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THE RISKS OF BANKRUPTCY IN INSURANCE COMPANIES,
STOCHASTIC STABILITY AND FAVORABLE GAMES
DR. COSTAS KYRITSIS
Software laboratory
National Technical
University of Athens
PROF. PETROS KIOCHOS
Department of Statistics
and Insurance Science
University of Pireas
Abstract
In this paper we make an analysis of the risks of Bankruptcy in Insurance
Companies for example due to an overaccumulation of accidents.
We apply the methods of stochastic differential equations and stochastic
stability of dynamical systems .We make use of popular models of
aggregate investment and growth in Insurance Companies based on the
geometric Brownian motion .We extract a theorem that gives necessary
and sufficient conditions of Bankrupcy . The result has a relevancy with
situtations met in the theory of games. It is given a strategic management
interpretation of the Bankruptcy theorem .
Key words:
Insurance, Investments, Ban Bankruptcy, Growth Models, Brownian
Motion, Stochastic differential equations, Stochastic stability of
dynamical systems.
Introduction.
There many Insurance Companies that have led to Bankruptcy . In
Greece for example there are too many Insurance Companies compared
to the population and to what hapens in other European countries . In the
United States a unique number in history of Banks and Insurance
Companies were led to failure during 1980-1990 .
Of course Insurance Companies apply a sytsem of policies and
strategies in order to avoid the risk of bankruptcy that provided there are
the resources to be applied ,are very effective .Can we model
quantitatively the aggregate investment in an Insurance Company and
analyse quantitatvely the risk of bankrupcy due to an overaccumulation
say of accidents? Is there a theorem of mathematcal nature that describes
the risk? Can we deduce from such a theorem methods and strategies to
avoid it? What are the relations of the implied strategies to
diversification , horizontal and defensive strategies?
In this paper we make use of popular models of aggregate
investment and growth in Insurance Companies and we analyse the risk
of Bankrupcy . We extract a theorem of Bankrupcy through stochastic
stability of dynamical systems formulated by stochastic differential
equations .We make use in paricular of the geometric Brownian motion .
The result has a very simple interpretation based on the theory of games.
We may oversimplify for the moment the investment decision of an
Insurance Company as a coin tossing. The coin is not fair (it is superfair)
with probability p of heads and q (p>q) of tails which accounts for the
profitability of the insurance business. But there is always the risk say
that too many car accidents may occur in this partcular year giving as a
result loss instead of profit for the company. This means that the result is
tails fo the Insurance Company Each coin tossing corresponds to an
accounting year. Although this simple game is superfair this does not
mean at all that the result is a favorable game for the Company. The
game is favorable if there is strategy that permits the invested assets to
converge to infinite ,with probability one, as time goes to infinite. For
example if the Company applies the policy of full leverage ,and is betting
at each coin tossing all the assets and the profits then with probability
one the company shall result to Bankrupcy . In the simple game and
stochastic dynamical sytem of coin tossing there are at least two atractors
,infinite and zero .In order to have that the bettor in coin tossing shall
have his fortune to go to infinite he must bet ecah time a percentage of
his profits and fortune of the amount of p-q (see [Breiman L.] . With this
simple example we get the idea of the main Bankruptcy theorem of the
paper and its relations to stochastic stability and game theory .
Although we focus on insurance Business because the risk in them
is inherent ,significant and anavoidable ,the same results can apply
somehow to the general bankrupcy problem in relation to the investment
and leverage decisions e.g in Stock Exchange Markets.
1. Stochastic stabitity and favorable games .
It has been stressed quite often the relation of economic behaviour and
decisions with the strategies of games and game theory . In [Owen G.]
for example it is presented a quite modern approach to the concepts of
games ,multi-stage games and their srategies . We want to stress the
difference of a superfair game and a favorable game. A multi-stage game
is superfair (fair ,subfair ) to a player if at each turn the lottery and is
odds which is related to every turn gives a positive (zero ,negative )
average value of profit for the palyer(see [Dubins L.E.,Savage L.J]). For
example coin tossing with probabilities p=q=1/2 of heads and tails is a
fair game ,while casino’s roulette ,because of the zero ,is subfair for the
gambler and superfair for the casino.
A favorable game on the other hand (see [Feller W.] p
248,249,262,346) is a multi-stage game (see [Owen G.] ,[Breiman L.]
when there is a stratetgy that permits the fortune Sn of the palyer to
converge to + as n converges to + . It is a common misconception to
assume that a fair game can be also a favorable game . For example if in
fair coin tossing the available capital of the one player is finite while of
the other infinite ,then it can be proved that the game cannot be not
favorable for the first palyer and if he bets at each time a fixed ammount
the Bankrupcy is certain (see [Karlin S.,Taylor H.M.] p49,92-94 ,108
and [Feller W.]chpt 14 p344 ) . Also in the theory of martingales that
arise from fair multi-stage games it is a celebrated and not easy result that
in the previous fair coin tossing even if both players have infinite fortune
the game cannot be favorable for any of them (see [ Karlin] theorem 4.1
p 266) . Of course if the coin tossing is superfair (p>q) then it can be
proved that the game is also favorable .But there are strategies that lead
to bankrupcy even in this superfiar and favorable game and it is far from
trivial to exctract the strategy that makes the fortune to converge to
infinite . In [Breimna L.] it is proved that the strategy that in a fixed
numper n of coin tossings maximizes the probability that starting with a
fortune x we result with a fortune y>x is to bet a;ways not a fixed
ammount but the fixed percentage p-q of the available fortune each
time.
We may understand coin tossing as a stochastic dynamical
system .Then the concept of being favorable is translated to the
existence of straegy or control that leads to an attractor which is the
infinite. The risk of bankrupcy is translated with the existence of an
atrractor which is the zero .For discrete time stochastic dymanical
systems and their stability see [ Azariadis C.]
and [Tong H.owell] .
Although we shall not formulate our bankruptct theorem with coin
tossing ,the previous simple concepts are usefull to grasp its meaning.
2.The law of large numbers ,geometric Brownian motion and
growth models of aggregate investments in insurance companies.
Most of the time series models of aggregate investment and
growth models of companies are first order linear stochastic difference
equations with constant ciefficient (ARMA time series) see [Berndt
,E.R.] chapter 6 p 233 .
The same holds for continuous time models of growth ,like the
neoclassical for example ,see [Mallaris A.G. Brock W.A.] chpt 3 p142
and [Oksendal B]chpt 5 p 59,60. The usual linear continuous time
stochastic model of growth of aggragate investment is the geometric
Brownian motion (see [Oksendal B]p 60 and [Karlin S.,Taylor H.M.]
p357,363,385.
The law of large numbers justifies the use of the
normal distribution and the «white noise» as the innovation term or
random fluctuation term in the stochastic differential equation .
Also an other justification comes from recent models of the life
insurance based on Ito’s diffusions like the geometric Brownian motion
,see [Janssen J.Skiadas C.H.] .
Of course strictly speaking the aggregate investment of an
insurance company is based and measured every the accounting year and
it can be formulated as a time series .This time series that represents also
the profits and reinvestment (leverage ,or increase of the equities) of the
copmany depends heavily on the stochastic processes and time series
that model the events to be insured (deaths,accidents ,diseases ,legal
events etc)
Life or death
time series can be considered as linear
autoregressive time series with variable coeficients and no noise at all .A
subfair lottery is defined every year which usually is averaged to a
constant premium by an rate of change of the value of manoey in
time.This means that the subfair rate is averaged among all years ( till
death) and becames say hihgly subfair the first years and probably
superfair the late years. The health time series is again a lotter based on a
multi state time series which is Markovian and non-statinary The
regression curve satisfies a linear difference equation with variable
coeficients . The premium becames constant after averaging the (subsuper )fair mean value for a zone of 4-9 years . In (tangible-inangible)
assets insurance the time series is again Markovian non-stationary and
the regression curve satisfies a linear difference equation with variable
coeficients . Actually there is no time series depending on the year of the
contract as the contract is renewed every year ,The probabilities and
regression curve depend on the age and state of the assets The
probabilities depend on the age of the assets and its depreciation thus it is
changing from year to year. There is no averaging of the premium which
is according to design subfair. We shall not enlarge on such a
formulation based on time series .Although it is equivalent with the
standard one of actuarial mathematics it would lead as away from the
goals of this paper .
According to the design of the actuarial mathematics of the
insurance company we have a superfair game for the insurance company
from year to year .Nevertheless as we shall see it is far from being a
favorable game without an appropriate investment strategy .

In this paper in order to simplify the mathematics and have them
in conformance with the area of most results of stochastic dynamical
systems ,we consider contunous time models and in particular as we said
the geometric Brownian motion .
Continuous time models have often simpler symbolic computation
.In addition sometimes the exact discrete time Maximum likelihood or
least squares estimators of the parameters are intractable while the
discrete approximation of continuous time maximum likelihood
estimators are feasible .For this reason we chose a continuos time nonlinear model. It is also a good opportunity to make explicit how the
somehow advanced research on stochastic differential equations can be
combined with very real ,elementary and practical applications.
The model for the accumulated total investment that we
chose is the (bilinear) geometric Brownian motion that is described by
the stochastic differential equation :
dX t  rX t dt  X t dBt
Where Bt is a Brownian motion and r, are constants .For the
definition of the stochastic differential equations and the geometric
Brownian motion see [Oksental] p121 Chpt. V p 60 ,exerc.7.9 ,p
121,example 5.1 p 60)
Some of the properties of this stochastic process are the next:
a) If r<(1/2)2
then Xt converges to 0 as t goes to infinite ,almost
sure.
The probability p to reach ever the value X starting from x0 <X is
P=(x0/X)a where a=1-(2r)/ 2
b)If r>(1/2) 2 then Xt converges to  as t goes to infinite almost
sure.
The average time T that it reaches X for the first time starting from x0
Is T=log(X/ x0)/(r-(1/2) 2)
c) The logarithm of this process X/ x 0 is an ordinary Brownian motion
with drift:
log(X/ x0)=(r-(1/2) 2)t +  dBt .
The regression curve is the next:
X  X 0 exp( rt )
The solution of this stochastic differential equation is given by the
formula:
X t  X 0 exp(( r  (1 / 2) 2 )t  Bt )
The previous properties a) and b) of the geometric Brownian motion is
our basic point for the condition of Bankruptcy. If the risk σ is too
large compared to the average rate of return per unit of time r ,
r<(1/2)2 , then we are lead to Bankruptcy with probability one
(almost surely) .If on the other hand the risk σ is small compared to
the average rate of return per unit of time r , r>(1/2) 2, then the
investments define by leverage and reinvestment a favorable
multistage game .
3.The basic theorem of bankruptcy .When is the superfair
investment game favorable for the insurance company?
We reformulate the previous mathematical results with financial
interpretation :
The Bankruptcy theorem
Let an insurance company and let us assume that the average rate of
return per unit of time of the aggregate investments in insurance
products ,follows the normal distribution with mean r and variance σ . If
the company’s policy is to apply leverage by reinvesting all profits then:
a) The company shall be led to Bankruptcy with probability equal to one
(almost surely goes to the attractor 0 ) if r<(1/2)2
b) The company’s equities shall accumulate without upper bound ,that is
the leverage by reinvestment of the profits makes a favorable game and
business (almost surely goes to the attractor +) if r>(1/2) 2
Proof: Since the r and σ are constant in time ,repeated reinvestment in
the long rum ,where the reinvestment steps are assumed infinitesimal,
follows a geometric Brownian motion, as we remarked in the previous
paragraph. Then the corresponding stochastic stability property of the
geometric Brownian motion ( see [Oksental] p121 Chpt. V p 60
,exerc.7.9 ,p 121,example 5.1 p 60) gives the result .
QED
We must remark that this theorem is of a different technique of the usual
statistical techniques to forecast Bankruptcy .The standard statistical
techniques are discriminant analysis etc (see e.g.[ Altman E.] [Beaver
W.])
4. Implications in strategic financial management and Total Quality
Managements of insurance business
From the previous Bankrupcy theorem we see that if the variance
or difussion coeficient σ is large enough compared to the average rate of
growth μ of the insurance company then the Bankruptcy is almost certain
in the long run . The rate of growth depends both on the average rate of
return and the average reinvestment rate (the latter depends of course
also on the divident ) Such a large variance may be due to the large
variance of the insured events (e.g car accidents ) in the particular
population of clients of the insurance company .We stress that although
the national population may have small variance ,the significant quantity
is the variance in the particular population of clients of the company .
The σ can quite easily estimated from the acconting department of the
company . The theory of estimators provides with formulas that estimate
σ even from samples .With some simplyfications it can be even
estimated with the usual formula of sample variance .
The insurance company has in such a situation many options to
cure the risk of bankruptcy. All of the strategies nevertheless have their
cost ,and this costs is paid each time by one or more of the stakeholders
(the company ,the clients etc) .It is matter of total Quality Management
to distribute the cost of avoiding bankruptcy in an appropriate way
among the stakholders
In particular the company can
1. Increase his anuall profit as far as the law may permit resulting to
an increase of the average rate of growth .This of course makes the
insurance products more expensive for the clients
2. Change the clients gradualy and put filters in Marketing that lead
to a more safe population of clients.This is related with Total Quality
Management especially in Marketing .
3. Use reinsurance . This is nevertheless quite costly for the company
4. Reduce the variance coeficient σ by reinvesting only part of the
profits to insurance products .The rest of the profits that are not
dividend can be invested to other products or industries (E.g
financial products with low risk as banking etc) .In other words
apply a horizontal defensive diversification strategy .
(See [Porter E. M.] chpt 10,13,14 ,[ Grant M.R.] chpt 14 )
5. Apply a combination of the previous strategies that with
appropriate Total Quality Mangement distributes the cost of
avoiding bankruptcy among the stakeholders.
If the insurance company is posessed say by a bank then the horizontal
defensive diversification strategy automatically occurs. This gives a
stable company but only together with the mother company.
Most of the smaller insurance companies apply reinsurance. Thus the
cost of avoiding bankruptcy is paid almost exclusively from the
company itself.
There are many managers of small insurance companies that resort to the
panacea of increasing the sales . This is hardly a corect solution and even
if too much effort is spent to increase sales the real danger of Bankruptcy
from high random fluctuations say of the accidents may still exist.
We must discriminate strictly between a short term optimal
strategy that may bring higher anual profit from a long term optimal
strategy that avoids Bankruptcy . It is after all much like the trading
tactics in stock exchange market. . A stock may seem very profitable
compared say to a bond ,and a short term optimal strategy may indicate
100% investments in the stock. Nevertheless a very high volatility of the
stock combined with a need to cash out for example at the end of the
year ,may lead to much damage compared to investment in the bond .
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