Mathematical and Statistical Methods for Actuarial Sciences and Finance

174 S. Levantesi and M. Menzietti

level. Section 5 deals with the profit analysis of the portfolio according to the profit
profile. Simulation results are analysed in Section 6, while some concluding remarks
are presented in Section 7.

2 Actuarial model for Enhanced Pensions

The probabilistic framework of an EP is defined consistently with a continuous and
inhomogeneous multiple state model (see Haberman and Pitacco [2]). LetS(t)rep-
resent the random state occupied by the insured at timet,foranyt≥0, wheretis
the policy duration and 0 the time of entry. The possible realisations ofS(t)are: 1 =
“active” (or healthy), 2 = “LTC disabled” or 3 = “dead”. We disregard the possibility
of recovery from the LTC state due to the usually chronic character of disability and
we assumeS( 0 )=1. Let us define transition probabilities and intensities:

Pij(t,u)=Pr{S(u)=j|S(t)=i} 0 ≤t≤u, i,j∈{ 1 , 2 , 3 }, (1)

μij(t)=lim

u→t

Pij(t,u)

u−t

t≥ 0 , i,j∈{ 1 , 2 , 3 }, i=j. (2)

EPs are single premium covers providing an annuity paid at an annual rateb 1 (t)when
the insured is healthy and an enhanced annuity paid at an annual rateb 2 (t)>b 1 (t)
when the insured is LTC disabled. Let us suppose all benefits to be constant with
time. Letωbe the maximum policy duration related to residual life expectancy at age
xand letv(s,t)=

∏t
h=s+ 1 v(h−^1 ,h)be the value at timesof a monetary unit at
timet; the actuarial value at time 0 of these benefits, ( 0 ,ω), is given by:

( 0 ,ω)=b 1 a 11 ( 0 ,ω)+b 2 a 12 ( 0 ,ω), (3)

where:aij(t,u)=

u−∑t− 1

s=t

Pij(t,s)v(s,t)for alli,j∈ 1 ,2. Assuming the equivalence

principle, the gross single premium paid int=0, Tis defined as:

T=

( 0 ,ω)

1 −α−β−γ[a 11 ( 0 ,ω)+a 21 ( 0 ,ω)]

, (4)

whereα,βandγrepresent the premium loadings for acquisition, premium earned
and general expenses, respectively.

3 Demographic scenarios

Long-term covers, such as the EPs, are affected by demographic trends (mortality and
disability). A risk source in actuarial evaluations is the uncertainty in future mortality
and disability; to represent such an uncertainty we adopt different projected scenarios.
We start from a basic scenario,HB, defined according to the most recent statistical
data about people reporting disability (see ISTAT [3]) and, consistent with this data,