The recent wide development and changes in insurance markets highlighted the necessity to map out the solvency analysis in a more complete framework. The approach we present in the paper comes up with an integrated analysis of the risk profile of an insurance business, taking into account the actual European directives about solvency assessment. The aim of the paper is to construct a methodology apt to incorporate properly the effect of the risk sources in calculating mathematical provisions related to a portfolio of insurance policies.Life insurance, financial risk, demographic risk, capital adequacy, reserves, conditional random processes

Cocozza, R

Di Lorenzo, E

Sibillo, M

English

Research Papers in Economics

Investment Management and Financial Innovations, 2/2004  95
 Methodological Problems in Solvency Assessment of an 
Insurance Company
1
Rosa Cocozza
2, Emilia Di Lorenzo
3, Marilena Sibillo
4
Abstract
The recent wide development and changes in insurance markets highlighted the necessity 
to map out the solvency analysis in a more complete framework. The approach we present in the 
paper comes up with an integrated analysis of the risk profile of an insurance business, taking into 
account the actual European directives about solvency assessment. The aim of the paper is to con-
struct a methodology apt to incorporate properly the effect of the risk sources in calculating 
mathematical provisions related to a portfolio of insurance policies. 
Jel classification: G22, G28, G13 
Key words: Life insurance, financial risk, demographic risk, capital adequacy, reserves, 
conditional random processes. 
1. Life insurance business: a risk management approach 
An insurance company is solvent “if it is able to fulfil its obligations under all contracts 
under all reasonably foreseeable circumstances” (IAIS 2002). Nevertheless, in order to come to a 
practicable definition, it is necessary to make clear under which situation the appropriateness of 
the assets to cover claims is to be considered. 
The question, referring to the evaluation purposes, does not have a comprehensive an-
swer, since it depends on many issues. Indeed, it is relevant whether the company is deemed as a 
closed operation or a going concern, thus including only written business (run–off basis) or also 
future new business (going–concern basis). Additionally, it depends on the aim of the evaluation 
that is the mere financial progress of the company or its ability to meet claims and other obliga-
tions under all but the most extreme circumstances. The first topic defines the relevant risk factors: 
on a going concern basis also fluctuations around the expected value of the new business will be 
taken into account (thus allowing for netting within different pools), while on a runoff basis only 
fluctuations within the single pool will be considered (thus secluding clearing). The second one 
defines the amplitude of the evaluation, which in probabilistic terms would correspond to the con-
fidence level choice (from intermediate levels to extreme events). Therefore, solvency evaluation 
is a process, whose logical paradigm should sequentially consist of three main steps: relevant risk 
recognition, risk measurement and definition of capital requirements to absorb potential losses. 
In general, the main risk for a firm is that revenues prove to be unable to cover expenses 
and, regard the valuation also the shareholders, to remunerate adequately capital invested. This is 
namely business risk. This very broad definition does embrace all risks and provide for measuring 
them through the variance of the expected profit over a time bucket (risk horizon, accounting pe-
riod etc.) or also over the entire duration of the business. Premiums and claims respectively are 
typical insurance revenues and expenses: therefore business risk stems from the potential inequal-
ity among these elements, with the further difficulty that revenues have to be estimated before ex-
penses because of the inverted cycle. Therefore, all the factors that can induce the inequality – and 
in fact to a loss – are relevant and define the whole risk subsystem. This approach does imply that 
                                                          
1 This research was partially supported by Italian MIUR (Project: PRIN 2002 Metodi e strumenti per l’analisi e la gestione 
dei rischi nei settori delle assicurazioni di persone e previdenziale. Although the paper is the result of a common study, the 
first section is written by R. Cocozza and the rest two are by E. Di Lorenzo and M. Sibillo. 
2 Ph.D., Associate professor of financial intermediaries at the University of Naples, Italy.
3 Professor of financial mathematics at the University of Naples, Italy. 
4 Professor of financial mathematics at the University of Salerno, Italy. Investment Management and Financial Innovations, 2/2004  96
the capability of covering expenses by means of premiums depends on the design of relevant cost 
drivers. For life business, the main puzzles are mortality and interest rates
1, which could impact of 
the single period performance (income), by the modification they produce on relevant costs and 
revenues. Therefore, concentrating on the pure premium components and performing an analysis 
on a run-off basis, the risk impact evaluation can start off on the breakdown of the income compo-
nents over a single accounting period. 
Profit (ʌt) can be outlined as the algebraic sum of: provisions at the beginning of the year 
(Rt-1), earned premiums (Pt)
2, investment income (Rt-1+Pt)(e
G-1), payments to policyholders (St)
and provisions at the end of the year (Rt), that is 
S t   Rt1  Pt  e
G  St  Rt >@ . (1) 
For an immediate temporary (n) unitary annuity, equation (1) can be specified as
3
 
 >@ , 1
1
1 1 1
1 1
x
t t x
x
t
x
t
x
t
x
t
x
t
x
t t
r p e r N
r N e r N
   
     
   
 
G
G S
 (2) 
where Nt
x is the actual number of survivors at age x+t and  x
t r  is the expected single re-
serve for each existing policy at the end of period t.
In this perspective accrued income can be seen as a function of the force of the total rate 
of return G for the period (t-1,t) and of the actual number of survivors Nt-1
x and of the probability 
px+t-1, and 
 0 1 1 1
1
!       

x
t t x
x
t x
t
t r p e r
N
G
w
wS
, (3) 
GSt
Gpxt1
 Nt1
x 1 rt
x  0, (4) 
wSt
wG
  Nt1
x rt
xe
G ! 0. (5) 
The first derivative (equation 3) gives the filter of the insurance risk, that is a risk indica-
tor for the event of an actual number of survivors diverse from the expected one. The indicator is 
always positive because a higher number of survivors in the preceding period forces the insurer to 
increase the corresponding provision, giving rise to a capitalised value larger than expected ex-
penses. This benefit, although always positive, shows a dynamics connected to the age (x) and the 
evaluation time (t) as shown by Figure 1. 
                                                          
1 For example, for temporary annuities, the minimal equilibrium is directly dependent on the single period differentials 
between the integral of the instantaneous total return on assets purchased with written premium and that of the original 
interest rate applied in premium rating and the logarithm of the actual number of survivors and the expected number of 
survivors (Cocozza et al., 2003b). 
2 For the sake of clarity, premiums are assumed to be earned at the beginning of the accounting period and payments to 
policyholders are assumed to be performed immediately before the end of the period. 
3 The aim of the paper is to outline a valuation methodology: we have decided to work all the examples through an annuity 
but any other kind of policy could serve our purpose. Investment Management and Financial Innovations, 2/2004  97
Fig. 1. Dynamics connected to the age and the evaluation time
The second derivative (equation 4) gives the longevity risk indicator, that is a proxy for 
the event that there is a shock in the mortality function. The negative sign accounts for the expense 
increase because of a larger number of survivors for the current period. The impact of the longev-
ity risk is proportional to the size of the portfolio and to the reserve value, thus increasing not only 
for younger policyholders and/or for evaluation time closer to the issue (as shown by Figure 2) but 
also for larger size portfolios. 
80 74 68 62 56 50 44 38 32 26
age ati ssue (x) 0
2
4
6
8
10
12
14
16
18
time (t)
1
2
3
4
5
6
7
8
9
10
l
o
n
g
e
v
i
t
y
 
r
i
s
k
 
f
i
l
t
e
r
1+r t
x
SIM 1991 n=20  G=9%
Fig. 2. The proportion of longevity risk impact th the size of the portfolio 
The third derivative (equation 5) gives the investment risk, since it gives the multiplier of 
the rate shock. The positive sign of the indicator accounts for the direct proportionality between 
the rate shock and the income over period t; naturally the higher the value of the reserve is, the 
stronger the impact appears to be (Figure 3). Investment Management and Financial Innovations, 2/2004  98
80
74
68
62
56
50
44
38
32
26
age at issue (x)
20 18 16 14 12 10 8 6 4 2
time (t)
0
1
2
3
4
5
6
7
8
9
10
i
n
v
e
s
t
m
e
n
t
 
r
i
s
k
 
f
i
l
t
e
r
SIM 1991 n =20 G=9%
Fig. 3. The value of the reserve and the impact ratio
It can be easily shown that the impact of financial and longevity risk is far larger than in-
surance risk since their relevance is filtered through the reserve value connected to the portfolio 
size. This implies that their impact is directly proportional to the actual number of survivors Nt-1
x
and, in a sense, to the number of issued policies c. In conclusion, the variation velocity of the 
profit is measured by means of the gradient 
St  
wSt
wNt1
x ,
wS t
wpxt1
,
wS t
wG
§
©
¨
·
¹
¸
since
  G S S ' ' '  # '    , , 1 1 t x
x
t t t p N . (6) 
Therefore, the total variability of the profit over the single time bucket can be divided into 
the main three components, thus giving the opportunity to separately identify the impact of a spe-
cific risk factor, also considering the time evolution of the indicators. 
2. The mathematical representation 
As already pointed out in the previous sections, in riskiness valuation concerning the li-
ability components, the joint effect of demographic and financial factors plays a fundamental role. 
In this sense it is necessary to correctly evaluate the impact of such factors and their interactions 
on the mathematical reserve. On the basis of the balance equation, we can argue that fluctuations 
of the rate of return are particularly relevant for the assets, while on the liabilities the interactions 
between demographic and financial components have a more and more marked effect. 
As a first step in solving the problem, we focus on the riskiness analysis concerning the 
mathematical reserve, for which we introduce a simple and a suitable measure, apt to quantify the 
synergy produced by the random fluctuation of interest and mortality (survival). 
Let us consider a portfolio of c identical n-year temporary life annuity-immediate, each 
policy being of 1 unit payable at the end of each year while the life aged (x) survives. Let us de-
note the random variable representing the future lifetime of the i-th insured (for each i) by Ti(x)Investment Management and Financial Innovations, 2/2004  99
and the curtate future lifetime of (x) by Ki(x), the is the number of complete future years lived by 
(x) in >0,n-x@ . 
The prospective loss at time t, for the i-th policy, is defined as the difference tL
(i)at time t
between the present value of future benefit payments and the present value of future premium 
payments, assuming that Ti(x)>t:
tL
(i)   e
 G s  ds
t
t j
³
j 1
Ki xt 
¦ , (7) 
where G (s) is the force of interest. 
The net premium reserve at time t is defined as the conditional expectation of 
tL
(i), given that Ti(x)>t.
  Let us denote by tL the prospective loss for the entire portfolio, according to the 
following notation: 

 ³
 

 
 
 ¦
r t
t
ds s t n
r
t x r
x
t e p t N
G
1
L , (8) 
where N
x(t) is the number of survivors at time t.
Since in our analysis tL is affected by two risk sources, G (s) and Ki(x), the regression 
function >E>tL_G(s)@@ (Frees, 1998) provides an average of tL over all values of G(s). In this way we 
consider a conditional mean value with respect to the fluctuations of the insured’s lifetime into the 
residual policy duration >t,n@. In this framework var>E>tL_G(s)@@ represents a measure of the volatil-
ity of tL arising from the uncertain behaviour of the interest rate. 
By means of this measure we quantify the financial risk, taking into account also the 
presence, even though averaged, of the demographic component. 
We observe that: 
 >@ >@ 





 
. , cov
var
var var
11
2
1
1
¦¦
¦
¦

 
 
 

 

 



 


¸
¸
¸
¹
· ³
¨
¨
¨
©
§ ³
 
 
»
»
»
¼
º
«
«
«
¬
ª ³
 
 
»
»
»
¼
º
«
«
«
¬
ª
»
»
»
¼
º
«
«
«
¬
ª ³
 
 


t n
r
du u t n
s
du u
t x s t x r x t
t n
r
ds s
t x r x t
t n
r
ds s
t x r
x
t
s t
t
r t
t
r t
t
r t
t
e e p p p c
e p p c
s e p t N E s E
G G
G
G
G G L
 (9) 
We can also observe that the residual risk component is quantified by the obvious difference be-
cause of the uncertainty of life durations 
var tL >@  var E tLG s  >@ > @. (10) Investment Management and Financial Innovations, 2/2004  100
3. How to quantify the uncertainty impact 
Let us assume that the force of interest in the period under consideration is governed by a 
deterministic low, r(t), deducible on the basis of the current relevant rates, corrected by a stochas-
tic process, X(t), apt to summarize the randomness affecting the rate evolution in time and to cap-
ture all its possible deviations from the deterministic pattern. So, it is natural to set (Di Lorenzo et 
al., 1999) 
G t   rt  Xt  . (11) 
In our example X(t) is an Ornstein-Uhlenbeck process, with parameters E>0 and V>0, and 
initial position X(0)=0, involved by the following stochastic differential equation 
dX t  EXt  dtVdW t  , (12) 
where W(t) is a standard Wiener process. 
On the basis of stochastic calculus principles and after some lines of algebra, it is possible 
to obtain the following covariance and variance functions of the process  Xs 
0
th
³ ds:
  
    >@
    >@ . 2 1 1
2
var
; 1 1
2
, min , cov
2 2 2 2
3
2
2
2
0
2 2 2
3
2
2
2
0 0
t h h t t
h t
k h t t k h k h t t
k t h t
e e e e e h ds s X
e e e e e e e
k h ds s X ds s X
E E E E E
E E E E E E E
E
V
E
V
E
V
E
V
        »
¼
º
«
¬
ª
      
   »
¼
º
«
¬
ª
   

       
 
³
³ ³
For a simpler representation of the formulas for the variance and the covariance functions 
of the evaluation stochastic factor in formula (9), we pose: 
cov Xs 
0
th
³ ds, Xs 
0
tk
³ ds
ª
¬
«
º
¼
» )h,k 
and
var Xs  ds
0
th
³
ª
¬
«
º
¼
» Q(h).
Considering, in our hypotheses, that the evaluation factor is log-normally distributed, we 
can write: 
      >@ . 1 , cov
, 2
1
0 0   
»
»
»
¼
º
«
«
«
¬
ª ³ ³   
 
k h k h ds s X ds s X
e e e e
k t h t
I Q QInvestment Management and Financial Innovations, 2/2004  101
Now we are able to compute the impact of the risk sources on a life annuity portfolio. As 
an exemplification we implement our model in the case of a portfolio consisting of c=1000 poli-
cies at issue, each one related to a life aged x, with duration n=20. With regard to the instantaneous 
interest rate, we fix a constant deterministic component r=0.06, and the parameters E=0.11, 
V=0.005 for the stochastic one. In figures 4 and 5 we can observe the behaviour of the two risk 
components as functions of the age x and the reserve valuation time t (and, implicitly, of the policy 
residual duration). 
Fig. 4.The Financial Risk (x=20,…,70; t=10,…,15) 
Fig. 5.The Insurance Risk (x=20,…,70; t=10,…,15) 
The financial risk, as function of x, decreases when t is fixed; the same behaviour appears 
when the roles of x and t are interchanged. 
The insurance risk, for every fixed value of x, increases with t until a certain valuation 
time; the younger the insured is, the longer the period of the increasing behaviour is. This phe-
nomenon depends on the reserve amount exposed to the demographic risk, taking into account the 
influence of the insureds’ age and the residual duration of the portfolio 
When t is fixed, the demographic risk generally increases with x, but for great values of t
it decreases when the age becomes great, as it is clear reasoning as in the previous case. Investment Management and Financial Innovations, 2/2004  102
4. Conclusions 
The application of risk factor analysis has given the opportunity to build up a methodology 
set able to evaluate the impact of different risk drivers, both in a deterministic and in a stochastic 
contexts. Therefore, it could be of valuable application for both external and internal controls. 
It has been shown that, in both the contexts, there is a significant and consistent evolution 
along time of main risk indicators, thus suggesting the opportunity to scale the capital requirement 
according to the size and time evolution of the business. 
Finally, it is necessary to underline that the extension of the analysis to other kind of poli-
cies could give rise to very different result as far the relevance and the time evolution of the risk 
indicators are concerned. 
References 
1. Babbel D.F., F.J. Fabozzi. Investment management for insurers. - New York: Wiley, 
1999. 
2. Babbel, D.F., J. Gold, C.B. Merrill. Fair value of liabilities: the financial economics per-
spective//North American Actuarial Journal 2002. N
06 (1) - pp.12-27. 
3. Cocozza, R. La gestione del rischio di tasso di interesse nelle assicurazioni del ramo vita. 
– Padova: CEDAM, 2000. 
4. Cocozza, R., E. Di Lorenzo. Solvency of life insurance companies: methodological is-
sues//Working paper 2003a. 
5. Cocozza, R., E. Di Lorenzo. Risk profiles of life insurance business: a combined ap-
proach//VI Italian-Spanish Conference on Financial Mathematics (Trieste), 2003b. 
6. Di Lorenzo, E., M. Sibillo, G. Tessitore. A Stochastic Model for Financial Evaluation: 
Applications to Actuarial Contracts//Applied Stochastic Models in Business and Industry 
1999. N
0 15 - pp. 269-275. 
7. Frees, E. Relative Importance of Risk Sources in Insurance Systems//North American Ac-
tuarial Journal 1998. N
0 2 (2) pp. 34-48. 
8. IAIS Solvency & Actuarial Issues Subcommittee. On Solvency, Solvency Assessments 
and Actuarial Issues. 2000. IAIS, March. 
9. IAIS Solvency & Actuarial Issues Subcommittee. Principles on capital adequacy and sol-
vency. 2002. IAIS, January.  
10. IASB Steering Committee. Insurance Issue paper. 1999. IASB, December 
11. Knight, K. Mathematical Statistics - Boca Raton, Florida: Chapman and Hall/CRC, 2000. 
12. Parker, G. Stochastic analysis of the interaction between investment and insurance 
risks//North American actuarial journal 1997. N
0 1 (2) – pp. 55-84. 
13. Vanderhoof, I.T., E.I. Altman (eds). The fair value of insurance liabilities - Boston. 
Dordrecht, London: Kluwer Academic Publishers. 1998. 

A Stochastic Model for Financial Evaluation: Applications to Actuarial Contracts//Applied Stochastic Models in Business and Industry

Actuarial Issues Subcommittee. On Solvency, Solvency Assessments and Actuarial Issues.

Actuarial Issues Subcommittee. Principles on capital adequacy and solvency.

Altman (eds). The fair value of insurance liabilities -

Committee. Insurance Issue paper.

Fair value of liabilities: the financial economics perspective//North

Investment management for insurers. -

La gestione del rischio di tasso di interesse nelle assicurazioni del ramo vita. –

Mathematical Statistics - Boca Raton, Florida: Chapman and Hall/CRC,

Relative Importance of Risk Sources in

Risk profiles of life insurance business: a combined approach//VI

Solvency of life insurance companies: methodological issues//Working paper 2003a.

Stochastic analysis of the interaction between investment and insurance risks//North American actuarial journal

Methodological problems in solvency assessment of an insurance company

Methodological problems in solvency assessment of an insurance company

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics