In addition to maximizing a certain objective function, insurers must satisfy a solvency constraint imposed by the regulator. A simplification is to approximate a q-quantile of aggregate claim amount of n i.i.d. risks by a bilinear function of n and
where the solvency coefficient has to be determined and Y is the generic individual claim severity variable. and are a mean and standard deviation of a randon variable. Using the approximation the solvency capital requirement SCR is deduced as
Then the solvency constraint function can be defined as follows
where is the expense rate as a percentage of gross written premium.
The one-period Nash equilibrium model in non-life insurance markets becomes
and are the minimum and the maximum premium, respectively.
3. Extension of the One-Period Model
This section deals with an extension of the one-period model considered from the perspective economic factors. Let m be number of periods. To consider a possible extension of the model with m -period, we assume that policyholders will react on the current economic situation i.e., if the economy is getting better, then they have interests to be insured. As before, we assume that the market has I insurers and n insureds. Let be economic factor in kth period and be a vector of economic weights in kth period with respect to the movement from insurer i to j. We introduce a transition matrix (see  ) describing insureds’ movement to insurers.
where denotes the probability for customers to switch from insurer i to j in kth period. th column corresponds to uninsured ones whose state can be called inactive. According to  (see, also  ), the transition probability can be modelled as
where is the Euclidean inner product. If the economy is deteriorate in kth period, some insureds don’t want to keep insurance contracts, therefore will decrease and will increase. In kth period, the portfolio size of insurer j for the next period is determined by the sum of renewed policies and businesses coming from other insurers. Hence
where . Let be a risk free rate and be a discount
factor. Based on  , the insurer j maximizes the present value of expected profit of renewing policies defined as
where , for , and . The solvency constraints for insurer j can be redefined as
Then, the strategy set of each player is
Now we give some similar results for m-period model.
Proposition 1. The m-period insurance game with I players whose objective functions and solvency constraints are defined by (1) and (2), respectively, admits a unique Nash premium equilibrium.
Proof: In a similar way as in  and by Theorem 1 in  , we can verify the existence of a Nash equilibrium. On the other hand, since for any , the function is strictly concave and differentiable with respect to , for it hold
Adding both inequalities, we have
Denoting by and taking the sum by , we obtain that
which guarantees the uniqueness of the equilibrium (cf. Theorem 2 in  ).
Proposition 2. Let be the premium equilibrium of the m-period insurance game with I players.
1) If all solvency constraints are either active or inactive, then for each player j and period k, the corresponding equilibrium depends on the parameters in the following way:
a) It increases with break-even premiums , solvency coefficient , loss volatility , expense rate , and risk free rate for and
b) Decreases with sensitivity parameter , capital for , and, portfolio size for .
2) If all constraint functions are inactive, then the premium equilibrium is a solution of the linear system of equations
Proof: The KKT conditions for the premium equilibrium of insurer j has the following form:
k-th component from the first equation of the system becomes
1) Let . Then . We consider two cases.
a) Let us assume that the solvency constraints are all inactive, i.e., . Then, insurer j’s premium equilibrium verifies , i.e.,
Let . In order to investigate the sensitivity depending on parameter z, let us define the function as
and consider the equation of the form . Under assumptions that partial derivatives of exist and are continuous at , and also
, by the implicit function theorem, there exists a function
defined in a neighbourhood of such that and . The derivative of is given by
In our case, we have
As a consequence, it holds
i) Let . Then
In other words, the function is increasing.
ii) Let z be the sensitivity coefficient . Then, we have
By using (4), we obtain that
Therefore, the function is decreasing.
b) If the solvency constraints are all active, then the premium equilibrium satisfies , for and consequently, one get
From (5), we can verify directly that is an increasing function of , , and for . Moreover, it is a decreasing function of for and for .
2) If all constraints are inactive at a Nash equilibrium , then taking into account and from (4) follows that
This system can be rewritten in matrix form as . As in  mentioned, we can see that the matrix is strictly diagonally dominant if the conditions are fulfilled. Under this condition is invertible and therefore .
Remark 1. If or , then the premium equilibrium is independent of those parameters.
Remark 2. For a game with one leader and followers with payoff functions and the strategy set , a Stackelberg equilibrium is the problem that consists in finding a vector , such that solves the problem
where is a Nash equilibrium for the game with the followers and given strategy for insurer 1 which is assumed to be a leader. In this case, it is not difficult to show the existence of Stackelberg equilibrium (cf.  ).
4. Numerical Experiments
In this section we show some numerical results dealing with sensitivity analysis presented in Proposition 2 in Section 3. Let us notice that the Nash equilibrium model can be reduced to the variational inequality problem which consists in finding such that
where . In order solve the problem (VI), we apply the hyperplane projection algorithm (see  and  ). We consider three player’s game and let .
1) Base case:
Table 1 shows that if we use the data from  in each period, then we get the same results.
2) Scenario 1:
Table 2 shows results for the case if elasticity parameter of first player increases up to 3.5 in three periods.
3) Scenario 2:
Table 3 presents results for the case if elasticity parameters of all players increase in Period 2. Then, premium equilibriums are changed only in Period 2.
4) Scenario 3:
In this case, we assume that break-even premium for player 1 in Period 1 and for player 3 in Period 3 are increasing and break-even premium for player 2 in Period 2 is decreasing. Then, premium equilibriums in Period 1 and Period 3 for players 1 and 3 are increasing, but premium equilibrium in Period 2 for player 2 is decreasing as compared with “Base case” (see Table 4).
5) Scenario 4:
Table 1. Basic case.
Table 2. Scenario 1.
Finally, we assume that . Let the economic factor be −3 (which means that the economy is deteriorated) in Period 1, 0 in Period 2 and 3 (which means that the economy is raised) in Period 3. If the economic factor is equal to −3, then the number of uninsured people (which corresponds to inactive state) increases up to . If economic factor is equal to 3, then the number of uninsured people (which corresponds to inactive state) decreases down to . The results are presented in Table 5.
Table 3. Scenario 2.
Table 4. Scenario 3.
Table 5. Scenario 4.
In this paper, we aim to investigate an extension of the one-period model in non-life insurance markets (cf.  ) by introducing a transition probability matrix depending on some economic factors. In the future, we concentrate on alternative ways of the extension including generalized Nash equilibrium (see, for instance  and  ) formulations. Moreover, it would be interesting to investigate in more detail about economic factors that influence in our model.
The research funding was provided by the “L2766-MON: Higher Education Reform” project financed by the Asian Development Bank and executed by the Ministry of Education, Culture, Science and Sports of Mongolia.
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