THE RISKS OF BANKRUPTCY IN INSURANCE COMPANIES, STOCHASTIC STABILITY AND FAVORABLE GAMES
Dr. Costas KyritsisProf. Petros Kiochos
Software laboratory Department of Statistics
National Technical and Insurance Science
University of AthensUniversity 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 (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, 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/ x0 is an ordinary Brownian motion with drift:
log(X/ x0)=(r-(1/2) 2)t + 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 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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