Managing demographic risk in enhanced pensions 175the Italian Life-Table SIM-1999. Actives’ mortalityμ 13 (t)is approximated by the
Weibull law, while transition intensitiesμ 12 (t)are approximated by the Gompertz law
(for details about transition intensities’ estimation see Levantesi and Menzietti [5]).
Disabled mortality intensityμ 23 (t)is expressed in terms ofμ 13 (t)according to the
time-dependent coefficientK(t),μ 23 (t)=K(t)μ 13 (t).ValuesofK(t), coming from
the experience data of an important reinsurance company, are well approximated by
the function exp(c 0 +c 1 t+c 2 t^2 ).
Mortality of projected scenarios has been modelled evaluating a different set of
Weibull parameters (α,β) for each ISTAT projection (low, main and highhypoth-
esis, see ISTAT [4]). Furthermore, the coefficientK(t)is supposed to be the same
for all scenarios. Regarding transition intensity,μ 12 (t), three different sets of Gom-
pertz parameters have been defined starting from a basic scenario to represent a 40%
decrease (Hp. a), a 10% decrease (Hp. b) and a 20% increase (Hp. c) in disability
trend, respectively. By combining mortality and disability projections we obtain nine
scenarios.
We assume that possible changes in demographic scenarios occur everykyears,
e.g., in numerical implementation 5 years is considered a reasonable time to capture
demographic changes. LetH(t)be the scenario occurring at timet(t= 0 ,k, 2 k,...).
It is modelled as a time-discrete stochastic process. LetP ̄(t)be the vector of scenario
probabilities at timetandM(t)the matrix of scenario transition probabilities between
tandt+k. The following equation holds:P ̄(t+k)=P ̄(t)·M(t).
We suppose that at initial time the occurring scenario is the central one. We assume that
the stochastic processH(t)is time homogeneous (M(t)=M,∀t) and the scenario
probability distribution,P ̄(t), is stationary after the first period, so thatP ̄(t)=P ̄,
∀t≥k. Note thatP ̄is the left eigenvector of the transition matrixMcorresponding
to the eigenvalue 1. Values ofP ̄are assigned assuming the greatest probability of
occurrence for the central scenario and a correlation coefficient between mortality
and disability equal to 75%:
P ̄=( 0 .01 0.03 0.16 0.03 0.54 0.03 0.16 0.03 0. 01 ).Further, we assume that transitions between strongly different scenarios are not pos-
sible in a single period and consistently with supposed correlation between mortality
and disability, some transitions are more likely than others.
Resulting scenarios’ transition probabilities are reported in the matrix below.