Cybernetics and Systems Analysis, Vol. 47, No. 1, 2011
MATHEMATICAL MODELS FOR INSURANCE BUSINESS OPTIMIZATION1
UDC 519.21
B. V. Norkin
Abstract. A new approach to the problem of approximate optimization of insurance business is proposed that lies in optimizing net income (dividends) under a constraint on the probability of ruin. The probability is then replaced by its exponential upper bound. This trick allows one to eliminate a complicated probabilistic constraint and to decompose the problem according to separate lines of business. Thus, problems of optimization of tariffs, insurance, portfolios, reinsurance treaties, and operational management are approximately solved. Keywords: actuarial mathematics, risk process, probability of ruin, optimization, optimal control, insurance portfolio, reinsurance, Monte Carlo method. INTRODUCTION The essence of insurance business lies in obtaining the maximum net profit with insurance reserves sufficient for covering possible insurance claims with a given degree of reliability [1, 2]. In a competitive environment, a company is striving to increase its market share. To improve business, flexible (new) insurance tariffs (prices), reinsurance of risks, new insurance products, extension of a dealer network, advertising, investments, etc. are used [3], all require expenditures that, of course, are provided owing to arrived premiums or a decrease in the initial reserve. Expenses decrease insurance reserves or incomes of a company and thereby exert influence on its reliability. Therefore, a fine balance is required between the problems of profit maximization and maintenance of the company stability. Insurance business is based on the law of large numbers, i.e., the larger the number of independent random quantities (risks) being summed, the more predictable their average. A competent insurance business is a science-intensive kind of activity in which complicated mathematical models of the theory of random processes, mathematical statistics, optimization, and optimal control are used. The traditional theory of the microeconomics of insurance is based on the concept of expected utility [2, 4]. Economic agents including insurance companies are guided by the principle of maximization of expected utility. Utility functions allow one to qualitatively explain many economic phenomena but are of little use for practical calculations since they are not measurable by physical or other direct methods. Another approach to the analysis of insurance activities is based on the investigation of profitability and risk (the probability of ruin). As a measure of profitability, the capital of a company at the end of the planning period [1, 3, 5] or the average profitability of insurance activities [6] can be used and, as a measure of risk, capital dispersion [1, 3, 5], probability of ruin [1, Ch. 10], its exponential upper estimates [6, 7], the Lundberg coefficient [2, Sec. 14.5], expectation of maximum total losses [2, Secs.13.6, 14.4, and 14.5], and other characteristics of the event of ruin can be used. In this approach, the cardinal problem is the computation or estimation of probability of ruin of an insurance company. An extensive literature [1, 2, 7–9] is devoted to this problem. 1
This work is supported under grant No. GP/F27/0088 of the President of Ukraine for young scientists.
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine,
[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 128–145, January–February 2011. Original article submitted July 2, 2009. 1060-0396/11/4701-0117
©
2011 Springer Science+Business Media, Inc.
117
In [6, 10], optimization problems of risk reinsurance were considered within the framework of the classical Cramer–Lundberg model of collective risk, namely, the average growth rate of the capital of a company under a constraint on the probability of ruin estimated over an infinite time interval was maximized. Parameters of reinsurance treaties were considered to be optimization variables. Since the computation of the probability of ruin is not a simple problem, its upper estimate in the constraint, namely, the exponential Cramer–Lundberg estimate was used. In [2], in choosing reinsurance parameters, the Lundberg coefficient was maximized that enters into the exponential probability estimate of ruin with a minus sign. In the present article, another optimization criterion is used, namely, the level of net profit (dividends) under constraints on the probability of ruin. We apply this approach to solve some insurance business management problems, namely, the choice of an initial insurance reserve, optimization of tariffs, insurance portfolios, reinsurance treaties, and optimal management of the dividend policy of a company. In this case, instead of the exact probability of ruin, its estimate is used. In the capacity of optimization variables, both parameters of reinsurance treaties and other activity characteristics of an insurance company are used such as policy prices, levels of investments in some sphere of activity of the company or other, etc. The sought-for quantities are the dividend level and other company management parameters calculated as functions of the initial capital. A similar approach to a company as a whole is considered in [7]. The obtained functions can be used for the current management of the company, namely, the current random value of capital is substituted in these functions and the obtained management parameters are used for day-to-day management of the company. However, under this management, the process becomes nonlinear, though the current management has been constructed proceeding from the assumption that, after the choice of new management parameters, they will not vary. Therefore, the necessity arises of repeated estimation of the probability of ruin of a new nonlinearly controlled risk process in which premiums depend on the current capital. It is no longer the classical risk process, and classical models of assessing the risk of ruin are not applicable to it, but the general Monte Carlo method and the method of successive approximations [11–14] are applicable. The repeated risk assessment is especially justified since, in optimizing company activity parameters, simplified estimates for the probability of ruin have been used. We note that the estimation of small probabilities of ruin requires an infinitude of tests in the Monte-Carlo method and, therefore, it makes sense to use it on parallel computing systems (multicore computers or clusters). Thus, our main innovations are the maximization of the net profit (dividends) that is the difference between insurance premiums and deductions for the insurance reserve, and the replacement of the constraint on the probability of ruin by the exponential Cramer–Lundberg estimate (this expedient was used earlier in [6]). In aggregate, these two expedients allow one to obtain an expression for the Lundberg coefficient in explicit form, to eliminate a complicated probabilistic constraint from the problem formulation, to decompose the problem of optimization of insurance portfolios, and to obtain analytical solutions in many cases. For the problem of optimal day-to-day management of the dividend policy of a company under a constraint on the probability of ruin, stationary strategies are used that are optimal for the current state and are calculated analytically. It is shown by numerical calculations with the help of the parallel Monte-Carlo method that an adaptive strategy obtained in this case greatly surpasses the stationary strategy optimal for the initial capital both in the level of total dividends and in the probability of ruin. An innovation also is the method of successive approximations for estimating the risk of ruin of a company when its premiums and dividends depend on its current capital. OPTIMIZATION OF INSURANCE ACTIVITY PARAMETERS UNDER A CONSTRAINT ON THE PROBABILITY OF RUIN Let, in the portfolio of a company, there be n i contracts of price p i of the ith type with Poisson intensity of arriving requests a i (per one contract) and a distribution function of independent requests Fi ( × ) , i = 1, 2, K, n . We assume that arrival times and sizes of all insurance claims are independent. Let there be j = 1,... , m current kinds of activity aimed at perfecting business with constant intensities x j and expenses b j ( x j ) and, hence, the total expenses for them per unit time m
are equal to
å b j ( x j ). We denote by x = ( x1 ,... , xm ) Î X
a vector, where X Ì R m is some set of feasible solutions. Any
i =1
activity leads to a change in the number of contracts in the portfolio of a company, i.e., n i ( x ) or the number of contracts of the ith type depends on the activity vector x. In particular, the number itself of contracts n j of the jth type can be considered as the optimization variable, and then b j ( n j ) expresses the cost of providing service of these contracts. Under these assumptions, the total insurance claims for each contract type over any finite time interval is a compound Poisson process 118
and, hence, the summarized insurance claims of a portfolio also form a compound Poisson process [2, Secs. 12.4 and 13.3] n
n
i =1
i =1
in which cumulative premiums c ( x ) = å p i n i ( x ) , intensities of arrival of requests a( x ) = å a i n i ( x ) , and distribution of requests
n
n
i =1
i =1
Fx ( × ) = å a i n i ( x )Fi ( × ) / å a i n i ( x )
(1)
depend on the vector x. The problem of decision-making lies in maximizing the current pure profit (net premiums and dividends) D under the following constraints on the business stability: (2) D ® max , xÎ X , D ³0
m æ n ö yT ç u, å p i n i ( x ) - å b j ( x j ) - D , a( x ), Fx ÷ £ e , ç i =1 ÷ j =1 è ø
(3)
~, F~ ) is the probability of ruin of the company over an interval of time [ 0, T £ ¥ ] , the parameter u where yT ( u, c~, a n
m
i =1
j =1
denotes the initial insurance reserve of the company, c~ = å p i n i ( x ) -å b j ( x j ) - D is the amount of the pure ~ = a( x ) is the intensity of the flow of independent requests, F~ = F is the cumulative premium per unit time, a x probability distribution of the size of one random request in the overall flow of requests, and e is the business reliability parameter, 0 < e < 1. Problem statement (2), (3) presumes that the company stably functions during a long time interval [ 0, T ] (infinite in the ideal case) and that parameters of its business under a fixed management x do not vary. The function yT ( × ) is not usually known in explicit form. It is specified by approximate formulas (approximations) or is found by solving an auxiliary computational problem, for example, the inverse Laplace transform or by solving an integral equation for the probability of ruin [9]. Thus, statement (2), (3) is a complicated computational problem. For the probability of ruin yT ( × ) of the complicated risk process being considered, the Cramer–Lundberg estimate [9] ~, F~ ) £ y ( u, c~, a ~, F~ ) £ e - Ru yT ( u, c~, a ¥ takes place, provided that there exists a positive solution (the Lundberg coefficient) R to the equation ¥
ò
¥
~ ~ e R y (1- F ( y )) dy = c~ / a.
0
~ (1- F~( y )) dy < c~ and insurance claims are uniformly bounded. If, in Such a solution undoubtedly exists if a ò 0
constraint (3), the probability of ruin yT ( × ) is replaced by its exponential estimate e - Ru , then this constraint is strengthened, i.e., the feasible region of the problem is decreased. Instead of problem (2), (3), let us consider the problem D®
max
,
(4)
xÎ X , D ³0 , R ³0
e - Ru £ e,
(5)
¥
n
m
0
i =1
j =1
a( x ) ò e R y (1- Fx ( y )) dy = å p i n i ( x ) - å b j ( x j ) - D .
(6)
If feasible solutions to this problem exist, then, in the optimal solution, the inequality constraint e - Ru £ e is obviously fulfilled as equality. This allows one to eliminate the quantities R and D from the problem statement, R = (1/ u ) ln (1/ e ),
119
n
¥
m
D = å p i n i ( x ) - å b j ( x j ) - a( x ) ò e ( y/ u ) ln (1/ e) (1- Fx ( y )) dy i =1
j =1
0
n
m
n
¥
i =1
j =1
i =1
0
= å p i n i ( x ) - å b j ( x j ) - å a i n i ( x ) ò e ( y/ u ) ln (1/ e) ¥ n æ m n n ö æ ö ´ ç 1- å a i n i ( x ) Fi ( × ) / å a i n i ( x ) ÷ dy = å ç p i - a i ò e ( y/ u ) ln (1/ e) (1- Fi ( y ))dy ÷ n i ( x ) - å b j ( x j ) . ç ÷ ç ÷ i =1 è j =1 i =1 è i =1 ø 0 ø
As a result, we arrive at the following problem: n
m
i =1
j =1
D ( x ) = å a i ( u, e )n i ( x ) - å b j ( x j ) ® max , where the coefficients
xÎ X
(7)
¥
a i ( u, e, p i ) = p i - a i I i ( u, e ) , I i ( u, e ) = ò e ( y/ u ) ln (1/ e) (1- Fi ( y )) d y
(8)
0
can be computed independently and before the solution of optimization problem (7). Statement 1. We assume that all integrals I i ( u, e ) in coefficients (8) are finite. If the value of the objective function in the optimal solution x* of problem (7) is non-negative, D ( x* ) ³ 0, then the collection {D* = D ( x* ), R * = (1/ u ) ln (1/ e ), x* } is the optimal solution of problem (4)–(6) and the pair {D* = D ( x* ), x* } is feasible for problem (2), (3). Proof. We denote by D** the upper bound of values of objective function (4) over feasible set (5), (6). If x* is an optimal solution of problem (7), then it is obvious that {D* = D ( x* ) , R * = (1/ u )ln (1/ e ), x* } is a feasible solution of problem (4)–(6). Therefore, the optimal value of D** in problem (4)–(6) it is no less than D ( x* ). We assume that D** > D ( x* ). Then there is a collection ( D¢ , R ¢ , x¢ ) feasible for problem (4)–(6) and such that we have D** ³ D¢ > D ( x* ) ³ 0. In this case, without loss of generality, we can consider that inequality (5) is changed into equality, i.e., we obtain R ¢ = (1/ u )ln (1/ e ) . Then the pair ( D¢, x¢ ) is feasible for problem (7), (8) and, hence, we have D¢ £ D ( x* ), contrary to the inequality D¢ > D ( x* ) established above. Thus, we have D ** = D ( x * ) and, hence, {D* = D ( x* )R * = (1/ u ) ln (1/ e ), x* } is an optimal solution to problem (4)–(6). By construction, R * = (1/ u ) ln (1/ e ) > 0 is a positive root of the equation ¥
n
m
0
i =1
j =1
a( x* ) ò e R y (1- Fx* ( y )) dy = å p i n i ( x* ) - å b j ( x*j ) - D* . Therefore, according to [9, Theorem 3.8], the Cramer–Lundberg inequality is satisfied for the probability of ruin y( × ) and, hence, { D* = D ( x* ), x* } is a feasible solution to problem (2), (3). The statement is proved. When a i ( u, e, p i ) < 0 , the corresponding addend in problem (7) is negative and this means that this kind of insurance is unprofitable with given parameters u, e, and p i . Thus, quantities p*i ( u, e ) = a i I i ( u, e ) specify the lower bound of the breakeven prices for contracts of the ith type for a given insurance reserve u and a reliability level e.
During the day-to-day operation of an insurance company, quantities p*i ( x t , e ) = a i I i ( x t , e ), where x t is the current
(at a moment t ) insurance reserve, reflect values of current breakeven insurance tariffs (premiums). The comparison of actual tariffs p i with guideposts p*i ( x t , e ) allows managers of the company to give correct accents to the extension or curtailment of some kind of insurance or other with allowance for the assumed risk level e. Comment. Problems (4)–(6) and (8), (9) provide only an approximate solution to the problem of optimization of dividends. Actual dividends received during a time T are less than DT since they are lost in the case of ruin but are greater than (1- e )DT since the probability of ruin is less than e. 120
TABLE 1 Results of Calculations Parameters i=1
i= 2
i= 3
i= 4
i= 5
~ =an a i i i
i= 6
i= 7
0.2
0.5
1
1.2
2
3
4
mi
3
2
3
2
1
0.3
0.1
ci = pi ni
0.8
1.2
3.5
2.8
2.4
1
0.5
a i mi / ci
0.7500
0.8333
0.8571
0.8571
0.8333
0.9000
0.8000
ILLUSTRATIVE NUMERICAL DATA AND APPROXIMATIONS
Let us illustrate the proposed approach to the optimization of the insurance business by a number of problems. In what follows, we will accompany theoretical results with numerical calculations. The initial data for calculations are taken from [6] and are presented in Table 1. ~ denotes the intensity of the Poisson flow of insurance claims for contracts of the ith type, c is the intensity Here, a i i of the flow of premiums, and distribution functions of requests are exponential with an average m i , Fi ( y ) = 1- e - y/ mi . A flow of requests can be simulated by a compound Poisson process with the intensity of arriving requests a = å i a i = 11.9, with the intensity of arriving premiums c = å i c i = 12. 2 , and with the distribution of requests F ( y ) = å i a i Fi ( y ) / å i a i , m = å i a i m i / å i a i = 0.8655. The corresponding insurance load r = c / ( am ) - 1 =
å i ci / å i a i m i - 1 = 0.1845.
The Cramer–Lundberg estimate for the probability of ruin is of the form
y( u ) £ e - Ru = e - 0.0773 × u , where R = 0.0773 is the Lundberg coefficient that is the solution of the equation ¥
a ò e R y (1- F ( y )) dy = c . 0
The exponential approximation F ( y ) = 1- e - y / m of the distribution of insurance claims is used in the approximation of de Vilder [9, Sec. III, §5]. In this case, the initial process with the parameters ( m , r, a ) is replaced by a process with the parameters 9m3 m 2m m m = 3 = 2.4151, r = 1 3 r = 0.2305, a = 2 a = 3.4135, 3m2 3m2 2m2 2
3
where m j = (1/ a ) å i a i m ij is the jth moment of the initial distribution of requests F ( y ) = (1/ a ) å i a i Fi ( y ) and ¥
m ij = ò y j dFi ( y ) is the jth moment of the distribution of requests Fi ( y ). In this case m1 = 0.8655, m2 = 3.3462 , and 0
m3 = 24.2445. The corresponding approximation of the probability of ruin is of the form ru
1 - (1+ r) m y (u ) = e = 0.8127e - 0.0776 × u . 1+ r
(9)
~ ~ In [2, p. 375], it is also proposed to use the exponential approximation of the distribution of requests F ( y ) = 1- e - y/ m with the average ~ m = m / ( 2m ) = 1.9330 under an invariable insurance load. In this case, the approximation of probability of 2
ruin is of the form
1
ru
1 - (1+ r) ~m ~ y( u ) = e = 0.8443e -0.0806 × u . 1+ r
(10)
If requests corresponding to concrete kinds of insurance are nonexponential, then the mentioned exponential approximations can be applied to concrete portfolio components. 121
SOME OPTIMIZATION PROBLEMS OF THE INSURANCE BUSINESS Problem 1 (estimation of breakeven insurance tariffs). We rewrite problem (7) in the form n
m
i =1
j =1
, å ( ci (x ) - a i ( x )I i ( u, e )) - å b j ( x j ) ® xmax ÎX where c i ( x ) = p i n i ( x ) and a i ( x ) = a i n i ( x ). The negativity of the addend ( c i ( x ) - a i ( x ) I i ( u, e )) implies the undoubted unprofitability of the corresponding kind of insurance for given ( u, e ). If Fi ( y ) = 1- e - y/ mi and u > m i ln(1/ e ), then we have ¥
¥
I i ( u, e ) = ò e ( y/ u ) ln (1/ e) (1- Fi ( y )) d y = ò e ( y/ u ) ln (1/ e) - y/ mi dy 0
=
1 e ( y/ u ) ln(1/ e) - y/ mi (1/ u ) ln (1/ e ) - 1/ m i
When u ³ ui (e ) =
0
¥
= 0
mi u mi . = u - m i ln (1/ e ) 1- ( m i / u ) ln (1/ e )
mi 1 ln , 1- a i m i / ci e
we have c i - a i I i ( u, e ) ³ 0 and, hence, u i ( e ) specifies a minimum insurance reserve level for which the corresponding kind of insurance can be breakeven. In the case of nonexponential request distributions, the computation of quantities I i ( u, e ) requires the use of numerical integration. For example, for the bound of the probability of insolvency e = 10-3 , the quantities u i (10-3 ) are, respectively, equal to 82.8931, 82.8931, 145.0629, 96.7086, 41.4465, 20.7233, and 3.4539. Problem 2 (estimation of a safe insurance reserve). The problem of estimation of the value of insurance reserves that provides a given safety level is the classical problem of actuarial mathematics [1]. Let the evolution of insurance reserves of a company be aggregately described by a classical risk process with intensity c of arrival of premiums, with the Poisson flow of insurance claims with intensity a, and with a distribution function of requests F ( y ) with average m, a m < c . Then problem (4)–(6) of maximization of dividends assumes the form D®
max ,
D ³0 , R ³0
(11)
e - Ru £ e,
(12)
a ò e R y (1- F ( y )) dy = c - D .
(13)
¥ 0
We assume that there is a solution u* ( e ) of the equation ¥
a ò e ( y/ u ) ln (1/ e) (1- F ( y )) dy = c .
(14)
0
Since the left side of Eq. (14) is monotone with respect to u, the solution u* ( e ) is unique. Such a solution undoubtedly exists if am < c and insurance claims are bounded. If the distribution of requests is exponential with an average m, then m 1 u* ( e ) = ln . 1- am / c e Statement 2. If u ³ u* ( e ), where u* ( e ) is a root of Eq. (14), then a solution of problem (11)–(13) exists and is as follows:
¥
D* ( u ) = c - a ò e ( y/ u ) ln (1/ e) (1- F ( y )) dy, R * ( u ) = (1/ u ) ln(1/ e ). 0
122
(15)
¥
Proof. The function I ( u ) = a ò e ( y/ u ) ln (1/ e) (1- F ( y )) dy is nonnegative, monotonically decreases with respect to u, 0
*
and I ( u ( e )) = c and, therefore, when u ³ u* ( e ) , we have 0 £ I ( u ) £ c and, hence, obtain 0 £ D * ( u ) £ c. As is obvious, R * ( u ) and D * ( u ) form a feasible solution to problem (11)–(13). For any feasible ( R , D ) , we have R ³ R * ( u ) and ¥
0 £ a ò eR
*
y
0
From this we obtain
¥
(1- F ( y )) dy £ a ò e R y (1- F ( y )) dy £ c . 0
¥
¥
0
0
D = c - a ò e R y (1- F ( y )) dy £ c - a ò e R *
*
y
(1- F ( y )) dy = D* ( u )
*
and, hence, the pair ( D ( u ), R ( u )) is optimal, which is what had to be proved. ~ The truth of the inequality u ³ u* ( e ) guarantees that the probability of ruin y ( u, D ) £ e . The value of u* ( e ) is an estimate of the minimal safe level of insurance reserves, and the value of ¥
c - D* = a ò e ( y/ u ) ln (1/ e) (1- F ( y )) dy 0
is the minimum level of obligatory deductions for the insurance reserve. If, at the current moment t, we have x t ³ u* ( e ), then the corresponding company is guaranteedly in safety at the level e. If x t < u* ( e ), then the required reliability level e is not guaranteed. Let the portfolio of contracts of a company be specified by Table 1 with exponential distributions of insurance claims. In aggregate form, the evolution of reserves of the company is described by the compound Poisson flow with the intensity a = å i a i = 12. 2, cumulative premiums c = å i c i = 11.9, and distribution of requests F ( y ) = å i a i Fi ( y ) / å i a i , m = å i ai m i
/ å i ai
= 0.8655. When u > (max m i ) ln (1/ e ) = 20.7233 , the left side of Eq. (14) is of the form i
¥
a ò e ( y/ u ) ln (1/ e) (1- F ( y )) dy = å i
0
and the equation
å i
ai 1/ m i - (1/ u ) ln(1/ e )
ai = å ci 1/ m i - (1/ u ) ln(1/ e ) i
has the solution u* (10-3 ) = 89.3883, which also is a solution to Eq. (14). Let the distribution of insurance claims be approximated by the exponential distribution F ( y ) = 1- e - y/ m with its average m; then the probability of ruin is of the form [9, Theorem 3.3] r( D ) u ì 1 ( r( D )) m + 1 ï , c - D > am , y ( u, D ) = í 1+ r( D ) e ï c - D £ am , î1
where r( D ) =
c-D -1. We formulate the problem of optimization of dividends in the form am D®
max ,
0 £D
