to finance the stabilization of compromised land

Development and validation of two clinical prediction models to inform clinical decision-making for lumbar spinal fusion surgery for degenerative disorders and rehabilitation following surgery: protocol for a prospective observational study

INTRODUCTION: Potential predictors of poor outcome will be measured at baseline: (1) preoperatively to develop a clinical prediction model to predict which patients are likely to have favourable outcome following lumbar spinal fusion surgery (LSFS) and (2) postoperatively to predict which patients are likely to have favourable long-term outcomes (to inform rehabilitation). METHODS AND ANALYSIS: Prospective observational study with a defined episode inception of the point of surgery. Electronic data will be collected through the British Spine Registry and will include patient-reported outcome measures (eg, Fear-Avoidance Beliefs Questionnaire) and data items (eg, smoking status). Consecutive patients (≥18 years) undergoing LSFS for back and/or leg pain of degenerative cause will be recruited. EXCLUSION CRITERIA: LSFS for spinal fracture, inflammatory disease, malignancy, infection, deformity and revision surgery. 1000 participants will be recruited (n=600 prediction model development, n=400 internal validation derived model; planning 10 events per candidate prognostic factor). The outcome being predicted is an individual's absolute risk of poor outcome (disability and pain) at 6 weeks (objective 1) and 12 months postsurgery (objective 2). Disability and pain will be measured using the Oswestry Disability Index (ODI), and severity of pain in the previous week with a Numerical Rating Scale (NRS 0-10), respectively. Good outcome is defined as a change of 1.7 on the NRS for pain, and a change of 14.3 on the ODI. Both linear and logistic (to dichotomise outcome into low and high risk) multivariable regression models will be fitted and mean differences or ORs for each candidate predictive factor reported. Internal validation of the derived model will use a further set of British Spine Registry data. External validation will be geographical using two spinal registries in The Netherlands and Switzerland. ETHICS AND DISSEMINATION: Ethical approval (University of Birmingham ERN_17-0446A). Dissemination through peer-reviewed journals and conferences