Mathematical and Statistical Methods for Actuarial Sciences and Finance

88 R. Cocozza et al.

The value of the business at timetis expressed by the portfolio surplusStat
that time, that is the stochastic differenceof the value of the assets and the liabilities
assessed at timet. In general we can write:

St=

∑

s

NsXse

∫t

sδudu, (1)

whereδuis the stochastic force of interest andXsis the difference between premiums
and benefits at times.
Assuming that the random variablesNsare mutually independent on the random
interestδsand denoting byFtthe information flow at timet,

St=E[St|Ft]=

∑

s

cXsE[ (^1) s]E[e
∫t
sδudu]. (2)
Formula (2) can be easily specialised in the case of a portfolio ofm-deferred life
annuities, with annual level premiumsPpayable at the beginning of each year for a
period ofnyears (n≤m) and constant annual instalments,R, paid at the end of each
year, payable if the insured is surviving at the corresponding payment date. It holds:
St=E[St]=

∑

s

cXsspxE[e

∫t

sδudu](3)

wherespxdenotes the probability that the individual agedxsurvives at the agex+s
and

Xs=

⎧

⎪⎨

⎪⎩

P if s<n

−R if s>m

0ifn<s<m.

(4)

As widely explained in the previous section,the surplus analysis provides useful tools
for the equilibrium appraisal, which can be synthesised by the following rough but
meaningful and simple relationship:

Prob(St> 0 )=ε. (5)

For a deeper understanding of the choice ofε, refer to [1]. From a more general
perspective, we can estimate the maximun lossSαof the surplus at a certain valuation
timetwith a fixed confidence levelα,definedas

Prob(St>Sα)=α, (6)

that is:
Sα=F−^1 ( 1 −α), (7)
Fbeing the cumulative distribution function ofSt.
In the following we will take advantage of a simulative procedure to calculate the
quantile surplus involved in (6), basing our analysis on the portfoliomean surplusat
timet.