A financial analysis of surplus dynamics for deferred life schemes 87- the ratio of the CVaR ofStminusSt− 1 to thet−1 result can be regarded as the
worst expected Return on Surplus (RoS) for the selected level of confidence.
Analogous conclusions can be inferred when the analysis is referred to a business line
and the surplus is enhanced by the allocated capital: the interpretation of the result is
similar and even clearer, as the last ratio is a proper worst expected return on equity.
From a methodological point of view, we would like to stress that the analysis
and, therefore, the simulation procedure could be performed with reference to all
the risk factors relevant to the time evolution of the portfolio. Many dynamics can
simultaneously contribute to the differentials that depend on risk factors linked to
both the assets in which premiums are invested and the value of liabilities for which
capitalised premiums are deferred. Together with the demographic dynamic, the most
important factor is the nature of the assets: if these are financial, the risks faced will
be mainly financial, they will depend directly on the asset type and will not have any
autonomous relevance. Besides, the crux of the problem is the difference between
the total rate of return on assets and the rate of interest originally applied in premium
calculation, so that it can be precisely addressed asinvestment risk, in order to highlight
the composite nature of relevant risk drivers. At the same time, other factors can
contribute to the difference such as the quality of the risk management process, with
reference to both diversification and risk pooling. This implies that the level of the
result and its variability is strictly dependent on individual company elements that
involve both exogenous and endogenous factors.
Since our focus is on the financial aspect of the analysis, we concentrate in the
following of the paper only on the question of the investment rate, excluding any
demographic component and risk evaluation from our analysis. Bearing in mind this
perspective, the rate actually used as a basis for the simulation procedure has to be
consistent with the underlying investment and the parameters used to describe the rate
process have to be consistent with the features of the backing asset portfolio. There-
fore, once we decide the strategy, the evaluation is calibrated to the expected value
and estimated variance of the proper return on asset as set by the investment portfo-
lio. In other words, if we adopt, for instance, a bond strategy the relevant parameters
will be estimate from the bond market, while if we adopt an equity investment, the
relevant values will derive from the equity market, and so on, once we have defined
the composition of the asset portfolio.
3 Surplus analysis
3.1 The mathematical framework
In the following we take into account a stochastic scenario involving the financial and
the demographic risk components affecting a portfolio of identical policies issued to
a cohort ofcinsureds agedxat issue.
We denote byXsthe stochastic cash flow referred to each contract at timesand by
Nsthe number of claims at times,{Ns}being i.i.d. and multinomial(c,E[ (^1) s]),where
the random variable (^1) stakes the value 1 if the insured event occurs, 0 otherwise.