90 R. Cocozza et al.
are independent and identically distributed. As far as the premiums are concerned,
we build up the cash flow mapping considering that premiums are paid periodically
at the beginning of each year of the deferment period. The market premium has a
global loading percentage of 7% compensating for expenses, safety and profit. Pure
premiums are computed by applying 2% as the policy rate and by using as lifetables
the Italian IPS55.
Since our analysis is focused on the financial aspect, the single local source of
uncertainty is the spot rate, which is a diffusion process described by a Vasicek model
dr(t)=k(μ−r(t))dt+σdW(t),r( 0 )=r 0 , (10)wherek,μ,σandr 0 are positive constants andμis the long-term rate. As informa-
tive filtration, we use the information set available at time 0. As a consequence, for
instance, in calculating the flows accrued up to timet, the starting valuer 0 for the
simulated trajectories is the value known at time 0. Analogously, in discounting the
flows of the period subsequent tot, the starting value of the simulated trajectories is
E[rt|F 0 ]. The parameter estimation is based on Euribor-Eonia data with calibration
set on 11/04/2007 (cf. [2]), since we make the hypothesis that the investment strategy
is based on a roll-over investment in short-term bonds, as we face an upward term
structure. The estimated values areμ= 4.10%,σ=0.5%andr 0 = 3.78%.
In order to evaluate the Expected Surplus and the CVaR in a simulation framework,
we consider the Vasicek model to describe the evolution in time of the global rate of
return on investments earned by the asset portfolio. Theα-quantile,qα,ofthesurplus
distribution is defined as:
Prob{S(t)<qα}= 1 −α. (11)In the simulation procedure we setα=99%. The expected( 1 −α)worst case is
given by the following:
E[worst cases( 1 −α)]=( 1 −α)−^1∫ 1
αqpdp, (12)qpbeing thep-quantileof the surplus distribution. The last equation is then the
average of the surplus value lower than theα-quantile,qα.
4Results
Recalling Section 2, the simulation results provide us with the expected value of
the surplus for each period, the first value at time 0 being the portfolio difference
between the pure premium and the market premium. Therefore, scrolling down the
table we can very easily see the evolution of the surplus over time together with the
corresponding CVaR.
As far as the time evolution is concerned, the surplus shows an increasing trend,
which is consistent with the positive effect of the financial leverage, since we invest