THE RISKS OF BANKRUPTCY
IN INSURANCE COMPANIES, STOCHASTIC
STABILITY AND FAVORABLE GAMES
Dr. Costas Kyritsis Prof.
Petros Kiochos
Software laboratory Department of Statistics
National Technical and Insurance Science
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
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 (sub-super )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 non-linear 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 :
Where
Bt is a Brownian motion and
r,s 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)s2 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)/ s2
b)If r>(1/2) s2 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) s2)
c) The logarithm of this process X/ x0 is an ordinary Brownian motion with
drift:
log(X/ x0)=(r-(1/2) s2)t
+ s dBt
.
The regression
curve is the next:
The solution of this stochastic differential equation
is given by the formula:
The previous properties a) and b) of the geometric Brownian motion is
our basic point for the condition of Bankruptcy. If the risk ó is too large compared to the average rate of
return per unit of time r , r<(1/2)s2 ,
then we are lead to Bankruptcy with probability one (almost surely) .If on the
other hand the risk ó is small compared to the average rate of
return per unit of time r , r>(1/2) s2, 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 ó . 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)s2
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) s2
Proof: Since the r and ó 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 ó is large enough compared to the average rate of growth
ì 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 ó can quite easily estimated from the acconting
department of the company . The theory of estimators provides with formulas
that estimate ó 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 ó
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 .
Bibliography .
[Altman E.1968] Financial Ratios ,Discriminant
Analysis and the Prediction of
Corporate Bankruptcy
The
Journal of Finance (Sept 1968
pp 589-609)
[Azariadis C.1993] Intertemporal Macroeconomics
Blackwell .
[Beaver W.1966] Financial ratios as Predictors of Failure
«Empirical Research in Accounting»
Supplement
to Journal of Accounting Research
1966 pp 71-110
[Berndt E.R.1991] The practice of Econometrics
classic
and contemporary
Addison Wesley .
[Bowers ,N Gerber H.,
Hickman J., Jones D.,
Nesbitt C.1986], Actuarial
Mathematics
The Society of
Actuaries ,Itasca ,
Illinois
.
[Breiman L.1961] Optimal gambling Systems
for
Favorable Games
Proc. 4th Berkeley Sympos.
on
Mathematical Statistics and Probability
University of
California Press Berkeley
1961 ,1,
pp 65-78
[R.G.Clarke ,B.D.Wilson
R.H.Daines,S.D.Nadault 1988] Strategic Financial Management
IRWIN
[Dubins L.E.,
Savage L.J,1965] How to gamble if
you must
New York McGraw-Hill
[Homan M.1989] A survey of administrations under the Insolvency Act 1986The
results of Administrations orders made in 1987
ICAEW London
[Feller W.1957] Probability Theory and its
Applications
Vol
1 , Wiley
[Janssen J.
Skiadas C.H.] Dynamic Modeling of
Life
Table Data
Applied Stoch. Models
and Data Analysis vol.
11 35-49
[Jordan C.1975] Life Contingencies
The Society of Actuaries
Chicago
Illinois .
[Grant M.R.1995] Contemporary Strategy
Analysis
Blackwell
[Karlin S.,
Taylor H.M.1975] A first course in stochastic
processes
2nd
edition Academic Press
[Kiochos P.1991] Techniques
of Actuarial Studies
(In
Greek)
Contemporary Publications
Athens.
[Kiochos
P.1996] Actuarial
mathematics
Interbooks,Athens
[Kiochos P.1996] Methodology
of Actuarial Studies
Interbooks.Athens
[Kiochos P.1997]
Methodolohy of Research
Stamoulis
P.Athens
[Kloeden P.E.
Platen E.
Schurtz H.1997] Numerical Solution of SDE
through
Computer Experiments
Springer .
[Mallaris A.G.
Brock W.A.1995] Stochastic methods in
Economics and Finance
North-Holland .
[Petauton P,1991] Theorie et Pratique de
l’Assurance
Vie
,Dumond Paris
[Porter E. M.] Competitive
advantage
The free press
[Oksendal B.1995] Stochastic Differential
Equations
Springer .
[Owen G.1995] Game
Theory
3d edition Academic Press.
[Tong H.owell, 1990] Non-linear Time Series
A Dynamic System Approach
Clarendon
Press .
[Turner J.R.] The
handbook of Project -Based
Management
McGraw-Hill