Mathematical foundation of the replicating portfolio approach

Abstract

In the last few years, the first theoretical foundations for replicating portfolios – probably the most prevailing technique for risk capital calculation in life insurance – have been given in a series of papers by Beutner, Pelsser and Schweizer. In these papers, the asymptotic behaviour of replicating portfolios concerning the approximation of the terminal value (TVL) and the fair value distribution of the liabilities (FVL) has been investigated in detail. We complement this line of research by providing results on approximations based on a finite number of replicating instruments. We do so by providing the link between the approximation error of the TVL distribution, the FVL distribution and the error in the resulting risk capital figure, either value at risk or some coherent risk measure. We further allow for a variety of practically relevant formulations of the replication problem, including cash flow matching approaches. In contrast to the existing literature, all our results apply to approaches both under the risk-neutral and the real-world measure. Our strongest bounds are due to the observation that in discrete time, the measure change from the real-world to the risk-neutral measure can be both bounded below and above by a suitable constant in the first period.

Keywords:

• Replicating portfolio
• Solvency II
• MCEV
• risk capital
• cash flow matching
• terminal value matching
• gain–loss ratio
• FTAP

Acknowledgements

We are very thankful for fruitful and thorough discussions with several practitioners from major life insurance companies. Special thanks go to Pierre Joos and Tobias Herwig for their support in putting things into context. We further thank Damir Filipović, Stefan Jaschke and Peter Hieber for valuable comments. We are also indebted to two anonymous referees whose comments helped to improve the paper, especially in terms of clarity and transparency of presentation.

Notes

No potential conflict of interest was reported by the authors.

1 More exactly, the distribution of own funds, including new business. However, this is of no further importance in the following.

2 More precisely, their analysis is focused on the solution of the problem . The objective function will be formally introduced in Section 2.4.

3 Some of these formulations and their connections have already been investigated in Natolski & Werner (2014). Further, yet another meaningful objective function, , which is not considered in detail here, has been proposed and analysed in detail in Natolski & Werner (2017) together with pros and cons compared to objective functions considered in this contribution.

4 Cash flows are actually linked to the balance sheet and thus happen only at regular discrete time intervals. As only static replication is considered, a continuous time framework does not offer a significant advantage over the discrete time approach.

5 We chose this time grid for notational convenience. One can choose any finite time grid. The results in this paper do not rely on the size of the time steps. In particular, when insurers are interested in studying the effects of instantaneous shocks on the MCEV, time 1 can be a placeholder for a small time step .

6 We use tilded variables to express the fact that the variable is discounted to time 0 by the numéraire.

7 The expected shortfall is also known as tail value at risk or average value at risk.

8 Let us refer to Natolski & Werner (2014), Natolski & Werner (2016), Natolski & Werner (2017) for some further comments on the choice of the norm, pros and cons from a numerical point of view and further connections between the different formulations.

9 A second line of argument was based on the fact that the first period does not represent year one, but some very small time after some instantaneous shock.

Additional information

Funding

The research leading to these results has received partial funding for the first author from Allianz Deutschland AG.