Financial and demographic risk impact on private PAYG pension system: the Italian case

Abstract

The aim of this paper is to analyze private pension systems financed by pay-as-you-go, with a focus on the pension funds of the Italian Professional Orders. The research centres on the financial and demographic risks and on their impact on the future evolution of the fund. It presents a model to investigate the dynamics of these types of pension funds which operate according to the pay-as-you-go rule: the two stochastic variables global asset return and new entrants variation rate are modelled by autoregressive processes. The numerical applications carried out, using the data provided by the Italian Chartered Accountants’ pension fund, show that the rate of variation of new entrants has a higher influence on the evolution of the fund with respect to the global asset return, despite the considerable invested wealth. Some proposal are developed to face the demographic "extinction risk" of the insured professional category.

Full text

Roberta Melis1, Alessandro Trudda2
FINANCIAL AND DEMOGRAPHIC RISK IMPACT ON PRIVATE PAYG
PENSION SYSTEM: THE ITALIAN CASE
The aim of this paper is to analyze private pension systems financed by payasyougo, with a
focus on the pension funds of the Italian Professional Orders. The research centres on the financial
and demographic risks and on their impact on the future evolution of the fund. It presents a model
to investigate the dynamics of the types of pension funds which operate according to the payasyou
go rule: the two stochastic variables global asset return and new entrants variation rate are mod
elled by autoregressive processes. The numerical applications are carried out using the data pro
vided by the Italian Chartered Accountants' pension fund.
Keywords: Pension Funds; PAYG system; demographic risk; stochastic new entrants; extinction risk.
JEL classification. G23; H55; J11.
Роберта Меліс, Алессандро Трудда
ВПЛИВ ФІНАНСОВИХ ТА ДЕМОГРАФІЧНИХ РИЗИКІВ
НА ПРИВАТНУ РОЗПОДІЛЬНУ ПЕНСІЙНУ СИСТЕМУ:
НА ПРИКЛАДІ ІТАЛІЇ
У статті проаналізовано приватні пенсійні системи на базі розподільних виплат
(PAYG), зосереджуючися на пенсійних фондах італійських профспілок. Дослідження
базується на фінансових і демографічних ризиках та їхньому впливі на подальший
розвиток фонду. Представлено модель дослідження динаміки таких типів пенсійних
фондів, які оперують за принципом розподільних виплат: оборот загальних активів за
двома стохастичними змінними та варіативність кількості нових членів змодельовано за
авторегресивними процесами. Обчислення здійснено з використанням даних, наданих
пенсійним фондом дипломованих бухгалтерів Італії.
Ключові слова: пенсійні фонди; система PAYG; демографічний ризик; ймовірні нові члени;
ризик погашення.
Форм. 7. Рис. 6. Табл. 1. Літ. 17.
Роберта Мелис, Алессандро Трудда
ВЛИЯНИЕ ФИНАНСОВЫХ И ДЕМОГРАФИЧЕСКИХ РИСКОВ
НА ЧАСТНУЮ РАСПРЕДЕЛИТЕЛЬНУЮ ПЕНСИОННУЮ
СИСТЕМУ: НА ПРИМЕРЕ ИТАЛИИ
В статье проанализировано частные пенсионные системы на базе
распределительных выплат (PAYG), сосредоточившись на пенсионных фондах
итальянских профсоюзов. Исследование основано на финансовых и демографических
рисках и их влиянии на последующее развитие фонда. Представлена модель исследования
динамики таких типов пенсионных фондов, которые оперируют по принципу
распределительных выплат: оборот общих активов по двум стохастическим переменным
и вариативность количества новых членов смоделированы в соответствии с
авторегрессивными процессами. Вычисления осуществлены с использованием данных,
предоставленных пенсионным фондом дипломированных бухгалтеров Италии.
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© Roberta Melis, Alessandro Trudda, 2012
1Department of Economics, Business and Regulation, University of Sassari, Italy.
2Department of Economics, Business and Regulation, University of Sassari, Italy.

Ключевые слова: пенсионные фонды; система PAYG; демографический риск; вероятные
новые члены; риск погашения.
1. Introduction. The function of a pension scheme is to redistribute financial
cash flows paid in the work period (contributions) and returned in the retirement time
(benefits). Pension schemes are usually classified by the rules used to calculate the
benefits (or the contributions) and by financing system. They can be divided into
either a defined contribution scheme or a defined benefit scheme. In a defined con
tribution scheme, the contributions are made to a pension fund without specifying
the benefits, which depend on the fund portfolio, whereas, in a defined benefit
scheme, a sponsor fixes the benefits in advance and contributions are adapted to
maintain the fund in balance.
From the financial point of view, these schemes can be classified as either a fund
ed system or a payasyougo (PAYG) system. In a funded system, contributions are
used to purchase assets, which are kept to pay for future benefits; the PAYG system is
based on the immediate use of contributions to pay current pensions. In a PAYG sys
tem an "intergenerational pact" is established, in which the retired generation is sus
tained by the active one.
Funded schemes are above all exposed to financial risks. For example, during the
financial crisis of 20082009, it is estimated that in Europe the pension funds have
reduced their wealth accumulation by 15.8 % (OECD, 2009). As highlighted by
BorschSupam (2010) financial crises have different impact from demographic ones.
A financial crisis is a shortterm shock of a few years, while demographic change is a
long term process lasting several decades.
In Europe, a PAYG system is generally used for public pension schemes, instead
private schemes (relative to first and second pillar) are managed as funded systems.
According to OECD (2009) in most countries of Europe the share of unfunded pen
sion income in total retirement income is more than 90%; among these Italy, France,
Belgium, Spain; while in some northern countries it is smaller: the Netherlands
(53,5%), UK (56.2%), Ireland (63.9%).
To manage the crisis due to the aging population some countries adopted para
metric reforms, rising retirement age and contribution rates, others, like Sweden,
Italy, Latvia and Poland introduced notional defined contribution (NDC) system,
where pension is linked to contributions paid during the working life.
Until 1995, in Italy, all the workers' pensions were administrated by the state: all
social security was publicly managed and the financial mechanism used was PAYG
(with a defined benefit). In 1995, a reform of the National Pension Plan was enforced
in Italy (L. 335/1995 – the socalled "Dini Reform") establishing the transition from
a defined benefit to a defined contribution scheme. Another relevant step of the
Reform was privatization of the retirement funds of Italian Professional Orders. In
Italy, each professional group recognized by a Board (such as lawyers, doctors,
accountants, engineers etc.) administers its own retirement fund. After privatization
these funds continued to operate according to a PAYG financing mechanism.
In this paper, we propose a model to describe the evolution of these "closed" pen
sion funds, according to the PAYG rule. The fund is defined closed because it has a
restriction on membership: only those who have a specific job or profession have a
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right to become the members of the fund. These types of funds show an anomaly as,
in general, PAYG systems are applied to an open group. However, this specific PAYG
system has a barrier to entry, as only those who pass a public examination can exer
cise a profession and consequently become members of a fund.
There is a broad literature emphasizing financial and demographic risks associ
ated with private pension funds operating in a funded scheme. Regarding financial
risks, the investment risk is linked to the randomness of the return rate; this derives
from changes at financial markets where a plan invests and occurs in deviations of the
real return rate from its expected values. This issue was discussed in numerous papers.
Among these Dufresne (1988), Haberman (1994), Cairn and Parker (1997) analyze
investment risk for a definedbenefit pension fund. Blake, Cairns and Dowd (2001,
2003) propose a model for a definedcontribution pension fund, with stochastic
wages and returns, in a discrete and continuous time respectively.
Regarding demographic risks, it is well known that the longevity risk, which
derives from improvements in the mortality trend and determines systematic devia
tions in the number of deaths from its expected value. This kind of risk must be faced
by all types of pension plans and life insurance products. On this topic an exhaustive
literary review can be found in Pitacco (2004).
In particular for a PAYG scheme, financial sustainability is related to the bal
ance between active and retired members. There is a further demographic risk source
to take into account: the risk relates to future monetary cash flows necessary to
ensure payments of future pensions. This risk is related to the demographic variable
"new entrants" and to their future contribution capacity. This is a relevant compo
nent because a balance between the number of contributors and the number of pen
sioners is necessary to sustainability of a fund. In this sense we have identified a fur
ther risk related to the demographic variable "new entrants", that we called "extinc
tion risk". We refer to as the possibility that in the shortterm changes, either in the
job market or regulatory actions in the world of professions produce a more or less
sudden reduction in enrolments to the professional order, causing a financial dise
quilibrium on the fund cash flows, and consequently to the related retirement fund
sustainability.
Over the past few years, a vast literature regarding demographic risks in public
PAYG system was developed. In this paper we do not examine public PAYG systems,
only private ones.
Actuarial literature on pension population is mostly based on deterministic mod
els (Bower et al, 1976, Winklevoss, 1993) which consider the population as stationary.
There are some contributions which take into account the fluctuation of the number
of new entrants in a pension fund. Mandle and Mazurova (1996) use spectral decom
position of stationary random sequences to investigate a pension scheme under ran
domly fluctuating rates of return and number of entrants. Iyer (2003) derives algebra
ic expressions for the variances of, and covariance among important aggregates that
characterize the development of a pension scheme taking into account the stochastic
variation of the new entrants. Colombo and Haberman (2005) analyze the impact of
the stochastic evolution of active membership population on the mismatch between
assets and liabilities of a (funded) defined benefit pension scheme. Menoncin (2005)
studies the allocation problem for a pension fund, which behaves according to a
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PAYG rule considering the total number of workers and pensioners as random vari
ables. Melis and Trudda (2010) propose a model for evolution of a PAYG pension
fund, with stochastic new entrants and global asset return, they adopt risk indicators
to monitor the solvency of a fund.
In a pure PAYG pension system revenues equal outlays each year (Towerbridge,
1952). Here we analyze a spurious PAYG scheme, yet being in a growth phase (con
tributors greater than pensioners) where there is the accumulation of partial reserves.
The active population evolves according to the process of new entrants, which can be
different for each fund. Therefore, the main problem is to analyze the flow of new
entrants into the fund, to find out if the number of future taxpayers is sufficient to
maintain a system in balance, and thus to ensure its solvency.
We study the impact of the stochastic component "new entrants" analyzing how
its variation rate influences future cash flows of a fund. The goal is to measure and to
compare the influence of the stochastic variable "new entrants" and the stochastic
variable global asset return, with respect to the evolution of the overall fund.
A numerical application to the data of Cassa Nazionale di Previdenza e
Assistenza a favore dei Dottori Commercialisti (CNPADC – the pension fund man
aging the pension system of Italian chartered accountants professional order) is
developed. Some comparisons between the influence of the financial variable ''global
asset return'' and the demographic variable "new entrants" are carried out to show the
most persuasive one by analyzing the impact of the single stochastic variable.
The work is organized as follows: Section 2 illustrates the single components of
the cash flows of the fund; in Section 3 a model to measure the fund evolution is
proposed; Section 4 presents a numerical application of the CNPADC data; final
ly, in Section 5 conclusions and proposals are drawn from the empirical investiga
tion.
2. Fund Evolution. As already stated, the purpose of the analysis is to study the
impact of the evolution of new entrants in a pension fund financed with PAYG. There
are different approaches to analyze future flows of new entrants for these types of pro
fessions. Here we propose a model for evolution of new entrants based on the analy
sis of the variable "new entrants variation rate".
In this section we study the fund evolution through the dynamical analysis of its
single components.
The amount of the total assets belonging to the pension scheme at a specific time
t represents the fund value. Excluding the fixed cost of management the evolution of
the fund can be represented as follows:
(1)
where C(t) and B(t) represent respectively the annual contribution income and pen
sion benefits paid at the beginning of the year tand r(t,t+1) indicates the global asset
return related to the period between time tand time (t+1).
The evolution of the annual contributions C(t) is analyzed starting in a generic time
zero and adopting a recursive dynamic year by year. The assumptions are as follows:
 αis the only entry age to the scheme;
 τis the retirement age; in this way τ – αis the length of the contribution period;
 ωis the extreme age; ω – τis the maximum length of the retirement period;
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( ) () () ()[]()[],1,11
++−+=+
ttrtBtCtftf

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 the scheme provides pensions only upon reaching the age of retirement (dis
ability or survivors' pensions are not considered);
 administrative costs are not included;
 in the theoretical framework for notation simplicity we consider the same con
tributory seniority for the members of the same age.
The analysis is simply extendible releasing to these hypotheses (in the applica
tions real seniority and administrative costs are considered).
In this way it is possible to study the amount of contributions C(t), paid at time
t, decomposing the total quantity by the age of contributors.
In a generic time tthe total contributions are calculated by the summation for all
the ages xbetween αand tof the number of contributors aged xat time tmultiplied
by the average wage at time tand the contribution rate.
If we separate the original population (members already in the scheme at time 0)
from new entrants (members entered in the scheme after time 0), we have the sum of
two summations to calculate the contributions, one for the contributions paid by the
original population, and the other one – by new entrants.
The active population evolves according to the force of mortality.
As stated in the introduction a PAYG system needs equilibrium between reve
nues and outlays in each year: it is necessary that active members inside a fund pay a
sufficient amount of contributions to cover benefit payments of the retired members.
Active members evolve according to the process of new entrants. It is assumed that
the random number of new entrants in the period kdepends on the number of new
entrants in the previous time multiplied by (1 + λk)being λkthe new entrants varia
tion rate at time k.
In a generic time tthe total contributions are calculated as follows:
(2)
where γis the fixed contribution rate, A(x, t) indicates the number of the original
members aged xat time t, A'(x, t) indicates the number of new members aged xat
time t, (xα)pαis the probability of a member aged αremains in the scheme for xα
years and wx,t indicates the average income for a member aged xat time t.
As we can see the total amount of contributions is directly influenced by the ran
dom variable ''new entrants'' (as well as the pension benefits when future active mem
bers become pensioners), depending by the value of the parameters.
The same approach is used here for the total pensions B(t). The pension benefit
is obtained considering the number of retired members and the average amount of
benefit. The benefit depends on the amount of the accumulated contributions and the
transformation coefficient based on the age, that is the annuitization coefficient used
for the conversion into annuity of the notional contribution amount accumulated by
each worker. In the scheme all the agents retire at the same age τ; furthermore, at τ
they have paid the same average amount of contribution and consequently they
receive the same amount of periodic benefit Bx,t, which represents the average pen
()
()
()
()
,1'
'
11
,
1
0,,0,
11
,,,,










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++=
=
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

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∑∑∏
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+=
−+
=
−−
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−
+=
−+
=
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α
α
α
α
ααα
τ
α
α
α
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γ
tx
t
x
tx
xt
i
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x
txtxtxtx
wApwpA
wAwAtC

sion of people aged xat time t. The total amount of the pension is expressed as fol
lows:
(3)
where Px,t represents the number of pensioners aged xin the generic time t. The total
amount of the pension benefits is influenced by the random variable ''new entrants''
in the long run, when active members become pensioners and when new entrants
reach the age x > τ. In other words, when the projection time is higher than the con
tribution period (t > τ– α), the survivor members that entered at time 1become pen
sioners: i.e., the new entrants at time 1A'
α,1 become pensioners Pτ,d in the period t =
d and so on in the following years.
3. The model. The model includes two stochastic variables: the demographic
variable "new entrants variation rate" and the financial variable ''global asset return''.
There are different approaches to analyze future flows of new entrants for these
types of funds. An approach consists of studying variables related to demographic
evolution of population, the development of education and the attractiveness of the
profession, through the analysis of the transition probabilities from states of the pop
ulation (university students, graduates, employment rates, active workers, members
of the pension fund). This method is useful for shortterm forecass (510 years). As
the aim is to study the fund dynamics in the long run, here we propose a model for
the evolution of new entrants based on the analysis of the variable "new entrants vari
ation rate". This indicates the variation that occurs in the number of new entrants
from one year to the next.
The rate of variation in the number of new entrants can be represented as an
autoregressive and moving average process of order (p, q). Indeed through an accu
rate analysis of real data for different retirement funds, we notice that for the analyzed
funds an ARMA(1,1) is well suited to describe the new entrants variation rate, split
ting the population into males and females. Therefore, we assume that new entrants
variation rate follows an ARMA (1,1) model, defined by linear difference equations
with constant coefficients, written as follows:
(4)
where ϕ1and ϑ1are respectively the autoregressive and the moving average parameters,
and Ztare normal mutually independent random variables with mean 0 and variance σ2z.
The global asset return can be represented in different ways according to the asset
composition of the fund's investment portfolio and the consequent risk associated. The
funds analyzed in this paper deal with first pillar pension schemes and therefore they
usually present prudential portfolios. The global asset returns associated usually show
limited variation around their historical trends. For this reason to represent the inter
est rate dynamics we use the following model (see Orlando and Trudda, 2004):
(5)
where the interest rate is the sum of a deterministic component rt+1 and a stochastic
one Xr(t+1), described by an autoregressive process of first order (AR1), expressed by
the following nonhomogeneous equation:
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ACTUAL PROBLEMS OF ECONOMICS, #7, 2012
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()
,
,, tx
x
tx BPtB
∑
=
=
ω
τ
,
1111 −−
−+=
tttt
ZZ
ϑλϕλ
() ()
,1
ˆ
1, 1
++=+
+tXrttr rt

(6)
which expresses the autoregressive dependence of order one, where ϕrand σαare the
parameters of the process and atare normal mutually independent random variables
(with mean 0 and variance 1).
This model is a discrete representation of the Vasicek model and it is suitable to
represent the global asset return on a risky asset portfolio, since the return can reach
a negative value, as there can be losses of capital. The choice of a mean reverting sto
chastic process is due to the fact that the analyzed funds are characterized by pru
dential portfolios composed with lowrisk assets (the heritage is in large part com
posed of real estate and liquidity and only in limited part of stock funds).
Taking the above into consideration, the general function of the fund (1) can be
represented as follows:
(7)
4. Numerical application. The application is carried out on the data provided by
the CNPADC, the pension fund of Italian Chartered Accountants, which is a defined
contribution pension fund, financed by a PAYG system. Data are available from 1976
to 2006. It is one of the Italian Professional Order retirement funds privatized by
Legislative Decree (D.Lgs) 509/1994. Until 1995 these organizations, managing the
social security of given categories of selfemployed professionals, were administrat
ed by the State that would step in, in case of insolvency. Since 1995 they have man
aged the security of a growing number of selfemployed without being sponsored by
the state. New funds, built by D.Lgs 103/1996, follow a fully funded financial
scheme.
The pension funds of Professional Orders are selfmanaged and they continue to
operate according to a PAYG financing mechanism, although they were privatized.
As already highlighted in the introduction this is an anomaly because private closed
schemes are usually funded. In this particular system financial selfsufficiency is cer
tainly guaranteed only at the initial phase, because there are many contributors and
absence of pensioners. In the medium run the performance depends on the accumu
lated capitals. In the long run it is necessary for financial sustainability of the pension
plan that the number of pensioners remains proportional to the number of workers. If
the ratio active/retired decreases, the increase in the financial burden can entail a sit
uation of financial disequilibrium. This is most relevant for the retirement funds of
each specific professional order for which, unlike in a public system, there is indeed
no intergroup compensation.
The social security of professional orders is not marginal within the Italian pen
sion system; about 1.3 mln. workers, equal to 5.6% of total employment, are regis
tered as contributors to the private pension system of Italian Professional Orders. The
subscribers share has increased notably for some funds (engineers and architects,
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() ()
1
1
+
+=+
trrr
atXtX
α
σϕ
() ()
()
()
()
()
[]
11,,
1
,1111
1
0,,
1
0,
ˆ
1
1'1
++
=
−−
=−−
−+
=−−
−
+= −
+++



+









−++++=+
∑
∏∑∑
trrttx
x
tx
xt
i
txiii
t
x
xtxtxt
tx
tx
atXrPB
wZZApwpAtftf
α
ω
τ
α
α
αααα
τ
α
σϕ
θλϕγ

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lawyers and chartered accountants), others remain stationary (notaries), while some
have reduced (accountants). Table 1 and Figure 1 show the number of contributors
and pensions of Italian Professional Orders Pension Funds in the last 3 years. The last
two columns of the table indicate respectively the financing mechanism of the fund
(PAYG or fully funded) and the type of benefit (defined benefit or defined contribu
tion).
Contribution 2008
Figure 1. Contributors of Italian Professional Orders Pension Funds
classified by profession (2008)
Table 1. Contributors and pensions of Italian Professional
Orders Pension Funds (20062008)
Fund
2006
2007
2008
Agents and
Commer cial
Representatives
Contributors
n.r.
271,093
271,002
PAYG
DB
Pensions
n.r.
112,167
111,658
Lawyers
Contributors
129,359
136,818
144,070
PAYG
DB
Pensions
22,997
23,697
24,358
Chartered
Accountants
Contributors
45,353
47,322
49,759
PAYG
DC
Pensions
4,619
4,833
4,946
Labour
Consultants
Contributors
21,684
22,225
22,897
PAYG
DB
Pensions
5,951
6,282
6,782
Pharmacis ts
Contributors
69,572
71,373
73,728
PAYG
DB
Pensions
27,060
27,298
27,431
Surveyors
Contributors
92,779
93,487
94,486
PAYG
DB
Pensions
22,219
23,786
24,774
Reporters
Contributors
17,344
17,681
18,163
PAYG
DB
Pensions
5,794
6,002
6,230
Engineers,
Ar c hi t e ct s
Contributors
131,095
138,124
143,851
PAYG
DB
Pensions
11,756
12,086
12,706
Doctors
Contributors
332,834
337,798
342,260
PAYG
DB
Pensions
80,770
81,390
82,501
Notaries
Contributors
5,312
5,312
5,312
PAYG
DB
Pensions
2,362
2,380
2,409
Accountants
Contributors
29,690
29,297
28,659
PAYG
DC
Pensions
5,431
5,751
6,268

The End of Table 1
The following assumptions are used:
 all the new members join the fund at the same average age α= 30 and they
retire at the same age τ = 65;
 for the actual members real entry age and contributory seniority is considered;
 a subjective contribution rate γis equal to 10,7% of annual professional income3;
 the evolution of the population is based on IPS55 male and female mortality
tables4;
 the pension benefit value is obtained by multiplying the accumulated contribu
tion and transformation coefficients based on the age5;
 administrative costs Atare considered resulting from 2005 balance sheet,
appreciated at the 3% annual rate. Thus the general equation of the fund becomes:
 professional incomes are appreciated at rate of inflation;
 the inflation rate is fixed at 2%.
Figure 2 shows the demographic structure of the CNPADC fund on January, 1st,
2006. There is a high component of young members: the main class of age is repre
sented by 3545 category. The fund is in a strong growth phase and thus the cash flows
of contributions C(t) are much higher than payments of pensions B(t). The analysis,
carried out on a time horizon of 40 years, demonstrates how the variable "new
entrants" affects the probability of default of the fund. The "hump" of 3545 year old
actual members will retire after 2030 years (20272037): the application shows how
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Fund
2006
2007
2008
Veterinar ians
Contributors
24,123
24,902
25,478
PAYG
DB
Pensions
5,996
5,980
5,963
Agricultural
experts
Contributors
1,096
1,121
1,048
FF
DC
Pensions
0
0
2
Biologists
Contributors
8,874
9,155
9,477
FF
DC
Pensions
105
153
219
Medical
attendants
Contributors
12,183
14,275
15,286
FF
DC
Pensions
141
214
300
Agriculturalists
Contributors
3,234
3,184
3,203
FF
DC
Pensions
221
247
277
Industrial experts
Contributors
13,639
13,828
14,093
FF
DC
Pensions
759
938
1187
Mul ticategoric al
Contributors
17,101
17,556
17,628
FF
DC
Pensions
452
522
680
Psychologists
Contributors
25,876
27,911
30,101
FF
DC
Pensions
577
719
885
Total contributors
981,148
1,282,462
1,310,501
Total pensions
197,210
314,445
319,576
Total members
1,178,358
1,596,907
1,630,077
3Subjective contribution is the contribution paid by a member of the CNPADC pension fund. It is calculated applying to
the professional annual income a contribution rate which varies electively between 10% and 17%. In 2005 the average
rate was 10.71% (Source CNPADC).
4IPS55 are projected life tables for Italian males and females, cohort 1955.
5The transformation coefficients of Italian Law have been employed: in particular the value corresponding to age 65 is
6.13%.
( ) () () () ()
[]
()
[]
1,11
++−−+=+
ttrtAtBtCtftf

in the long run the fund could go to zero in the case that the future flows of new
entrants were not sufficiently high. The demographic scheme is very similar to that of
other most important analysed funds (architects, engineers and doctors).
From this demographical situation a dynamic fund evolution has been devel
oped.
Active population
Figure 2. Actual contributors classified by age and gender
10 ths. Monte Carlo simulations have been run with both stochastic components
(interest rates and new entrants).
Figure 3 shows the percentiles of frequency distribution and the expected value
(value is expressed in million euros).
Fund value
Figure 3. Fund dynamics with stochastic interest rates and new entrants
(percentiles and expected value)
The chart indicates there is a probability for the lower percentiles that the
fund could reach its peak in 2035, then taking a downward trend until reaching
the default. Developing the projections in a longterm horizon (40 years) we can
observe that, if there is not a sufficient number of new entrants and therefore a
sufficient flow of contributions, the fund value will tend to decrease rapidly, as
the contributions and the investment performance are not sufficient to cover
pensions and administrative costs and the fund will tend to achieve zero and so
default.
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In order to analyze the single influence of two processes, two further simulations
have been considered using only one stochastic variable and fixing the other one on
the expected value. The results are illustrated in Figures 4 and 5. In the former case
(fixed new entrants and stochastic interest rates) the fund reaches a peak after about
30 years and then begins to decrease for all the percentiles. In the latter (fixed inter
est rates and stochastic new entrants) the fund continues to increase, and begins to
decrease in about 2035, only at lower percentiles. The comparison shows how in our
application the new entrants variation rate has a stronger influence on the fund value
than the global asset return.
Fund value
Figure 4. Evolution of the fund, with deterministic new entrants
and stochastic interest rates
Fund value
Figure 5. Evolution of the fund with deterministic interest rates
and stochastic new entrants
To compare the influence of the new entrance on the default probability, the
Value at Risk (VaR) at 95% confidence level of the pension fund value for 4 cases is
calculated: in the first case, we consider interest rates and new entrants both stochas
tic; in the other two cases we considered only one variable as stochastic and the other
one as fixed on its expected value; finally we considered the case in which there are
no new entrants: the results are shown in Figure 6. As we can see, the "new entrants"
influence is stronger than the one of financial variable in this case as well.
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To evaluate the "extinction risk" of the profession, we considered also the case of
total absence of new entrants. In this case we verify that the fund would quickly short
fall.
Fund value VaR 95%
Figure 6. Fund value VaR at 95%
5. Conclusions. The paper examines private pension funds financed by PAYG,
developing a stochastic model to describe the evolution of the fund. Some applica
tions were carried out on the pension funds of the Italian professional orders.
The most relevant risk sources of these funds were analyzed focusing on the demo
graphic risk related to insufficient number of new members into the fund to ensure the
payment of future pensions; this hypothesis has to be considered, given the potential
barrier at labour markets associated with these funds. The empirical analysis shows how
the new entrants variation rate has a stronger influence on the fund value with respect
to the global asset return. This kind of risk, that we called "extinction risk" of the insured
professional category, is not an insurable risk, but it can be diversified and shared among
the whole system of the different professions' pension funds. In the long run this strat
egy would reduce this risk for a single professional group, through funding gained from
all the professional categories. Using our model it is possible to monitor the financial
sustainability in the mediumlong run, in function of the random variables considered.
In this way the management has at its disposal a control instrument to forecast finan
cial disequilibrium situations and to promptly activate the actions to rebalance the fund
with respect to a shock in a determinant variable of the fund.
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Стаття надійшла до редакції 06.12.2011.
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