A financial analysis of surplus dynamics for deferred life schemes 89We focus on capturing the impact on the financial position at timet– numerically
represented by the surplus on that date – of the financial uncertainty, which constitutes
a systematic risk source, and is thus independent of the portfolio size. In fact in this
case the pooling effect does not have any consequences, in contrast to the effect of
specific risk sources, as the accidental deviations of mortality.
Formally the valuation of the mean surplus can be obtained observing that it
is possible to construct a proxy of the cumulative distribution function ofStsince
(cf. [4])
lim
c→∞P
(∣
∣
∣
∣
Ns
c−E[ (^1) s]
∣
∣
∣
∣≥
)
= 0 ,
hence, when the number of policies tends to infinity,St/cconverges in distribution
to the random variable
t=
∑
sXsE[ (^1) s]e
∫t
sδudu. (8)
In the case of the portfolio ofm-deferred life annuities described above, we set:
xs=
{
P ifs<n
−Rifs>m, ys={
R ifs>m
−Pifs<n,
so, making explicit the surplus’ formalisation, we can write
St=∑ci= 1⎛
⎝
min(Kxi,t)
∑s= 0xse∫t
sδudu−min(Kxi,T)
∑s=t+ 1yse−∫s
tδudu⎞
⎠
whereKxidenotes the curtate future lifetime of theith insured agedxat issue andT
is the contract maturity (T≤ω−x,ωbeing the ultimate age). We can point out that
the second term on the right-hand side represents the mathematical provision at time
t.
So, remembering the homogeneity assumptions about the portfolio components,
formula 8 can be specialised as follows:
t=E[⎛
⎝
min∑(Kx,t)s= 0xse∫t
sδudu−min∑(Kx,T)s=t+ 1yse−∫s
tδudu⎞
⎠|{δu}u≥ 0 ]= (9)=
∑
s≤txsspxE[e∫t
sδudu]−∑
s>tysspxE[e−∫s
tδudu].3.2 The computational application
As the computational application of the preceding model we consider a portfolio of
unitary 20-year life annuities with a deferment period of 10 years issued to a group of
1000 male policyholders aged 40. The portfolio is homogeneous, since it is assumed
that policyholders have the same risk characteristics and that the future lifetimes