Managing demographic risk in enhanced pensions 177distribution. We calculate RBC requirements with different time horizons and confi-
dence levels. Let us define the finite time ruin probability as the probability of being
ina ruinstate inat least one ofthe timepoints1,2...,T,foragivenU( 0 )=u:
!u( 0 ,T)= 1 −Pr{T
⋂
t= 1U(t)≥ 0∣
∣
∣U( 0 )=u}
. (6)
RBC requirements for the time horizon( 0 ,T)with a( 1 −)confidence level are
defined as follows:
RBC 1 −( 0 ,T)=in f{
U( 0 )≥ 0
∣
∣
∣! 0 (^0 ,T)<
}
. (7)
Note that the risk reserve must be not negative for allt∈( 0 ,T).
To make data comparable, results are expressed as a ratio between RBC require-
ments and total single premium income
rbc 1 −( 0 ,T)=RBC 1 −( 0 ,T)
( 0 ,ω)N 1 ( 0 ).
An alternative method to calculate RBC requirements is based on the Value-at-Risk
(VaR)oftheU-distribution in the time horizon( 0 ,T)with a( 1 −)confidence level:
VaR 1 −( 0 ,T)=−U(T),whereU(t)is the-th quantile of theU-distribution at
timet. Hence RBC requirements are given by:
RBCVaR 1 −( 0 ,T)=VaR 1 −( 0 ,T)v( 0 ,T). (8)If an initial capitalU( 0 )is given, theRBC 1 VaR−( 0 ,T)requirements increase by the
amountU( 0 ). Values are reported in relative terms as
rbcVaR 1 −( 0 ,T)=RBC 1 VaR−( 0 ,T)
( 0 ,ω)N 1 ( 0 ).
5 Profit analysis
In this section we analyse the annual profit,Y(t), emerging from the management
of the portfolio. In order to capture the profit sources,Y(t)can be broken down into
insurance profit,YI(t), and profit coming from investment income on shareholders’
funds (which we call “patrimonial profit”),YP(t).
YI(t)=( 1 +i(t− 1 ,t))[V(t− 1 )+PT(t)−E(t)−B(t)]−V(t) (9)YP(t)=U(t− 1 )i(t− 1 ,t) (10)The following relation holds:Y(t)=YI(t)+YP(t). The sequence{Y(t)}t≥ 1 is called
profit profile. Letρbe the rate of return on capital required by the shareholders; the
present value of future profits discounted at rateρ(withρ>i),Y( 0 ,T)is given by:
Y( 0 ,T)=
∑T
t= 1Y(t)vρ( 0 ,t), (11)