### Delong, Ł., Szatkowski, M., 2019, One-year premium risk and emergence pattern of ultimate loss based on conditional distribution

Working Paper,
04-09-2019

We study the relation between one-year premium risk and ultimate premium
risk. In practice, the one-year risk is usually related to the ultimate risk by using a so-called
emergence pattern formula introduced by England et al. (2012) and Bird, Cairns
(2011). We postulate to define the emergence pattern of the ultimate loss based on the
conditional distribution of the best estimate of the ultimate loss given the ultimate loss,
where the conditional distribution is derived from the multivariate distribution of the claims
development process. We start with investigating Gaussian Incremental Loss Ratio, Hertig's
Lognormal and Over-Dispersed Poisson claims development models. We derive the true
emergence pattern formulas in these models and prove that they are different from the
emergence pattern postulated by England et al. (2012), Bird, Cairns (2011). We assume
that the risk is measured with Value-at-Risk. We identify that the true one-year risk can be
significantly under and overestimated if the emergence pattern formula from England et al.
(2012), Bird, Cairns (2011) is applied. We show that the ratio of the true one-year risk to
the ultimate risk varies across the claims development models and depends on the confidence
level. We prove that the one-year risk is lower than the ultimate risk only if a suficiently
high confidence level is used. Moreover, in a general claims development model we illustrate
that the one-year risk can be higher than the ultimate risk at all high confidence levels and
the distributions of the one-year risk and the ultimate risk can have different tail behaviour.

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### Delong, Ł., Dhaene, J., Barigou, K., 2019, Fair valuation of insurance liability cash-flow streams in continuous time: Theory

Published Paper,
05-03-2019

We investigate fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. We first consider one-period hedge-based valuations, where in the first step, an optimal dynamic hedge for the liability is set up, based on the assets traded in the market and a quadratic hedging objective, while in the second step, the remaining part of the claim is valuated
via an actuarial valuation. Then, we extend this approach to a multi-period setting by backward iterations for a given discrete-time step $h$, and consider the continuous-time limit for $h\to 0$. We formally derive a partial differential equation for the valuation operator which satisfies the continuous-time limit of the multi-period, discrete-time iterations and prove that this valuation operator is actuarial and market-consistent. We show that our continuous-time fair valuation operator has a natural decomposition into the best estimate of the liability and a risk margin. The dynamic hedging strategy associated with the continuous-time fair valuation operator is also established. Finally, the valuation operator and the hedging strategy allow us to study the dynamics of the net asset value of the insurer.

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### Delong, Ł., 2013, Backward Stochastic Differential Equations with Jumps and their Actuarial and Financial Applications

Book,
10-04-2013

Backward Stochastic Differential Equations with jumps can be used to solve problems in both finance and insurance.

This book will make BSDEs more accessible to those who are interested in applying these equations to actuarial and financial problems. It will be beneficial to students and researchers in mathematical finance, risk measures, portfolio optimization as well as actuarial practitioners.

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