86 R. Cocozza et al.
order to exemplify the potential of the model. In this paper deferred schemes are
selected considering that they can be regarded as the basis for many life insurance
policies and pension plans. Nevertheless, the model can be applied, given the neces-
sary adjustments, to any kind of contract as well as to non-homogeneous portfolios.
The rest of the paper is organised as follows. Section 2 introduces the logical
background of the model itself, while Section 3 detaches the mathematical framework
and the computational application. Section 4 comments on the numerical results
obtained and Section 5 concludes.
2 The model
As stated [1], the surplus of the policy is identified by the difference between the
present value of the future net outcomes of the insurer and the (capitalised) flows paid
by the insureds. This breakdown is evaluated year by year with the intent to compile
a full prospectiveaccount of the surplus dynamics. In the case of plain portfolio
analysis, the initial surplus is given by the loadings applied to pure premiums; in the
case of a business line analysis, the initial surplus, set as stated, is boosted by the
initial capital allocated to the business line or the product portfolio.
The initial surplus value, in both cases, can be regarded as the proper initial capital
whose dynamic has to be explored with the aim of setting a general scheme of dis-
tributable and undistributable earnings. More specifically, given that at the beginning
of the affair the initial surplus is set asS 0 , the prospective futuret-outcomes, defined
asSt, can be evaluated by means of simulated results to assess worst cases given a
certain level of probability or a confidence interval.
The build up of these results, by means of the selected model and of Monte Carlo
simulations (see Section 3), provides us with a complete set of future outcomes at
the end of each periodt. These values do not depend on the amount of the previous
distributed earnings. Those results with an occurrence probability lower than the
threshold value (linked to the selected confidence interval) play the role of worst
cases scenarios and their average can be regarded as theexpected worst occurrence
corresponding to a certain level of confidence when it is treated as a Conditional
Value-at-Risk (CVaR). Ultimately, foreach period of time, we end up with a complete
depiction of the surplus by means of a full set of outcomes, defined by both expected
values and corresponding CVaR.
The results we obtain for each period can therefore be used as a basis for the
evaluation of the distributable earnings, with the final aim of assessing the distributable
surplus share. If the CVaR holds for the expected worst occurrence given a level of
confidence, its interpretation is pragmatically straightforward: it is the expected worst
value of the surplus for the selected confidence level. So for anyt-period, the CVaR
estimates the threshold surplus at the confidence level selected and automatically sets
the maximum distributable earnings of the preceding period. In other words, the CVaR
ofStcan be regarded as the maximum distributable amount ofSt− 1 ; at the same time:
- the ratio of the CVaR ofSttoSt− 1 can be regarded as the distributable share (DS)
of thet−1 result; and