arXiv subjects

Thomas Bernhardt

Publications and source records attributed to Thomas Bernhardt.

7 recordsLinked to original sources

arXiv2025

A note on bequest preferences in utility maximisation for modern tontines

In this short note, we address two issues in the literature about modern tontines with bequest and utility maximisation: how to verify optimal controls and the decreasing allocation of funds in the tontine. We want to raise awareness in the actuarial community about the dual approach to solve optimal control problems when working with power utilities. Additionally, we point out that bequest preferences should be time-dependent or otherwise yield unrealistic investment strategies. We base our attempt at modelling bequest preferences on common sense rules like 100% payback upon death at the start that vanishes over time. Our modelling shows that the resulting investment strategy almost linearly adjusts the allocation in the tontine from 0% to 100% over time.

q-fin.RM↗

arXiv2023

On Watanabe's characterisation and change of intensity \`{a} la Girsanov for Cox processes

We discuss the equivalence of definitions for conditional Poisson processes, Cox processes, and stochastic intensities of point processes on the real line. We show that Watanabe's characterisation of conditional Poisson processes in terms of local martingales is necessary and sufficient. Additionally, we consider conditions enabling the measure change method a la Girsanov to alter the intensity of Cox processes to a desired new target intensity, e.g. for the probability reference approach in filtering. Such a measure change exists if a corresponding stochastic exponential is a proper martingale. We show that this holds if the new locally integrable target intensity is the product of the original intensity and another non-negative process.

math.PR↗

arXiv2021

Wealth heterogeneity in a closed pooled annuity fund

The stability of income payments in a pooled annuity fund is studied. In those funds, members receive a fluctuating income depending on their experienced mortality in exchange for their pension savings. The focus is on describing the influence of different initial savings on the ability of the fund to provide a stable income in retirement. Because of this, members coincide in their characteristics except for their initial savings. We identify a term, which we dub ``implied number of homogeneous members'', that directly links the initial savings to the size of the income fluctuations. Our main contribution is the analysis of this term and the development of a criterion to answer the question of whether or not a given group of same-aged people should pool their funds together.

q-fin.RM↗

arXiv2020

Quantifying the trade-off between income stability and the number of members in a pooled annuity fund

The number of people who receive a stable income for life from a closed pooled annuity fund is studied. Income stability is defined as keeping the income within a specified tolerance of the initial income in a fixed proportion of future scenarios. The focus is on quantifying the effect of the number of members, which drives the level of idiosyncratic longevity risk in the fund, on the income stability. To do this, investment returns are held constant and systematic longevity risk is omitted. An analytical expression that closely approximates the number of fund members who receive a stable income is derived and is seen to be independent of the mortality model. An application of the result is to calculate the length of time for which the pooled annuity fund can provide the desired level of income stability

q-fin.RM↗

arXiv2020

On the Continuity of the Root Barrier

We show that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero.

math.PR↗

arXiv2019

Modern tontine with bequest: innovation in pooled annuity products

We introduce a new pension product that offers retirees the opportunity for a lifelong income and a bequest for their estate. Based on a tontine mechanism, the product divides pension savings between a tontine account and a bequest account. The tontine account is given up to a tontine pool upon death while the bequest account value is paid to the retiree's estate. The values of these two accounts are continuously re-balanced to the same proportion, which is the key feature of our new product. Our main research question about the new product is what proportion of pension savings should a retiree allocate to the tontine account. Under a power utility function, we show that more risk averse retirees allocate a fairly stable proportion of their pension savings to the tontine account, regardless of the strength of their bequest motive. The proportion declines as the retiree becomes less risk averse for a while. However, for the least risk averse retirees, a high proportion of their pension savings is optimally allocated to the tontine account. This surprising result is explained by the least risk averse retirees seeking the potentially high value of the bequest account at very old ages.

q-fin.PM↗

arXiv2019

Discretionary stopping of stochastic differential equations with generalised drift

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem's value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem's data. Next, we derive the complete explicit solution to the problem that arises when the state process is a skew geometric Brownian motion and the reward function is the one of a financial call option. In this case, we show that the optimal stopping strategy can take several qualitatively different forms, depending on parameter values. Furthermore, the explicit solution to this special case reveals that the so-called "principle of smooth fit" does not hold in general for this type of optimal stopping problems in standard senses that this can be formulated.

math.PR↗